Control Method

Design Methods

for

Control Systems

Okko H. BosgraDelft University of TechnologyDelft, The Netherlands

Huibert KwakernaakUniversity of TwenteEnschede, The Netherlands

Gjerrit MeinsmaUniversity of TwenteEnschede, The Netherlands

Notes for a course of theDutch Institute of Systems and ControlWinter term 2001–2002

ii

Preface

Part of these notes were developed for a course of the Dutch Network on Systems and Controlwith the title “Robust control andH∞ optimization,” which was taught in the Spring of 1991.These first notes were adapted and much expanded for a course with the title “Design Methodsfor Control Systems,” first taught in the Spring of 1994. They were thoroughly revised for theWinter 1995–1996 course. For the Winter 1996–1997 course Chapter 4 was extensively revisedand expanded, and a number of corrections and small additions were made to the other chapters.In the Winter 1997–1998 edition some material was added to Chapter 4 but otherwise there wereminor changes only. The changes in the 1999–2000 version were limited to a number of minorcorrections. In the 2000–2001 version an index and an appendix were added and Chapter 4 wasrevised. A couple of mistakes were corrected for the 2001–2002 issue.

The aim of the course is to present a mature overview of several important design techniquesfor linear control systems, varying from classical to “post-modern.” The emphasis is on ideas,methodology, results, and strong and weak points, not on proof techniques.

All the numerical examples were prepared using MATLAB . For many examples and exercisesthe Control Toolbox is needed. For Chapter 6 the Robust Control Toolbox or theµ-Tools toolboxis indispensable.

iii

iv

Contents

1. Introduction to Feedback Control Theory 11.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Basic feedback theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3. Closed-loop stability . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4. Stability robustness . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.5. Frequency response design goals. . . . . . . . . . . . . . . . . . . . . . . . . . 271.6. Loop shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7. Limits of performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.8. Two-degrees-of-freedom feedback systems . .. . . . . . . . . . . . . . . . . . 471.9. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521.10. Appendix: Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2. Classical Control System Design 592.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.2. Steady state error behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.3. Integral control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.4. Frequency response plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.5. Classical control system design . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.6. Lead, lag, and lag-lead compensation . . . . . . . . . . . . . . . . . . . . . . . . 812.7. The root locus approach to parameter selection . . . . . . . . . . . . . . . . . . 872.8. The Guillemin-Truxal design procedure. . . . . . . . . . . . . . . . . . . . . . 892.9. Quantitative feedback theory (QFT) . .. . . . . . . . . . . . . . . . . . . . . . 922.10. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3. LQ, LQG and H2 Control System Design 1033.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.2. LQ theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043.3. LQG Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1153.4. H2 optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1243.5. Feedback system design byH2 optimization . . . . . . . . . . . . . . . . . . . . 1273.6. Examples and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1333.7. Appendix: Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

4. Multivariable Control System Design 1534.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1534.2. Poles and zeros of multivariable systems . . . . . . . . . . . . . . . . . . . . . . 1584.3. Norms of signals and systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1674.4. BIBO and internal stability . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 1744.5. MIMO structural requirements and design methods . . . . .. . . . . . . . . . . 177

v

Contents

4.6. Appendix: Proofs and Derivations . . . . . . . . . . . . . . . . . . . . . . . . . 192

5. Uncertainty Models and Robustness 1975.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1975.2. Parametric robustness analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 1985.3. The basic perturbation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2085.4. The small gain theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2105.5. Stability robustness of the basic perturbation model . . . . .. . . . . . . . . . . 2135.6. Stability robustness of feedback systems. . . . . . . . . . . . . . . . . . . . . . 2215.7. Structured singular value robustness analysis .. . . . . . . . . . . . . . . . . . 2325.8. Combined performance and stability robustness. . . . . . . . . . . . . . . . . . 2405.9. Appendix: Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

6. H∞-Optimization and µ-Synthesis 2496.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2496.2. The mixed sensitivity problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 2506.3. The standardH∞ optimal regulation problem . . . . . . . . . . . . . . . . . . . 2586.4. Frequency domain solution of the standard problem . . . . . . . . . . . . . . . . 2646.5. State space solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2726.6. Optimal solutions to theH∞ problem . . . . . . . . . . . . . . . . . . . . . . . 2756.7. Integral control and high-frequency roll-off . .. . . . . . . . . . . . . . . . . . 2796.8. µ-Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2886.9. An application ofµ-synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 2926.10. Appendix: Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

A. Matrices 311A.1. Basic matrix results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311A.2. Three matrix lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

Index 324

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1. Introduction to Feedback Control Theory

Overview– Feedback is an essential element of automatic control sys-tems. The primary requirements for feedback control systems are stabil-ity, performance and robustness.

The design targets for linear time-invariant feedback systems may bephrased in terms of frequency response design goals and loop shaping.The design targets need to be consistent with the limits of performanceimposed by physical realizability.

Extra degrees of freedom in the feedback system configuration intro-duce more flexibility.

1.1. Introduction

Designing a control system is a creative process involving a number of choices and decisions.These choices depend on the properties of the system that is to be controlled and on the re-quirements that are to be satisfied by the controlled system. The decisions imply compromisesbetween conflicting requirements. The design of a control system involves the following steps:

1. Characterize the system boundary, that is, specify the scope of the control problem and ofthe system to be controlled.

2. Establish the type and the placement of actuators in the system, and thus specify the inputsthat control the system.

3. Formulate a model for the dynamic behavior of the system, possibly including a descrip-tion of its uncertainty.

4. Decide on the type and the placement of sensors in the system, and thus specify the vari-ables that are available for feedforward or feedback.

5. Formulate a model for the disturbances and noise signals that affect the system.

6. Specify or choose the class of command signals that are to be followed by certain outputs.

7. Decide upon the functional structure and the character of the controller, also in dependenceon its technical implementation.

1


8. Specify the desirable or required properties and qualities of the control system.

In several of these steps it is crucial to derive useful mathematical models of systems, signals andperformance requirements. For the success of a control system design the depth of understandingof the dynamical properties of the system and the signals often is more important than thea prioriqualifications of the particular design method.

The models of systems we consider are in general linear and time-invariant. Sometimes theyare the result of physical modelling obtained by application of first principles and basic laws.On other occasions they follow from experimental or empirical modelling involving experimen-tation on a real plant or process, data gathering, and fitting models using methods for systemidentification.

Some of the steps may need to be performed repeatedly. The reason is that they involve de-sign decisions whose consequences only become clear at later steps. It may then be necessaryor useful to revise an earlier decision. Design thus is a process of gaining experience and devel-oping understanding and expertise that leads to a proper balance between conflicting targets andrequirements.

The functional specifications for control systems depend on the application. We distinguishdifferent types of control systems:

Regulator systems. The primary function of a regulator system is to keep a designated outputwithin tolerances at a predetermined value despite the effects of load changes and otherdisturbances.

Servo or positioning systems. In a servo system or positioning control system the system isdesigned to change the value of an output as commanded by a reference input signal, andin addition is required to act as a regulator system.

Tracking systems. In this case the reference signal is not predetermined but presents itself as ameasured or observed signal to be tracked by an output.

Feedback is an essential element of automatic control. This is why§ 1.2 presents an elemen-tary survey of a number of basic issues in feedback control theory. These includerobustness,linearity andbandwidth improvement,anddisturbance reduction.

Stability is a primary requirement for automatic control systems. After recalling in§ 1.3 vari-ous definitions of stability we review several well known ways of determining stability, includingthe Nyquist criterion.

In view of the importance of stability we elaborate in§ 1.4 on the notion of stability ro-bustness. First we recall several classical and more recent notions of stability margin. Morerefined results follow by using the Nyquist criterion to establish conditions for robust stabilitywith respect to loop gain perturbations and inverse loop gain perturbations.

For single-input single-output feedback systems realizing the most important design targetsmay be viewed as a process of loop shaping of a one-degree-of-freedom feedback loop. Thetargets include

targets

• closed-loop stability,• disturbance attenuation,• stability robustness,

within the limitations set by

limitations

{ • plant capacity,• corruption by measurement noise.

2

1.2. Basic feedback theory

Further design targets, which may require a two-degree-of-freedom configuration, are

further targets

{ • satisfactory closed-loop response,• robustness of the closed-loop response.

Loop shaping and prefilter design are discussed in§ 1.5. This section introduces various impor-tant closed-loop system functions such as the sensitivity function, the complementary sensitivityfunction, and the input sensitivity function.

Certain properties of the plant, in particular its pole-zero pattern, impose inherent restrictionson the closed-loop performance. In§ 1.7 the limitations that right-half plane poles and zerosimply are reviewed. Ignoring these limitations may well lead to unrealistic design specifications.These results deserve more attention than they generally receive.

112 and 2-degree-of-freedom feedback systems, designed for positioning and tracking, are

discussed in Section 1.8.


1.2.1. Introduction

In this section feedback theory is introduced at a low conceptual level1. It is shown how thesimple idea of feedback has far-reaching technical implications.

Example 1.2.1 (Cruise control system). Figure 1.1 shows a block diagram of an automobilecruise control system, which is used to maintain the speed of a vehicle automatically at a constantlevel. The speedv of the car depends on the throttle openingu. The throttle opening is controlledby the cruise controller in such a way that the throttle opening isincreasedif the differencevr − vbetween the reference speedvr and the actual speed is positive, anddecreasedif the differenceis negative.

This feedback mechanism is meant to correct automatically any deviations of the actual ve-hicle speed from the desired cruise speed.

vr vr − v u

throttleopening

referencespeed

vcruisecontroller

car

Figure 1.1.: Block diagram of the cruise control system

For later use we set up a simple model of the cruising vehicle that accounts for the majorphysical effects. By Newton’s law

mv(t) = Ftotal(t), t ≥ 0, (1.1)

wherem is the mass of the car, the derivativev of the speedv its acceleration, andFtotal the totalforce exerted on the car in forward direction. The total force may be expressed as

Ftotal(t) = cu(t)− ρv2(t). (1.2)1This section has been adapted from Section 11.2 of Kwakernaak and Sivan (1991).

3


The first termcu(t) represents the propulsion force of the engine, and is proportional to thethrottle openingu(t), with proportionality constantc. The throttle opening varies between 0(shut) and 1 (fully open). The second termρv2(t) is caused by air resistance. The friction forceis proportional to the square of the speed of the car, withρ the friction coefficient. Substitutionof Ftotal into Newton’s law results in

mv(t) = cu(t)− ρv2(t), t ≥ 0. (1.3)

If u(t) = 1, t ≥ 0, then the speed has a corresponding steady-state valuevmax, which satisfies0 = c− ρv2

max. Hence,vmax = √c/ρ. Defining

w = v

vmax(1.4)

as the speed expressed as a fraction of the top speed, the differential equation reduces to

Tw(t) = u(t)−w2(t), t ≥ 0, (1.5)

whereT = m/√ρc. A typical practical value forT is T = 10 [s].

We linearize the differential equation (1.5). To a constant throttle settingu0 corresponds asteady-state cruise speedw0 such that 0= u0 − w2

0. Let u = u0 + u andw = w0 + w, with|w| � w0. Substitution into (1.5) while neglecting second-order terms yields

T ˙w(t) = u(t)− 2w0w(t). (1.6)

Omitting the circumflexes we thus have the first-order linear differential equation

w = −1θw+ 1

Tu, t ≥ 0, (1.7)

with

θ = T2w0

. (1.8)

The time constantθ strongly depends on the operating conditions. If the cruise speed increasesfrom 25% to 75% of the top speed thenθ decreases from 20 [s] to 6.7 [s]. �

Exercise 1.2.2 (Acceleration curve). Show that the solution of the scaled differential equation(1.5) for a constant maximal throttle position

u(t) = 1, t ≥ 0, (1.9)

and initial conditionw(0) = 0 is given by

w(t) = tanh(tT), t ≥ 0. (1.10)

Plot the scaled speedw as a function oft for T = 10 [s]. Is this a powerful car? �

1.2.2. Feedback configurations

To understand and analyze feedback we first consider the configuration of Fig. 1.2(a). The signalris an external control input. The “plant” is a given system, whose output is to be controlled. Oftenthe function of this part of the feedback system is to provide power, and its dynamical properties

4


r

r

e

e

u

u

forwardcompensator

forwardcompensator

returncompensator

y

yplant

plant

(a)

(b)

Figure 1.2.: Feedback configurations: (a) General. (b) Unit feedback

r re eφ

ψ

γ

plant

y

(a) (b)return compensator

Figure 1.3.: (a) Feedback configuration with input-output maps. (b) Equiva-lent unit feedback configuration

are not always favorable. The outputy of the plant is fed back via thereturn compensatorandsubtracted from the external inputr . The differencee is called theerror signaland is fed to theplant via theforward compensator.

The system of Fig. 1.2(b), in which the return compensator is a unit gain, is said to haveunitfeedback.

Example 1.2.3 (Unit feedback system). The cruise control system of Fig. 1.1 is a unit feedbacksystem. �

For the purposes of this subsection we reduce the configuration of Fig. 1.2(a) to that ofFig. 1.3(a), where the forward compensator has been absorbed into the plant. The plant is repre-sented as an input-output-mapping system with input-output (IO) mapφ, while the return com-pensator has the IO mapψ. The control inputr , the error signale and the output signaly usuallyall are time signals. Correspondingly,φ andψ are IO maps of dynamical systems, mapping timesignals to time signals.

The feedback system is represented by the equations

y = φ(e), e= r −ψ(y). (1.11)

These equations may or may not have a solutioneandy for any given control inputr . If a solution

5


exists, the error signale satisfies the equatione = r −ψ(φ(e)), or

e+ γ(e) = r. (1.12)

Hereγ = ψ ◦ φ, with ◦ denoting map composition, is the IO map of the series connection of theplant followed by the return compensator, and is called theloop IO map. Equation (1.12) reducesthe feedback system to a unit feedback system as in Fig. 1.3(b). Note that becauseγ maps timefunctions into time functions, (1.12) is afunctionalequation for the time signale. We refer to itas thefeedback equation.

1.2.3. High-gain feedback

Feedback is most effective if the loop IO mapγ has “large gain.” We shall see that one ofthe important consequences of this is that the map from the external inputr to the outputy isapproximately the inverseψ−1 of the IO mapψ of the return compensator. Hence, the IO mapfrom the control inputr to the control system outputy is almost independent of the plant IO map.

Suppose that for a given class of external input signalsr the feedback equation

e+ γ(e) = r (1.13)

has a solutione. Suppose also that for this class of signals the “gain” of the mapγ is large, thatis,

‖γ(e)‖ � ‖e‖, (1.14)

with ‖ · ‖ some norm on the signal space in whiche is defined. This class of signals generallyconsists of signals that are limited in bandwidth and in amplitude. Then in (1.13) we may neglectthe first term on the left, so that

γ(e) ≈ r. (1.15)

Since by assumption‖e‖ � ‖γ(e)‖ this implies that

‖e‖ � ‖r‖. (1.16)

In words: If the gain is large then the errore is small compared with the control inputr . Goingback to the configuration of Fig. 1.3(a), we see that this implies thatψ(y) ≈ r , or

y ≈ ψ−1(r ), (1.17)

whereψ−1 is theinverseof the mapψ (assuming that it exists).Note that it is assumed that the feedback equation has a bounded solution2 efor every bounded

r . This is not necessarily always the case. Ife is bounded for every boundedr then the closed-loop system by definition is BIBO stable3. Hence, the existence of solutions to the feedbackequation is equivalent to the (BIBO) stability of the closed-loop system.

Note also that generally the gain may only be expected to be large for aclassof error signals,denotedE . The class usually consists of band- and amplitude-limited signals, and depends onthe “capacity” of the plant.

2A signal is bounded if its norm is finite. Norms of signals are discussed in§ 4.3. See also§ 1.3.3A system is BIBO (bounded-input bounded-output) stable if every bounded input results in a bounded output (see§ 1.3).

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Example 1.2.4 (Proportional control of the cruise control system). A simple form of feedbackthat works reasonably well but not more than that for the cruise control system of Example 1.2.1is proportional feedback. This means that the throttle opening is controlled according to

u(t)− u0 = g[r (t)−w(t)], (1.18)

with the gain g a constant andu0 a nominal throttle setting. Denotew0 as the steady-statecruising speed corresponding to the nominal throttle settingu0, and writew(t) = w0 + w(t) asin Example 1.2.1. Settingr (t) = r (t)−w0 we have

u(t) = g[r(t)− w(t)]. (1.19)

Substituting this into the linearized equation (1.7) (once again omitting the circumflexes) we have

w = −1θw+ g

T(r −w), (1.20)

that is,

w = −(

1θ

+ gT

)w+ g

Tr. (1.21)

Stability is ensured as long as

1θ

+ gT> 0. (1.22)

After Laplace transformation of (1.21) and solving for the Laplace transform ofw we identifythe closed-loop transfer functionHcl from

w =gT

s+ 1θ+ g

T︸︷︷︸Hcl(s)

r. (1.23)

We follow the custom of operational calculus not to distinguish between a time signal and itsLaplace transform.

Figure 1.4 gives Bode magnitude plots of the closed-loop frequency responseHcl(jω), ω ∈ R,for different values of the gaing. If the gaing is large thenHcl(jω) ≈ 1 for low frequencies. Thelarger the gain, the larger the frequency region is over which this holds. �

1.2.4. Robustness of feedback systems

The approximate identityy ≈ ψ−1(r ) (1.17)remainsvalid as long as the feedback equation hasa bounded solutione for everyr and the gain is large. The IO mapψ of the return compensatormay often be implemented with good accuracy. This results in a matching accuracy for the IOmap of the feedback system as long as the gain is large, even if the IO map of the plant is poorlydefined or has unfavorable properties. The fact that

y ≈ ψ−1(r ) (1.24)

in spite of uncertainty about the plant dynamics is calledrobustnessof the feedback system withrespect to plant uncertainty.

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|Hcl|(log scale)

1

g1

g2g3

ω (log scale)1θ+ g

T

Figure 1.4.: Magnitude plots of the closed-loop frequency response functionfor three values of the gain withg1 < g2 < g3.

Example 1.2.5 (Cruise control system). The proportional cruise feedback control system ofExample 1.2.4 is a first-order system, like the open-loop system. The closed-loop time constantθcl follows by inspection of (1.23) as

1θcl

= 1θ

+ gT. (1.25)

As long asg � Tθ

the closed-loop time constantθcl approximately equalsTg . Hence,θcl doesnot depend much on the open-loop time constantθ, which is quite variable with the speed of thevehicle. Forg � T

θwe have

Hcl(jω) ≈gT

jω+ gT

≈ 1 for |ω| ≤ gT. (1.26)

Hence, up to the frequencygT the closed-loop frequency response is very nearly equal to the unitgain. The frequency response of the open-loop system is

H(jω) =1T

jω+ 1θ

≈ θ

Tfor |ω| < 1

θ. (1.27)

The open-loop frequency response function obviously is much more sensitive to variations in thetime constantθ than the closed-loop frequency response. �

1.2.5. Linearity and bandwidth improvement by feedback

Besides robustness, several other favorable effects may be achieved by feedback. They includelinearity improvement, bandwidth improvement, and disturbance reduction.

Linearity improvementis a consequence of the fact that if the loop gain is large enough,the IO map of the feedback system approximately equals the inverseψ−1 of the IO map of thereturn compensator. If this IO map is linear, so is the IO map of the feedback system, with goodapproximation, no matter how nonlinear the plant IO mapφ is.

Also bandwidthimprovement is a result of the high gain property. If the return compensatoris a unit gain, the IO map of the feedback system is close to unity over those frequencies forwhich the feedback gain is large. This increases the bandwidth.

8


r e z zφ

ψ

δ

d

dy

disturbance

plant

return compenator (a) (b)

Figure 1.5.: (a) Feedback system with disturbance. (b) Equivalent unit feed-back configuration in the absence of the control inputr .

Example 1.2.6 (Bandwidth improvement of the cruise control system). In Example 1.2.5 thetime constant of the closed-loop proportional cruise control system is

θcl = θ

1+ gθT

. (1.28)

For positive gaing the closed-loop time constant is smaller than the open-loop time constantθ

and, hence, the closed-loop bandwidth is greater than the open-loop bandwidth. �

Exercise 1.2.7 (Steady-state linearity improvement of the proportional cruise control sys-tem). The dynamics of the vehicle are given by

Tw = u −w2. (1.29)

For a given steady-state solution(u0, w0), with w0 = √u0, consider the proportional feedback

scheme

u − u0 = g(r −w). (1.30)

Calculate the steady-state dependence ofw−w0 on r − w0 (assuming thatr is constant). Plotthis dependence forw0 = 0.5 andg = 10.

To assess the linearity improvement by feedback compare this plot with a plot ofw − w0

versusu − u0 for the open-loop system. Comment on the two plots. �

1.2.6. Disturbance reduction

A further useful property of feedback is that the effect of (external)disturbancesis reduced. Itfrequently happens that in the configuration of Fig. 1.2(a) external disturbances affect the outputy. These disturbances are usually caused by environmental effects.

The effect of disturbances may often be modeled by adding adisturbance signal dat theoutput of the plant as in Fig. 1.5(a). For simplicity we study the effect of the disturbance in theabsence of any external control input, that is, we assumer = 0. The feedback system then isdescribed by the equationsz = d + y, y = φ(e), ande = −ψ(z). Eliminating the outputy andthe error signale we havez = d + φ(e) = d + φ(−ψ(z)), or

z= d − δ(z), (1.31)

9


whereδ = (−φ) ◦ (−ψ). The mapδ is also called aloop IO map,but it is obtained by “breakingthe loop” at a different point compared with when constructing the loop IO mapγ = ψ ◦ φ.

The equation (1.31) is a feedback equation for the configuration of Fig. 1.5(b). By analogywith the configuration of Fig. 1.3(b) it follows that if the gain islarge in the sense that‖δ(z)‖ �‖z‖ then we have

‖z‖ � ‖d‖. (1.32)

This means that the outputzof the feedback system is small compared with the disturbanced, sothat the effect of the disturbance is much reduced. All this holds provided the feedback equation(1.31) has at all a bounded solutionz for any boundedd, that is, provided the closed-loop systemis BIBO stable.

Example 1.2.8 (Disturbance reduction in the proportional cruise control system). Theprogress of the cruising vehicle of Example 1.2.1 may be affected by head or tail winds andup- or downhill grades. These effects may be represented by modifying the dynamical equation(1.3) to mv = cu− ρv2 + d, with d the disturbing force. After scaling and linearization as inExample 1.2.1 this leads to the modification

w = −1θw− 1

Tu+ d (1.33)

of (1.7). Under the effect of the proportional feedback scheme (1.18) this results in the modifi-cation

w = −(

1θ

+ gT

)w+ g

Tr + d (1.34)

of (1.21). Laplace transformation and solution forw (while settingr = 0) shows that the effectof the disturbance on the closed-loop system is represented by

wcl = 1

s+ 1θcl

d. (1.35)

From (1.33) we see that in the open-loop system the effect of the disturbance on the output is

wol = 1

s+ 1θ

d. (1.36)

This signalwol actually is the “equivalent disturbance at the output” of Fig. 1.6(a). Comparisonof (1.34) and (1.35) shows that

wcl =s+ 1

θ

s+ 1θcl︸︷︷︸

S(s)

wol. (1.37)

S is known as thesensitivity functionof the closed-loop system. Figure 1.6 shows the Bodemagnitude plot of the frequency response functionS(jω). The plot shows that the open-loopdisturbances are attenuated by a factor

θcl

θ= 1

1+ gθT

(1.38)

10

1.3. Closed-loop stability

until the angular frequency 1/θ. After a gradual rise of the magnitude there is no attenuation oramplification of the disturbances for frequencies over 1/θcl.

The disturbance attenuation is not satisfactory at very low frequencies. In particular, constantdisturbances (that is, zero-frequency disturbances) are not completely eliminated becauseS(0) 6=0. This means that a steady head wind or a long uphill grade slow the car down. In§ 2.3 it isexplained how this effect may be overcome by applyingintegral control. �

1.2.7. Pitfalls of feedback

As we have shown in this section, feedback may achieve very useful effects. It also has pitfalls:

1. Naıvely making the gain of the system large may easily result in anunstablefeedbacksystem. If the feedback system is unstable then the feedback equation has no boundedsolutions and the beneficial effects of feedback are nonexistent.

2. Even if the feedback system is stable then high gain may result in overly large inputs to theplant, which the plant cannot absorb. The result is reduction of the gain and an associatedloss of performance.

3. Feedback implies measuring the output by means of an output sensor. The associatedmeasurement errorsandmeasurement noisemay cause loss of accuracy.

We return to these points in§ 1.5.


1.3.1. Introduction

In the remainder of this chapter we elaborate some of the ideas of Section 1.2 for linear time-invariant feedback systems. Most of the results are stated for single-input single-output (SISO)systems but from time to time also multi-input multi-output (MIMO) results are discussed.

We consider thetwo-degree-of-freedomconfiguration of Fig. 1.7. A MIMO or SISO plantwith transfer matrixP is connected in feedback with a forward compensator with transfer matrixC. The function of the feedback loop is to provide stability, robustness, and disturbance atten-uation. The feedback loop is connected in series with a prefilter with transfer matrixF. Thefunction of the prefilter is to improve the closed-loop response to command inputs.

The configuration of Fig. 1.7 is said to have two degrees of freedom because both the com-pensatorC and the prefilterF are free to be chosen by the designer. When the prefilter is replaced

|S(jω)|(log scale)

1

1/θ 1/θcl

θcl/θ

ω (log scale)

Figure 1.6.: Magnitude plot of the sensitivity function of the proportionalcruise control system

11


prefilterF

compensatorC

plantP

referenceinput

plantinput

errorsignal

outputr zue

Figure 1.7.: Two-degree-of-freedom feedback system configuration

(a) (b)

rr zz uue

C

C PP

Figure 1.8.: Single-degree-of-freedom feedback system configurations

with the unit system as in Fig. 1.8(a) the system is called asingle-degree-of-freedomfeedbacksystem. Also the configuration of Fig. 1.8(b), with a return compensator instead of a forwardcompensator, is called a single-degree-of-freedom system.

In § 1.8 we consider alternative two-degree-of-freedom configurations, and study which isthe most effective configuration.

1.3.2. Stability

In the rest of this section we discuss the stability of the closed-loop control system of Fig. 1.7.We assume that the overall system, including the plant, the compensator and the prefilter, has astate representation

x(t) = Ax(t)+ Br(t), (1.39)z(t)

u(t)e(t)

= Cx(t)+ Dr(t). (1.40)

The command signalr is the external input to the overall system, while the control system outputz, the plant inputu and the error signale jointly form the output. The signalx is the state of theoverall system.A, B, C, andD are constant matrices of appropriate dimensions.

The state representation of the overall system is formed by combining the state space repre-sentations of the component systems. We assume that these state space representations includeall the important dynamic aspects of the systems. They may be uncontrollable or unobservable.Besides the reference inputr the external input to the overall system may include other exogenoussignals such as disturbances and measurement noise.

Definition 1.3.1 (Stability of a closed-loop system). The feedback system of Fig. 1.7 (or anyother control system) isstableif the state representation (1.39–1.40) isasymptotically stable,thatis, if for a zero input and any initial state the state of the system asymptotically approaches thezero state as time increases. �

12


Given the state representation (1.39–1.40), the overall system is asymptotically stable if andonly if all the eigenvalues of the matrixA have strictly negative real part.

There is another important form of stability.

Definition 1.3.2 (BIBO stability). The system of Fig. 1.7 is calledBIBO stable(bounded-input-bounded-output stable) if every bounded inputr results in bounded outputsz, u, ande for anyinitial condition on the state. �

To know what “bounded” means we need a norm for the input and output signals. A signalis said to be bounded if its norm is finite. We discuss the notion of the norm of a signal at somelength in§ 4.3. For the time being we say that a (vector-valued) signalv(t) is bounded if thereexists a constantM such that|vi(t)| ≤ M for all t and for each componentvi of v.

Exercise 1.3.3 (Stability and BIBO stability).

1. Prove that if the closed-loop system is stable in the sense of Definition 1.3.1 then it is alsoBIBO stable.

2. Conversely, prove that if the system is BIBO stable and has no unstable unobservablemodes4 then it is stable in the sense of Definition 1.3.1.

3. Often BIBO stability is defined so that bounded input signals are required to result inbounded output signals forzeroinitial conditions of the state. With this definition, Part (1)of this exercise obviously still holds. Conversely, prove that if the system is BIBO stablein this sense and has no unstable unobservable and uncontrollable modes5 then it is stablein the sense of Definition 1.3.1.

�

We introduce a further form of stability. It deals with the stability of interconnected systems,of which the various one- and two-degree-of-freedom feedback systems we encountered are ex-amples. Stability in the sense of Definition 1.3.1 is independent of the presence or absence ofinputs and outputs. BIBO stability, on the other hand, is strongly related to the presence andchoice of input and output signals.Internalstability is BIBO stability but decoupled from a par-ticular choice of inputs and outputs. We define the notion of internal stability of an interconnectedsystems in the following manner.

Definition 1.3.4 (Internal stability of an interconnected system). In each “exposed intercon-nection” of the interconnected system, inject an “internal” input signalvi (with i an index), anddefine an additional “internal” output signalwi just after the injection point. Then the system issaid to beinternally stableif the system whose input consists of the joint (external and internal)inputs and whose output is formed by the joint (external and internal) outputs is BIBO stable.�

To illustrate the definition of internal stability we consider the two-degree-of-freedom feed-back configuration of Fig. 1.9. The system has the external inputr , and the external outputz.Identifying five exposed interconnections, we include five internal input-output signal pairs asshown in Fig. 1.10. The system is internally stable if the system with input (r , v1, v2, v3, v4, v5)and output (z, w1, w2, w3, w4, w5) is BIBO stable.

Exercise 1.3.5 (Stability and internal stability).4A state systemx = Ax+ Bu, y = Cx+ Du has an unobservable mode if the homogeneous equationx = Ax has a

nontrivial solutionx such thatCx= 0. The mode is unstable if this solutionx(t) does not approach 0 ast → ∞.5The state systemx = Ax+ Bu, y = Cx+ Duhas an uncontrollable mode if the state differential equationx = Ax+ Bu

has a solutionx that is independent ofu.

13


r zC PF

Figure 1.9.: Two-degree-of-freedom feedback system

r zC PF

v1w1

v2w2

v3w3

v4w4

v5w5

Figure 1.10.: Two-degree-of-freedom system with internal inputs and out-puts added

1. Prove that if the system of Fig. 1.9 is stable then it is internally stable.

2. Conversely, prove that if the system is internally stable and none of the component sys-tems has any unstable unobservable modes then the system is stable in the sense of Defi-nition 1.3.1.Hint: This follows from Exercise 1.3.3(b).

�

When using input-output descriptions, such as transfer functions, then internal stability isusually easier to check than stability in the sense of Definition 1.3.1. If no unstable unobservableand uncontrollable modes are present then internal stability is equivalent to stability in the senseof Definition 1.3.1.

We say that a controllerstabilizesa loop, or,is stabilizing, if the closed loop with that con-troller is stable or internally stable.

1.3.3. Closed-loop characteristic polynomial

For later use we discuss the relation between the state and transfer functions representations ofthe closed-loop configuration of Fig. 1.9.

The characteristic polynomialχ of a system with state space representation

x(t) = Ax(t)+ Bu(t), (1.41)

y(t) = Cx(t)+ Du(t), (1.42)

is the characteristic polynomial of itssystem matrix A,

χ(s) = det(sI − A). (1.43)

The roots of the characteristic polynomialχ are the eigenvalues of the system. The system isstable if and only if the eigenvalues all have strictly negative real parts, that is, all lie in the openleft-half complex plane.

14


The configuration of Fig. 1.9 consists of the series connection of the prefilterF with thefeedback loop of Fig. 1.11(a).

Exercise 1.3.6 (Stability of a series connection). Consider two systems, one with statex andone with statez and letχ1 andχ2 denote their respective characteristic polynomials. Prove thatthe characteristic polynomial of the series connection of the two systems with state

[ xz

]is χ1χ2.

From this it follows that the eigenvalues of the series connection consist of the eigenvalues of thefirst system together with the eigenvalues of the second system. �

We conclude that the configuration of Fig. 1.9 is stable if and only if both the prefilter andthe feedback loop are stable. To study the stability of the MIMO or SISO feedback loop of

C Pe u y

(a)

L

(b)

Figure 1.11.: MIMO or SISO feedback systems

Fig. 1.11(a) represent it as in Fig. 1.11(b).L = PC is the transfer matrix of the series connectionof the compensator and the plant.L is called theloop transfer matrix. We assume thatL isproper, that is,L(∞) exists.

Suppose that the series connectionL has the characteristic polynomialχ. We callχ theopen-loop characteristic polynomial. It is proved in§ 1.10 that the characteristic polynomial of theclosed-loop system of Fig. 1.11 is

χcl(s) = χ(s)det[I + L(s)]

det[I + L(∞)]. (1.44)

We callχcl theclosed-loop characteristic polynomial.In the SISO case we may write the loop transfer function as

L(s) = R(s)Q(s)

, (1.45)

with R andQ polynomials, whereQ = χ is the open-loop characteristic polynomial. Note thatwe allownocancellation between the numerator polynomial and the characteristic polynomial inthe denominator. It follows from (1.44) that within the constant factor 1+ L(∞) the closed-loopcharacteristic polynomial is

χ(s)(1+ L(s)) = Q(s)[1 + R(s)Q(s)

] = Q(s)+ R(s). (1.46)

We return to the configuration of Fig. 1.11(a) and write the transfer functions of the plant and thecompensator as

P(s) = N(s)D(s)

, C(s) = Y(s)X(s)

. (1.47)

D and X are the open-loop characteristic polynomials of the plant and the compensator, re-spectively. N and Y are their numerator polynomials. Again we allow no cancellation be-tween the numerator polynomials and the characteristic polynomials in the denominators. Since

15


R(s)= N(s)Y(s) andQ(s)= D(s)X(s)we obtain from (1.46) the well-known result that withina constant factor the closed-loop characteristic polynomial of the configuration of Fig. 1.11(a) is

D(s)X(s)+ N(s)Y(s). (1.48)

With a slight abuse of terminology this polynomial is often referred to as the closed-loop char-acteristic polynomial. The actual characteristic polynomial is obtained by dividing (1.48) by itsleading coefficient6.

Exercise 1.3.7 (Hidden modes). Suppose that the polynomialsN andD have a common poly-nomial factor. This factor corresponds to one or several unobservable or uncontrollable modes ofthe plant. Show that the closed-loop characteristic polynomial also contains this factor. Hence,the eigenvalues corresponding to unobservable or uncontrollable modes cannot be changed byfeedback. In particular, any unstable uncontrollable or unobservable modes cannot be stabilized.

The same observation holds for any unobservable and uncontrollable poles of the compen-sator. �

The stability of a feedback system may be tested by calculating the roots of its characteristicpolynomial. The system is stable if and only if each root has strictly negative real part. TheRouth-Hurwitz stability criterion,which is reviewed in Section 5.2, allows to test for stabilitywithout explicitly computing the roots. Anecessarybut not sufficient condition for stability isthat all the coefficients of the characteristic polynomial have the same sign. This condition isknown asDescartes’ rule of signs.

1.3.4. Pole assignment

The relation

χ = DX + NY (1.49)

for the characteristic polynomial (possibly within a constant) may be used for what is known aspole assignmentor pole placement. If the plant numerator and denominator polynomialsN andD are known, andχ is specified, then (1.49) may be considered as an equation in the unknownpolynomialsX andY. This equation is known as theBezoutequation. If the polynomialsN andD have a common nontrivial polynomial factor that is not a factor ofχ then obviously no solutionexists. Otherwise, a solution always exists.

The Bezout equation (1.49) may be solved by expanding the various polynomials as powersof the undeterminate variable and equate coefficients of like powers. This leads to a set of linearequations in the coefficients of the unknown polynomialsX andY, which may easily be solved.The equations are known as theSylvesterequations (Kailath 1980).

To set up the Sylvester equations we need to know the degrees of the polynomialsX andY. Suppose thatP = N/D is strictly proper7, with degD = n and degN < n given. We try tofind a strictly proper compensatorC = Y/X with degrees degX = m and degY = m− 1 to bedetermined. The degree ofχ= DX+ NY is n+ m, so that by equating coefficients of like powerswe obtainn + m+ 1 equations. Setting this number equal to the number 2m+ 1 of unknowncoefficients of the polynomialsY andX it follows thatm= n. Thus, we expect to solve the poleassignment problem with a compensator of the same order as the plant.

6That is, the coefficient of the highest-order term.7A rational function or matrixP is strictly proper if lim |s|→∞ P(s)= 0. A rational functionP is strictly proper if and

only if the degree of its numerator is less than the degree of its denominator.

16


Example 1.3.8 (Pole assignment). Consider a second-order plant with transfer function

P(s) = 1s2. (1.50)

Because the compensator is expected to have order two we need to assign four closed-loop poles.We aim at a dominant pole pair at1

2

√2(−1± j) to obtain a closed-loop bandwidth of 1 [rad/s],

and place a non-dominant pair at 5√

2(−1± j). Hence,

χ(s) = (s2 + s√

2+ 1)(s2 + 10√

2s+ 100)

= s4 + 11√

2s3 + 121s2 + 110√

2s+ 100. (1.51)

Write X(s) = x2s2 + x1s+ x0 andY(s) = y1s+ y0. Then

D(s)X(s)+ N(s)Y(s) = s2(x2s2 + x1s+ x0)+ (y1s+ y0)

= x2s4 + x1s3 + x0s2 + y1s+ y0. (1.52)

Comparing (1.51) and (1.52) the unknown coefficients follow by inspection, and we see that

X(s) = s2 + 11√

2s+ 121, (1.53)

Y(s) = 110√

2s+ 100. (1.54)

�

Exercise 1.3.9 (Sylvester equations). More generally, suppose that

P(s) = bn−1sn−1 + bn−2sn−2 + · · · + b0

ansn + an−1sn−1 + · · · + a0, (1.55)

C(s) = yn−1sn−1 + yn−2sn−2 + · · · + y0

xnsn + xn−1sn−1 + · · · + x0, (1.56)

χ(s) = χ2ns2n + χ2n−1s2n−1 + · · · + χ0. (1.57)

Show that the equationχ = DX + NY may be arranged as

an 0 · · · · · · 0an−1 an 0 · · · 0· · · · · · · · · · · · · · ·· · · · · · · · · · · · · · ·a0 a1 · · · · · · an

0 a0 a1 · · · an−1

· · · · · · · · · · · · · · ·· · · · · · · · · · · · · · ·0 · · · · · · 0 a0

︸︷︷︸A

xn

xn−1

· · ·· · ·· · ·· · ·x0

︸︷︷︸x

+

0 · · · · · · · · · 00 · · · · · · · · · 0

bn−1 0 0 · · · 0bn−2 bn−1 0 · · · 0· · · · · · · · · · · · · · ·b0 b1 · · · · · · bn−1

0 b0 b1 · · · bn−2

· · · · · · · · · · · · · · ·0 · · · · · · 0 b0

︸︷︷︸B

yn−1

yn−2

· · ·· · ·· · ·· · ·y0

︸︷︷︸y

=

χ2n

χ2n−1

· · ·· · ·· · ·· · ·χ0

︸︷︷︸c

(1.58)

17


0

Im

Re

ω = −∞

ω = ∞

ω = 0

ω

−1 k

Figure 1.12.: Nyquist plot of the loop gain transfer functionL(s) = k/(1+sθ)

This in turn may be represented as

[A B

] [xy

]= c (1.59)

and solved forx andy. If the polynomialsD andN are coprime8 then the square matrix[A B

]is nonsingular. �

1.3.5. Nyquist criterion

In classical control theory closed-loop stability is often studied with the help of theNyquiststability criterion,which is a well-known graphical test. Consider the simple MIMO feedbackloop of Fig. 1.11. The block marked “L” is the series connection of the compensatorK and theplant P. The transfer matrixL = PK is called theloop gain matrix— or loop gain,for short —of the feedback loop.

For a SISO system,L is a scalar function. Define theNyquist plot9 of the scalar loop gainLas the curve traced in the complex plane by

L(jω), ω ∈ R. (1.60)

Because for finite-dimensional systemsL is a rational function with real coefficients, the Nyquistplot is symmetric with respect to the real axis. Associated with increasingω we may define apositive direction along the locus. IfL is proper10 and has no poles on the imaginary axis thenthe locus is a closed curve. By way of example, Fig. 1.12 shows the Nyquist plot of the loop gaintransfer function

L(s) = k1+ sθ

, (1.61)

with k andθ positive constants. This is the loop gain of the cruise control system of Example 1.2.5with k = gθ/T.

We first state the best known version of the Nyquist criterion.

8That is, they have no nontrivial common factors.9The Nyquist plot is discussed at more length in§ 2.4.3.

10A rational matrix functionL is proper if lim |s|→∞ L(s) exists. For a rational functionL this means that the degree ofits numerator is not greater than that of its denominator.

18


Summary 1.3.10 (Nyquist stability criterion for SISO open-loop stable systems). Assumethat in the feedback configuration of Fig. 1.11 the SISO systemL is open-loop stable. Thenthe closed-loop system is stable if and only if the Nyquist plot ofL does not encircle the point−1. �

It follows immediately from the Nyquist criterion and Fig. 1.12 that ifL(s) = k/(1+ sθ) andthe block “L” is stable then the closed-loop system is stable for all positivek andθ.

Exercise 1.3.11 (Nyquist plot). Verify the Nyquist plot of Fig. 1.12. �

Exercise 1.3.12 (Stability of compensated feedback system). Consider a SISO single-degree-of-freedom system as in Fig. 1.8(a) or (b), and define the loop gainL = PK. Prove that if boththe compensator and the plant are stable andL satisfies the Nyquist criterion then the feedbacksystem is stable. �

The result of Summary 1.3.10 is a special case of thegeneralized Nyquist criterion. Thegeneralized Nyquist principle applies to a MIMO unit feedback system of the form of Fig. 1.11,and may be phrased as follows:

Summary 1.3.13 (Generalized Nyquist criterion). Suppose that the loop gain transfer functionL of the MIMO feedback system of Fig. 1.11 is proper such thatI + L(j∞) is nonsingular(this guarantees the feedback system to be well-defined) and has no poles on the imaginary axis.Assume also that the Nyquist plot of det( I + L) does not pass through the origin. Then

the number of unstable closed-loop poles=

the number of times the Nyquist plot of det( I + L) encircles the origin clockwise11

+the number of unstable open-loop poles.

It follows that the closed-loop system is stable if and only if the number of encirclements ofdet( I + L) equals the negative of the number of unstable open-loop poles. �

Similarly, the “unstable open-loop poles” are the right-half plane eigenvalues of the systemmatrix of the state space representation of the open-loop system. This includes any uncontrollableor unobservable eigenvalues. The “unstable closed-loop poles” similarly are the right-half planeeigenvalues of the system matrix of the closed-loop system.

In particular, it follows from the generalized Nyquist criterion that if the open-loop systemis stable then the closed-loop system is stable if and only if the number of encirclements is zero(i.e., the Nyquist plot of det( I + L) doesnot encircle the origin).

For SISO systems the loop gainL is scalar, so that the number of times the Nyquist plot ofdet( I + L)= 1+ L encircles the origin equals the number of times the Nyquist plot ofL encirclesthe point−1.

The condition that det( I + L) has no poles on the imaginary axis and does not pass throughthe origin may be relaxed, at the expense of making the analysis more complicated (see forinstance Dorf (1992)).

The proof of the Nyquist criterion is given in§ 1.10. More about Nyquist plots may be foundin § 2.4.3.

11This means the number of clockwise encirclements minus the number of anticlockwise encirclements. I.e., this numbermay be negative.

19


1.3.6. Existence of a stable stabilizing compensator

A compensator that stabilizes the closed-loop system but by itself is unstable is difficult to handlein start-up, open-loop, input saturating or testing situations. There are unstable plants for whicha stable stabilizing controller does not exist. The following result was formulated and proved byYoula, Bongiorno, and Lu (1974); see also Anderson and Jury (1976) and Blondel (1994).

Summary 1.3.14 (Existence of stable stabilizing controller). Consider the unit feedback sys-tem of Fig. 1.11(a) with plantP and compensatorC.

The plant possesses theparity interlacing propertyif it has an even number of poles (countedaccording to multiplicity) between each pair of zeros on the positive real axis (including zeros atinfinity.)

There exists a stable compensatorC that makes the closed-loop stable if and only if the plantP has the parity interlacing property. �

If the denominator of the plant transfer functionP has degreen and its numerator degreemthen the plant hasn poles andm (finite) zeros. Ifm< n then the plant is said to haven− mzerosat infinity.

Exercise 1.3.15 (Parity interlacing property). Check that the plant

P(s) = s(s− 1)2

(1.62)

possesses the parity interlacing property while

P(s) = (s− 1)(s− 3)s(s− 2)

(1.63)

does not. Find a stabilizing compensator for each of these two plants (which for the first plant isitself stable.) �

1.4. Stability robustness

1.4.1. Introduction

In this section we consider SISO feedback systems with the configuration of Fig. 1.13. We discusstheir stability robustness,that is, the property that the closed-loop system remains stable underchanges of the plant and the compensator. This discussion focusses on theloop gain L= PC,with P the plant transfer function, andC the compensator transfer function. For simplicity weassume that the system isopen-loop stable,that is, bothP andC represent the transfer functionof a stable system.

We also assume the existence of anominalfeedback loop with loop gainL0, which is theloop gain that is supposed to be valid under nominal circumstances.

1.4.2. Stability margins

The closed-loop system of Fig. 1.13 remains stable under perturbations of the loop gainL as longas the Nyquist plot of the perturbed loop gain does not encircle the point−1. Intuitively, thismay be accomplished by “keeping the Nyquist plot of the nominal feedback system away fromthe point−1.”

The classicgain marginandphase marginare well-known indicators for how closely theNyquist plot approaches the point−1.

20


e u yC P

Figure 1.13.: Feedback system configuration

Gain margin The gain margin is the smallest positive numberkm by which the Nyquist plot mustbe multiplied so that it passes through the point−1. We have

km = 1|L(jωr )| , (1.64)

whereωr is the angular frequency for which the Nyquist plot intersects the negative realaxis furthest from the origin (see Fig. 1.14).

Phase margin The phase margin is the extra phaseφm that must be added to make the Nyquistplot pass through the point−1. The phase marginφm is the angle between the negativereal axis andL(jωm), whereωm is the angular frequency where the Nyquist plot intersectsthe unit circle closest to the point−1 (see again Fig. 1.14).

−1 1

L

ω

Im

Re

reciprocal of thegain margin

modulus margin

phase margin

Figure 1.14.: Robustness margins

In classical feedback system design, robustness is often specified by establishing minimum valuesfor the gain and phase margin. Practical requirements arekm > 2 for the gain margin and 30◦ <φm< 60◦ for the phase margin.

The gain and phase margin do not necessarily adequately characterize the robustness. Fig-ure 1.15 shows an example of a Nyquist plot with excellent gain and phase margins but where arelatively smalljoint perturbation of gain and phase suffices to destabilize the system. For thisreason Landau, Rolland, Cyrot, and Voda (1993) introduced two more margins.

Modulus margin 12 The modulus marginsm is the radius of the smallest circle with center−1that is tangent to the Nyquist plot. Figure 1.14 illustrates this. The modulus margin verydirectly expresses how far the Nyquist plot stays away from−1.

12French:marge de module.

21


ω

-10

Im

Re

L

Figure 1.15.: This Nyquist plot has good gain and phase margins but a smallsimultaneous perturbation of gain and phase destabilizes the system

Delay margin 13 The delay marginτm is the smallest extra delay that may be introduced in theloop that destabilizes the system. The delay margin is linked to the phase marginφm bythe relation

τm = minω∗

φ∗ω∗. (1.65)

Hereω∗ ranges over all nonnegative frequencies at which the Nyquist plot intersects theunit circle, andφ∗ denotes the corresponding phaseφ∗ = argL(jω∗). In particularτm ≤ φm

ωm.

A practical specification for the modulus margin issm> 0.5. The delay margin should be at leastof the order of 1

2B , whereB is the bandwidth (in terms of angular frequency) of the closed-loopsystem.

Adequate margins of these types are not only needed for robustness, but also to achieve asatisfactory time response of the closed-loop system. If the margins are small, the Nyquist plotapproaches the point−1 closely. This means that the stability boundary is approached closely,manifesting itself by closed-loop poles that are very near to the imaginary axis. These closed-loop poles may cause an oscillatory response (called “ringing” if the resonance frequency is highand the damping small.)

Exercise 1.4.1 (Relation between robustness margins). Prove that the gain marginkm and thephase marginφm are related to the modulus marginsm by the inequalities

km ≥ 11− sm

, φm ≥ 2 arcsinsm

2. (1.66)

This means that ifsm ≥ 12 thenkm ≥ 2 andφm ≥ 2 arcsin1

4 ≈ 28.96◦ (Landau, Rolland, Cyrot,and Voda 1993). The converse is not true in general. �

1.4.3. Robustness for loop gain perturbations

The robustness specifications discussed so far are all rather qualitative. They break down whenthe system is not open-loop stable, and, even more spectacularly, for MIMO systems. We intro-duce a more refined measure of stability robustness by considering the effect of plant perturba-tions on the Nyquist plot more in detail. For the time being the assumptions that the feedbacksystem is SISO and open-loop stable are upheld. Both are relaxed later.13French:marge de retard.

22


−1 0 Re

Im

L(jω)L0(jω)

L

L0

Figure 1.16.: Nominal and perturbed Nyquist plots

Naturally, we suppose the nominal feedback system to be well-designed so that it is closed-loop stable. We investigate whether the feedback systemremainsstable when the loop gain isperturbed from the nominal loop gainL0 to the actual loop gainL.

By the Nyquist criterion, the Nyquist plot of the nominal loop gainL0 does not encircle thepoint−1, as shown in Fig. 1.16. The actual closed-loop system is stable if also the Nyquist plotof the actual loop gainL does not encircle−1.

It is easy to see by inspection of Fig. 1.16 that the Nyquist plot ofL definitely does notencircle the point−1 if for all ω ∈ R the distance|L(jω)− L0(jω)| between any pointL(jω) andthe corresponding pointL0(jω) is lessthan the distance|L0(jω)+ 1| of the pointL0(jω) and thepoint−1, that is, if

|L(jω)− L0(jω)| < |L0(jω)+ 1| for all ω ∈ R. (1.67)

This is equivalent to

|L(jω)− L0(jω)||L0(jω)| · |L0(jω)|

|L0(jω)+ 1| < 1 for all ω ∈ R. (1.68)

Define thecomplementary sensitivity function T0 of the nominal closed-loop system as

T0 = L0

1+ L0. (1.69)

T0 bears its name because its complement

1− T0 = 11+ L0

= S0 (1.70)

is thesensitivity function. The sensitivity function plays an important role in assessing the effectof disturbances on the feedback system, and is discussed in Section 1.5.

GivenT0, it follows from (1.68) that if

|L(jω)− L0(jω)||L0(jω)| · |T0(jω)| < 1 for all ω ∈ R (1.71)

then the perturbed closed-loop system is stable.The factor|L(jω)− L0(jω)|/|L0(jω)| in this expression is therelativesize of the perturbation

of the loop gainL from its nominal valueL0. The relation (1.71) shows that the closed-loopsystem is guaranteed to be stable as long as the relative perturbations satisfy

|L(jω)− L0(jω)||L0(jω)| <

1|T0(jω)| for all ω ∈ R. (1.72)

23


The larger the magnitude of the complementary sensitivity function is, the smaller is the allowableperturbation.

This result is discussed more extensively in Section 5.6, where also its MIMO version is de-scribed. It originates from Doyle (1979). The stability robustness condition has been obtainedunder the assumption that the open-loop system is stable. In fact, it also holds for open-loopunstable systems,providedthe number of right-half plane poles remains invariant under pertur-bation.

Summary 1.4.2 (Doyle’s stability robustness criterion). Suppose that the closed-loop systemof Fig. 1.13 is nominally stable. Then it remains stable under perturbations that do not affect thenumber of open-loop unstable poles if

|L(jω)− L0(jω)||L0(jω)| <

1|T0(jω)| for all ω ∈ R, (1.73)

with T0 the nominal complementary sensitivity function of the closed-loop system. �

Exercise 1.4.3 (No poles may cross the imaginary axis). Use the general form of the Nyquiststability criterion of Summary 1.3.13 to prove the result of Summary 1.4.2. �

Doyle’s stability robustness condition is asufficientcondition. This means that there maywell exist perturbations that do not satisfy (1.73) but nevertheless do not destabilize the closed-loop system. This limits the applicability of the result. With a suitable modification the conditionis also necessary, however. Suppose that the relative perturbations are known to bounded in theform

|L(jω)− L0(jω)||L0(jω)| ≤ |W(jω)| for all ω ∈ R, (1.74)

with W a given function. Then the condition (1.73) is implied by the inequality

|T0(jω)| < 1|W(jω)| for all ω ∈ R. (1.75)

Thus, if the latter condition holds, robust stability is guaranteed for all perturbations satisfying(1.74). Moreover, (1.75) is not only sufficient but alsonecessaryto guarantee stability forall per-turbations satisfying (1.74) (Vidysagar 1985). Such perturbations are said to “fill the uncertaintyenvelope.”

Summary 1.4.4 (Stability robustness). Suppose that the closed-loop system of Fig. 1.13 isnominally stable. It remains stable under all perturbations that do not affect the number of open-loop unstable poles satisfying the bound

|L(jω)− L0(jω)||L0(jω)| ≤ |W(jω)| for all ω ∈ R, (1.76)

with W a given function, if and only if

|T0(jω)| < 1|W(jω)| for all ω ∈ R. (1.77)

�

Again, the MIMO version is presented in Section 5.6. The result is further discussed in§ 1.5.

24


1.4.4. Inverse loop gain perturbations

According to the Nyquist criterion, the closed-loop system remains stable under perturbationas long as under perturbation the Nyquist plot of the loop gain does not cross the point−1.Equivalently, the closed-loop system remains stable under perturbation as long as theinverse1/L of the loop gain does not cross the point−1. Thus, the sufficient condition (1.67) may bereplaced with the sufficient condition∣∣∣∣ 1

L(jω)− 1

L0(jω)

∣∣∣∣ <∣∣∣∣ 1L0(jω)

+ 1

∣∣∣∣ for all ω ∈ R. (1.78)

Dividing by the inverse 1/L0 of the nominal loop gain we find that a sufficient condition forrobust stability is that∣∣∣∣∣

1L( jω) − 1

L0( jω)1

L0( jω)

∣∣∣∣∣ <∣∣∣∣∣

1L0( jω)

+ 11

L0( jω)

∣∣∣∣∣ = |1+ L0(jω)| = 1|S0(jω)| (1.79)

for all ω ∈ R. This in turn leads to the following conclusions.

Summary 1.4.5 (Inverse loop gain stability robustness criterion). Suppose that the closed-loop system of Fig. 1.13 is nominally stable. It remains stable under perturbations that do notaffect the number of open-loop right-half plane zeros of the loop gain if∣∣∣∣∣

1L( jω) − 1

L0( jω)1

L0( jω)

∣∣∣∣∣ < 1|S0(jω)| for all ω ∈ R, (1.80)

with S0 the nominal sensitivity function of the closed-loop system. �

Exercise 1.4.6 (Reversal of the role of the right half plane poles and the right-half planezeros). Note that the role of the right-half plane poles has been taken by the right-half planezeros. Explain this by deriving a stability condition based on theinverseNyquist plot, that is, thepolar plot of 1/L. �

Again the result may be generalized to a sufficient and necessary condition.

Summary 1.4.7 (Stability robustness under inverse perturbation). Suppose that the closed-loop system of Fig. 1.13 is nominally stable. It remains stable under all perturbations that do notaffect the number of right-half plane zeros satisfying the bound∣∣∣∣∣

1L( jω) − 1

L0( jω)1

L0( jω)

∣∣∣∣∣ ≤ |W(jω)| for all ω ∈ R, (1.81)

with W a given function, if and only if

|S0(jω)| < 1|W(jω)| for all ω ∈ R. (1.82)

�

Thus, for robustnessboth the sensitivity functionS and its complementT are important.Later it is seen that for practical feedback design the complementary functionsSandT need tobe made small in complementary frequency regions (for low frequencies and for high frequencies,respectively).

We illustrate these results by an example.

25


Example 1.4.8 (Frequency dependent robustness bounds). Consider a SISO feedback loopwith loop gain

L(s) = k1+ sθ

, (1.83)

with k andθ positive constants. The nominal sensitivity and complementary sensitivity functionsS0 andT0 are

S0(s) = 11+ L0(s)

= 1+ sθ0

1+ k0 + sθ0, (1.84)

T0(s) = L0(s)1+ L0(s)

= k0

1+ k0 + sθ0. (1.85)

with k0 andθ0 the nominal values ofk andθ, respectively. Figure 1.17 displays doubly logarith-mic magnitude plots of 1/S0 and 1/T0.

ω (log scale)

ω (log scale)1+k0θ0

1+k0θ0

1+k0k0

1+ k0

1

1

1θ0

1|T0|

(log scale)

1|S0|

(log scale)

Figure 1.17.: Magnitude plots of 1/T0 and 1/S0.

By the result of Summary 1.4.4 we conclude from the magnitude plot of 1/T0 that for lowfrequencies (up to the bandwidth(k0 + 1)/θ0) relative perturbations of the loop gainL of relativesize up to 1 and slightly larger are permitted while for higher frequencies increasingly largerperturbations are allowed without danger of destabilization of the closed-loop system.

By the result of Summary 1.4.7, on the other hand, we conclude from the magnitude plotof 1/S0, that for low frequencies (up to the frequency 1/θ0) relative perturbations of the loopgain up to 1+ k0 are permitted. For high frequencies (greater than the bandwidth) the allowablerelative size of the perturbations drops to the value 1. �

Exercise 1.4.9 (Landau’s modulus margin and the sensitivity function).

1. In Subsection 1.4.2 the modulus marginsm is defined as the distance from the point−1 tothe Nyquist plot of the loop gainL:

sm = infω∈R

|1+ L(jω)|. (1.86)

Prove that 1/sm is the peak value of the magnitude of the sensitivity functionS.

26

1.5. Frequency response design goals

2. If the Nyquist plot of the loop gainL approaches the point−1 closely then so does that ofthe inverse loop gain 1/L. Therefore, the number

rm = infω∈R

∣∣∣∣1+ 1L(jω)

∣∣∣∣ (1.87)

may also be viewed as a robustness margin. Prove that 1/rm is the peak value of themagnitude of the complementary sensitivity functionT.

�


1.5.1. Introduction

In this section we translate the design targets for a linear time-invariant two-degree-of-freedomfeedback system as in Fig. 1.18 into requirements on various closed-loop frequency responsefunctions. The design goals are

r e u zPCF

Figure 1.18.: Two-degree-of-freedom feedback system

• closed-loop stability,

• disturbance attenuation,

• satisfactory closed-loop command response,

• stability robustness, and

• robustness of the closed-loop response,

within the limitations set by

• plant capacity, and

• corruption by measurement noise.

We discuss these aspects one by one for single-input-single-output feedback systems.

1.5.2. Closed-loop stability

Suppose that the feedback system of Fig. 1.13 is open-loop stable. By the Nyquist stabilitycriterion, for closed-loop stability the loop gain should be shaped such that the Nyquist plot ofLdoes not encircle the point−1.

27


e

v

zL

Figure 1.19.: Feedback loop with disturbance

1.5.3. Disturbance attenuation and bandwidth — the sensitivity function

To study disturbance attenuation, consider the block diagram of Fig. 1.19, wherev representsthe equivalent disturbance at the output of the plant. In terms of Laplace transforms, the signalbalance equation may be written asz = v− Lz. Solution forz results in

z= 11+ L︸︷︷︸

S

v = Sv, (1.88)

whereS is the sensitivity function of the closed-loop system. The smaller|S(jω)| is, withω ∈ R,the more the disturbances are attenuated at the angular frequencyω. |S| is small if the magnitudeof the loop gainL is large. Hence, for disturbance attenuation it is necessary to shape the loopgain such that it is large over those frequencies where disturbance attenuation is needed.

Making the loop gainL large over a large frequency band easily results in error signalseandresulting plant inputsu that are larger than the plant can absorb. Therefore,L can only be madelarge over a limited frequency band. This is usually a low-pass band, that is, a band that rangesfrom frequency zero up to a maximal frequencyB. The numberB is called thebandwidthof thefeedback loop. Effective disturbance attenuation is only achieved up to the frequencyB.

The larger the “capacity” of the plant is, that is, the larger the inputs are the plant can handlebefore it saturates or otherwise fails, the larger the maximally achievable bandwidth usuallyis. For plants whose transfer functions have zeros with nonnegative real parts, however, themaximally achievable bandwidth is limited by the location of the right-half plane zero closest tothe origin. This is discussed in Section 1.5.

Figure 1.20(a) shows an “ideal” shape of the magnitude of the sensitivity function. It is smallfor low frequencies and approaches the value 1 at high frequencies. Values greater than 1 andpeaking are to be avoided. Peaking easily happens near the point where the curve crosses overthe level 1 (the 0 dB line).

The desired shape for the sensitivity functionS implies a matching shape for the magnitudeof the complementary sensitivity functionT = 1− S. Figure 1.20(b) shows a possible shape14 forthe complementary sensitivityT corresponding to the sensivity function of Fig. 1.20(a). WhenS is as shown in Fig. 1.20(a) thenT is close to 1 at low frequencies and decreases to 0 at highfrequencies.

It may be necessary to impose further requirements on the shape of the sensitivity functionif the disturbances have a distinct frequency profile. Consider for instance the situation that theactual disturbances enter the plant internally, or even at the plant input, and that the plant ishighly oscillatory. Then the equivalent disturbance at the output is also oscillatory. To attenuatethese disturbances effectively the sensitivity function should be small at and near the resonance

14Note thatSandT are complementary, not|S| and|T|.

28


(a) (b)

(log

scal

e)

(log

scal

e)

angular frequency (log scale) angular frequency (log scale)

|S| |T|

BB

11

Figure 1.20.: (a) “Ideal” sensitivity function. (b) A corresponding comple-mentary sensitivity function.

frequency. For this reason it sometimes is useful to replace the requirement thatSbe small withthe requirement that

|S(jω)V(jω)| (1.89)

be small over a suitable low frequency range. The shape of the weighting functionV reflects thefrequency contents of the disturbances. If the actual disturbances enter the system at the plantinput then a possible choice is to letV = P.

Exercise 1.5.1 (Plant input disturbance for oscillatory plant). Consider an oscillatory second-order plant with transfer function

P(s) = ω20

s2 + 2ζ0ω0s+ ω20

. (1.90)

Choose the compensator

C(s) = k

ω20

s2 + 2ζ0ω0s+ ω20

s(s+ α). (1.91)

Show that the sensitivity functionS of the closed-loop system is independent of the resonancefrequencyω0 and the relative dampingζ0. Selectk andα such that a well-behaved high-passsensitivity function is obtained.

Next, select the resonance frequencyω0 well within the closed-loop bandwidth and take therelative dampingζ0 small so that the plant is quite oscillatory. Demonstrate by simulation thatthe closed-loop response to disturbances at the plant input reflects this oscillatory nature eventhough the closed-loop sensitivity function is quite well behaved. Show that this oscillatorybehavior also appears in the response of the closed-loop system to a nonzero initial condition ofthe plant. �

29


1.5.4. Command response — the complementary sensitivity function

The response of the two-degree-of-freedom configuration of Fig. 1.21 to the command signalrfollows from the signal balance equationz= PC(−z+ Fr ). Solution forz results in

z= PC1+ PC

F︸︷︷︸H

r. (1.92)

Theclosed-loop transfer function Hmay be expressed as

H = L1+ L︸︷︷︸

T

F = T F, (1.93)

with L = PC the loop gain andT the complementary sensitivity function.Adequate loop shaping ideally results in a complementary sensitivity functionT that is close

to 1 up to the bandwidth, and transits smoothly to zero above this frequency. Thus, without aprefilter F (that is, withF = 1), the closed-loop transfer functionH ideally is low-pass with thesame bandwidth as the frequency band for disturbance attenuation.

Like for the sensitivity function, the plant dynamics impose limitations on the shape thatTmay assume. In particular, right-half plane plant poles constrain the frequency above whichTmay be made to roll off. This is discussed in Section 1.5.

If the shape and bandwidth ofT are satisfactory then no prefilterF is needed. If the closed-loop bandwidth isgreaterthan necessary for adequate command signal response then the prefilterF may be used to reduce the bandwidth of the closed-loop transfer functionH to prevent overlylarge plant inputs. If the bandwidth islessthan required for good command signal response theprefilter may be used to compensate for this. A better solution may be to increase the closed-loopbandwidth. If this is not possible then probably the plant capacity is too small.

1.5.5. Plant capacity — the input sensitivity function

Any physical plant has limited “capacity,” that is, can absorb inputs of limited magnitude only.The configuration of Fig. 1.21 includes both the disturbancesv and the measurement noisem. Interms of Laplace transforms we have the signal balanceu = C(Fr − m− v− Pu). This may besolved foru as

u = CI + CP︸︷︷︸

M

(Fr − m− v). (1.94)

The functionM determines the sensitivity of the plant input to disturbances and the commandsignal. It is sometimes known as theinput sensitivity function.

If the loop gainL = CP is large then the input sensitivityM approximately equals theinverse1/P of the plant transfer function. If the open-loop plant has zeros in the right-half complexplane then 1/P is unstable. For this reason the right-half plane open-loop plant zeros limit theclosed-loop bandwidth. The input sensitivity functionM may only be made equal to 1/P upto the frequency which equals the magnitude of the right-half plane plant zero with the smallestmagnitude.

The input sensitivity functionM is connected to the complementary sensitivity functionT bythe relation

T = M P. (1.95)

30


r

m

u

v

e zPCF

Figure 1.21.: Two-degree-of-freedom system with disturbances and mea-surement noise

By this connection, for a fixed plant transfer functionP design requirements on the input sen-sitivity function M may be translated into corresponding requirements on the complementarysensitivityT, and vice-versa.

To prevent overly large inputs, generallyM should not be too large. At low frequencies a highloop gain and correspondingly large values ofM are prerequisites for low sensitivity. If theselarge values are not acceptable, the plant capacity is inadequate, and either the plant needs to bereplaced with a more powerful one, or the specifications need to be relaxed. At high frequencies— that is, at frequencies above the bandwidth —M should decrease as fast as possible. This isconsistent with the robustness requirement thatT decrease fast.

Except by Horowitz (Horowitz 1963) the term “plant capacity” does not appear to be usedwidely in the control literature. Nevertheless it is an important notion. The maximum bandwidthdetermined by the plant capacity may roughly be estimated as follows. Consider the response ofthe plant to the step inputa1(t), t ∈ R, with a the largest amplitude the plant can handle beforeit saturates or otherwise fails, and1 the unit step function. Letθ be half the time needed until theoutput either reaches

1. 85% of its steady-state value, or

2. 85% of the largest feasible output amplitude,

whichever is less. Then the angular frequency 1/θ may be taken as an indication of the largestpossible bandwidth of the system.

This rule of thumb is based on the observation that the first-order step response 1− e−t/θ,t ≥ 0, reaches the value 0.865 at time 2θ.

Exercise 1.5.2 (Input sensitivity function for oscillatory plant). Consider a stabilizing feed-back compensator of the form (1.91) for the plant (1.90), withk andα selected as suggestedin Exercise 1.5.1. Compute and plot the resulting input sensitivity functionM and discuss itsbehavior. �

1.5.6. Measurement noise

To study the effect of measurement noise on the closed-loop output we again consider the con-figuration of Fig. 1.21. By solving the signal balancez = v+ PC(Fr − m− z) for the outputzwe find

z= 11+ PC︸︷︷︸

S

v+ PC1+ PC︸︷︷︸

T

Fr − PC1+ PC︸︷︷︸

T

m. (1.96)

31


This shows that the influence of the measurement noisem on the control system output is de-termined by the complementary sensitivity functionT. For low frequencies, where by the otherdesign requirementsT is close to 1, the measurement noise fully affects the output. This empha-sizes the need for good, low-noise sensors.

1.5.7. Stability robustness

In Section 1.4 it is seen that for stability robustness it is necessary to keep the Nyquist plot “awayfrom the point−1.” The target is to achieve satisfactory gain, phase, and modulus margins.

Alternatively, as also seen in Section 1.4, robustness for loop gain perturbations requires thecomplementary sensitivity functionT to be small. For robustness for inverse loop gain perturba-tions, on the other hand, the sensitivity functionSneeds to be small.

By complementarityT and S cannot be simultaneously small, at least notvery small. Thesolution is to makeT andS small in different frequency ranges. It is consistent with the otherdesign targets to have the sensitivityS small in thelow frequencyrange, andT small in thecomplementary high frequency range.

The fasterT decreases with frequency — this is calledroll-off — the more protection theclosed-loop system has against high-frequency loop perturbations. This is important becauseowing to neglected dynamics — also known asparasitic effects— high frequency uncertainty isever-present.

Small values of the sensitivity function for low frequencies, which are required for adequatedisturbance attenuation, ensure protection against perturbations of the inverse loop gain at lowfrequencies. Such perturbations are often caused by load variations and environmental changes.

In thecrossover regionneitherS nor T can be small. The crossover region is the frequencyregion where the loop gainL crosses the value 1 (the zero dB line.) It is the region that is mostcritical for robustness. Peaking ofSandT in this frequency region is to be avoided. Good gain,phase and modulus margins help to ensure this.

1.5.8. Performance robustness

Feedback system performance is determined by the sensitivity functionS, the complementarysensitivity functionT, the input sensitivity functionM, and the closed-loop transfer functionH,successively given by

S= 11+ L

, T = L1+ L

, (1.97)

M = C1+ L

= SC, H = L1+ L

F = T F. (1.98)

We consider the extent to which each of these functions is affected by plant variations. For sim-plicity we suppose that the system environment is sufficiently controlled so that the compensatortransfer functionC and the prefilter transfer functionF are not subject to perturbation. Inspectionof (1.97–1.98) shows that under this assumption we only need to study the effect of perturbationson S and T: The variations inM are proportional to those inS, and the variations inH areproportional to those inT.

Denote byL0 thenominalloop gain, that is, the loop gain that is believed to be representativeand is used in the design calculations. Correspondingly,S0 andT0 are the nominal sensitivityfunction and complementary sensitivity function.

32


It is not difficult to establish that when the loop gain changes from its nominal valueL0 toits actual valueL the correspondingrelative changeof the reciprocal of the sensitivity functionSmay be expressed as

1S − 1

S0

1S0

= S0 − SS

= T0L − L0

L0. (1.99)

Similarly, the relative change of the reciprocal of the complementary sensitivity function may bewritten as

1T − 1

T0

1T0

= T0 − TT

= S0L0 − L

L= S0

1L − 1

L0

1L0

. (1.100)

These relations show that for the sensitivity functionSto be robust with respect to changes in theloop gain we desire the nominal complementary sensitivity functionT0 to be small. On the otherhand, for the complementary sensitivity functionT to be robust we wish the nominal sensitivityfunction S0 to be small. These requirements are conflicting, becauseS0 andT0 add up to 1 andtherefore cannot simultaneously be small.

The solution is again to have each small in a different frequency range. As seen before,normal control system design specifications requireS0 to be small at low frequencies (below thebandwidth). This causesT to be robust at low frequencies, which is precisely the region where itsvalues are significant. Complementarily,T0 is required to be small at high frequencies, causingS to be robust in the high frequency range.

Exercise 1.5.3 (Formulas for relative changes). Prove (1.99–1.100). �

Exercise 1.5.4 (MIMO systems). Suppose that the configuration of Fig. 1.21 represents a MIMOrather than a SISO feedback system. Show that the various closed-loop system functions encoun-tered in this section generalize to the following matrix system functions:

• thesensitivity matrix S= ( I + L)−1, with L = PC the loop gain matrixand I an identitymatrix of suitable dimensions;

• thecomplementary sensitivity matrix T= I − S= L( I + L)−1 = ( I + L)−1L;

• the input sensitivity matrix M= ( I + CP)−1C = C( I + PC)−1;

• theclosed-loop transfer matrix H= ( I + L)−1LF = T F.�

1.5.9. Review of the design requirements

We summarize the conclusions of this section as follows:

• The sensitivitySshould be small at low frequencies to achieve

– disturbance attenuation,

– good command response, and

– robustness at low frequencies.

• The complementary sensitivityT should be small at high frequencies to prevent

33


– exceeding the plant capacity,

– adverse effects of measurement noise, and

– loss of robustness at high frequencies.

• In the intermediate (crossover) frequency region peaking of bothS and T should beavoided to prevent

– overly large sensitivity to disturbances,

– excessive influence of the measurement noise, and

– loss of robustness.

1.6. Loop shaping

1.6.1. Introduction

The design of a feedback control system may be viewed as a process ofloop shaping. Theproblem is to determine the feedback compensatorC as in Fig. 1.18 such that the loop gainfrequency response functionL(jω), ω ∈ R, has a suitable shape. The design goals of the previoussection may be summarized as follows.

Low frequencies. At low frequencies we needSsmall andT close to 1. Inspection of

S= 11+ L

, T = L1+ L

(1.101)

shows that these targets may be achieved simultaneously by making the loop gainL large,that is, by making|L(jω)| � 1 in the low-frequency region.

High frequencies. At high frequencies we needT small andS close to 1. This may be ac-complished by making the loop gainL small, that is, by making|L(jω)| � 1 in the high-frequency region.

Figure 1.22 shows how specifications on the magnitude ofS in the low-frequency region and thatof T in the high-frequency region result in bounds on the loop gainL.

(logscale)

ω(log scale)

L(jω)

0 dB

upper bound forhigh frequencies

lower bound forlow frequencies

Figure 1.22.: Robustness bounds onL in the Bode magnitude plot

34

1.6. Loop shaping

Crossover region. In the crossover region we have|L(jω)| ≈ 1. In this frequency region it isnot sufficient to consider the behavior of the magnitude ofL alone. The behavior of themagnitude and phase ofL in this frequency region together determine how closely theNyquist plot approaches the critical point−1.

The more closely the Nyquist plot ofL approaches the point−1 the more

S= 11+ L

(1.102)

peaks. If the Nyquist plot ofL comes very near the point−1, so does the inverse Nyquistplot, that is, the Nyquist plot of 1/L. Hence, the more closely the Nyquist plot ofLapproaches−1 the more

T = L1+ L

= 1

1+ 1L

(1.103)

peaks.

Gain and phase of the loop gain are not independent. This is clarified in the next subsection.

1.6.2. Relations between phase and gain

A well-known and classical result of Bode (1940) is that the magnitude|L(jω)|, ω ∈ R, and thephase argL(jω), ω ∈ R, of a minimum phase15 linear time-invariant system with real-rational16

transfer functionL are uniquely related. If on a log-log scale the plot of the magnitude|L(jω)|versusω has an approximately constant slopen [decade/decade] then

argL(jω) ≈ n× π

2. (1.104)

Thus, if |L(jω)| behaves like 1/ω then we have a phase of approximately−π/2 [rad] = −90◦,while if |L(jω)| behaves like 1/ω2 then the phase is approximately−π [rad] = −180◦.

Exercise 1.6.1 (Bode’s gain-phase relationship). Why (1.104) holds may be understood fromthe way asymptotic Bode magnitude and phase plots are constructed (see§ 2.4.2). Make itplausible that between any two break points of the Bode plot the loop gain behaves as

L(jω) ≈ c(jω)n, (1.105)

with c a real constant. Show thatn follows from the numbers of poles and zeros ofL whosemagnitude is less than the frequency corresponding to the lower break point. What isc? �

More precisely Bode’s gain-phase relationship may be phrased as follows (Bode 1945).

Summary 1.6.2 (Bode’s gain-phase relationship). Let L be a minimum phase real-rationalproper transfer function. Then magnitude and phase of the corresponding frequency responsefunction are related by

argL(jω0) = 1π

∫ ∞

−∞

d log|Lω0(ju)|du

W(u) du, ω0 ∈ R, (1.106)

15A rational transfer function is minimum phase if all its poles and zeros have strictly negative real parts (see Exer-cise 1.6.3).

16That is,L(s) is a rational function ins with real coefficients.

35


with log denoting the natural logarithm. The intermediate variableu is defined by

u = logω

ω0. (1.107)

W is the function

W(u) = logcoth|u|2

= log

∣∣∣∣∣ωω0

+ 1ωω0

− 1

∣∣∣∣∣ . (1.108)

Lω0, finally, is given by

Lω0(ju) = L(jω), u = logω

ω0. (1.109)

Lω0 is the frequency response functionL defined on the logaritmically transformed and scaledfrequency axisu = logω/ω0. �

In (1.106) the first factor under the integral sign is the slope of the magnitude Bode plot (in[decades/decade]) as previously expressed by the variablen. W is a weighting function of theform shown in Fig. 1.23.W has most of it weight near 0, so that argL(jω0) is determined by the

u

W

4

2

00−1

1

1

1/e ω/ω0 e

Figure 1.23.: Weighting functionW in Bode’s gain-phase relationship

behavior ofLω0 near 0, that is, by the behavior ofL nearω0. If log |Lω0| would have a constantslope then (1.104) would be recovered exactly, and argL(jω0) would be determined byn alone.If the slope of log|L| varies then the behavior of|L| in neighboring frequency regions also affectsargL(jω0).

The Bode gain-phase relationship leads to the following observation. Suppose that the generalbehavior of the Nyquist plot of the loop gainL is as in Fig. 1.14, that is, the loop gain is greaterthan 1 for low frequencies, and enters the unit disk once (at the frequency±ωm) without leavingit again. The frequencyωm at which the Nyquist plot of the loop gainL crosses the unit circleis at the center of the crossover region. For stability the phase of the loop gainL at this pointshould be between−180◦ and 180◦. To achieve a phase margin of at least 60◦ the phase should bebetween−120◦ and 120◦. Figure 1.24 (Freudenberg and Looze 1988) illustrates the bounds onthe phase in the crossover region. Since|L| decreases at the point of intersection argL generallymay be expected to be negative in the crossover region.

If at crossover the phase ofL is, say,−90◦ then by Bode’s gain-phase relationship|L| de-creases at a rate of about 1 [decade/decade]. To avoid instability the rate of decrease cannot be

36

1.6. Loop shaping

(logscale)

ω(log scale)

ω(log scale)

L(jω)

0 dB

180◦

−180◦argL(jω)

allowablephase functions

0◦

upper bound forhigh frequencies

lower bound forlow frequencies

Figure 1.24.: Allowable regions for gain and phase of the loop gainL

greater than 2 [decades/decade]. Hence, for robust stability in the crossover region the mag-nitude |L| of the loop gain cannot decreases faster than at a rate somewhere between 1 and 2[decade/decade]. This bound on the rate of decrease of|L| in turn implies that the crossoverregion cannot be arbitrarily narrow.

Bode’s gain-phase relationship holds for minimum phase systems. For non-minimum phasesystems the trade-off between gain attenuation and phase lag is even more troublesome. LetL benon-minimum phase but stable17. ThenL = Lm · Lz, whereLm is minimum phase andLz is anall-pass function18 such that|Lz(jω)| = 1 for allω. It follows that|L(jω)| = |Lm(jω)| and

argL(jω) = argLm(jω)+ argLz(jω) ≤ argLm(jω). (1.110)

As Lz only introduces phase lag, the trade-off between gain attenuation and limited phase lagis further handicapped. Non-minimum phase behavior generally leads to reduction of the open-loop gains — compared with the corresponding minimum phase system with loop gainLm — orreduced crossover frequencies. The effect of right-half plane zeros — and also that of right-halfplane poles — is discussed at greater length in Section 1.7.

Exercise 1.6.3 (Minimum phase). Let

L(s) = k(s− z1)(s− z2) · · · (s− zm)

(s− p1)(s− p2) · · · (s− pn)(1.111)

17That is,L has right-half plane zeros but no right-half plane poles.18That is,|Lz(jω)| is constant for allω ∈ R.

37


be a rational transfer function with all its polesp1, p2, · · · , pn in the left-half complex plane.Then there exists a well-defined corresponding frequency response functionL(jω), ω ∈ R.

Changing the sign of the real part of theith zerozi of L leaves the behavior of the magni-tude|L(jω)|, ω ∈ R, of L unaffected, but modifies the behavior of the phase argL(jω), ω ∈ R.Similarly, changing the sign of the gaink does not affect the magnitude. Prove that under suchchanges the phase argL(jω), ω ∈ R, isminimalfor all frequencies if all zeroszi lie in the left-halfcomplex plane andk is a positive gain.

This is why transfer functions whose poles and zeros are all in the left-half plane and havepositive gain are calledminimum phasetransfer functions. �

1.6.3. Bode’s sensitivity integral

Another well-known result of Bode’s pioneering work is known asBode’s sensitivity integral

Summary 1.6.4 (Bode’s sensitivity integral). Suppose that the loop gainL has no poles in theopen right-half plane19. Then if L has at least two more poles than zeros the sensitivity function

S= 11+ L

(1.112)

satisfies∫ ∞

0log|S(jω)| dω = 0. (1.113)

�

The assumption thatL is rational may be relaxed (see for instance Engell (1988)). Thestatement thatL should have at least two more poles than zeros is sometimes phrased as “thepole-zero excess ofL is at least two20.” If the pole-zero excess is one, the integral on the left-hand side of (1.113) is finite but has a nonzero value. IfL has right-half plane poles then (1.113)needs to be modified to∫ ∞

0log|S(jω)| dω = π

∑i

Re pi . (1.114)

The right-hand side is formed from the open right-half plane polespi of L, included accordingto their multiplicity. The proof of (1.114) may be found in§ 1.10.

We discuss some of the implications of Bode’s sensitivity integral. Suppose for the timebeing thatL has no open right-half plane poles, so that the sensitivity integral vanishes. Thenthe integral over all frequencies of log|S| is zero. This means that log|S| both assumes negativeand positive values, or, equivalently, that|S| both assumes values less than 1 and values greaterthan 1.

For the feedback system to be useful,|S| needs to be less than 1 over an effective low-frequency band. Bode’s sensitivity integral implies thatif, this can be achieved at all then it isat the cost of disturbanceamplification(rather than attenuation) at high frequencies. Figure 1.25illustrates this. If the open-loop system has right-half plane poles this statement still holds. Ifthe pole-zero excess of the plant is zero or one then disturbance attenuation is possible over allfrequencies.

19That is, no poles with strictly positive real part.20In adaptive control the expression is “L has relative degree greater than or equal to 2.”

38

1.6. Loop shaping

1

0

|S|

B

ω (log scale)

Figure 1.25.: Low frequency disturbance attenuation may only be achievedat the cost of high frequency disturbance amplification

Exercise 1.6.5 (Bode integral).

1. Suppose that the loop gain isL(s)= k/(1+ sθ), with k andθ positive constants. Calculatethe sensitivity functionSand plot its magnitude. Does Bode’s theorem apply?

2. Repeat this for the loop gainL(s) = k/(1+ sθ)2. First check that the closed-loop systemis stable for all positivek andθ.

�

Freudenberg and Looze (1988) use Bode’s sensitivity integral to derive a lower bound onthe peak value of the sensitivity function in the presence of constraints on the low-frequencybehavior ofS and the high-frequency behavior ofL. Suppose thatL is real-rational, minimumphase and has a pole-zero excess of two or more. LetωL andωH be two frequencies such that0< ωL < ωH . Assume thatSandL are bounded by

|S(jω)| ≤ α < 1, 0 ≤ ω ≤ ωL, (1.115)

and

|L(jω)| ≤ ε(ωH

ω

)k+1, ω ≥ ωH, (1.116)

with 0< ε < 0.5 andk ≥ 0. Then the peak value of the sensitivity function is bounded by

supωl ≤ω≤ωH

|S(jω)| ≥ 1ωH − ωL

(ωL log

1α

− 3εωH

2k

). (1.117)

The proof is given in§ 1.10.This result is an example of the more general phenomenon that bounds on the loop gain

in different frequency regions interact with each other. Control system design therefore involvestrade-offs between phenomena that occur in different frequency regions. The interaction becomesmore severe as the bounds become tighter. Ifα or ε decrease or ifωL or k increase then the lowerbound for the peak value of the sensitivity function increases. Also if the frequenciesωL andωH

are required to be closely together, that is, the crossover region is required to be small, then thisis paid for by a high peak value ofS in the crossover region.

39


The bounds (1.115) and (1.116) are examples of the bounds indicated in Fig. 1.24. Theinequality (1.117) demonstrates that stiff bounds in the low- and high-frequency regions maycause serious stability robustness problems in the crossover frequency range.

The natural limitations on stability, performance and robustness as discussed in this sectionare aggravated by the presence of right-half plane plant poles and zeros. This is the subject ofSection 1.7.

Exercise 1.6.6 (Bode’s integral for the complementary sensitivity). Let T be the complemen-tary sensitivity function of a stable feedback system that has integrating action of at least ordertwo21. Prove that∫ ∞

0log

∣∣∣∣T( 1jω

)∣∣∣∣ dω = π∏

i

Re1zi, (1.118)

with the zi the right-half plane zeros of the loop gainL (Middleton 1991; Kwakernaak 1995).What does this equality imply for the behavior ofT? �

1.7. Limits of performance

1.7.1. Introduction

In this section22 we present a brief review of several inherent limitations of the behavior of thesensitivity functionSand its complementT which result from the pole-zero pattern of the plant.

In particular, right-half plane zeros and poles play an important role. The reason is that if thethe plant has right-half plane poles, it is unstable, which imposes extra requirements on the loopgain. If the plant has right-half plane zeros, its inverse is unstable, which imposes limitations onthe way the dynamics of the plant may be compensated.

More extensive discussions on limits of performance may be found in Engell (1988) for theSISO case and Freudenberg and Looze (1988) for both the SISO and the MIMO case.

1.7.2. Freudenberg-Looze equality

A central result is theFreudenberg-Looze equality, which is based on the Poisson integral formulafrom complex function theory. The result that follows was originally obtained by Freudenbergand Looze (1985) and Freudenberg and Looze (1988).

C Pe u y

Figure 1.26.: SISO feedback system

Summary 1.7.1 (Freudenberg-Looze equality). Suppose that the closed-loop system ofFig. 1.26 is stable, and that the loop gain has a right-half plane zeroz = x + jy with x > 0.

21This means that 1/L(s) behaves asO(s2) for s→ 0, (see§ 2.3).22The title has been taken from Engell (1988) and Boyd and Barratt (1991).

40


Then the sensitivity functionS= 1/(1+ L) must satisfy∫ ∞

−∞log(|S(jω)|) x

x2 + (y− ω)2dω = π log|B−1

poles(z)|. (1.119)

Bpoles is theBlaschke product

Bpoles(s) =∏

i

pi − spi + s

, (1.120)

formed from the open right-half plane polespi of the loop gainL = PC. The overbar denotesthe complex conjugate. �

The proof is given in§ 1.10. It relies on the Poisson integral formula from complex functiontheory.

1.7.3. Trade-offs for the sensitivity function

We discuss the consequences of the Freudenberg-Looze relation (1.119), which holds at anyright-half plane zeroz = x + jy of the loop gain, and, hence, at any right-half plane zero ofthe plant. The presentation follows Freudenberg and Looze (1988) and Engell (1988). We firstrewrite the equality in the form∫ ∞

0log(|S(jω)|) wz(ω) dω = log|B−1

poles(z)|, (1.121)

with wz the function

wz(ω) = 1π

(x

x2 + (ω− y)2+ x

x2 + (ω+ y)2

). (1.122)

We arrange (1.121) in turn as∫ ∞

0log(|S(jω)|) dWz(ω) = log|B−1

poles(z)|, (1.123)

with Wz the function

Wz(ω) =∫ ω

0wz(η) dη = 1

πarctan

ω− yx

+ 1π

arctanω+ y

x. (1.124)

The functionWz is plotted in Fig. 1.27 for different values of the ratioy/x of the imaginary partto the real part. The plot shows thatWz increases monotonically from 0 to 1. Its steepest increaseis about the frequencyω = |z|.Exercise 1.7.2 (Weight function). Show that the weight functionWz represents the extra phaselag contributed to the phase of the plant by the fact that the zeroz is in the right-half plane, thatis, Wz(ω) is the phase of

z+ jωz− jω

. (1.125)

�

41


1

0.5

00 1 2 3 4 5

Wz

ω/|z|

(a)

(b)

Figure 1.27.: The functionWz for values of argz increasing from (a) 0 to (b)almostπ/2

Exercise 1.7.3 (Positivity of right-hand side of the Freudenberg-Looze equality). Prove thatlog|B−1

poles(z)| is positive for anyz whose real part is positive. �

The comments that follow hold for plants that have at least one right-half plane zero.The functionwz is positive and also log|B−1

poles(z)| is positive. Hence, (1.121) implies that iflog|S| is negative over some frequency range so that, equivalently,|S| is less than 1, then nec-essarily|S| is greater than 1 over a complementary frequency range. This we already concludedfrom Bode’s sensitivity integral.

The Freudenberg-Loozeequality strengthens the Bode integral because of the weighting func-tion wz included in the integrand. The quantitydWz(jω) = wz(jω)dω may be viewed as aweighted length of the frequency interval. The weighted length equals the extra phase added bythe right-half plane zerozover the frequency interval. The larger the weighted length is, the morethe interval contributes to the right-hand side of (1.123). The weighting function determines towhat extent small values of|S| at low frequencies need to be compensated by large values at highfrequencies. We argue that if|S| is required to be small in a certain frequency band—in particu-

|S|

µ

1

ε0

ω1

ω (log scale)

Figure 1.28.: Bounds on|S|

lar, a low-frequency band—it necessarily peaks in another band. Suppose that we wish|S(jω)|to be less than a given small numberε in the frequency band [0, ω1], with ω1 given. We shouldlike to know something about the peak valueµ of |S| in the complementary frequency range.Figure 1.28 shows the numbersε andµ and the desired behavior of|S|. Define the boundingfunction

b(ω) ={ε for |ω| ≤ ω1,

µ for |ω| > ω1.(1.126)

42


Then|S(jω)| ≤ b(ω) for ω ∈ R and the Freudenberg-Looze equality together imply thatb needsto satisfy∫ ∞

0log(b(ω)) dWz(ω) ≥ log|B−1

poles(z)|. (1.127)

Evaluation of the left-hand side leads to

Wz(ω1) logε+ (1− Wz(ω1)) logµ ≥ log|B−1poles(z)|. (1.128)

Resolution of (1.128) results in the inequality

µ ≥ (1ε)

Wz(ω1 )1−Wz(ω1 ) · ∣∣B−1

poles(z)∣∣ 1

1−Wz(ω1 ) . (1.129)

We note this:

• For a fixed zeroz = x+ jy and fixedω1 > 0 the exponents in this expression are positive.By 1.7.3 we have|B−1

poles(z)| ≥ 1. Hence, forε < 1 the peak valueµ is greater than 1.Moreover, the smallerε is, the larger is the peak value. Thus, small sensitivity at lowfrequencies is paid for by a large peak value at high frequencies.

• For fixedε, the two exponents increase monotonically withω1, and approach∞ asω1

goes to∞. The first exponent (that of 1/ε) crosses the value 1 atω =√

x2 + y2 = |z|.Hence, if the width of the band over which sensitivity is required to be small is greaterthan the magnitude of the right-half plane zeroz, the peak value assumes excessive values.

The Freudenberg-Looze equality holds forany right-half plane zero, in particular the one withsmallest magnitude. Therefore, if excessive peak values are to be avoided, the width of theband over which sensitivity may be made small cannot be extended beyond the magnitude of thesmallestright-half plane zero.

The number

|B−1poles(z)| =

∏i

∣∣∣∣ pi + zpi − z

∣∣∣∣ (1.130)

is replaced with 1 if there are no right-half plane poles. Otherwise, it is greater than 1. Hence,right-half plane poles make the plant more difficult to control.

The number (1.130) is large if the right-half plane zeroz is close to any of the right-half planepolespi. If this situation occurs, the peak values of|S| are correspondingly large, and, as a result,the plant is difficult to control. The Freudenberg-Looze equality holds for any right-half planezero. Therefore, plants with a right-half plane zero close to a right-half plane pole are difficultto control. The situation is worst when a right-half plane zero coincides with a right-half planepole—then the plant has either an uncontrollable or an unobservable unstable mode.

We summarize the qualitative effects of right-half plane zeros of the plant on the shape of thesensitivity functionS (Engell 1988).

Summary 1.7.4 (Effect of right-half plane open-loop zeros).

1. The upper limit of the band over which effective disturbance attenuation is possible isconstrained from above by the magnitude of the smallest right-half plane zero.

2. If the plant has unstable poles, the achievable disturbance attenuation is further impaired.This effect is especially pronounced when one or several right-half plane pole-zero pairsare close.

43


3. If the plant has no right-half plane zeros the maximally achievable bandwidth is solelyconstrained by the plant capacity. As seen in the next subsection the right-half plane polewith largest magnitude constrains thesmallestbandwidth that is required.

The trade-off between disturbance attenuation and amplification is subject to Bode’s sen-sitivity integral.

�

We consider an example.

u

φ

Figure 1.29.: Double inverted pendulum system

Example 1.7.5 (Double inverted pendulum). To illustrate these results we consider the doubleinverted pendulum system of Fig. 1.29. Two pendulums are mounted on top of each other. Theinput to the system is the horizontal positionu of the pivot of the lower pendulum. The measuredoutput is the angleφ that the lower pendulum makes with the vertical. The pendulums have equallengthsL, equal massesm and equal moments of inertiaJ (taken with respect to the center ofgravity).

The transfer functionP from the inputu to the angleφ may be found (Kwakernaak andWestdijk 1985) to be given by

P(s) = 1L

s2(−(3K + 1)s2 + 3)(K2 + 6K + 1)s4 − 4(K + 2)s2 + 3

, (1.131)

with K the ratioK = J/(mL2). For a pendulum whose mass is homogeneously distributed alongits lengthK is 1

3. If we furthermore letL = 1 then

P(s) = s2(−2s2 + 3)289 s4 − 28

3 s2 + 3. (1.132)

This plant transfer function has zeros at 0, 0, and±1.22474, and poles at±0.60507 and±1.62293. The plant has two right-half plane poles and one right-half plane zero.

By techniques that are explained in Chapter 6 we may calculate the transfer functionC ofthe compensator that makes the transfer matricesSandT stable and at the same time minimizesthepeak valueof supω∈R |S(jω)| of the sensitivity functionS of the closed-loop system. Thiscompensator transfer function is

C(s) = 1.6292(s+ 1.6229)(s+ 0.6051)(s− 0.8018)

(s+ 1.2247)s2. (1.133)

44


Figure 1.30 shows the magnitude plot (a) of the corresponding sensitivity functionS. This mag-nitude does not depend on the frequencyω and has the constant value 21.1178. Note that thecompensator has a double pole at 0, which in the closed loop cancels against the double zero at0 of the plant. The corresponding double closed-loop pole at 0 actually makes the closed-loopsystem unstable. It causes the plant inputu to drift.

100

10

1

10−4 10−2 100 102 104 106

40 dB

20 dB

0 dB

(a)

(b)(c)

(d)

ω

|S|

Figure 1.30.: Sensitivity functions for the double pendulum system

The peak value 21.1178 for the magnitude of the sensitivity function is quite large (as com-pared with 1). No compensator exists with a smaller peak value. To reduce the sensitivity ofthe feedback system to low frequency disturbances it is desirable to make|S| smaller at lowfrequencies. This is achieved, for instance, by the compensator with transfer function

C(s) = 1.6202(s+ 1.6229)(s+ 0.6051)(s− 0.8273)

(s+ 1.2247)(s+ 0.017390)(s− 0.025715). (1.134)

Figure 1.30 shows the magnitude (b) of the corresponding sensitivity function. This compensatorno longer cancels the double zero at 0. As a result, the sensitivitySat zero frequency is 1, whichmeans that constant disturbances are not attenuated. The reason is structural: Because the planttransfer functionP equals zero at frequency 0, the plant is unable to compensate for constantdisturbances. Actually, the zeros at 0 play the role of right-half plane zeros, except that theybound the frequencies whereSmay be made small frombelowrather than from above.

The plot of Fig. 1.30 also shows that compared with the compensator (1.133) the compen-sator (1.134) reduces the sensitivity to disturbances up to a frequency of about 0.01. By furthermodifying the compensator this frequency may be pushed up, at the price of an increase in thepeak value of the magnitude ofS. Figure 1.30 shows two more magnitude plots (c) and (d). Thecloser the magnitude 1.2247 of the right-half plane plant zero is approached the more the peakvalue increases.

There exist compensators that achieve magnitudes less than 1 for the sensitivity in the fre-quency range, say, between 0.01 and 0.1. The cost is a further increase of the peak value.

The compensators considered so far all result in sensitivity functions that do not approachthe value 1 as frequency increases to∞. The reason is that the loop gain does not approach 0.This undesirable phenomenon, which results in large plant input amplitudes and high sensitivityto measurement noise, is removed in Example 1.7.9. �

Exercise 1.7.6 (Interlacing property). Check that the double inverted pendulum does not havethe parity interlacing property of§ 1.3.6 (p. 20). Hence, no stabilizing compensator exists thatby itself is stable. �

Exercise 1.7.7 (Lower bound for the peak value of S).

45


1. Define‖S‖∞ as the peak value of|S|, that is,‖S‖∞ = supω∈R |S(jω)|. Use (1.129) toprove that if the closed-loop system is stable then

‖S‖∞ ≥ ∣∣B−1poles(z)

∣∣ , (1.135)

whereBpoles is the Blaschke product formed from the right-half plane poles of the plant,andz any right-half plane zero of the plant.

2. Check that the compensator (1.133) actually achieves this lower bound.�

1.7.4. Trade-offs for the complementary sensitivity function

Symmetrically to the results for the sensitivity function well-defined trade-offs hold for the com-plementary sensitivity function. The role of the right-half plane zeros is now taken by the right-half plant open-looppoles,and vice-versa. This is seen by writing the complementary sensitivityfunction as

T = L1+ L

= 1

1+ 1L

. (1.136)

Comparison with Freudenberg-Looze equality of 1.7.1 leads to the conclusion that for any right-half plane open-loop polep = x+ jy we have (Freudenberg and Looze 1988)∫ ∞

−∞log(|T(jω)|) x

x2 + (y− ω)2dω = π log|B−1

zeros(p)|, (1.137)

with Bzerosthe Blaschke product

Bzeros(s) =∏

i

zi − szi + s

(1.138)

formed from the open right-half planezeros zi of L.We consider the implications of the equality (1.137) on the shape ofT. Whereas the sensitivity

S is required to be small atlow frequencies,T needs to be small athigh frequencies. By anargument that is almost dual to that for the sensitivity function it follows that if excessive peakingof the complementary sensitivity function at low and intermediate frequencies is to be avoided,|T| may only be made small at frequencies thatexceedthe magnitude of the open-loop right-halfplanepole with largestmagnitude. Again, close right-half plane pole-zero pairs make thingsworse.

We summarize the qualitative effects of right-half plane zeros of the plant on the shape achiev-able for the complementary sensitivity functionT (Engell 1988).

Summary 1.7.8 (Effect of right-half plane open-loop poles).

1. The lower limit of the band over which the complementary sensitivity function may bemade small is constrained from below by the magnitude of the largest right-half planeopen-loop pole. Practically, the achievable bandwidth is always greater than this magni-tude.

2. If the plant has right-half plane zeros, the achievable reduction ofT is further impaired.This effect is especially pronounced when one or several right-half plane pole-zero pairsare very close.

46

1.8. Two-degrees-of-freedom feedback systems

0 dB

0 dB

ω (log scale)

ω (log scale)

magnitude of the smallestright-half plane pole

magnitude of the smallestright-half plane pole

magnitude of the smallestright-half plane zero

magnitude of the smallestright-half plane zero

|S|(log scale)

|T|(log scale)

Figure 1.31.: Right-half plane zeros and poles constrainSandT

�

Figure 1.31 summarizes the difficulties caused by right-half plane zeros and poles of the planttransfer functionP. Scan only be small up to the magnitude of the smallest right-half plane zero.T can only start to roll off to zero at frequencies greater than the magnitude of the largest right-half plane pole. The crossover region, whereSandT assume their peak values, extends over theintermediate frequency range.

Example 1.7.9 (Double inverted pendulum). We return to the double inverted pendulum ofExample 1.7.5. For robustness to plant uncertainty and reduction of the susceptibility to mea-surement noise it is necessary that the loop gain decreases to zero at high frequencies. Corre-spondingly, the complementary sensitivity function also decreases to zero while the sensitivityfunction approaches 1. The compensator

C(s) = − 1.5136(s+ 1.6229)(s+ 0.60507)(s− 0.82453)(s+ 1.2247)(s+ 0.017226)(s− 0.025394)(1+ 0.00061682s)

, (1.139)

whose transfer function is strictly proper, accomplishes this. Figure 1.32 shows that for lowfrequencies the magnitude plot (e) of the corresponding sensitivity function closely follows themagnitude plot (c) of Fig. 1.30, which is repeated in Fig. 1.32. At high frequencies the magnitudeof Sdrops off to 1, however, starting at a frequency of about 100.

The lowest frequency at which|S| may start to drop off to 1 coincides with the lowest fre-quency at which the complementary sensitivity may be made to start decreasing to zero. This, inturn, is determined by the magnitude 1.6229 of the right-half plane plant pole with largest mag-nitude. The magnitude plot (f) in Fig. 1.32 shows that making|S| drop off at a lower frequencythan in (e) causes the peak value to increase. �


In § 1.3 we introduced the two-degrees-of-freedom configuration of Fig. 1.33. The function of

47


100

10

1

10−4 10−2 100 102 104 106

40 dB

20 dB

0 dB

(c)

(e)(f)

ω

|S|

Figure 1.32.: More sensitivity functions for the double pendulum system

F C Pr z

Figure 1.33.: Two-degrees-of-freedom feedback system configuration

the precompensatorF is to improve the closed-loop response to command inputsr .Figure 1.34 shows two other two-degrees-of-freedom configurations. In this section we study

whether it makes a difference which configuration is chosen. We restrict the discussion to SISOsystems.

In the configuration of Fig. 1.33 we write the plant and compensator transfer functions in thepolynomial fraction form

P = ND, C = Y

X. (1.140)

The feedback loop is stable if and only if the roots of the closed-loop characteristic polynomialDcl = DX + NY are all in the open left-half complex plane.

In this same configuration, the closed-loop transfer functionH from the command signalr tothe control system outputz is

H = PC1+ PC

F = NYDcl

F. (1.141)

The prefilter transfer functionF is available to compensate for any deficiencies of the uncom-pensated closed-loop transfer function

H0 = NYDcl

. (1.142)

Right-half plane zeros of this uncompensated transfer function are a handicap for the compensa-tion. Right-half plane roots ofN (that is, open-loop right-half plane zeros) and right-half planeroots ofY may well be present. Such zeros cannot be canceled by corresponding poles ofFbecause this would make the precompensator, and, hence, the whole control system, unstable.

Next consider the two-degrees-of-freedom configuration of Fig. 1.34(a). We now have for

48


(a)

(b)

F

C

C1

C2

P

P

r

r z

zu

Figure 1.34.: Further two-degrees-of-freedom feedback system configura-tions

the closed-loop transfer function

H = P1+ PC

F = N XDcl︸︷︷︸H0

F. (1.143)

Inspection shows that the open-loop plant zeros re-occur in the uncompensated closed-loop trans-fer functionH0 but that instead of the roots ofY (the compensator zeros) now the roots ofX (thecompensator poles) appear as zeros. Hence, the precompensator design problem for this config-uration is different from that for the configuration of Fig. 1.33. In fact, if the compensator hasright-half plane poles or zeros, or both, it is impossible to achieve identical overall closed-looptransfer functions for the two configurations.

Comparison of (1.142) and (1.143) suggests that there may exist a configuration such that thenumerator of the uncompensated closed-loop transfer function is independent of the compensator.To investigate this, consider the configuration of Fig. 1.34(b).C1 andC2 have the polynomialfractional representations

C1 = Y1

X1, C2 = Y2

X2. (1.144)

To match the closed-loop characteristics of the configurations of Figs. 1.33 and 1.34(a) we needC = C1C2. This implies thatX1X2 = X andY1Y2 = Y. The closed-loop transfer function now is

H = PC1

1+ PC= N X2Y1

Dcl. (1.145)

Inspection shows that the numerator ofH is independent of the compensator if we letX2Y1 = 1,so that

C1 = Y1

X1= 1

X1X2= 1

X, C2 = Y2

X2= Y1Y2 = Y. (1.146)

The closed-loop transfer function now is

H = NDcl

. (1.147)

49


FX

C0

YX

P

r

zeu

Figure 1.35.: 112-degrees-of-freedom control system

The corresponding configuration of Fig. 1.34(b) has two disadvantages:

1. The configuration appears to require the implementation of a block with the purely polyno-mial transfer functionC2(s) = Y(s), which is physically impossible (unlessY is of degreezero).

2. The configuration actually has only one degree of freedom. The reason is that one de-gree of freedom has been used to make the numerator of the closed-loop transfer functionindependent of the compensator.

The first difficulty may be remedied by noticing that from the block diagram we have

u = C1r − C1C2z = 1X

r + YX

e. (1.148)

This implies

Xu= r + Ye. (1.149)

This input-output relation — withr andeas inputs andu as output — may be implemented by astate realization of order equal to the degree of the polynomialX.

The second disadvantange may be overcome by modifying (1.149) to

Xu= Fr + Ye, (1.150)

with F a polynomial of degree less than or equal to that ofX. This still allows implementationby a state realization of order equal to the degree ofX (see Exercise 1.8.1). The compensatoris represented in the block diagram of Fig. 1.35. The combined blockC0 is jointly realized as asingle input-output-state system. The closed-loop transfer function is

H = N FDcl

. (1.151)

The design of the prefilter amounts to choosing the polynomialF. This might be called a 112-degrees-of-freedomcontrol system23. By application of a further prefilterF0 as in Fig. 1.36 theclosed-loop transfer function becomes

H = N FDcl

F0. (1.152)

23Half degrees of freedom were introduced in control engineering terminology by Grimble (1994), though with a differ-ent connotation than the one used here.

50


F0

C0

FX

YX

P

r

zeu

Figure 1.36.: 212-degrees-of-freedom configuration

This results in a 212-degrees-of-freedom control system.An application is described in Example 2.9.5 in§ 2.9.5.

Exercise 1.8.1 (Realization of the 112-degrees-of-freedom compensator). Represent the poly-

nomialsX, F andY as

X(s) = sn + an−1sn−1 + an−2sn−2 + · · · + a0, (1.153)

F(s) = bnsn + bn−1sn−1 + bn−2sn−2 + · · · + b0, (1.154)

Y(s) = cnsn + cn−1sn−1 + cn−2sn−2 + · · · + c0. (1.155)

1. Show that the 112-degrees-of-freedom compensatorXu = Fr + Yemay be realized as inFig. 1.37.

2. Find a state representation for the compensator.

3. Prove that the feedback system of Fig. 1.35, with the dashed block realized as in Fig. 1.37,has the closed-loop characteristic polynomialDX + NY.

�

b0

a0

c0 b1

a1

c1 bn−1

an−1

cn−1 bn cn

x0 x1 xn−1

r

e

u1s

1s

1s

Figure 1.37.: Realization of the 112-degrees-of-freedom compensatorXu =

Fr + Ye

51


1.9. Conclusions

It is interesting to observe a number of “symmetries” or “dualities” in the results reviewed inthis chapter (Kwakernaak 1995). For good performance and robustness the loop gainL of awell-designed linear feedback system should be

• large at low frequencies and

• small at high frequencies.

As the result, the sensitivity functionS is

• small at low frequencies and

• approximately equal to 1 at high frequencies.

The complementary sensitivity functionT is

• approximately equal to 1 at low frequencies and

• small at high frequencies.

Such well-designed feedback systems are

• robust with respect to perturbations of the inverse loop gain at low frequencies, and

• robust with respect to perturbations of the loop gain at high frequencies.

Furthermore,

• right-half plane open-loop zeros limit the frequency up to whichSmay be made small atlow frequencies, and

• right-half plane open-loop poles limit the frequency from whichT may be made small athigh frequencies.

Note that to a large extent performance and robustness go hand in hand, that is, the requirementsfor good performance imply good robustness, and vice-versa. This is also true for the criticalcrossover region, where peaking of bothS and T is to be avoided, both for performance androbustness.

1.10. Appendix: Proofs

In this section we collect a number of proofs for Chapter 1.

1.10.1. Closed-loop characteristic polynomial

We first prove (1.44) in Subsection 1.3.3.

Proof 1.10.1 (Closed-loop characteristic polynomial). Let

x = Ax+ Bu, y = Cx+ Du, (1.156)

be a state realization of the blockL in the closed-loop system of Fig. 1.11. It follows thatL(s) = C(sI −A)−1 + D. From u = −y we obtain with the output equation thatu = −Cx− Du, so thatu = −( I +D)−1Cx. Since by assumptionI + D = I + L(j∞) is nonsingular the closed-loop system is well-defined.

52


Substitution ofu into the state differential equation shows that the closed-loop system is described by thestate differential equation

x = [ A− B( I + D)−1C]x. (1.157)

The characteristic polynomialχcl of the closed-loop system hence is given by

χcl(s) = det[sI − A+ B( I + D)−1C]

= det(sI − A) · det[I + (sI − A)−1B( I + D)−1C]. (1.158)

Using the well-known determinant equality det( I + MN) = det( I + N M) it follows that

χcl(s) = det(sI − A) · det[I + ( I + D)−1C(sI − A)−1B]

= det(sI − A) · det[( I + D)−1] · det[I + D + C(sI − A)−1B]

= det(sI − A) · det[( I + D)−1] · det[I + L(s)]. (1.159)

Denoting the open-loop characteristic polynomial as det(sI − A) = χ(s) we thus have

χcl(s)χ(s)

= det[I + L(s)]det[I + L(j∞)]

. (1.160)

1.10.2. The Nyquist criterion

The proof of the generalized Nyquist criterion of Summary 1.3.13 in Subsection 1.3.5 relies on theprincipleof the argumentof complex function theory24.

Summary 1.10.2. Principle of the argument LetR be a rational function, andC a closed contour in thecomplex plane as in Fig. 1.38. As the complex numbers traverses the contourC in clockwise direction, its

0

ImIm

ReRe

R

R(C )C

pole of R

zero ofR

Figure 1.38.: Principle of the argument. Left: a closed contourC in thecomplex plane. Right: the imageR(C ) of C under a rational functionR.

imageR(s) underR traverses a closed contour that is denoted asR(C ), also shown in Fig. 1.38. Then asstraverses the contourC exactly once in clockwise direction,

(the number of timesR(s) encircles the origin in clockwise direction ass traversesC )=

(the number of zeros ofR insideC ) − (the number of poles ofR insideC ).

24See Henrici (1974). The generalized form in which we state the principle may be found in Postlethwaite and MacFar-lane (1979).

53


We prove the generalized Nyquist criterion of Summary 1.3.13.

Proof of the generalized Nyquist criterion.We apply the principle of the argument to (1.160), where wechoose the contourC to be the so-calledNyquist contouror D-contourindicated in Fig. 1.39. The radiusρof the semicircle is chosen so large that the contour encloses all the right-half plane roots of bothχcl andχol.Then by the principle of the argument the number of times that the image of det( I + L) encircles the originequals the number of right-half plane roots ofχcl (i.e., the number of unstable closed-loop poles) minus thenumber of right-half plane roots ofχol (i.e., the number of unstable open-loop poles). The Nyquist criterionfollows by letting the radiusρ of the semicircle approach∞. Note that asρ approaches∞ the image of thesemicircle under det( I + L) shrinks to the single point det( I + L(j∞)).

Im

Reρ0

Figure 1.39.: Nyquist contour

1.10.3. Bode’s sensitivity integral

The proof of Bode’s sensitivity integral is postponed until the next subsection. Accepting it as true we useit to derive the inequality (1.117) of Subsection 1.6.3.

Proof 1.10.3 (Lower bound for peak value of sensitivity). If the open-loop system is stable then we haveaccording to Bode’s sensitivity integral∫ ∞

0log |S(jω)| dω = 0. (1.161)

From the assumption that|S(jω)| ≤ α < 1 for 0≤ ω ≤ ωL it follows that if 0< ωL < ωH < ∞ then

0 =∫ ∞

0log |S(jω)| dω

=∫ ωL

0log |S(jω)| dω+

∫ ωH

ωL

log |S(jω)| dω+∫ ∞

ωH

log |S(jω)| dω

≤ ωL logα+ (ωH − ωL) supωL≤ω≤ωH

log|S(jω)| +∫ ∞

ωH

log |S(jω)| dω. (1.162)

As a result,

(ωH − ωL) supωL≤ω≤ωH

log |S(jω)| ≥ ωL log1α

−∫ ∞

ωH

log |S(jω)| dω. (1.163)

Next consider the following sequence of (in)equalities on the tail part of the sensitivity integral∣∣∣∣∫ ∞

ωH

log|S(jω)| dω

∣∣∣∣ ≤∫ ∞

ωH

| log |S(jω)|| dω

≤∫ ∞

ωH

| log S(jω)| dω =∫ ∞

ωH

| log[1+ L(jω)]| dω. (1.164)

54


From the inequalities (4.1.38) and (4.1.35) of Abramowitz and Stegun (1965) we have for any complexnumberz such that 0≤ |z| ≤ 0.5828

| log(1+ z)| ≤ − log(1− |z|) ≤ | log(1− |z|)| ≤ 3|z|2. (1.165)

The assumption that

|L(jω)| ≤ ε(ωH

ω

)k+1for ω > ωH (1.166)

with 0< ε < 0.5, implies that|L(jω)| ≤ ε < 0.5 for ω > ωH. With this it follows from (1.164) and (1.165)that ∣∣∣∣

∫ ∞

ωH

log|S(jω)|dω∣∣∣∣ ≤

∫ ∞

ωH

32

|L(jω)| dω ≤∫ ∞

ωH

3ε2

(ωH

ω

)k+1dω = 3ε

2ωH

k. (1.167)

The final step of of the proof is to conclude from (1.163) that

supωL≤ω≤ωH

log |S(jω)| ≥ 1ωH − ωL

(ωL log

1α

− 3εωH

2k

). (1.168)

This is what we set out to prove.

1.10.4. Limits of performance

The proof of the Freudenberg-Looze equality of Summary 1.7.1 relies on Poisson’s integral formula fromcomplex function theory.

Summary 1.10.4 (Poisson’s integral formula). Let F be a functionC → C that is analytic in the closedright-half plane25 and is such that

limR→∞

|F(Rejθ)|R

= 0 (1.169)

for all θ ∈ [− π2 , − π

2 ]. Then the value ofF(s) at any points = x + jy in the open right-half plane26 can berecovered from the values ofF(jω), ω ∈ R, by the integral relation

F(s)= 1π

∫ ∞

−∞F(jω)

xx2 + (y− ω)2

dω. (1.170)

A sketch of the proof of Poisson’s integral formula follows.

Proof of Poisson’s integral formula.We present a brief proof of Poisson’s integral formula based on ele-mentary properties of the Laplace and Fourier transforms (see for instance Kwakernaak and Sivan (1991)).Since by assumption the functionF is analytic in the closed right-half plane, its inverse Laplace transformf is zero on(−∞, 0). Hence, fors= x+ jy we may write

F(s)=∫ ∞

−∞f (t)e−(x+jy)t dt =

∫ ∞

−∞f (t)e−x|t| e−jyt dt. (1.171)

For x> 0 the function e−x|t| , t ∈ R, is the inverse Fourier transform of the frequency function

1x − jω

+ 1x + jω

= 2xx2 + ω2

, ω ∈ R. (1.172)

25For rationalF this means thatF has no poles in Re(s)≥ 0.26That is, for Re(s) > 0.

55


Thus, we have

F(s) =∫ ∞

−∞f (t)e−jyt

∫ ∞

−∞

2xx2 + ω2

ejωt dω2π

dt

= 1π

∫ ∞

−∞

xx2 + ω2

∫ ∞

−∞f (t)e−j(y−ω)t dt︸︷︷︸

F(j(y− ω))

dω (1.173)

By replacing the integration variableω with y− ω we obtain the desired result

F(s)= 1π

∫ ∞

−∞F(jω)

xx2 + (y− ω)2

dω. (1.174)

We next consider the Freudenberg-Looze equality of Summary 1.7.1.

Proof 1.10.5 (Freudenberg-Looze equality). The proof of the Freudenberg-Looze equality of Sum-mary 1.7.1 follows that of Freudenberg and Looze (1988). We first writeL as L = N/D, with N andD coprime polynomials27. Then

S= DD + N

. (1.175)

Since by assumption the closed-loop system is stable, the denominatorD + N has all its roots in the openleft-half plane. Hence,S is analytic in the closed right-half plane. Moreover, any right-half plane polez ofL is a root ofD and, hence, a zero ofS.

We should like to apply Poisson’s formula to the logarithm of the sensitivity function. Because of theright-half plane rootspi of D, however, logS is not analytic in the right-half plane, and Poisson’s formulacannot be used. To remedy this wecancelthe right-half plane zeros ofSby considering

S= B−1polesS. (1.176)

Application of Poisson’s formula to logSyields

log S(s) = 1π

∫ ∞

−∞log(S(jω))

xx2 + (y− ω)2

dω (1.177)

for any open right-half plane points = x + jy. Taking the real parts of the left- and right-hand sides wehave

log |S(s)| = 1π

∫ ∞


x2 + (y− ω)2dω (1.178)

Now replaces with a right-half plane zeroz = x+ jy of L, that is, a right-half plane zero ofN. Then

S(z)= 11+ L(z)

= 1, (1.179)

so thatS(z) = B−1poles(z). Furthermore,|Bpoles(jω)| = 1 for ω ∈ R, so that|S(jω)| = |S(jω)| for ω ∈ R.

Thus, after settings= z we may reduce (1.178) to

log |B−1poles(z)| = 1

π

∫ ∞


x2 + (y− ω)2dω, (1.180)

which is what we set out to prove.

Bode’s sensitivity integral (1.114) follows from Proof 1.10.5.

27That is,N andD have no common factors.

56


Proof 1.10.6 (Bode’s sensitivity integral). The starting point for the proof of Bode’s sensitivity integral(1.114) is (1.178). Settingy = 0, replacingSwith B−1

polesS, and multiplying on the left and the right byπxwe obtain (exploiting the fact that|Bpoles| = 1 on the imaginary axis)∫ ∞

−∞log(|S(jω)|) x2

x2 + ω2dω = πx log|S(x)| + πx log|B−1

poles(x)|. (1.181)

Letting x approach∞, the left-hand side of this expression approaches the Bode integral, while under theassumption thatL has pole excess two the quantityx log |S(x)| approaches 0. Finally,

limx→∞

x log|B−1poles(x)| = lim

x→∞x log

∏i

∣∣∣∣ pi + xpi − x

∣∣∣∣ = limx→∞

Re∑

i

x log1+ pi

x

1− pix

= 2∑

i

Re pi . (1.182)

This completes the proof.

57


58

2. Classical Control System Design

Overview– Classical criteria for the performance of feedback controlsystems are the error constants and notions such as bandwidth and peak-ing of the closed-loop frequency response, and rise time, settling timeand overshoot of the step response.

The graphical tools Bode, Nyquist, Nichols plots, andM-, N- androot loci belong to the basic techniques of classical and modern control.

Important classical control system design methods consist of loopshaping by lag compensation (including integral control), lead compen-sation and lag-lead compensation. Quantitative feedback design (QFT)allows to satisfy quantitative bounds on the performance robustness.

2.1. Introduction

In this chapter we review methods for the design of control systems that are known under thename ofclassical control theory. The main results in classical control theory emerged in theperiod 1930–1950, the initial period of development of the field of feedback and control engi-neering. The methods obtained a degree of maturity during the fifties and continue to be of greatimportance for the practical design of control systems, especially for the case of single-input,single-output linear control systems. Much of what now is called modern robust control theoryhas its roots in these classical results.

The historical development of the “classical” field started with H. Nyquist’s stability criterion(Nyquist 1932), H. S. Black’s analysis of the feedback amplifier (Black 1934), H. W. Bode’sfrequency domain analysis (Bode 1940), and W. R. Evans’ root locus method (Evans 1948). Toan extent these methods are of aheuristicnature, which both accounts for their success and fortheir limitations. With these techniques a designer attempts to synthesize a compensation networkor controller that makes the closed-loop system perform as required. The terminology in use inthat era (with expressions such as “synthesize,” “compensation,” and “network”) is from the fieldof amplifier circuit design (Boyd and Barratt 1991).

In this chapter an impression is given of some of the classical highlights of control. Thepresentation is far from comprehensive. More extensive introductions may be found in classicaland modern textbooks, together with an abundance of additional material. Well-known sourcesare for instance Bode (1945), James, Nichols, and Philips (1947), Evans (1954), Truxal (1955),

59


Savant (1958), Horowitz (1963), Ogata (1970), Thaler (1973), Franklin, Powell, and Emami-Naeini (1986), D’Azzo and Houpis (1988), Van de Vegte (1990), Franklin, Powell, and Emami-Naeini (1991), and Dorf (1992).

In § 2.2 (p. 60) we discuss the steady-state error properties of feedback control systems. Thisnaturally leads to review of the notion of integral control in§ 2.3 (p. 64).

The main emphasis in classical control theory is on frequency domain methods. In§ 2.4(p. 69) we review various important classical graphic representations of frequency responses:Bode, Nyquist and Nichols plots.

The design goals and criteria of classical control theory are considered in§ 2.5 (p. 79). In§ 2.6(p. 81) the basic classical techniques of lead, lag and lag-lead compensation are discussed. A briefsurvey of root locus theory as a method for parameter selection of compensators is presented in§ 2.7 (p. 87). The historically interesting Guillemin-Truxal design procedure is considered in§ 2.8(p. 89). In the 1970squantitative feedback theory(QFT) was initiated by Horowitz (1982). Thispowerful extension of the classical frequency domain feedback design methodology is explainedin § 2.9 (p. 92).

All the design methods aremodel-based.They rely on an underlying and explicit modelof the plant that needs to be controlled. The experimental Ziegler-Nichols rules for tuning aPID-controller mentioned in§ 2.3 (p. 64) form an exception.

2.2. Steady state error behavior

2.2.1. Steady-state behavior

One of the fundamental reasons for adding feedback control to a system is thatsteady-stateerrorsare reduced by the action of the control system. Consider the typical single-loop controlsystem of Fig. 2.1. We analyze the steady-state behavior of this system, that is, the asymptoticbehavior in the time domain fort → ∞ when the reference inputr is a polynomial time functionof degreen. Thus,

r (t) = tn

n!1(t), t ≥ 0, (2.1)

where1 is theunit stepfunction,1(t) = 1 for t ≥ 0 and1(t) = 0 for t < 0. Forn = 0 we have astep of unit amplitude, forn = 1 the input is a ramp with unit slope and forn = 2 it is a parabolawith unit second derivative.

The Laplace transform of the reference input isr (s) = 1/sn+1. The control error is the signalε defined by

ε(t) = r (t)− y(t), t ≥ 0. (2.2)

The steady-state error,if it exists,is

ε∞ = limt→∞ ε(t). (2.3)

r zC PF

Figure 2.1.: Single-loop control system configuration

60


The Laplace transform of the control error is

ε(s) = r (s)− y(s) = 1sn+1

− H(s)sn+1

= 1− H(s)sn+1

. (2.4)

The function

H = PCF1+ PC

= L1+ L

F (2.5)

is the closed-loop transfer function.L = PC is the loop gain.Assuming that the closed-loop system is stable, so that all the poles ofH are in the left-half

plane, we may apply the final value theorem of Laplace transform theory. It follows that thesteady-state error,if it exists, is given by

ε(n)∞ = lims↓0

sε(s) = lims↓0

1− H(s)sn

, (2.6)

with n the order of the polynomial reference input. Assume for the moment that no prefilter isinstalled, that is, ifF(s) = 1. Then

1− H(s) = 11+ L(s)

, (2.7)

and the steady-state error is

ε(n)∞ = lims↓0

1sn[1 + L(s)]

. (2.8)

This equation allows to compute the steady-state error of the response of the closed-loop systemto the polynomial reference input (2.1).

2.2.2. Type k systems

A closed-loop system with loop gainL is of type k if for some integerk the limit lims↓0 skL(s)exists and is nonzero. The system is of typek if and only if the loop gainL has exactlyk polesin the origin. If the system is of typek then

lims↓0

snL(s)

= ∞ for 0 ≤ n< k,6= 0 for n = k,= 0 for n> k.

(2.9)

Consequently, from (2.8) we have for a typek system without prefilter (that is, ifF(s) = 1)

lims↓0

ε(n)∞

= 0 for 0≤ n< k,6= 0 for n = k,= ∞ for n> k.

(2.10)

Hence, if the system is of typek and stable then it has a zero steady-state error for polynomialreference inputs of order less thank, a nonzero finite steady-state error for an input of orderk,and an infinite steady-state error for inputs of order greater thank.

A type 0 system has a nonzero but finite steady-state error for a step reference input, and aninfinite steady-state error for ramp and higher-order inputs.

A type 1 system has zero steady-state error for a step input, a finite steady-state error for aramp input, and infinite steady-state error for inputs of order two or higher.

A type 2 system has zero steady-state error for step and ramp inputs, a finite steady-state errorfor a second-order input, and infinite steady-state error for inputs of order three or higher.

Figure 2.2 illustrates the relation between system type and steady-state errors.

61


0 10 20 300

0.5

1

0 10 20 300

0.5

1

0 5 10 15 200

10

20

0 5 10 15 200

10

20

0 5 100

50

100

0 5 100

50

100

Nonzero position error Zero position error

Nonzero velocity error Zero velocity error

Nonzero acceleration error Zero acceleration error

type 0

type 1

type 1

type 2

type 2

type 3

timetime

Figure 2.2.: System type and steady-state errors

Exercise 2.2.1 (Type k systems with prefilter). The results of this subsection have been provedif no prefilter is used, that is,F(s) = 1. What condition needs to be imposed onF in case aprefilter is used? �

2.2.3. Error constants

If the system is of type 0 then the steady-state error for step inputs is

ε(0)∞ = 11+ L(0)

= 11+ Kp

. (2.11)

The numberKp = L(0) is theposition error constant.For a type 1 system the steady-state error for a ramp input is

ε(1)∞ = lims↓0

1s[1 + L(s)]

= lims↓0

1sL(s)

= 1Kv. (2.12)

The numberKv = lims↓0 sL(s) is thevelocity constant.The steady-state error of a type 2 system to a second-order input is

ε(2)∞ = lims↓0

1s2L(s)

= 1Ka. (2.13)

The numberKa = lims↓0 s2L(s) is theacceleration constant.Table 2.1 summarizes the various steady-state errors. In each case, the larger the error constant

is the smaller is the corresponding steady-state error.

62


Table 2.1.: Steady-state errors

Input

System step ramp parabola

type 0 11+ Kp

∞ ∞

type 1 0 1Kv

∞

type 2 0 0 1Ka

The position, velocity and acceleration error provide basic requirements that must be satisfiedby servomechanism systems depending upon their functional role.

The steady-state error results are robust in the sense that if the coefficients of the transferfunction of the system vary then the error constants also vary but the zero steady-state errorproperties are preserved — as long as the system does not change its type. Thus, for a type 1system the velocity error constant may vary due to parametric uncertainty in the system. As longas the type 1 property is not destroyed the system preserves its zero steady-state position error.

Exercise 2.2.2 (Steady-state response to polynomial disturbance inputs). Consider the feed-back loop with disturbance input of Fig. 2.3. Suppose that the closed-loop system is stable, andthat the disturbance is polynomial of the form

v(t) = tn

n!1(t), t ≥ 0. (2.14)

Show that the steady-state response of the output is given by

zn∞ = lim

t→∞ z(t) = lims↓0

1sn[1 + L(s)]

. (2.15)

This formula is identical to the expression (2.8) for the steady-state error of the response to apolynomial reference input.

It follows that a typek system has a zero steady-state response to polynomial disturbances oforder less thank, a nonzero finite steady-state response for polynomial disturbances of orderk,and an infinite steady-state response for polynomial disturbances of order greater thank. �

Exercise 2.2.3 (Steady-state errors and closed-loop poles and zeros). Suppose that theclosed-loop transfer functionGcl of (2.5) is expanded in factored form as

Gcl(s) = k(s+ z1)(s+ z2) · · · (s+ zm)

(s+ p1)(s+ p2) · · · (s+ pn). (2.16)

1. Prove that the position error constant may be expressed as

Kp = k∏m

j=1 zj∏nj=1 pj − k

∏mj=1 zj

. (2.17)

2. Suppose thatKp = ∞, that is, the system has zero position error. Prove that the velocityand acceleration error constants may be expressed as

1Kv

=n∑

j=1

1pj

−m∑

j=1

1zj

(2.18)

63


and

1Ka

= 12

(m∑

j=1

1

z2j

−n∑

j=1

1

p2j

− 1K2v

), (2.19)

respectively.Hint: Prove that 1/Kv = −G′cl(0) and 1/Ka = −G′′

cl(0)/2, with the primedenoting the derivative. Next differentiate lnGcl(s) twice with respect tos at s = 0, withGcl given by (2.16).

These results originate from Truxal (1955).The relations (2.18) and (2.19) represent the connection between the error constants and the

system response characteristics. We observe that the further the closed-loop poles are from theorigin, the larger the velocity constantKv is. The velocity constant may also be increased byhaving closed-loop zeros close to the origin. �

2.3. Integral control

Integral control is a remarkably effective classical technique to achieve low-frequency disturbanceattenuation. It moreover has a useful robustness property.

Disturbance attenuation is achieved by making the loop gain large. The loop gain may bemade large at low frequencies, and indeed infinite at zero frequency, by including a factor 1/s inthe loop gainL(s) = P(s)C(s). If P(s) has no “natural” factor 1/s then this is accomplished byincluding the factor in the compensator transfer functionC by choosing

C(s) = C0(s)s

. (2.20)

The rational functionC0 remains to be selected. The compensatorC(s) may be considered asthe series connection of a system with transfer functionC0(s) and another with transfer function1/s. Because a system with transfer function 1/s is an integrator, a compensator of this type issaid to haveintegrating action.

If the loop gainL(s) has a factor 1/s then in the terminology of§ 2.2 (p. 60) the system is oftype 1. Its response to a step reference input has a zero steady-state error.

Obviously, if L0(s) contains no factors then the loop gain

L(s) = L0(s)s

(2.21)

is infinite at zero frequency and very large at low frequencies. As a result, the sensitivity functionS, which is given by

S(s) = 11+ L(s)

= 1

1+ L0(s)s

= ss+ L0(s)

, (2.22)

is zero at zero frequency and, by continuity, small at low frequencies. The fact thatS is zeroat zero frequency implies that zero frequency disturbances, that is, constant disturbances, arecompletely eliminated. This is calledconstant disturbance rejection.

Exercise 2.3.1 (Rejection of constant disturbances). Make the last statement clearer by prov-ing that if the closed-loop system of Fig. 2.3 has integrating action and is stable, then its responsez (from any initial condition) to a constant disturbancev(t) = 1, t ≥ 0, has the property

limt→∞ z(t) = 0. (2.23)

Hint: Use Laplace transforms and the final value property. �

64


e

v

zL

Figure 2.3.: Feedback loop

The constant disturbance rejection property isrobustwith respect to plant and compensatorperturbations as long as the perturbations are such that

• the loop gain still contains the factor 1/s, and

• the closed-loop system remains stable.

In a feedback system with integrating action, the transfer function of the series connection of theplant and compensator contains a factor 1/s. A system with transfer function 1/s is capable ofgenerating a constant output with zero input. Hence, the series connection may be thought of ascontaining amodelof the mechanism that generates constant disturbances, which are preciselythe disturbances that the feedback system rejects. This notion has been generalized (Wonham1979) to what is known as theinternal model principle. This principle states that if a feedbacksystem is to reject certain disturbances then it should contain a model of the mechanism thatgenerates the disturbances.

Exercise 2.3.2 (Type k control). The loop gain of a typek system contains a factor 1/sk, with ka positive integer. Prove that if a typek closed-loop system as in Fig. 2.3 is stable then it rejectsdisturbances of the form

v(t) = tn

n!, t ≥ 0, (2.24)

with n any nonnegative integer such thatn ≤ k − 1. “Rejects” means that for such disturbanceslim t→∞ z(t) = 0. �

Compensators with integrating action are easy to build. Their effectiveness in achieving lowfrequency disturbance attenuation and the robustness of this property make “integral control” apopular tool in practical control system design. The following variants are used:

• Pure integral control,with compensator transfer function

C(s) = 1sTi. (2.25)

The single design parameterTi (called thereset time) does not always allow achievingclosed-loop stability or sufficient bandwidth.

• Proportional and integral control,also known asPI control, with compensator transferfunction

C(s) = g

(1+ 1

sTi

), (2.26)

gives considerably more flexibility.

65


• PID (proportional-integral-derivative)control, finally, is based on a compensator transferfunction of the type

C(s) = g

(sTd + 1+ 1

sTi

). (2.27)

Td is thederivative time. The derivative action may help to speed up response but tends tomake the closed-loop system less robust for high frequency perturbations.

Derivative action technically cannot be realized. In any case it would be undesirable be-cause it greatly amplifies noise at high frequencies. Therefore the derivative termsTd in(2.27) in practice is replaced with a “tame” differentiator

sTd

1+ sT, (2.28)

with T a small time constant.

Standard PID controllers are commercially widely available. In practice they are often tunedexperimentally with the help of the rules developed by Ziegler and Nichols (see for instanceFranklin, Powell, and Emami-Naeini (1991)). The Ziegler-Nichols rules (Ziegler and Nichols1942) were developed under the assumption that the plant transfer function is of a well-dampedlow-pass type. When tuning a P-, PI- or PID-controller according to the Ziegler-Nichols rulesfirst a P-controller is connected to the plant. The controller gaing is increased until undampedoscillations occur. The corresponding gain is denoted as asg0 and the period of oscillation asT0.Then the parameters of the PID-controller are given by

P-controller: g = 0.5g0, Ti = ∞, Td = 0,

PI-controller: g = 0.45g0, Ti = 0.85T0, Td = 0,

PID-controller: g = 0.6g0, Ti = 0.5T0, Td = 0.125T0.

The corresponding closed-loop system has a relative damping of about 0.25, and its closed-loopstep response to disturbances has a pak value of about 0.4. Normally experimental fine tuning isneeded to obtain the best results.

e u

v

zPID-controller

nonlinearplant

Figure 2.4.: Nonlinear plant with integral control

Integral control also works for nonlinear plants. Assume that the plant in the block diagramof Fig. 2.4 has the reasonable property that for every constant inputu0 there is a unique constantsteady-state outputw0, and that the plant may maintain any constant outputw0. The “integralcontroller” (of type I, PI or PID) has the property that it maintains a constant outputu0 if andonly if its input e is zero. Hence, if the disturbance is a constant signalv0 then the closed-loopsystem is in steady-state if and only if the error signale is zero. Therefore, if the closed-loopsystem is stable then it rejects constant disturbances.

66


0

Ti ↓ 0Im

Re− 1θ

−12θ

Figure 2.5.: Root loci for the cruise control system. The×s mark the open-loop poles

Example 2.3.3 (Integral control of the cruise control system). The linearized cruise controlsystem of Example 1.2.1 (p. 3) has the linearized plant transfer function

P(s) =1T

s+ 1θ

. (2.29)

If the system is controlled with pure integral control

C(s) = 1sTi

(2.30)

then the loop gain and sensitivity functions are

L(s) = P(s)C(s) =1

TTi

s(s+ 1θ), S(s) = 1

1+ L(s)= s(s+ 1

θ)

s2 + 1θs+ 1

TTi

. (2.31)

The roots of the denominator polynomial

s2 + 1θ

s+ 1TTi

(2.32)

are the closed-loop poles. Sinceθ, T andTi are all positive these roots have negative real parts,so that the closed-loop system is stable. Figure 2.5 shows the loci of the roots asTi variesfrom ∞ to 0 (see also§ 2.7 (p. 87)). Write the closed-loop denominator polynomial (2.32) ass2 + 2ζ0ω0s+ω2

0, with ω0 the resonance frequency andζ0 the relative damping. It easily followsthat

ω0 = 1√TTi

, ζ0 = 1/θ2ω0

=√

TTi

2θ. (2.33)

The best time response is obtained forζ0 = 12

√2, or

Ti = 2θ2

T. (2.34)

If T = θ = 10 [s] (corresponding to a cruising speed of 50% of the top speed) thenTi = 20 [s].It follows thatω0 = 1/

√200≈ 0.07 [rad/s]. Figure 2.6 shows the Bode magnitude plot of the

resulting sensitivity function. The plot indicates that constant disturbance rejection is obtained aswell as low-frequency disturbance attenuation, but that the closed-loop bandwidth is not greaterthan the bandwidth of about 0.1 [rad/s] of the open-loop system.

67


IncreasingTi decreases the bandwidth. DecreasingTi beyond 2θ2/T does not increase thebandwidth but makes the sensitivity functionS peak. This adversely affects robustness. Band-width improvement without peaking may be achieved by introducing proportional gain. (SeeExercise 2.6.2, p. 84.) �

101

100

10−1

10−2

10−3 10−2 10−1 100

20 dB

20 dB

-20 dB

-40 dB

|S|

angular frequency [rad/s]

Figure 2.6.: Magnitude plot of the sensitivity function of the cruise controlsystem with integrating action

Exercise 2.3.4 (PI control of the cruise control system). Show that by PI control constantdisturbance rejection and low-frequency disturbance attenuation may be achieved for the cruisecontrol system with satisfactory gain and phase margins for any closed-loop bandwidth allowedby the plant capacity. �

Exercise 2.3.5 (Integral control of a MIMO plant). Suppose that Fig. 2.3 (p. 65) represents astable MIMO feedback loop with rational loop gain matrixL such that alsoL−1 is a well-definedrational matrix function. Prove that the feedback loop rejects every constant disturbance if andonly if L−1(0) = 0. �

disturbancev

C Py

Figure 2.7.: Plant with input disturbance

Exercise 2.3.6 (Integral control and constant input disturbances). Not infrequently distur-bances enter the plant at the input, as schematically indicated in Fig. 2.7. In this case the transferfunction from the disturbancesv to the control system outputz is

R= P1+ PC

. (2.35)

Suppose that the system has integrating action, that is, the loop gainL = PC has a pole at 0.Prove that constant disturbances at the input are rejected if and only the integrating action islocalized in the compensator. �

68

2.4. Frequency response plots


2.4.1. Introduction

In classical as in modern control engineering the graphical representation of frequency responsesis an important tool in understanding and studying the dynamics of control systems. In this sectionwe review three well-known ways of presenting frequency responses: Bode plots, Nyquist plots,and Nichols plots.

Consider a stable linear time-invariant system with inputu, outputy and transfer functionL.A sinusoidal inputu(t) = usin(ωt), t ≥ 0, results in the steady-state sinusoidal outputy(t) =ysin(ωt + φ), t ≥ 0. The amplitudey and phaseφ of the output are given by

y = |L(jω)| u, φ = argL(jω). (2.36)

The magnitude|L(jω)| of the frequency response functionL(jω) is thegainat the frequencyω.Its argument argL(jω) is thephase shift.

Write

L(s) = k(s− z1)(s− z2) · · · (s− zm)

(s− p1)(s− p2) · · · (z− pn), (2.37)

with k a constant,z1, z2, · · · , zm the zeros, andp1, p2, · · · , pn the poles of the system. Then for any

0

0

resonance peak

p1

p2

p3 z1

Im p1

Im p1

ωpolezero

Re|L|

jω

Figure 2.8.: Frequency response determined from the pole-zero pattern

s= jω the magnitude and phase ofL(jω)may be determined by measuring the vector lengths andangles from the pole-zero pattern as in Fig. 2.8. The magnitude ofL follows by appropriatelymultiplying and dividing the lengths. The phase ofL follows by adding and subtracting theangles.

A pole pi that is close to the imaginary axis leads to a short vector lengths− pi at values ofω in the neighborhood of Impi. This results in large magnitude|L(jω)| at this frequency, andexplains why a pole that is close to the imaginary axis leads to a resonance peak in the frequencyresponse.

2.4.2. Bode plots

A frequency response functionL(jω),ω ∈ R, may be plotted in two separate graphs, magnitude asa function of frequency, and phase as a function of frequency. When the frequency and magnitudescales are logarithmic the combined set of the two graphs is called theBode diagramof L.Individually, the graphs are referred to as theBode magnitude plotand theBode phase plot.Figure 2.9 shows the Bode diagrams of a second-order frequency response function for differentvalues of the relative damping.

69


10-4 10-2 100 102 104

10

10

10

10

10

-4

2

-2

-6

-8

100

10-4 10-2 100 102 104

-145

-90

-180

-45

0

0 dB

-40 dB

-80 dB

-100 dB

-120 dB

40 dB

ζ0 = .01

ζ0 = .01

ζ0 = .1

ζ0 = .1

ζ0 = 1

ζ0 = 1

ζ0 = 10

ζ0 = 10ζ0 = 100

ζ0 = 100

ph

ase

[deg

]m

ag

nitu

de

normalized angular frequencyω/ω0

Figure 2.9.: Bode diagram of the transfer functionω20/(s

2 + 2ζ0ω0 +ω20) for

various values ofζ0

The construction of Bode diagrams for rational transfer functions follows simple steps. Writethe frequency response function in the factored form

L(jω) = k(jω− z1)(jω− z2) · · · (jω− zm)

(jω− p1)(jω− p2) · · · (jω− pn). (2.38)

It follows for the logarithm of the amplitude and for the phase

log|L(jω)| = log|k| +m∑

i=1

log|jω− zi | −n∑

i=1

log|jω− pi|, (2.39)

argL(jω) = argk +m∑

i=1

arg(jω− zi )−n∑

i=1

arg(jω− pi). (2.40)

The asymptotes of the individual terms of (2.39) and (2.40) may be drawn without computation.For a first-order factors+ω0 we have

log|jω+ ω0| ≈{

log|ω0| for 0 ≤ ω � |ω0|,logω for ω � |ω0|, (2.41)

arg(jω+ ω0) ≈{

arg(ω0) for 0 ≤ ω � |ω0|,90◦ for ω � |ω0|. (2.42)

The asymptotes for the doubly logarithmic amplitude plot are straight lines. The low frequencyasymptote is a constant. The high frequency asymptote has slope 1 decade/decade. If the ampli-tude is plotted in decibels then the slope is 20 dB/decade. The amplitude asymptotes intersect atthe frequency|ω0|. The phase moves from arg(ω0) (0◦ if ω0 is positive) at low frequencies to90◦ (π/2 rad) at high frequencies. Figure 2.10 shows the amplitude and phase curves for the firstorder factor and their asymptotes (forω0 positive).

70


10-3 10-2 10-1 100 101 102 103

100

102

10-3 10-2 10-1 100 101 102 103

0

90

0 dB

20 dB

40 dB

60 dB

20 dB/decade

ph

ase

[deg

]m

ag

nitu

de


Figure 2.10.: Bode amplitude and phase plots for the factors+ ω0. Dashed:low- and high-frequency asymptotes.

Factors corresponding to complex conjugate pairs of poles or zeros are best combined to asecond-order factor of the form

s2 + 2ζ0ω0s+ ω20. (2.43)

Asymptotically,

log|(jω)2 + 2ζ0ω0(jω)+ω20| ≈

{2 log|ω0| for 0 ≤ ω � |ω0|,2 logω for ω� |ω0|, (2.44)

arg((jω)2 + 2ζ0ω0(jω)+ω20) ≈

{0 for 0≤ ω � |ω0|,180◦ for ω � |ω0|. (2.45)

The low frequency amplitude asymptote is again a constant. In the Bode magnitude plot the highfrequency amplitude asymptote is a straight line with slope 2 decades/decade (40 dB/decade).The asymptotes intersect at the frequency|ω0|. The phase goes from 0◦ at low frequencies to180◦ at high frequencies. Figure 2.11 shows amplitude and phase plots for different values of therelative dampingζ0 (with ω0 positive).

Bode plots of high-order transfer functions, in particular asymptotic Bode plots, are obtainedby adding log magnitude and phase contributions from first- and second-order factors.

The “asymptotic Bode plots” of first- and second-order factors follow by replacing the lowfrequency values of magnitude and phase by the low frequency asymptotes at frequencies belowthe break frequency, and similarly using the high-frequency asymptotes above the break fre-quency. High-order asymptotic Bode plots follow by adding and subtracting the asymptotic plotsof the first- and second-order factors that make up the transfer function.

As shown in Fig. 2.12 the gain and phase margins of a stable feedback loop may easily beidentified from the Bode diagram of the loop gain frequency response function.

Exercise 2.4.1 (Complex conjugate pole pair). Consider the factors2 + 2ζ0ω0s+ ω20. The

positive numberω0 is the characteristic frequency andζ0 the relative damping.

71


10-2 10-1 100 101 10210-2

100

102

104

10-2 10-1 100 101 1020

90

180

0 dB

60 dB

80 dB

-60 dB

ζ0 = 1

ζ0 = .5

ζ0 = .2

ζ0 = .05

ζ0 = 1

ζ0 = .5

ζ0 = .2

ζ0 = .05

40 dB/decade

ph

ase

[deg

]m

ag

nitu

de


Figure 2.11.: Bode plots for a second-order factors2 + 2ζ0ω0s+ ω20 for dif-

ferent values of the relative dampingζ0. Dashed: low- and high-frequencyasymptotes

1. Prove that for|ζ0| < 1 the roots of the factor are the complex conjugate pair

ω0

(−ζ0 ± j

√1− ζ2

0

). (2.46)

For |ζ0| ≥ 1 the roots are real.

2. Prove that in the diagram of Fig. 2.13 the distance of the complex conjugate pole pair tothe origin isω0, and that the angleφ equals arccosζ0.

3. Assume that 0< ζ0 < 1. At which frequency has the amplitude plot of the factor(jω)2 +2ζ0ω0(jω)+ ω2

0, ω ∈ R, its minimum (and, hence, has the amplitude plot of its reciprocalits maximum)? Note that this frequency is not preciselyω0.

�

Exercise 2.4.2 (Transfer function and Bode plot). Consider the loop gainL whose Bode dia-gram is given in Fig. 2.14.

1. Use Bode’s gain-phase relationship (§ 1.6, p. 34) to conclude from the Bode plot that theloop gain is (probably) minimum-phase, and, hence, stable. Next argue that the corre-sponding closed-loop system is stable.

2. Fit the Bode diagram as best as possible by a rational pole-zero representation.

The conclusion of (a) is correct as long as outside the frequency range considered the frequencyresponse behaves in agreement with the low- and high-frequency asymptotes inferred from theplot. This is especially important for the low-frequency behavior. Slow dynamics that changethe number of encirclements of the Nyquist plot invalidate the result. �

72


0 dB

magnitude

phasephase

ω1 ω2

0◦

−180◦

Loop gain

gain

frequency(log scale)

margin

margin (dB)

Figure 2.12.: Gain and phase margins from the Bode plot of the loop gain

0

jω0

√1− ζ2

−ζω0

φ

Re

Im

Figure 2.13.: Complex conjugate root pair of the factors2 + 2ζ0ω0 + ω20

10-2 10-1

100

101

102

-100

-50

0

50

10-2 10-1

100

101

102

-90

-120

-150

-180

Ga

in[d

B]

Ph

ase

[deg

]

Frequency [rad/s]

Figure 2.14.: Bode diagram of a loop gainL

73


2.4.3. Nyquist plots

-1.5 -1 -0.5 0 0.5 1 1.5 2-3

-2.5

-2

-1.5

-1

-0.5

0

Im

Re

ω/ω0 = 10

ω/ω0 = 1.4ω/ω0 = .5

ω/ω0 = .7

ω/ω0 = 1

ζ0 = .2

ζ0 = .5

ζ0 = 1

ζ0 = 10

Figure 2.15.: Nyquist plots of the transfer functionω20/(s

2 + 2ζ0ω0s+ ω20)

for different values of the relative dampingζ0

In § 1.3 (p. 11) we already encountered theNyquist plot,which is a polar plot of the frequencyresponse function with the frequencyω as parameter. If frequency is not plotted along the locus— a service that some packages fail to provide — then the Nyquist plot is less informative thanthe Bode diagram. Figure 2.15 shows the Nyquist plots of the second-order frequency responsefunctions of Fig. 2.9.

Normally the Nyquist plot is only sketched for 0≤ ω <∞. The plot for negative frequenciesfollows by mirroring with respect to the real axis.

If L is strictly proper then forω → ∞ the Nyquist plot approaches the origin. WriteL interms of its poles and zeros as in (2.37). Then asymptotically

L(jω) ≈ k(jω)n−m

for ω → ∞. (2.47)

If k is positive then the Nyquist plot approaches the origin at an angle−(n − m)× 90◦. Thenumbern− m is called thepole-zero excessor relative degreeof the system.

In control systems of typek the loop gainL has a pole of orderk at the origin. Hence, at lowfrequencies the loop frequency response asymptotically behaves as

L(jω) ≈ c(jω)k

for ω ↓ 0, (2.48)

with c a real constant. Ifk = 0 then forω ↓ 0 the Nyquist plot ofL approaches a point on thereal axis. Ifk > 0 andc is positive then the Nyquist plot goes to infinity at an angle−k × 90◦.Figure 2.16 illustrates this.

74


-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

k = 0k = 1

k = 2 k = 3

L(s) = 1sk(1+ s)

ω

ω

ω

ω

Im

Re

Figure 2.16.: Nyquist plots of the loop gain for different values of systemtype

Exercise 2.4.3 (Nyquist plots). Prove the following observations.

1. The shape of the Nyquist plot ofL(s)= 1/(1+ sT) is a circle whose center and radius areindependent ofT.

2. The shape of the Nyquist plot ofL(s) = 1/(1+ sT1)(1 + sT2) only depends on the ratioT1/T2. The shape is the same forT1/T2 = α andT2/T1 = α.

3. The shape of the Nyquist plot ofL(s) = ω20/(ω

20 + 2ζω0s+ s2) is independent ofω0.

�

2.4.4. M- and N-circles

Consider a simple unit feedback loop with loop gainL as in Fig. 2.17. The closed-loop transferfunction of the system equals the complementary sensitivity function

H = T = L1+ L

. (2.49)

M- andN-circlesare a graphical tool — typical for the classical control era — to determine theclosed-loop frequency response function from the Nyquist plot of the loop gainL.

r yL

Figure 2.17.: Unit feedback loop

75


An M-circle is the locus of pointsz in the complex plane where the magnitude of the complexnumber

z1+ z

(2.50)

is constant and equal toM. An M-circle has center and radius

center

(M2

1− M2,0

), radius

∣∣∣∣ M1− M2

∣∣∣∣ . (2.51)

An N-circle is the locus of points in the complex plane where the argument of the number (2.50)is constant and equal to arctanN. An N-circle has center and radius

center

(−1

2,

12N

), radius

12

√1+ 1

N2. (2.52)

Figure 2.18 shows the arrangement ofM- andN-circles in the complex plane.The magnitude of the closed-loop frequency response and complementary sensitivity function

T may be found from the points of intersection of the Nyquist plot of the loop gainL with theM-circles. Likewise, the phase ofT follows from the intersections with theN-circles.

-4 -2 2-3

-3

3

-2

2

-4 2-3

-3

3

-2

-1

-1

0

1

1

2

0

-1

-1

0

1

1 -2 0

6dB

3dB

2dB

1dB .5dB -.5dB -1dB

-2dB

-3dB

-6dB

15o

-15o

30o

-30o

60o

-60o

5o

-5o

M-circles N-circles

ImIm

Re Re

Figure 2.18.:M- circles (left) andN-circles (right)

Figure 2.18 includes a typical Nyquist plot of the loop gainL. These are some of the featuresof the closed-loop response that are obtained by inspection of the intersections with theM-circles:

• The height of the resonance peak is the maximum value ofM encountered along theNyquist plot.

• The resonance frequencyωm is the frequency where this maximum occurs.

• The bandwidth is the frequency at which the Nyquist plot intersects the 0.707 circle (the−3 dB circle).

76


These and other observations provide useful indications how to modify and shape the loop fre-quency response to improve the closed-loop properties. TheM- and N-loci are more often in-cluded in Nichols plots (see the next subsection) than in Nyquist plots.

Exercise 2.4.4 ( M- and N-circles). Verify the formulas (2.51) and (2.52) for the centers andradii of theM- andN-circles. �

2.4.5. Nichols plots

The linear scales of the Nyquist plot sometimes obscure the large range of values over whichthe magnitude of the loop gain varies. Also, the effect of changing the compensator frequencyresponse functionC or the plant frequency response functionP on the Nyquist plot of the loopgainL = PC cannot always easily be predicted.

Both difficulties may be overcome by plotting the loop gain in the form of aNichols plot(James, Nichols, and Philips 1947). A Nichols plot is obtained by plotting the log magnitudeof the loop gain frequency response function versus its phase. In these coordinates, theM- enN-circles transform toM- and N- loci. The phase–log magnitude plane together with a set ofM- andN-loci is called aNichols chart. In Fig. 2.19 Nichols plots are given of the second-orderfrequency response functions whose Bode diagrams and Nyquist plots are shown in Figs. 2.9(p. 70) and 2.15 (p. 74), respectively.

In a Nichols diagram, gain change corresponds to a vertical shift and phase change to ahorizontal shift. This makes it easy to assess the effect of changes of the compensator frequencyresponse functionC or the plant frequency response functionP on the loop gainL = PC.

Exercise 2.4.5 (Gain and phase margins in the Nichols plot). Explain how the gain marginand phase margin of a stable feedback loop may be identified from the Nichols plot. �

Exercise 2.4.6 (Lag-lead compensator). Consider a compensator with the second-order trans-fer function

C(s) = (1+ sT1)(1+ sT2)

(1+ sT1)(1+ sT2)+ sT12. (2.53)

T1, T2 andT12 are time constants. The corresponding frequency response function is

C(jω) = (1−ω2T1T2)+ jω(T1 + T2)

(1− ω2T1T2)+ jω(T1 + T2 + T12), ω ∈ R. (2.54)

By a proper choice of the time constants the network acts as a lag network (that is, subtractsphase) in the lower frequency range and as a lead network (that is, adds phase) in the higherfrequency range.

Inspection of the frequency response function (2.54) shows that numerator and denominatorsimultaneously become purely imaginary at the frequencyω = 1/

√T1T2. At this frequency

the frequency response function is real. This frequency is the point where the character of thenetwork changes from lag to lead, and where the magnitude of the frequency response is minimal.

1. Plot the Bode, Nyquist, and Nichols diagrams of this frequency response function.

2. Prove that the Nyquist plot ofC has the shape of a circle in the right half of the complexplane with its center on the real axis. SinceC(0) = C(j∞) = 1 the plot begins and endsin the point 1.

�

77


-360 0

-40

-30

-20

-10

0

10

20

6 dB

3 dB

1 dB

-1 dB

-3 dB

-6 dB

-12 dB

-20 dB

-40 dB

.5 dB.25 dB

0 dB

-270 -180 -90

ωω0

= .2

ωω0

= .5

ωω0

= 1

ωω0

= 2

ωω0

= 5ζ0 = .01

ζ0 = .1

ζ0 = 1ζ0 = 2

op

en

-lo

op

ga

in[d

B]

open-loop phase [deg]

Figure 2.19.: Nichols plots of the transfer functionω20/(s

2 + 2ζ0ω0s+ ω20)

for different values ofζ0

78

2.5. Classical control system design

2.5. Classical control system design

2.5.1. Design goals and criteria

For SISO systems we have the following partial list of typical classical performance specifica-tions. Consider the feedback loop of Fig. 2.20. These are the basic requirements for a well-

rPC

y

Figure 2.20.: Basic feedback system

designed control system:

1. The transient response is sufficiently fast.

2. The transient response shows satisfactory damping.

3. The transient response satisfies accuracy requirements, often expressed in terms of theerror constants of§ 2.2 (p. 60).

4. The system is sufficiently insensitive to external disturbances and variations of internalparameters.

These basic requirements may be further specified in terms of both a number offrequency-domainspecificationsand certaintime-domain specifications.

Figures 2.12 (p. 73) and 2.21 illustrate several important frequency-domain quantities:

0 dB-3 dB

|H|

ωr

M

frequencyB

Figure 2.21.: Frequency-domain performance quantities

Gain margin. The gain margin — see§ 1.4 (p. 20) — measures relative stability. It is definedas the reciprocal of the magnitude of the loop frequency responseL, evaluated at the fre-quencyωπ at which the phase angle is−180 degrees. The frequencyωπ is called thephasecrossover frequency.

Phase margin. The phase margin — again see§ 1.4 — also measures relative stability. It isdefined as 180◦ plus the phase angleφ1 of the loop frequency responseL at the frequencyω1 where the gain is unity. The frequencyω1 is called thegain crossover frequency.

79


Bandwidth. The bandwidthB measures the speed of response in frequency-domain terms. Itis defined as the range of frequencies over which the closed-loop frequency responseHhas a magnitude that is at least within a factor1

2

√2 = 0.707 (3 dB) of its value at zero

frequency.

Resonance peak. Relative stability may also be measured in terms of the peak valueM of themagnitude of the closed-loop frequency responseH (in dB), occurring at theresonancefrequencyωr .

Figure 2.22 shows five important time-domain quantities that may be used for performance spec-ifications for the response of the control system output to a step in the reference input:

100 %

50 %

0 %10 %

90 %

FVEPO

Td

Tr

Ts

time

Step response

Figure 2.22.: Time-domain quantities

Delay time Td. delay time measures the total average delay between reference and output. Itmay for instance be defined as time where the response is at 50% of the step amplitude.

Rise time Tr . The rise time expresses the “sharpness” of the leading edge of the response. Var-ious definitions exist. One definesTR as the time needed to rise from 10% to 90% of thefinal value.

Percentage overshoot PO. This quantity expresses the maximum difference (in % of thesteady-state value) between the transient and the steady-state response to a step input.

Settling time Ts. The settling time is often defined as time required for the response to a stepinput to reach and remain within a specified percentage (typically 2 or 5%) of its finalvalue.

Final value of error FVE. The FVE is the steady-state position error.

This list is not exhaustive. It includes no specifications of thedisturbance attenuating properties.These specifications can not be easily expressed in general terms. They should be consideredindividually for each application.

Horowitz (1963, pp. 190–194) lists a number of quasi-empirical relations between the timedomain parametersTd, Tr , Ts and the overshoot on the one hand and the frequency domainparametersB, M and the phase at the frequencyB on the other. The author advises to use themwith caution.

Exercise 2.5.1. Cruise control system Evaluate the various time and frequency performanceindicators for the integral cruise control system design of Example 2.3.3 (p. 67). �

80

2.6. Lead, lag, and lag-lead compensation

2.5.2. Compensator design

In the classical control engineering era the design of feedback compensation to a great extentrelied on trial-and-error procedures. Experience and engineering sense were as important as athorough theoretical understanding of the tools that were employed.

In this section we consider the basic goals that may be pursued from a classical point of view.In the classical view the following series of steps leads to a successful control system design:

• Determine the plant transfer functionP based on a (linearized) model of the plant.

• Investigate the shape of the frequency responseP(jω), ω ∈ R, to understand the propertiesof the system fully.

• Consider the desired steady-state error properties of the system (see§ 2.2, p. 60). Choosea compensator structure — for instance by introducing integrating action or lag compen-sation — that provides the required steady-state error characteristics of the compensatedsystem.

• Plot the Bode, Nyquist or Nichols diagram of the loop frequency response of the compen-sated system. Adjust the gain to obtain a desired degree of stability of the system.M- andN-circles are useful tools. The gain and phase margins are measures for the success of thedesign.

• If the specifications are not met then determine the adjustment of the loop gain frequencyresponse function that is required. Use lag, lead, lag-lead or other compensation to realizethe necessary modification of the loop frequency response function. The Bode gain-phaserelation sets the limits.

The graphic tools essential to go through these steps that were developed in former time now areintegrated in computer aided design environments.

The design sequence summarizes the main ideas of classical control theory developed in theperiod 1940–1960. It is presented in terms ofshaping loop transfer functionsfor single-input,single-output systems.

In § 2.6 (p. 81) we consider techniques for loop shaping using simple controller structures— lead, lag, and lead-lag compensators. In§ 2.8 (p. 89) we discuss the Guillemin-Truxal designprocedure. Section 2.9 (p. 92) is devoted to Horowitz’s Quantitative Feedback Theory (Horowitzand Sidi 1972), which allows to impose and satisfy quantitative bounds on the robustness of thefeedback system.


2.6.1. Introduction

In this section we discuss the classical techniques of lead, lag, and lag-lead compensation. Anextensive account of these techniques is given by Dorf (1992).

2.6.2. Lead compensation

Making the loop gainL large at low frequencies — by introducing integrating action or makingthe static gain large — may result in a Nyquist plot that shows unstable behavior. Even if the

81


-2 -1.5 -1 -0.5 0-1

-0.5

0

0.5

1

1 dB -1 dB3 dB -3 dB

6 dB -6 dB

12 dB -12 dB

Im

Re

unco

mpe

nsat

edco

mpe

nsat

ed

Figure 2.23.: Nyquist plot of uncompensated and compensated plant

closed-loop system is stable the gain and phase margins may be unacceptably small, resulting innearly unstable, oscillatory behavior.

Figure 2.23 shows an instance of this. To obtain satisfactory stability we may reshape the loopgain in such a way that its Nyquist plot remains outside anM-circle that guarantees sufficientclosed-loop damping. A minimal value ofM = 1.4 (3 dB) might be a useful choice.

The required phase advance in the resonance frequency region may be obtained by utilizing aphase-advance network in series with the plant. The network may be of first order with frequencyresponse function

C(jω) = α1+ jωT

1+ jωαT. (2.55)

For 0< α < 1 we obtain a lead compensator and forα > 1 a lag compensator. In the first case thecompensator creates phase advance, in the second it creates extra phase lag. Figure 2.24 showsthe Bode diagrams.

Over the frequency interval(1/T,1/αT) the phase advance compensator has the characterof a differentiating network. By makingα sufficiently small the compensator may be given thecharacter of a differentiator over a large enough frequency range.

Phase lead compensation, also used in PD control, increases the bandwidth and, hence, makesthe closed-loop system faster. Keeping the Nyquist plot away from the critical point−1 has theeffect of improving the transient response.

Phase lead compensation results in an increase of the resonance frequency. If very smallvalues ofα are used then the danger of undesired amplification of measurement noise in the loopexists. The bandwidth increase associated with makingα small may aggravate the effect of highfrequency parasitic dynamics in the loop.

The characteristics of phase-lead compensation are reviewed in Table 2.2. An application oflead compensation is described in Example 2.6.3 (p. 85).

Exercise 2.6.1 (Specifics of the first-order lead or lag compensator). Inspection of Fig. 2.24shows that the maximum amount of phase lead or lag that may be obtained with the compensator

82


-50

0

10-2

100

102

1040

50

100

-100 0

50

100

10-4

10-2

100

102

-100

-50

0

α= .001

α = .1

α = .1

α = .01

α = .01

α = .001

α = .0001

α = .0001

α = 10000

α = 10000

α = 1000

α = 1000

α = 100

α = 100

α = 10

α = 10

normalized angular frequencyωTnormalized angular frequencyωT

Lead compensation Lag compensation

mag

nitu

de[d

B]

phas

e[d

eg]

Figure 2.24.: Log magnitude and phase of lead and lag compensatorsC(s)=α 1+sT

1+sαT

(2.55) is determined byα. Also the width of the frequency window over which significant phaselead or lag is achieved depends onα. Finally, the low frequency gain loss (for lead compensation)or gain boost (for lag compensation) depend onα.

1. Prove that the peak phase lead or lag occurs at the normalized frequency

ωpeakT = 1/√α, (2.56)

and that the peak phase lead or lag equals

φmax = arctan12

∣∣∣∣ 1√α

− √α

∣∣∣∣ . (2.57)

2. Show that the width of the window over which phase lead or lag is effected is roughly∣∣log10α∣∣ decades.

3. Show that the low frequency gain loss or boost is 20∣∣log10α

∣∣ dB.

Figure 2.25 shows plots of the peak phase lead or lag, the window width, and the low-frequencygain loss or boost. �

First-order phase advance compensation is not effective against resonant modes in the plantcorresponding to second order dynamics with low damping. The rapid change of phase from 0to −180 degrees caused by lightly damped second-order dynamics cannot adequately be coun-tered. This requires compensation by a second order filter (called anotch filter) with zeros nearthe lightly damped poles and stable poles on the real line at a considerable distance from theimaginary axis.

83


10

10

-4

-4

10

10

-3

-3

10

10

-2

-2

10

10

-1

-1

10

10

0

0

0

50

100

0

2

4

0

40

80

peak value ofphase orlag [deg]

lead or lagwindow width

[decades]

low frequencygain loss

or boost [dB]

α or 1/α

Figure 2.25.: Peak phase lead or lag

2.6.3. Lag compensation

The loop gain may be increased at low frequencies by a lag compensator. If the time constantTin

C(jω) = α1+ jωT

1+ jωαT(2.58)

is chosen such that 1/T is much greater than the resonance frequencyωR of the loop gain thenthere is hardly any additional phase lag in the crossover region. In the limitα→ ∞ the compen-sator frequency response function becomes

C(jω) = 1+ 1jωT

. (2.59)

This is a compensator with proportional and integral action.Increasing the low frequency gain by lag compensation reduces the steady-state errors. It also

has the effect of decreasing the bandwidth, and, hence, making the closed-loop system slower.On the other hand, the effect of high frequency measurement noise is reduced. Table 2.2 reviewsand summarizes the characteristics of lead and lag compensation.

Lag compensation is fully compatible with phase-lead compensation as the two compensa-tions affect frequency regions that are widely apart.

Exercise 2.6.2 (Phase lag compensation). An example of phase lag compensation is the inte-gral compensation scheme for the cruise control system of Example 2.3.3 (p. 67). The first-orderplant requires a large gain boost at low frequencies for good steady-state accuracy. This gain isprovided by integral control. As we also saw in Example 2.3.3 (p. 67) pure integral control limitsthe bandwidth. To speed up the response additional phase lead compensation is needed.

To accomplish this modify the pure integral compensation scheme to the PI compensator

C(s) = k1+ sTi

sTi. (2.60)

This provides integrating action up to the frequency 1/Ti. At higher frequencies the associated90◦ phase lag vanishes. A suitable choice for the frequency 1/Ti is, say, half a decade below thedesired bandwidth.

84


Table 2.2.: Characteristics of lead and lag compensation designCompensation Phase-lead Phase-lag

Method Addition of phase-lead angle nearthe crossover frequency

Increase the gain at low frequencies

Purpose Improve the phase margin and thetransient response

Increase the error constants whilemaintaining the phase margin andtransient response properties

Applications When a fast transient response isdesired

When error constants are specified

Results Increases the system bandwidth Decreases the system bandwidthAdvantages Yields desired response Suppresses high frequency noise

Speeds dynamic response Reduces the steady-state errorDisadvantages Increases the bandwidth and thus

the susceptibility to measurementnoise

Slows down the transient response

Not applicable If the phase decreases rapidly nearthe crossover frequency

If no low frequency range existswhere the phase is equal to thedesired phase margin

Suppose that the desired bandwidth is 0.3 [rad/s]. SelectTi as recommended, and choose thegain k such that the loop gain crossover frequency is 0.3 [rad/s]. Check whether the resultingdesign is satisfactory. �

2.6.4. Lag-lead compensation

We illustrate the design of a lag-lead compensator by an example. Note the successive designsteps.

Example 2.6.3 (Lag-lead compensator). Consider the simple second-order plant with transferfunction

P(s) = ω20

s2 + 2ζ0ω0s+ ω20

, (2.61)

with ω0 = 0.1 [rad/s] andζ0 = 0.2. The system is poorly damped. The design specifications are

• Constant disturbance rejection by integral action.

• A closed-loop bandwidth of 1 [rad/s].

• Satisfactory gain and phase margins.

Step 1: Lag compensation. To achieve integral control we introduce lag compensation of theform

C0(s) = k01+ sTi

sTi. (2.62)

The phase lag compensation may be extended to 1 decade below the desired bandwidthby choosing 1/Ti = 0.1 [rad/s], that is,Ti = 10 [s]. Lettingk0 = 98.6 makes sure that thecrossover frequency of the loop gain is 1 [rad/s]. Figure 2.26 shows the Bode diagram ofthe resulting loop gain. Inspection reveals a negative phase margin, so that the closed-loopsystem is unstable.

85


10-2

10-1

100

101

102

-100

0

100

10-2

10-1

100

101

102

-90

-180

-270

-360

0g

ain

[db

]p

ha

se[d

eg]

frequency [rad/s]

Lag compensation only

Lag compensation only

Lead-lag compensation

Lead-lag compensation

Lead-lag compensation with HF roll-off

Lead-lag compensation with HF roll-off

Figure 2.26.: Bode diagrams of the loop gain

Step 2: Phase lead compensation. We stabilize the closed loop by lead compensation of theform

C1(s) = k1α1+ sT

1+ sαT. (2.63)

Phase advance is needed in the frequency region between, say, 0.1 and 10 [rad/s]. In-spection of Fig. 2.24 or 2.25 and some experimenting leads to the choiceα = 0.1 andT = 3 [rad/s]. Settingk1 = 3.3 makes the crossover frequency equal to 1 [rad/s]. The re-sulting Bode diagram of the loop gain is included in Fig. 2.26. The closed-loop system isstable with infinite gain margin (because the phase never goes below−180◦) and a phasemargin of more than 50◦.

Figure 2.27 shows the Bode magnitude plot of the closed-loop frequency response functionand of the closed-loop step response. They are quite adequate.

Step 3. High-frequency roll-off. For measurement noise reduction and high-frequency robust-ness we provide high-frequency roll-off of the compensator by including additional lagcompensation of the form

C2(s) = ω21

s2 + 2ζ1ω1s+ ω21

. (2.64)

Settingω1 = 10 [rad/s] andζ1 = 0.5 makes the roll-off set in at 10 [rad/s] without unnec-essary peaking and without appreciable effect in the crossover region. The correspondingloop gain is shown in Fig. 2.26. The gain margin is now about 17 dB and the phase mar-gin about 45◦. Figure 2.27 shows the extra roll-off of the closed-loop frequency response.Enhancing high-frequency roll-off slightly increases the overshoot of the closed-loop stepresponse.

�

86

2.7. The root locus approach to parameter selection

10-2

100

102

-200

0

200

0 5 10 15 200

0.5

1

1.5

10-2

100

102

-200

-100

0

100

0 5 10 15 200

0.5

1

1.5

frequency [rad/s]

frequency response

frequency response step response

time [s]

step response

lag-lead com-pensation withextra HF roll-off

pensationlag-lead com-

mag

nitu

de[d

B]

mag

nitu

de[d

B]

ampl

itude

ampl

itude

Figure 2.27.: Closed-loop frequency and step responses

2.7. The root locus approach to parameter selection

2.7.1. Introduction

The root locus technique was conceived by Evans (1950) — see also Evans (1954). It consists ofplotting the loci of the roots of the characteristic equation of the closed-loop system as a functionof a proportional gain factor in the loop transfer function. This graphical approach yields a clearpicture of the stability properties of the system as a function of the gain. It leads to a designdecision about the value of the gain.

The root locus method is not a complete design procedure. First the controller structure,including its pole and zero locations, should be chosen. The root locus method then allows toadjust the gain. Inspection of the loci often provides useful indications how to revise the choiceof the compensator poles and zeros.

2.7.2. Root loci rules

We review the basic construction rules for root loci. Let the loop transfer function of a feedbacksystem be given in the form

L(s) = k(s− z1)(s− z2) · · · (s− zm)

(s− p1)(s− p2) · · · (s− pn). (2.65)

For physically realizable systems the loop transfer functionL is proper, that is,m≤ n. The rootsz1, z2, · · · , zm of the numerator polynomial are theopen-loop zerosof the system. The rootsp1, p2, · · · , pn of the denominator polynomial are theopen-loop poles. The constantk is thegain.

87


The closed-loop poles are those values ofs for which 1+ L(s) = 0, or, equivalently,

(s− p1)(s− p2) · · · (s− pn)+ k(s− z1)(s− z2) · · · (s− zm) = 0. (2.66)

Under the assumption thatm ≤ n there are preciselyn closed-loop poles. Theroot loci are theloci of the closed-loop poles ask varies from 0 to+∞.

Computer calculations based on subroutines for the calculation of the roots of a polynomialare commonly used to provide accurate plots of the root loci. The graphical rules that followprovide useful insight into the general properties of root loci.

Summary 2.7.1 (Basic construction rules for root loci).

1. Fork = 0 the closed-loop poles coincide with the open-loop polesp1, p2, · · · , pn.

2. If k → ∞ thenmof the closed-loop poles approach the (finite) open-loop zerosz1, z2, · · · ,zm. The remainingn− m closed-loop poles tend to infinity.

3. There are as many locus branches as there are open-loop poles. Each branch starts fork = 0 at an open-loop pole location and ends fork = ∞ at an open-loop zero (which thusmay be at infinity).

4. If m< n thenn− m branches approach infinity along straight line asymptotes. The direc-tions of the asymptotes are given by the angles

αi = 2i + 1n− m

π [rad] i = 0,1, · · · ,n− m− 1. (2.67)

Thus, forn − m = 1 we haveα = π, for n − m = 2 we haveα = ±π/2, and so on. Theangles are evenly distributed over [0,2π].

5. All asymptotes intersect the real axis at a single point at a distances0 from the origin, with

s0 = (sum of open-loop poles)− (sum of open-loop zeros)n− m

. (2.68)

6. As we consider real-rational functions only the loci are symmetrical about the real axis.

-4 0-4

0

4

4 4 4-4 0-4

0

4

-4 0-4

0

4

Re Re Re

Im

(a) (b) (c)

Figure 2.28.: Examples of root loci:(a) L(s) = k

s(s+2) , (b) L(s) = k(s+2)s(s+1) , (c) L(s) = k

s(s+1)(s+2)

7. Those sections of the real axis located to the left of an odd total number of open-loop polesand zeros on this axis belong to a locus.

88

2.8. The Guillemin-Truxal design procedure

8. There may exist points where a locus breaks away from the real axis and points wherea locus arrives on the real axis. Breakaway points occur only if the part of the real axislocated between two open-loop poles belongs to a locus. Arrival points occur only if thepart of the real axis located between two open-loop zeros belongs to a locus.

�

Figure 2.28 illustrates several typical root loci plots.The root locus method has received much attention in the literature subsequent to Evans’

pioneering work. Its theoretical background has been studied by Föllinger (1958), Berman andStanton (1963), Krall (1961), Krall (1963), Krall (1970), and Krall and Fornaro (1967). Theapplication of the root locus method in control design is described in almost any basic controlengineering book — see for instance Dorf (1992), Franklin, Powell, and Emami-Naeini (1986),Franklin, Powell, and Emami-Naeini (1991), and Van de Vegte (1990).

Exercise 2.7.2 (Root loci). Check for each of the root locus diagrams of Fig. 2.28 which of therules (a)–(h) of Summary 2.7.1 applies. �


2.8.1. Introduction

A network-theory oriented approach to the synthesis of feedback control systems was proposed byTruxal (1955). The idea is simple. Instead of designing a compensator on the basis of an analysisof the open-loop transfer function the closed-loop transfer functionH is directly chosen such thatit satisfies a number of favorable properties. Next, the compensator that realizes this behavior iscomputed. Generally an approximation is necessary to arrive at a practical compensator of loworder.

2.8.2. Procedure

Let H be the chosen closed-loop transfer function. For unit feedback systems as in Fig. 2.29 withplant and compensator transfer functionsP andC, respectively, we have

H = PC1+ PC

. (2.69)

Solving for the compensator transfer functionC we obtain

r yC P

Figure 2.29.: Unit feedback system

C = 1P

H1− H

. (2.70)

The determination of the desiredH is not simple. SometimesH may be selected on the basis ofa preliminary analysis of the behavior of a closed-loop system with a low-order compensator.

89


A classical approach is to consider the steady-state errors in selecting the closed-loop system.Suppose that the closed-loop system is desired to be of typek (see§ 2.2, p. 60). Then the loopgainL needs to be of the form

L(s) = N(s)skD(s)

, (2.71)

with N andD polynomials that have no roots at 0. It follows that

H(s) = L(s)1+ L(s)

= N(s)skD(s)+ N(s)

= amsm + · · · + aksk + bk−1sk−1 + · · · + b0

cnsn + · · · + cksk + bk−1sk−1 + · · · + b0. (2.72)

Conversely, choosing the firstk coefficientsbj in the numerator polynomial equal to that of thedenominator polynomial ensures the system to be of typek.

This still leaves considerable freedom to achieve other goals. Suppose that we select theclosed-loop transfer function as

H(s) = b0

sn + bn−1sn−1 + · · · + b0, (2.73)

which implies a zero steady-state error for step inputsr (t) = 1(t).

Exercise 2.8.1 (Zero steady-state error). Prove this. �

One way to choose the coefficientsb0, b1, · · · , bn−1 is to place the closed-loop poles evenlydistributed on the left half of a circle with center at the origin and radiusω0. This yields closed-loop responses with a desired degree of damping. The resulting polynomials are known asBut-terworth polynomials. For the normalized caseω0 = 1 the reference step responses are given inFig. 2.30(a). Table 2.3 shows the coefficients for increasing orders. For generalω0 the polyno-mials follow by substitutings := s/ω0.

Another popular choice is to choose the coefficients such that the integral of the time multi-plied absolute error∫ ∞

0t|e(t)| dt (2.74)

is minimal, with e the error for a step input (Graham and Lathrop 1953). The resulting stepresponses and the corresponding so-calledITAE standard formsare shown in Fig. 2.30(b) andTable 2.4, respectively. The ITAE step responses have a shorter rise time and less overshoot thanthe corresponding Butterworth responses.

2.8.3. Example

We consider the Guillemin-Truxal design procedure for the cruise control system of Exam-ple 2.3.3 (p. 67). The plant has the first-order transfer function

P(s) =1T

s+ 1θ

, (2.75)

with T = θ = 10 [s]. We specify the desired closed-loop transfer function

H(s) = ω20

s2 + 1.4ω0 + ω20

. (2.76)

90


0 5 10 15 200

0.5

1

1.5

Butterworth step responses

0 5 10 15 200

0.5

1

1.5

normalized timenormalized time

ITAE step responses

n = 1n = 1

n = 8n = 8

Figure 2.30.: Step responses for Butterworth (left) and ITAE (right) denom-inator polynomials

Table 2.3.: Normalized denominator polynomials for Butterworth pole pat-terns

Order Denominator

1 s+ 1

2 s2 + 1.4s+ 1

3 s3 + 2.0s2 + 2.0s+ 1

4 s4 + 2.6s3 + 3.4s2 + 2.6s+ 1

5 s5 + 3.24s4 + 5.24s3 + 5.24s2 + 3.24s+ 1

6 s6 + 3.86s5 + 7.46s4 + 9.14s3 + 7.46s2 + 3.86s+ 1

7 s7 + 4.49s6 + 10.1s5 + 14.6s4 + 14.6s3 + 10.1s2 + 4.49s+ 1

8 s8 + 5.13s7 + 13.14s6 + 21.85s5 + 25.69s4 + 21.85s3 + 13.14s2 + 5.13s+ 1

Table 2.4.: Normalized denominator polynomials for ITAE criterion

Order Denominator

1 s+ 1

2 s2 + 1.4s+ 1

3 s3 + 1.75s2 + 2.15s+ 1

4 s4 + 2.1s3 + 3.4s2 + 2.7s+ 1

5 s5 + 2.8s4 + 5.0s3 + 5.5s2 + 3.4s+ 1

6 s6 + 3.25s5 + 6.60s4 + 8.60s3 + 7.45s2 + 3.95s+ 1

7 s7 + 4.475s6 + 10.42s5 + 15.08s4 + 15.54s3 + 10.64s2 + 4.58s+ 1

8 s8 + 5.20s7 + 12.80s6 + 21.60s5 + 25.75s4 + 22.20s3 + 13.30s2 + 5.15s+ 1

The denominator is a second-order ITAE polynomial. The numerator has been chosen for a zeroposition error, that is, type 1 control. It is easy to find that the required compensator transfer

91


function is

C(s) = 1P(s)

H(s)1− H(s)

= ω20T(s+ 1

θ)

s(s+ 1.4s). (2.77)

The integrating action is patent. As seen in Example 2.3.3 (p. 67) the largest obtainable bandwidthwith pure integral control is about 1/

√200≈ 0.07 [rad/s]. For the Guillemin-Truxal design we

aim for a closed-loop bandwidthω0 = 1/√

2 ≈ 0.7 [rad/s].Figure 2.31 shows the resulting sensitivity function and the closed-loop step response. It

confirms that the desired bandwidth has been obtained. Inspection of (2.75) and (2.77) shows

10-1

100

101

10-2

10-1

100

0 5 100

0.5

1

1.5

Step response|H|

mag

nitu

de

angular frequency [rad/s] time [s]

Figure 2.31.: Closed-loop transfer functionH and closed-loop step responseof a Guillemin-Truxal design for the cruise control system

that in the closed loop the plant pole at−1/θ is canceled by a compensator zero at the samelocation. This does not bode well for the design, even though the sensitivity function and theclosed-loop step response of Fig. 2.31 look quite attractive. The canceling pole at−1/θ is alsoa closed-loop pole. It causes a slow response (with the open-loop time constantθ) to nonzeroinitial conditions of the plant and slow transients (with the same time constant) in the plant input.

This cancelation phenomenon is typical for na¨ıve applications of the Guillemin-Truxalmethod. Inspection of (2.70) shows that cancelation may be avoided by letting the closed-looptransfer functionH have a zero at the location of the offending pole. This constrains the choiceof H, and illustrates what is meant by the comment that the selection of the closed-loop transferfunction is not simple.

2.9. Quantitative feedback theory (QFT)

2.9.1. Introduction

Quantitative feedback theory (QFT) is a term coined by Horowitz (1982) (see also Horowitzand Sidi (1972)). A useful account is given by Lunze (1989). The method is deeply rooted inclassical control. It aims at satisfying quantatitive bounds that are imposed on the variations inthe closed-loop transfer function as a result of specified variations of the loop gain. The designmethod relies on the graphical representation of the loop gain in the Nichols chart.

2.9.2. Effect of parameter variations

Nichols plots may be used to study the effect of parameter variations and other uncertainties inthe plant transfer function on a closed-loop system.

92


In particular, it may be checked whether the closed-loop system remains stable. By theNyquist criterion, closed-loop stability is retained as long as the loop gain does not cross thepoint −1 under perturbation. In the Nichols chart, the critical point that is to be avoided is thepoint (−180◦, 0 dB), located at the heart of the chart.

The effect of the perturbations on the closed-loop transfer function may be assessed by study-ing the width of the track that is swept out by the perturbations among theM-loci.

Example 2.9.1 (Uncertain second-order system). As an example we consider the plant withtransfer function

P(s) = gs2(1+ sθ)

. (2.78)

Nominally g = 1 andθ = 0. Under perturbation the gaing varies between 0.5 and 2. Theparasitic time constant may independently vary from 0 to 0.2 [s]. We assume that a preliminary

10-2

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0 5 10

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102

-100

-50

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0 5 100

0

0.5

0.5

1

1

1.5

1.5

[dB]

[dB]

ω [rad/s]

ω [rad/s] time [s]

time [s]

nominal|T|

perturbed|T| perturbed step responses

nominal step response

Figure 2.32.: Nominal and perturbed complementary sensitivity functionsand step responses for the nominal design

study has led to a tentative design in the form of a lead compensator with transfer function

C(s) = k + Tds1+ T0s

, (2.79)

with k = 1, Td = √2 [s] andT0 = 0.1 [s]. The nominal system has closed-loop poles−0.7652±

j0.7715 and−8.4697. The closed-loop bandwidth is 1 [rad/s]. Figure 2.32 shows the nominaland perturbed complementary sensitivity function and closed-loop step response. Figure 2.33shows the Nichols plot of the nominal loop gainL0 = P0C, with P0(s) = 1/s2. The figurealso shows with the uncertainty regions caused by the parameter variations at a number of fixedfrequencies. These diagrams are constructed by calculating the loop gainL(jω) with ω fixed as

93


-360 -270 -180 -90 0-30

-20

-10

0

10

20

30

6 dB

3 dB

1 dB

0.5 dB

0.25 dB

0 dB

-1 dB

-3 dB

-6 dB

-12 dB

-20 dB

ω = 0.2

ω = 0.5

ω = 1

ω = 2

ω = 5

ω = 10A

B

C

D

open-loop phase (degrees)

op

en

-loo

pg

ain

(dB

)

Figure 2.33.: Nominal Nichols plot and uncertainty regions. (Shaded: for-bidden region)

94


a function of the uncertain parametersg andθ along the edges of the uncertainty regions. Thecorners of the uncertainty regions, as marked in the plot forω= 10, correspond to extreme valuesof the parameters as follows:

A: θ = 0, g = 0.5B: θ = 0.2, g = 0.5,C: θ = 0.2, g = 2,D: θ = 0, g = 2.

Inspection shows that no perturbation makes the Nichols plot cross over the center of the chart.This means that the closed-loop system remains stable under all perturbations. �

2.9.3. Stability and performance robustness

Robust stabilityof the closed-loop system is guaranteed if perturbations do not cause the Nicholsplot of the loop gain to cross over the center of the chart.

In the QFT approach, destabilization caused byunmodeled perturbationsis prevented byspecifying aforbidden regionabout the origin for the loop gain as in Fig. 2.33. The forbiddenregion is a region enclosed by anM-locus, for instance the 6 dB locus. If the Nichols plot ofL never enters the forbidden region, not even under perturbation, then the modulus margin isalways greater than 6 dB. Besides providing stability robustness, the guaranteed distance ofLfrom the critical point prevents ringing.

In the QFT approach, in the simplest situationperformance robustnessis specified in the formof bounds on the variation of the magnitude of the closed-loop frequency response functionH.Typically, for each frequencyω the maximally allowable variation1(ω) of |H(jω)|, called thetolerance band,is specified. SinceH = T F, with T the complementary sensitivity function andF the prefilter transfer function, it follows after taking logarithms that

log|H| = log|T| + log|F|. (2.80)

For simplicity we suppress the angular frequencyω. Inspection of (2.80) shows that ifF isnot subject to uncertainty then robust performance is obtained if and only if for each frequencylog|T| varies by at most1 on the uncertainty region. Whether this condition is satisfied may beverified graphically by checking in the Nichols chart whether the uncertainty region fits betweentwo M-loci whose values differ by less than1.

In the next subsection we discuss how to design the feedback loop such thatT satisfies thestability and performance robustness conditions.

Example 2.9.2 (Performance robustness of the design example). Inspection of the plots ofFig. 2.33 reveals that the perturbations sweep out a very narrow band of variations of|T| atfrequencies less than 0.2, a band with a width of about 5 dB at frequency 1, a band with a widthof about 10 dB between the frequencies 2 and 10, while the width of the band further increasesfor higher frequencies. This is borne out by Fig. 2.32. �

2.9.4. QFT design of robust feedback systems

A feedback system design may easily fail to satisfy the performance robustness specifications.This often may be remedied by re-shaping the loop gainL.

Changing the compensator frequency responseC(jω) for some frequencyω amounts toshift-ing the loop gainL(jω) at that same frequency in the Nichols plot. By visual inspection the shape

95


of the Nichols plot ofL may be adjusted by suitable shifts at the various frequencies so that theplot fits the tolerance bounds onT.

Part of the technique is to preparetemplatesof the uncertainty regions at a number of fre-quencies (usually not many more than five), and shifting these around in the Nichols chart. Thetranslations needed to shift the Nichols plot to make it fit the tolerance requirements are achievedby a frequency dependent correction of the compensator frequency responseC. Note that chang-ing the loop gain by changing the compensator frequency response function does not affect theshapes of the templates.

The procedure is best explained by an example.

Example 2.9.3 (QFT design). We continue the second-order design problem of the previousexamples, and begin by specifying the tolerance band1 for a number of critical frequencies asin Table 2.5. The desired bandwidth is 1 rad/s.

Table 2.5.: Tolerance band specifications.frequency tolerance band1

0.2 0.5 dB1 2 dB2 5 dB5 10 dB10 18 dB

Determination of the performance boundaries. The first step of the procedure is to trace foreach selected critical frequency the locus of thenominalpoints such that the tolerance bandis satisfied with the tightest fit. This locus is called theperformance boundary.Points onthe performance boundary may for instance be obtained by fixing the nominal point at acertain phase, and shifting the template up or down until thelowestposition is found wherethe tolerance band condition is satisfied.

Determination of the robustness boundaries. Next, by shifting the template around the for-bidden region so that it touches it but does not enter it therobustness boundaryis tracedfor each critical frequency.

A feedback design satisfies the performance bounds and robustness bounds if for each criticalfrequency the corresponding value of the loop gain is on or above the performance boundaryand to the right of or on the robustness boundary. If it is on the boundaries then the boundsare satisfied tightly. Figure 2.34 shows the performance boundaries thus obtained for the criticalfrequencies 1, 2 and 5 rad/s to the right in the Nichols chart. The performance boundary for thefrequency .1 rad/s is above the portion that is shown and that for 10 rad/s below it. The robustnessboundaries are shown for all five critical frequencies to the right of the center of the chart.

Inspection shows that the nominal design satisfies the specifications for the critical frequen-ciesω = 2, 5 and 10 rad/s, but not forω = 1 rad/s, and also forω = 0.2 it may be shown that thespecifications are not satisfied.

Loop gain shaping. The crucial step in the design is to shape the loop gain such that

1. at each critical frequencyω the corresponding loop gainL(jω) is on or above thecorresponding performance boundary;

2. at each critical frequencyω the corresponding loop gainL(jω) is to the right of thecorresponding stability robustness boundary.

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20

30

6 dB

3 dB

1 dB

0.5 dB

0.25 dB

0 dB

-1 dB

-3 dB

-6 dB

-12 dB

-20 dB

ω = 0.2

ω = 0.2

ω = 0.5

ω = 1

ω = 1

ω = 1

ω = 2

ω = 2

ω = 5

ω = 5

ω = 5

ω = 10

ω = 10

open-loop phase [degrees]

open

-loo

pga

in[d

B]

Figure 2.34.: Performance and robustness boundaries

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-180

30 dB

-30 dB

20 dB

-20 dB

10 dB

-10 dB

0

-90 0

5

ω = 0.2

ω = 0.5

ω = 1

ω = 2

ω = 5

ω = 0.2ω = 0.2

ω = 1

ω = 2

ω = 5

ω = 10

ω = 2

ω = 1

ω = 1

ω = 2

ω = 2

ω = 5

ω = 5

ω = 10

ω = 0.2

ω = 1

ω = 2

ω = 5

ω = 10

ω = 1

ω = 5

ω = 10

T1 = 1T1 = 3

T1 = 9

T2 = 0.02

(a)

open-loop phase [degrees]

open

-loo

pga

in[d

B]

Figure 2.35.: Redesign of the loop gain

This target should be achieved with a compensator transfer function of least complexityand without overdesign (that is, the loop gain should beon the boundaries rather thanabove or to the right). This stage of the design requires experience and intuition, and is theleast satisfactory from a methodical point of view.

In the problem at hand a design may be found in the following straightforward manner. Thevertical line (a) in Fig. 2.35 is the Nichols plot of the nominal plant transfer functionP(s)= 1/s2.Obviously phase lead is needed. This is provided with a compensator with transfer function

C(s) = 1+ sT1. (2.81)

The curves markedT1 = 1, T1 = 3 andT1 = 9 represent the corresponding loop gainsL = PC.The loop gains forT1 = 3 andT1 = 9 satisfy the requirements; the latter with wide margins. WechooseT1 = 3.

To reduce the high-frequency compensator gain we modify its transfer function to

C(s) = 1+ sT1

1+ sT2. (2.82)

The resulting loop gain forT2 = 0.02 is also included in Fig. 2.35. It very nearly satisfies therequirements1. Figure 2.36 gives plots of the resulting nominal and perturbed step responses andcomplementary sensitivity functions. The robustness improvement is evident. �

Exercise 2.9.4 (Performance and robustness boundaries). Traditional QFT relies on shiftingpaper templates around on Nichols charts to determine the performance and robustness bounds

1The requirements may be completely satisfied by adding a few dB to the loop gain.

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[dB]

[dB]

nominal|T| nominal step response

perturbed|T| perturbed step responses



time [s]

time [s]

Figure 2.36.: Nominal and perturbed complementary sensitivity functionsand step responses of the revised design

such as in Fig. 2.34. Think of ways to do this using routines from the MATLAB Control Toolbox.Practice these ideas by re-creating Fig. 2.35. �

2.9.5. Prefilter design

Once the feedback compensaor has been selected the QFT design needs to be completed with thedesign of the prefilter.

Example 2.9.5 (Prefilter design). We continue the design example, and complete it as a 212-

degree-of-freedom design as proposed in§ 1.8 (p. 47). Figure 2.37 shows the block diagram.The choice of the numerator polynomialF provides half a degree of freedom and the rational

r

eu zP

C0

F0

YX

FX

Figure 2.37.: 212-degree-of-freedom feedback system

transfer functionF0 of the rational prefilter constitutes another degree of freedom. The closed-

99


loop transfer function (from the reference inputr to the controlled outputz) is

H = N FDcl

F0. (2.83)

P = N/D is the plant transfer function andDcl = DX + NY the closed-loop characteristic poly-nomial.

In the problem at handN(s) = 1 and D(s) = s2. The compensatorY(s) = 3s + 1,X(s) = 0.02s+ 1 constructed in Example 2.9.3 (p. 96) results in the closed-loop characteris-tic polynomial

Dcl(s) = 0.02s3 + s2 + 3s+ 1. (2.84)

Its roots are−0.3815,−2.7995, and−46.8190. Completing the design amounts to choosing thecorrect polynomialF and transfer functionF0 to provide the necessary compensation in

H(s) = F(s)0.02(s+ 0.3815)(s+ 2.7995)(s+ 46.8190)

F0(s). (2.85)

The pole at−0.3815 slows the response down, so we cancel it by selecting the polynomialF— whose degree can be at most 1 — asF(s) = s/0.3815+ 1. To reduce the bandwidth to thedesired 1 rad/s and to obtain a critically damped closed-loop step response we let

F0(s) = ω20

s2 + 2ζ0ω0s+ ω20

, (2.86)

with ω0 = 1 rad/s andζ = 12

√2.

Figure 2.38 displays the ensuing nominal and perturbed step responses and closed-loop trans-fer functions. Comparison with Fig. 2.33 makes it clear that the robustness has been drasticallyimproved. �

2.9.6. Concluding remark

The QFT method has been extended to open-loop unstable plants and non-minimum phase plants,and also to MIMO and nonlinear plants. Horowitz (1982) provides references and a review.Recently a MATLAB toolbox for QFT design has appeared2.

2.10. Concluding remarks

This chapter deals with approaches to classical compensator design. The focus is on compensa-tion by shaping the open-loop frequency response.

The design goals in terms of shaping the loop gain are extensively considered in the classicalcontrol literature. The classical design techniques cope with them in anad hocand qualitativemanner. It requires profound experience to handle the classical techniques, but if this experienceis available then for single-input single-output systems it is not easy to obtain the quality ofclassical design results by the analytical control design methods that form the subject of the laterchapters of this book.

If the design problem has a much more complex structure, for instance with multi-input multi-output plants or when complicated uncertainty descriptions and performance requirements apply,then the analytical techniques are the only reliable tools. Even in this case a designer needsconsiderable expertise and experience with classical techniques to appreciate and understand thedesign issues involved.

2Quantitative Feedback Theory Toolbox, The MathWorks Inc., Natick, MA, USA, 1995 release.

100

2.10. Concluding remarks

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nominal closed-looptransfer function|H|

nominal closed-loopstep response

perturbed closed-looptransfer functions|H|

perturbed closed-loopstep responses

ω [rad/s]

ω [rad/s]

time [rad/s]

time [rad/s]

Figure 2.38.: Nominal and perturbed step responses and closed-loop transferfunctions of the final QFT design

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102

3. LQ, LQG and H2 Control System Design

Overview– LQ and LQG design methods convert control system designproblems to an optimization problem with quadratic time-domain per-formance criteria. Disturbances and measurement noise are modeled asstochastic processes.

TheH2 formulation of the same method eliminates the stochastic el-ement. It permits a frequency domain view and allows the introductionof frequency dependent weighting functions.

MIMO problems can be handled almost as easily as SISO problems.

3.1. Introduction

The application of optimal control theory to the practical design of multivariable control sys-tems attracted much attention during the period 1960–1980. This theory considers linear finite-dimensional systems represented in state space form, with quadratic performance criteria. Thesystem may be affected by disturbances and measurement noise represented as stochastic pro-cesses, in particular, by Gaussian white noise. The theoretical results obtained for this class ofdesign methods are known under the general name ofLQG theory. Standard references are An-derson and Moore (1971), Kwakernaak and Sivan (1972) and Anderson and Moore (1990). Thedeterministic part is calledLQ theory.

In the period since 1980 the theory has been further refined under the name ofH2 theory(Doyle, Glover, Khargonekar, and Francis 1989), in the wake of the attention for the so-calledH∞ control theory.

In the present chapter we present a short overview of a number of results of LQG andH2

theory with an eye to using them for control system design. LQ theory is basic to the wholechapter, and is dealt with at some length in Section 3.2 (p. 104). Besides presenting the solutionto the LQ problem we discuss its properties, the choice of the weighting matrices, and how toobtain systems with a prescribed degree of stability. Using the notion of return difference andthe associated return difference equality we discuss the asymptotic properties and the guaranteedgain and phase margins associated with the LQ solution. The section concludes with a subsectionon the numerical solution of Riccati equations.

Section 3.3 (p. 115) deals with the LQG problem. The LQ paradigm leads to state feedback.By using optimal observers — Kalman filters — compensators based on output feedback may be

103

3. LQ, LQG andH2 Control System Design

designed. For well-behaved plants — specifically, plants that have no right-half plane zeros —the favorable properties of the LQ solution may asymptotically be recovered by assuming verysmall measurement noise. This is known asloop transfer recovery.

In Section 3.4 (p. 124) it is demonstrated that LQG optimization amounts to minimizationof theH2-norm of the closed-loop system. This interpretation removes the stochastic ingredientfrom the LQG framework, and reduces the role of the intensity matrices that describe the whitenoise processes in the LQG problem to that of design parameters. TheH2 interpretation naturallyleads toH2 optimization with frequency dependent weighting functions. These permit a greatdeal of extra flexibility and makeH2 theory a tool for shaping closed-loop system functions. Auseful application is the design of feedback systems with integral control.

Multi-input multi-output systems are handled almost (but not quite) effortlessly in the LQGandH2 framework. In Section 3.6 (p. 133) we present both a SISO and a MIMO design example.

In Section 3.7 (p. 139) a number of proofs for this chapter are collected.

3.2. LQ theory

3.2.1. Introduction

In this section we describe the LQ paradigm. The acronym refers to Linear systems withQuadratic performance criteria. Consider a linear time-invariant system represented in state spaceform as

x(t) = Ax(t)+ Bu(t),z(t) = Dx(t),

t ≥ 0. (3.1)

For eacht ≥ 0 the statex(t) is ann-dimensional vector, the inputu(t) a k-dimensional vector,and the outputz(t) anm-dimensional vector.

We wish to control the system from any initial statex(0) such that the outputz is reduced toa very small value as quickly as possible without making the inputu unduly large. To this endwe introduce the performance index

J =∫ ∞

0[zT(t)Qz(t)+ uT(t)Ru(t)] dt. (3.2)

Q andRare symmetric weighting matrices, that is,Q = QT andR= RT. Often it is adequate tolet the two matrices simply be diagonal.

The two termszT(t)Qz(t) and uT(t)Ru(t) are quadratic forms in the components of theoutput z and the inputu, respectively. The first term in the integral criterion (3.2) measuresthe accumulated deviation of the output from zero. The second term measures the accumulatedamplitude of the control input. It is most sensible to choose the weighting matricesQ andRsuchthat the two terms are nonnegative, that is, to takeQ andRnonnegative-definite1. If the matricesare diagonal then this means that their diagonal entries should be nonnegative.

The problem of controlling the system such that the performance index (3.2) is minimal alongall possible trajectories of the system is theoptimal linear regulator problem.

1An n × n symmetric matrixR is nonnegative-definite ifxT Rx≥ 0 for everyn-dimensional vectorx. R is positive-definite if xT Rx> 0 for all nonzerox.

104

3.2. LQ theory

3.2.2. Solution of the LQ problem

There is a wealth of literature on the linear regulator problem. The reason why it attracted somuch attention is that its solution may be represented infeedbackform. An optimal trajectory isgenerated by choosing the input fort ≥ 0 as

u(t) = −Fx(t). (3.3)

This solution requires that the statex(t) be fully accessible for measurement at all times. Wereturn to this unreasonable assumption in§ 3.3 (p. 115). Thek × n state feedback gain matrix Fis given by

F = R−1BT X. (3.4)

The symmetricn× n matrix X is the nonnegative-definite solution of thealgebraic matrix Riccatiequation(ARE)

AT X + X A+ DT QD− X BR−1BT X = 0. (3.5)

The proof is sketched in§ 3.7 (p. 139), the appendix to this chapter. The solution of the algebraicRiccati equation is discussed in§ 3.2.9 (p. 114).

We summarize a number of well-known important facts about the solution of the LQ problem.An outline of the proof is given in§ 3.7.1 (p. 142).

Summary 3.2.1 (Properties of the solution of the optimal linear regulator problem).Assumptions:

• The system (3.1) is stabilizable2 and detectable3. Sufficient for stabilizability is that thesystem is controllable. Sufficient for detectability is that it is observable.

• The weighting matricesQ andR are positive-definite.

The following facts are well documented (see for instance Kwakernaak and Sivan (1972) andAnderson and Moore (1990)).

1. The algebraic Riccati equation (ARE)

AT X + X A+ DT QD− X BR−1BT X = 0 (3.6)

has a unique nonnegative-definite symmetric solutionX. If the systemx(t) = Ax(t),z(t) = Dx(t) is observable thenX is positive-definite. There are finitely many other solu-tions of the ARE.

2. The minimal value of the performance index (3.2) isJmin = xT(0)Xx(0).

3. The minimal value of the performance index is achieved by the feedback control law

u(t) = −Fx(t), t ≥ 0, (3.7)

with F = R−1BT X.

2That is, there exists a state feedbacku(t)= −Fx(t) such that the closed-loop systemx(t)= (A− BF)x(t) is stable.3That is, there exists a matrixK such that the systeme(t)= (A− K D)e(t) is stable.

105


4. The closed-loop system

x(t) = (A− BF)x(t), t ≥ 0, (3.8)

is stable, that is, all the eigenvalues of the matrixA− BF have strictly negative real parts.�

The reasons for the assumptions may be explained as follows. If the system is not stabilizablethen obviously it cannot be stabilized. If it is not detectable then there exist state feedback con-trollers that do not stabilize the system but hide the instability from the output—hence, stabilityof the optimal solution is not guaranteed.Rneeds to be positive-definite to prevent infinite inputamplitudes. IfQ is not positive-definite then there may be unstable closed-loop modes that haveno effect on the performance index.

Exercise 3.2.2 (Cruise control system). The linearized dynamics of the vehicle of Exam-ple 1.2.1 (p. 3) may be described by the equationx(t) = −ax(t)+ au(t), wherea = 1/θ is apositive constant. Without loss of generality we may takea = 1.

Consider finding the linear state feedback that minimizes the criterion∫ ∞

0[x2(t)+ ρu2(t)] dt (3.9)

with ρ a positive constant. Determine the ARE, find its positive solution, and compute the opti-mal state feedback gain.

Compute the closed-loop pole of the resulting optimal feedback system and check that theclosed-loop system is always stable. How does the closed-loop pole vary withρ? Explain.

Plot the closed-loop response of the statex(t) and inputu(t) to the initial statex(0) = 1 independence onρ. Explain the way the closed-loop responses vary withρ. �

3.2.3. Choice of the weighting matrices

The choice of the weighting matricesQ and R is a trade-off between control performance (Qlarge) and low input energy (R large). Increasing bothQ and R by the same factor leaves theoptimal solution invariant. Thus, only relative values are relevant. TheQ and R parametersgenerally need to be tuned until satisfactory behavior is obtained, or until the designer is satisfiedwith the result.

An initial guess is to choose bothQ andR diagonal

Q =

Q1 0 0 · · · 00 Q2 0 · · · 0· · · · · · · · · · · · · · ·0 · · · · · · 0 Qm

, R=

R1 0 0 · · · 00 R2 0 · · · 0· · · · · · · · · · · · · · ·0 · · · · · · 0 Rk

, (3.10)

whereQ andR have positive diagonal entries such that

√Qi = 1

zmaxi

, i = 1,2, · · · ,m,√

Ri = 1umax

i

, i = 1,2, · · · , k. (3.11)

The numberzmaxi denotes the maximally acceptable deviation value for theith component of the

outputz. The other quantityumaxi has a similar meaning for theith component of the inputu.

Starting with this initial guess the values of the diagonal entries ofQ andR may be adjustedby systematic trial and error.

106

3.2. LQ theory

3.2.4. Prescribed degree of stability

By including a time-dependent weighting function in the performance index that grows expo-nentially with time we may force the optimal solutions to decay faster than the correspondingexponential rate. The modified performance index is

Jα =∫ ∞

0e2αt[zT(t)Qz(t)+ uT(t)Ru(t)] dt, (3.12)

with α a real number. Define

xα(t) = x(t)eαt, uα(t) = u(t)eαt, t ≥ 0. (3.13)

These signals satisfy

xα(t) = (A+ α I )xα(t)+ Buα(t) (3.14)

and

Jα =∫ ∞

0[zTα(t)Qzα(t)+ uT

α(t)Ruα(t)] dt. (3.15)

Consequently, the minimizinguα is

uα(t) = −R−1BT Xαxα(t), (3.16)

or

u(t) = −Fαx(t), (3.17)

with Fα = R−1BT Xα. Xα is the positive-definite solution of the modified algebraic Riccati equa-tion

(AT + α I )Xα + Xα(A+ α I )+ DT QD− XαBR−1BT Xα = 0. (3.18)

The stabilizing property of the optimal solution of the modified problem implies that

Re λi(A+ α I − BFα) < 0, i = 1,2, · · · ,n, (3.19)

with λi (A+ α I − BFα) denoting theith eigenvalue. Application of the control law (3.17) to thesystem (3.1) creates a closed-loop system matrixAα = A− BFα. It follows from (3.19) that itseigenvalues satisfy

Re λi( Aα) < −α. (3.20)

Thus, choosingα positive results in an optimal closed-loop system with aprescribed degree ofstability.

Exercise 3.2.3 (Cruise control system). Modify the criterion (3.9) to∫ ∞

0e2αt[x2(t)+ ρu2(t)] dt, (3.21)

with α a positive constant. Rework Exercise 3.2.2 (p. 106) while explaining and interpreting theeffect ofα. �

107


(sI−A)−1B (sI−A)−1B

F F

− −

u ux xv

(a) (b)

Figure 3.1.: State feedback

3.2.5. Return difference equality and inequality

Figure 3.1(a) shows the feedback connection of the systemx = Ax+ Buwith the state feedbackcontrolleru = −Fx. If the loop is broken as in Fig. 3.1(b) then the loop gain is

L(s) = F(sI − A)−1B. (3.22)

The quantity

J(s) = I + L(s) (3.23)

is known as thereturn difference, J(s)u is the difference between the signalu in Fig. 3.1(b) andthe “returned” signalv = −L(s)u.

Several properties of the closed-loop system may be related to the return difference. Consider

detJ(s) = det[I + L(s)] = det[I + F(sI − A)−1B]. (3.24)

Using the well-known matrix equality det( I + MN) = det( I + N M) we obtain

detJ(s) = det[I + (sI − A)−1BF]

= det(sI − A)−1 det(sI − A+ BF)

= det(sI − A+ BF)det(sI − A)

= χcl(s)χol(s)

. (3.25)

The quantitiesχol(s)= det(sI − A) andχcl(s)= det(sI − A+ BF) are the open- and the closed-loop characteristic polynomial, respectively. We found the same result in§ 1.3 (p. 11) in the formχcl(s) = χol(s)det[I + L(s)].

Suppose that the gain matrixF is optimal as in Summary 3.2.1. It is proved in§ 3.7.2 (p. 143)by manipulation of the algebraic Riccati equation (3.6) that the corresponding return differencesatisfies the equality

JT(−s)RJ(s)= R+ GT(−s)QG(s). (3.26)

G(s) = D(sI − A)−1B is theopen-loop transfer matrixof the system (3.1).The relation (3.26) is known as thereturn difference equalityor as theKalman-Jakubovic-

Popov (KJP) equality, after its discoverers. In Subsection 3.2.6 (p. 109) we use the returndifference equality to study the root loci of the optimal closed-loop poles.

By settings= jω, with ω ∈ R, we obtain thereturn difference inequality4

JT(−jω)RJ(jω) ≥ R for all ω ∈ R. (3.27)

In Subsection 3.2.7 (p. 112) we apply the return difference inequality to establish a well-knownrobustness property of the optimal state feedback system.

4If P and Q are symmetric matrices of the same dimensions thenP ≥ Q means thatxT Px ≥ xT Qx for every realn-dimensional vectorx.

108

3.2. LQ theory

3.2.6. Asymptotic performance weighting

For simplicity we first consider the case that (3.1) is a SISO system. To reflect this in the notationwe rewrite the system equations (3.1) in the form

x(t) = Ax(t)+ bu(t),

z(t) = dx(t),(3.28)

with b a column vector andd a row vector. Similarly, we represent the optimal state feedbackcontroller as

u(t) = − f x(t), (3.29)

with f a row vector. The open-loop transfer functionG(s) = d(sI − A)−1b, the loop gainL(s) = f (sI − A)−1b and the return differenceJ(s) = 1 + L(s) now all are scalar functions.Without loss of generality we consider the performance index

J =∫ ∞

0[z2(t)+ ρu2(t)] dt, (3.30)

with ρ a positive number. This amounts to settingQ = 1 andR= ρ.Under these assumptions the return difference equality (3.26) reduces to

J(−s)J(s)= 1+ 1ρ

G(s)G(−s). (3.31)

From (3.27) we have

J(s) = χcl(s)χol(s)

, (3.32)

with χcl the closed-loop characteristic polynomial andχol the open-loop characteristic polyno-mial. We furthermore write

G(s) = kψ(s)χol(s)

. (3.33)

The constantk is chosen such that the polynomialψ is monic5. From (3.31–3.33) we now obtain

χcl(−s)χcl(s) = χol(−s)χol(s)+ k2

ρψ(−s)ψ(s). (3.34)

The left-hand sideχcl(−s)χcl(s) of this relation defines a polynomial whose roots consists ofthe closed-loop poles (the roots ofχcl(s)) together with theirmirror imageswith respect to theimaginary axis (the roots ofχcl(−s)). It is easy to separate the two sets of poles, because bystability the closed-loop poles are always in the left-half complex plane.

From the right-hand side of (3.34) we may determine the following facts about the loci of theclosed-loop poles as the weightρ on the input varies.

Infinite weight on the input term. If ρ→ ∞ then the closed-loop poles and their mirror imagesapproach the roots ofχol(s)χol(−s). This means that the closed-loop poles approach

5That is, the coefficient of the term of highest degree is 1.

109


• those open-loop poles that lie in the left-half complex plane (the “stable” open-looppoles), and

• the mirror images of those open-loop poles that lie in the right-half complex plane(the “unstable” open-loop poles).

If the open-loop system is stable to begin with then the closed-loop poles approach theopen-loop poles as the input is more and more heavily penalized. In fact, in the limitρ → ∞ all entries of the gainF become zero—optimal control in this case amounts to nocontrol at all.

If the open-loop system is unstable then in the limitρ→ ∞ the least control effort is usedthat is needed to stabilize the system but no effort is spent on regulating the output.

Vanishing weight on the input term. As ρ ↓ 0 the closed-loop poles the open-loop zeros (theroots ofψ) come into play. Suppose thatq− open-loop zeros lie in the left-half complexplane or on the imaginary axis, andq+ zeros in the right-half plane.

• If ρ ↓ 0 thenq− closed-loop poles approach theq− left-half plane open-loop zeros.

• A further q+ closed-loop poles approach the mirror images in the left-half plane oftheq+ right-half plane open-loop zeros.

• The remainingn − q− − q+ closed-loop poles approach infinity according to a But-terworth pattern of ordern − q− − q+ (see§ 2.7, p. 87).

Just how small or largeρ should be chosen depends on the desired or achievable bandwidth.We first estimate the radius of the Butterworth pole configuration that is created asρ decreases.Taking leading terms only, the right-hand side of (3.34) reduces to

(−s)nsn + k2

ρ(−s)qsq, (3.35)

with q the degree ofψ, and, hence, the number of open-loop zeros. From this the 2(n− q) rootsthat go to infinity asρ ↓ 0 may be estimated as the roots of

s2(n−q) + (−1)n−q k2

ρ= 0. (3.36)

Then− q left-half plane roots are approximations of the closed-loop poles. They form a Butter-worth pattern of ordern − q and radius

ωc =(

k2

ρ

) 12(n−q)

. (3.37)

If the plant has no right-half plane zeros then this radius is an estimate of the closed-loop band-width. The smallerρ is the more accurate the estimate is. The bandwidth we refer to is thebandwidth of the closed-loop system withz as output.

If the plant has right-half plane open-loop zeros then the bandwidth is limited to the magnitudeof the right-half plane zero that is closest to the origin. This agrees with the limits of performanceestablished in§ 1.7 (p. 40).

Exercise 3.2.4 (Closed-loop frequency response characteristics). Verify these statements byconsidering the SISO configuration of Fig. 3.2. LetL(s) = f (sI − A)−1b andG(s) = d(sI −A)−1b.

110

3.2. LQ theory

( )sI A b− −1

f

d−

ur

xz

Figure 3.2.: State feedback with reference inputr and outputz

1. Prove that

u = 11+ L(s)

r, z= G(s)1+ L(s)︸︷︷︸

H(s)

r. (3.38)

It follows that the closed-loop transfer functionH is

H(s) = kψ(s)χcl(s)

. (3.39)

2. Assume that the open-loop transfer functionG has no right-half plane zeros. Prove that asρ ↓ 0 the closed-loop transfer function behaves as

H(s) ≈ kBn−q(s/ωc)

. (3.40)

Bk denotes a Butterworth polynomial of orderk (see Table 2.3, p. 91). Hence, the closed-loop bandwidth isωc.

3. Next assume that the open-loop transfer function has a single (real) right-half plane zeroζ. Prove that asymptotically the closed-loop transfer function behaves as

H(s) ≈ s− ζ

s+ ζ

kBn−q(s/ωc)

. (3.41)

Argue that this means that asymptotically the bandwidth isζ (that is, the frequency re-sponse functionH(jω)/H(0)− 1,ω ∈ R, is small over the frequency range [0, ζ]).

�

For the MIMO case the situation is more complex. The results may be summarized as follows.Let R = ρR0, with ρ a positive number, andR0 a fixed positive-definite symmetric matrix. Westudy the root loci of the closed-loop poles as a function ofρ.

• As ρ → ∞ the closed-loop poles approach those open-loop poles that lie in the left-halfplane and the mirror images in the left-half plane of the right-half plane open-loop poles.

• If ρ ↓ 0 then those closed-loop poles that remain finite approach the left-half plane zerosof detGT(−s)QG(s).

If the open-loop transfer matrixG(s) = D(sI − A)−1B is square then we define the zerosof detG(s) as the open-loop zeros. In this case the closed-loop poles approach the left-half plane open-loop zeros and the left-half plane mirror images of the right-half planeopen-loop zeros.

111


The closed-loop poles that do not remain finite asρ ↓ 0 go to infinity according to severalButterworth patterns of different orders and different radii. The number of patterns andtheir radii depend on the open-loop plant (Kwakernaak 1976).

Understanding the asymptotic behavior of the closed-loop poles provides insight into the proper-ties of the closed-loop systems. We note some further facts:

• As ρ ↓ 0 the gain matrixF approaches∞, that is, some or all of its entries go to infinity.

• Assume that the open-loop transfer matrixD(sI − A)−1B is square, and that all its zerosare in the left-half plane. Then as we saw the closed-loop bandwidth inreases withoutbound asρ ↓ 0. Correspondingly, the solutionX of the Riccati equation approaches thezero matrix.

( )sI A B− −1

F

−

u xv

Figure 3.3.: Loop gain for state feedback

3.2.7. Guaranteed gain and phase margins

If the state feedback loop is opened at the plant input as in Fig. 3.3 then the loop gain isL(s) =F(sI − A)−1B. For the single-input case discussed in Subsection 3.2.6 (see Eqn. (3.31)) thereturn difference inequality (3.27) takes the form

|1+ L(jω)| ≥ 1, ω ∈ R. (3.42)

This inequality implies that the Nyquist plot of the loop gain stays outside the circle with centerat−1 and radius 1. Figure 3.4 shows two possible behaviors of the Nyquist plot.

ReRe

Im Im

(a) (b)

Figure 3.4.: Examples of Nyquist plots of optimal loop gains.(a): Open-loop stable plant. (b): Open-loop unstable plant with one right-half plane pole.

= unit circle

Inspection shows that the modulus margin of the closed-loop system is 1. The gain margin isinfinite and the phase margin is at least 60◦. More precisely, for open-loop stable systems the gain

112

3.2. LQ theory

may vary between 0 and∞ without destabilizing the system. For open-loop unstable systems itmay vary between12 and∞.

Exercise 3.2.5 (Phase margin). Prove that the phase margin is at least 60◦. �

The guaranteed stability margins are very favorable. Some caution in interpreting these resultsis needed, however. The margins only apply to perturbations at the point where the loop is broken,that is, at the plant input. The closed-loop system may well be very sensitive to perturbations atany other point.

The SISO results may be generalized to the multi-input case. Suppose that the loop gainsatisfies the return difference inequality (3.27). Assume that the loop gainL(s) is perturbed toW(s)L(s), with W a stable transfer matrix. It is proved in Subsection 3.7.3 (p. 144) of§ 3.7, theappendix to this chapter, that the closed-loopremainsstable provided

RW(jω)+ WT(−jω)R> R, ω ∈ R. (3.43)

If both R andW are diagonal then this reduces to

W(jω)+ WT(−jω) > I, ω ∈ R. (3.44)

This shows that if theith diagonal entryWi of W is real then it may have any value in the interval( 1

2,∞) without destabilizing the closed-loop system. If theith diagonal entry isWi(jω) = ejφ

then the closed-loop system remains stable as long as the angleφ is less thanπ3 , that is, 60◦.Thus, the SISO results applies to each input channel separately.

3.2.8. Cross term in the performance index

In the optimal regulator problem for the stabilizable and detectable system

x(t) = Ax(t)+ Bu(t),z(t) = Dx(t),

t ≥ 0, (3.45)

we may consider the generalized quadratic performance index

J =∫ ∞

0

[zT(t) uT(t)

][Q SST R

][z(t)u(t)

]dt. (3.46)

We assume that[Q SST R

](3.47)

is positive-definite. Definev(t) = u(t)+ R−1STz(t). Then minimization ofJ is equivalent tominimizing

J =∫ ∞

0[zT(t)(Q− SR−1ST)z(t)+ vT(t)Rv(t)] dt (3.48)

for the system

x(t) = (A− BR−1ST D)x(t)+ Bv(t). (3.49)

113


The condition that (3.47) be positive-definite is equivalent to the condition that bothR andQ−SR−1ST be positive-definite (see Exercise 3.2.7, p. 114). Thus we satisfy the conditions ofSummary 3.2.1. The Riccati equation now is

AT X + X A+ DT QD− (X B+ DTS)R−1(BT X + ST D) = 0. (3.50)

The optimal input for the system (3.45) isu(t) = −Fx(t), with

F = R−1(BT X + ST D). (3.51)

Exercise 3.2.6 (LQ problem for system with direct feedthrough). The system

x(t) = Ax(t)+ Bu(t),z(t) = Dx(t)+ Eu(t),

t ≥ 0, (3.52)

has what is called “direct feedthrough” because of the term withu in the outputz. Show that theproblem of minimizing

J =∫ ∞

0[zT(t)Qz(t)+ uT(t)Ru(t)] dt (3.53)

for this system may be converted into the cross term problem of this subsection. �

Exercise 3.2.7 (Positive-definiteness and Schur complement). Prove that the condition thatthe matrix (3.47) be positive-definite is equivalent to either of the following two conditions:

1. Both R and Q − SR−1ST are positive-definite.Q − SR−1ST is called theSchur comple-mentof R. Hint:[

Q SST R

]=[

I SR−1

0 I

][Q− SR−1ST 0

0 R

][I 0

R−1ST I

]. (3.54)

2. Both Q and R− STQ−1Sare positive-definite.R− ST Q−1S is the Schur complement ofQ.

�

3.2.9. Solution of the ARE

There are several algorithms for the solution of the algebraic Riccati equation (3.6) or (3.50).For all but the simplest problem recourse needs to be taken to numerical computer calculation.Equation (3.50) is the most general form of the Riccati equation. By redefiningDT QD asQ andDTSasS the ARE (3.50) reduces to

AT X + X A+ Q− (X B+ S)R−1(BT X + ST) = 0. (3.55)

The most dependable solution method relies on theHamiltonian matrix

H =[

A− BR−1ST −BR−1BT

−Q+ SR−1ST −(A− BR−1ST)T

](3.56)

associated with the LQ problem. Under the assumptions of Summary 3.2.1 (p. 105) or§ 3.2.7(p. 112) the Hamiltonian matrixH has no eigenvalues on the imaginary axis. Ifλ is an eigenvalueof the 2n × 2n matrix H then−λ is also an eigenvalue. Hence,H has exactlyn eigenvalues

114

3.3. LQG Theory

with negative real part. Let the columns of the real 2n × n matrix E form a basis for then-dimensional space spanned by the eigenvectors and generalized eigenvectors ofH correspondingto the eigenvalues with strictly negative real parts. Partition

E =[

E1

E2

], (3.57)

with E1 andE2 both square. It is proved in§ 3.7.4 (p. 145) that

X = E2E−11 (3.58)

is the desired solution of the algebraic Riccati equation.E may efficiently be computed by Schur decomposition (Golub and Van Loan 1983)

H = UTUH (3.59)

of the Hamiltonian matrixH . U is unitary, that is,UUH = UHU = I . I is a unit matrix andthe superscript H denotes the complex-conjugate transpose.T is upper triangular, that is, allentries below the main diagonal ofT are zero. The diagonal entries ofT are the eigenvalues ofH . The diagonal entries ofT may be arranged in any order. In particular, they may be orderedsuch that the eigenvalues with negative real part precede those with positive real parts. PartitionU = [U1 U2], whereU1 andU2 both haven columns. Then the columns ofU1 span the samesubspace as the eigenvectors and generalized eigenvectors corresponding to the eigenvalues ofH with negative real parts. Hence, we may takeE = U1.

This is the algorithm implemented in most numerical routines for the solution of algebraicRiccati equations. An up-to-date account of the numerical aspects of the solution of the AREmay be found in Sima (1996).

Exercise 3.2.8 (“Cruise control system”). Use the method of this subsection to solve the AREthat arises in Exercise 3.2.2 (p. 106). �

3.2.10. Concluding remarks

The LQ paradigm would appear to be useless as a design methodology because full state feedbackis almost never feasible. Normally it simply is too costly to install the instrumentation needed tomeasure all the state variables. Sometimes it is actually impossible to measure some of the statevariables.

In Section 3.3 (p. 115) we see how instead of using state feedback control systems may bedesigned based on feedback of selected output variables only. The idea is to reconstruct the stateas accurately as possible using an observer or Kalman filter. By basing feedback onestimatesofthe state several of the favorable properties of state feedback may be retained or closely recovered.

3.3. LQG Theory

3.3.1. Introduction

In this section we review what is known as LQG theory. LQG stands for Linear Quadratic Guas-sian. By including Gaussian white noise in the LQ paradigm linear optimal feedback systemsbased onoutput feedbackrather than state feedback may be found.

115


We consider the system

x(t) = Ax(t)+ Bu(t)+ Gv(t)y(t) = Cx(t)+w(t)z(t) = Dx(t)

t ∈ R. (3.60)

Themeasured output yis available for feedback. As in§ 2.3 (p. 64) the outputz is thecontrolledoutput.The noise signalv models theplant disturbancesandw themeasurement noise.

The signalsv andw are vector-valued Gaussian white noise processes with

Ev(t)vT(s) = Vδ(t − s)Ev(t)wT(s) = 0Ew(t)wT(s) = Wδ(t − s)

t, s∈ R. (3.61)

V andW are nonnegative-definite symmetric constant matrices, called theintensity matricesofthe two white noise processes. We do not go into the theory of stochastic processes in generaland that of white noise in particular, but refer to texts such as Wong (1983) and Bagchi (1993).The initial statex(0) is assumed to be a random vector.

The various assumptions define the statex(t), t ∈ R, and the controlled outputz(t), t ∈ R, asrandom processes. As a result, also the quadratic error expression

zT(t)Qz(t)+ uT(t)Ru(t), t ≥ 0, (3.62)

is a random process. The problem of controlling the system such that the integrated expectedvalue∫ T

0E[zT(t)Qz(t)+ uT(t)Ru(t)] dt (3.63)

is minimal is thestochastic linear regulator problem. The time interval [0, T] at this point istaken to be finite but eventually we consider the case thatT → ∞. At any timet the entire pastmeasurement signaly(s), s≤ t, is assumed to be available for feedback. Figure 3.5 clarifies thesituation.

y

w

v

zu

Plant

Controller

+ +

Figure 3.5.: LQG feedback

3.3.2. Observers

Consider the observed system

x(t) = Ax(t)+ Bu(t),y(t) = Cx(t),

t ∈ R. (3.64)

116

3.3. LQG Theory

This is the system (3.60) but without the state noisev and the measurement noisew. The statex of the system (3.64) is not directly accessible because only the outputy is measured. We mayreconstruct the state with arbitrary precision by connecting anobserverof the form

˙x(t) = Ax(t)+ Bu(t)+ K[y(t)− Cx], t ∈ R. (3.65)

The signalx is meant to be an estimate of the statex(t). It satisfies the state differential equationof the system (3.64) with an additional input termK[y(t)− Cx(t)] on the right-hand side.K is theobserver gain matrix. It needs to be suitably chosen. Theobservation error y(t)− Cx(t) is thedifference between the actual measured outputy(t) and the outputy(t) = Cx(t) as reconstructedfrom the estimated statex(t). The extra input termK[y(t)− Cx(t)] on the right-hand side of(3.65) provides a correction that is active as soon as the observation error is nonzero. Figure 3.6

y

K

zu Plant

Observer

Plantmodel

+

−

Figure 3.6.: Structure of an observer

shows the structure of the observer. Define

e(t) = x(t)− x(t) (3.66)

as thestate estimation error. Differentiation ofe yields after substitution of (3.65) and (3.64)that the error satisfies the differential equation

e(t) = (A− KC)e(t), t ∈ R. (3.67)

If the system (3.64) is detectable then there always exists a gain matrixK such that the errorsystem (3.67) is stable. If the error system is stable thene(t)→ 0 ast → ∞ for any initial errore(0). Hence,

x(t)t→∞−→ x(t), (3.68)

so that the estimated state converges to the actual state.

3.3.3. The Kalman filter

Suppose that we connect the observer

˙x(t) = Ax(t)+ Bu(t)+ K[y(t)− Cx(t)], t ∈ R. (3.69)

to the noisy system

x(t) = Ax(t)+ Bu(t)+ Gv(t),y(t) = Cx(t)+w(t),

t ∈ R. (3.70)

117


Differentiation ofe(t) = x(t)− x(t) leads to the error differential equation

e(t) = (A− KC)e(t)− Gv(t)+ Kw(t), t ∈ R. (3.71)

Owing to the two noise terms on the right-hand side the error now no longer converges to zero,even if the error system is stable. Suppose that the error system is stable. It is proved in§ 3.7.7(p. 149) that ast → ∞ theerror covariance matrix

Ee(t)eT(t) (3.72)

converges to a constant steady-state valueY that satisfies the linear matrix equation

(A− KC)Y + Y(A− KC)T + GVGT + KW KT = 0. (3.73)

This type of matrix equation is known as aLyapunov equation. It is made plausible in Subsec-tion 3.7.5 that as a function of the gain matrixK the steady-state error covariance matrixY isminimal if K is chosen as

K = YCTW−1. (3.74)

“Minimal” means here that ifY is the steady-state error covariance matrix corresponding to anyother observer gainK then Y ≥ Y. This inequality is to be taken in the sense thatY − Y isnonnegative-definite.

A consequence of this result is that the gain (3.74) minimizes the steady-state mean squarestate reconstruction error limt→∞ EeT(t)e(t). As a matter of fact, the gain minimizes theweighted mean square construction error limt→∞ EeT(t)Wee(t) for any nonnegative-definiteweighting matrixWe.

Substitution of the optimal gain matrix (3.74) into the Lyapunov equation (3.73) yields

AY+ Y AT + GVGT − YCTW−1CY = 0. (3.75)

This is another matrix Riccati equation. The observer

˙x(t) = Ax(t)+ Bu(t)+ K[y(t)− Cx], t ∈ R, (3.76)

with the gain chosen as in (3.74) and the covariance matrixY the nonnegative-definite solutionof the Riccati equation (3.75) is the famousKalman filter(Kalman and Bucy 1961).

We review several properties of the Kalman filter. They are the duals of the properties listedin Summary 3.2.1 (p. 105) for the Riccati equation associated with the regulator problem6 .

Summary 3.3.1 (Properties of the Kalman filter).Assumptions:

• The system

x(t) = Ax(t)+ Gv(t),y(t) = Cx(t),

t ∈ R, (3.77)

is stabilizable and detectable.6The optimal regulator and the Kalman filter are dual in the following sense. Given the regulator problem of§ 3.2

(p. 104), replaceA with AT, B with CT, D with GT, Q with V, andR with W. Then the regulator Riccati equation(3.6) becomes the observer Riccati equation (3.75), its solutionX becomesY, the state feedback gainF is thetranspose of the observer gainK, and the closed-loop system matrixA − BF is the transpose of the error systemmatrix A− KC. By matching substitutions the observer problem may be transposed to a regulator problem.

118

3.3. LQG Theory

• The noise intensity matricesV andW are positive-definite.

The following facts follow from Summary 3.2.1 (p. 105) by duality:

1. The algebraic Riccati equation

AY+ Y AT + GVGT − YCTW−1CY = 0 (3.78)

has a unique nonnegative-definite symmetric solutionY. If the system (3.77) is control-lable rather than just stabilizable thenY is positive-definite.

2. The minimal value of the steady-state weighted mean square state reconstruction errorlim t→∞ EeT(t)Wee(t) is7 tr YWe.

3. The minimal value of the mean square reconstruction error is achieved by the observergain matrixK = YCTW−1.

4. The error system

e(t) = (A− KC)e(t), t ∈ R, (3.79)

is stable, that is, all the eigenvalues of the matrixA− KC have strictly negative real parts.As a consequence also the observer

˙x(t) = Ax(t)+ Bu(t)+ K[y(t)− Cx(t)], t ∈ R, (3.80)

is stable.

�

The reasons for the assumptions may be explained as follows. If the system (3.77) is notdetectable then no observer with a stable error system exists. If the system is not stabilizable(with v as input) then there exist observers that are not stable but are immune to the state noisev. Hence, stability of the error system is not guaranteed.W needs to be positive-definite toprevent the Kalman filter from having infinite gain. IfV is not positive-definite then there maybe unstable modes that are not excited by the state noise and, hence, are not stabilized in the errorsystem.

Exercise 3.3.2 (Asymptotic results). The asymptotic results for the regulator problem may be“dualized” to the Kalman filter.

1. Define the “observer return difference”

Jf (s) = I + C(sI − A)−1K. (3.81)

Prove that

detJf (s) = χ f (s)

χol(s), (3.82)

whereχol(s)= det(sI − A) is the system characteristic polynomial andχ f (s) = det(sI −A+ KC) the observer characteristic polynomial.

7The quantity trM =∑mi=1 Mii is called thetraceof them× m matrix M with entriesMij , i, j = 1,2, · · · ,m.

119


2. Prove that the return differenceJf of the Kalman filter satisfies

Jf (s)W JTf (−s) = W + M(s)V MT(−s), (3.83)

whereM is the open-loop transfer matrixM(s) = C(sI − A)−1G.

3. Consider the SISO case withV = 1 andW = σ. Jf andM are now scalar functions. Provethat

χ f (s)χ f (−s) = χol(s)χol(−s)[1 + 1σ

M(s)M(−s)]. (3.84)

4. Write

M(s) = gζ(s)χol(s)

, (3.85)

with ζ a monic polynomial andg a constant.

• Prove that asσ → ∞ the optimal observer poles approach the open-loop poles thatlie in the left-half plane and the mirror images of the open-loop poles that lie in theright-half plane.

• Prove that asσ ↓ 0 the optimal observer poles that do not go to∞ approach theopen-loop zeros that lie in the left-half plane and the mirror images of the open-loopzeros that lie in the right-half plane.

• Establish the asymptotic pattern of the optimal observer poles that approach∞ asσ ↓ 0.

�

3.3.4. Kalman filter with cross correlated noises

A useful generalization of the Kalman filter follows by assuming cross correlation of the whitenoise processesv andw. Suppose that

E

[v(t)w(t)

] [vT(s) wT(s)

] =[

V UUT W

]δ(t − s), t, s∈ R. (3.86)

Assume that[V U

UT W

](3.87)

is positive-definite, and, as before, that the systemx(t) = Ax(t)+ Gv(t), y(t) = Cx(t) is stabi-lizable and detectable. Then the optimal observer gain is

K = (YCT + GU)W−1, (3.88)

where the steady-state error covariance matrixY is the positive-definite solution of the Riccatiequation

AY+ Y AT + GVGT − (YCT + GU)W−1(CY+ UTGT) = 0. (3.89)

Exercise 3.3.3 (Cross correlated noises). Prove this result. �

120

3.3. LQG Theory

3.3.5. Solution of the stochastic linear regulator problem

The stochastic linear regulator problem consists of minimizing∫ T

0E[zT(t)Qz(t)+ uT(t)Ru(t)] dt (3.90)

for the system

x(t) = Ax(t)+ Bu(t)+ Gv(t)y(t) = Cx(t)+w(t)z(t) = Dx(t)

t ∈ R. (3.91)

We successively consider the situation of no state noise, state feedback, and output feedback.

No state noise. From§ 3.2 (p. 139) we know that if the disturbancev is absent and the statex(t)may be directly and accurately accessed for measurement, then forT → ∞ the performanceindex is minimized by the state feedback law

u(t) = −Fx(t), (3.92)

with the feedback gainF as in Summary 3.2.1 (p. 105).

State feedback. If the white noise disturbancev is present then the state and input cannot bedriven to 0, and the integrated generalized square error (3.90) does not converge to a finite numberasT → ∞. It is proved in Subsection 3.7.6 (p. 148) that the state feedback law (3.92) minimizestherateat which (3.90) approaches∞, that is, it minimizes

limT→∞

1T

∫ T

0E[zT(t)Qz(t)+ uT(t)Ru(t)] dt. (3.93)

This limit equals thesteady-state mean square errorindexsteady-state mean square error

limt→∞ E[zT(t)Qz(t)+ uT(t)Ru(t)]. (3.94)

Hence, the state feedback law minimizes the steady-state mean square error.

Output feedback. We next consider the situation that the statecannotbe accessed for mea-surement. The state may be optimally estimated, however, with the help of the Kalman filter.Then the solution of the stochastic linear regulator problem withoutput feedback(rather thanstate feedback) is to replace the statex(t) in the state feedback law (3.92) with theestimatedstatex(t). Thus, the optimal controler is given by

˙x(t) = Ax(t)+ Bu(t)+ K[y(t)− Cx(t)],u(t) = −Fx(t),

t ∈ R. (3.95)

The controller minimizes the steady-state mean square error (3.94) under output feedback. Thefeedback gainF and the observer gainK follow from Summaries 3.2.1 (p. 105) and 3.3.1 (p. 118),respectively. Figure 3.7 shows the arrangement of the closed-loop system.

Using the estimated state as if it were the actual state is known ascertainty equivalence.It divorces state estimation and control input selection. This idea is often referred to as theseparation principle.

121


− y

zuPlant

ObserverF

Figure 3.7.: Observer based feedback control

The closed-loop system that results from interconnecting the plant (3.91) with the compen-sator (3.95) is stable — under the assumptions of Summaries 3.2.1 and 3.3.1, of course. This ismost easily recognized as follows. Substitution ofu(t) = −Fx(t) into x(t) = Ax(t)+ Bu(t)+Gv(t) yields with the further substitutionx(t) = x(t)+ e(t)

x(t) = (A− BF)x(t)− BFe(t)+ Gv(t). (3.96)

Together with (3.71) we thus have[x(t)e(t)

]=[

A− BF −BF0 A− KC

][x(t)e(t)

]+[

Gv(t)−Gv(t)+ Kw(t)

]. (3.97)

The eigenvalues of this system are the eigenvalues of the closed-loop system. Inspection showsthat these eigenvalues consist of the eigenvalues ofA− BF (theregulator poles) together with theeigenvalues ofA− KC (theobserver poles). If the plant (3.91) has ordern then the compensatoralso has ordern. Hence, there are 2n closed-loop poles.

Exercise 3.3.4 (Closed-loop eigenvalues).

1. Prove that the eigenvalues of (3.97) are the eigenvalues of the closed-loop system.

2. Show (most easily by a counterexample) that the fact that the observer and the closed-loopsystem are stable does not mean that the compensator (3.95) by itself is stable.

�

Exercise 3.3.5 (Compensator transfer function). The configuration of Fig. 3.7 may be re-arranged as in Fig. 3.8. Show that the equivalent compensatorCe has the transfer matrixCe(s) = F(sI − A+ BF + KC)−1K. �

Ce P− y

zu

Figure 3.8.: Equivalent unit feedback configuration

3.3.6. Asymptotic analysis and loop transfer recovery

In this subsection we study the effect of decreasing the intensityW of the measurement noise.Suppose thatW = σW0, with W0 a fixed symmetric positive-definite weighting matrix andσ a

122

3.3. LQG Theory

positive number. We investigate the asymptotic behavior of the closed-loop system asσ ↓ 0.Before doing this we need to introduce two assumptions:

• The disturbancev is additive to the plant inputu, that is,G = B. This allows the tightestcontrol of the disturbances.

• The open-loop plant transfer matrixC(sI− A)−1B is square, and its zeros all have negativereal parts.

Breaking the loop at the plant input as in Fig. 3.9 we obtain the loop gain

Lσ(s) = Ce(s)P(s) = F(sI − A+ BF + KσC)−1KσC(sI − A)−1B. (3.98)

(Compare Exercise 3.3.5, p. 122.) To emphasize the dependence onσ the observer gain and the

− y

zuPlant

ObserverFx

Figure 3.9.: Breaking the loop

loop gain are each provided with a subscript. Asσ ↓ 0 the gainKσ approaches∞. At the sametime the error covariance matrixYσ approaches the zero matrix. This is the dual of the conclusionof Subsection 3.2.6 (p. 109) that the state feedback gainF goes to∞ and the solutionX of theRiccati equation approaches the zero matrix as the weight on the input decreases.

The fact thatYσ ↓ 0 indicates that in the limit the observer reconstructs the state with completeaccuracy. It is proved in§ 3.7.7 (p. 149) that asσ ↓ 0 the loop gainLσ approaches the expression

L0(s) = F(sI − A)−1B. (3.99)

The asymptotic loop gainL0 is precisely the loop gain for full state feedback. Accordingly, theguaranteed gain and phase margins of Subsection§ 3.2.7 (p. 112) are recouped. This is calledloop transfer recovery(LTR).

The term loop transfer recovery appears to have been coined by Doyle and Stein (1981).Extensive treatments may be found in Anderson and Moore (1990) and Saberi, Chen, and Sannuti(1993). We use the method in the design examples of§ 3.6 (p. 127).

Exercise 3.3.6 (Dual loop transfer recovery). Dual loop recovery provides an alternative ap-proach to loop recovery. Dual loop recovery results when the loop is broken at the plantoutputy rather than at the input (Kwakernaak and Sivan 1972,§ 5.6). In this case it is necessary toassume thatD = C, that is, the controlled output is measured. Again we needC(sI − A)−1B tobe square with left-half plane zeros only. We letR= ρR0 and consider the asymptotic behaviorfor ρ ↓ 0.

1. Make it plausible that the loop gain approaches

L0(s) = C(sI − A)−1K. (3.100)

123


2. Show that the corresponding return differenceJ0(s)= I + L0(s) satisfies the return differ-ence inequality

J0(jω)W JT0 (−jω) ≥ W, ω ∈ R. (3.101)

3. Show that gain and phase margins apply that agree with those found in Subsection 3.2.7(p. 112).

�

3.4. H2 optimization

3.4.1. Introduction

In this section we define the LQG problem as a special case of a larger class of problems, whichlately has become known asH2 optimization. Most importantly, this approach allows to removethe stochastic ingredient of the LQG formulation. In many applications it is difficult to establishthe precise stochastic properties of disturbances and noise signals. Very often in the applicationof the LQG problem to control system design the noise intensitiesV and W play the role ofdesign parametersrather than that they model reality.

The stochastic element is eliminated by recognizing that the performance index for the LQGproblem may be represented as asystem norm—theH2-norm. To introduce this point of view,consider the stable system

x(t) = Ax(t)+ Bv(t),y(t) = Cx(t),

t ∈ R. (3.102)

The system has the transfer matrixH(s) = C(sI − A)−1B. Suppose that the signalv is whitenoise with covariance functionEv(t)vT(s) = Vδ(t − s). Then the outputy of the system is astationary stochastic process with spectral density matrix

Sy( f ) = H(j2π f )V HT(−j2π f ), f ∈ R. (3.103)

As a result, the mean square output is

EyT(t)y(t) = tr∫ ∞

−∞Sy( f ) d f = tr

∫ ∞

−∞H(j2π f )V H∼(j2π f ) d f. (3.104)

Here we introduce the notationH∼(s) = HT(−s). The quantity

‖H‖2 =√

tr∫ ∞

−∞H(j2π f )H∼(j2π f ) d f (3.105)

is called theH2-normof the system. If the white noisev has intensityV = I then the mean squareoutputEyT(t)y(t) equals precisely the square of theH2-norm of the system.

H2 refers to the space of square integrable functions on the imaginary axis whose inverseFourier transform is zero for negative time.

Exercise 3.4.1 (Lyapunov equation). Prove that theH2-norm of the stable system (3.102) isgiven by‖H‖2

2 = tr CYCT, where the matrixY is the unique symmetric solution of the LyapunovequationAY+ Y AT + BBT = 0. �

124

3.4. H2 optimization

3.4.2. H2 optimization

In this subsection we rewrite the time domain LQG problem into an equivalent frequency domainH2 optimization problem. While the LQG problem requires state space realizations, theH2-optimization problem is in terms of transfer matrices. To simplify the expressions to come weassume thatQ = I andR= I , that is, the LQG performance index is

limt→∞ E[zT(t)z(t)+ uT(t)u(t)]. (3.106)

This assumption causes no loss of generality because by scaling and transforming the variablesz andu the performance index may always be brought into this form.

For the open-loop system

x = Ax+ Bu+ Gv, (3.107)

z = Dx, (3.108)

y = Cx+w (3.109)

we have in terms of transfer matrices

z = D(sI − A)−1G︸︷︷︸P11(s)

v+ D(sI − A)−1B︸︷︷︸P12(s)

u, (3.110)

y = C(sI − A)−1G︸︷︷︸P21(s)

v+ C(sI − A)−1B︸︷︷︸P22(s)

u+w. (3.111)

Interconnecting the system as in Fig. 3.10 with a compensatorCe we have the signal balance

Ce P−

+

+y

z

u v

w

Figure 3.10.: Feedback system with stochastic inputs and outputs

u = −Cey = −Ce(P21v+ P22u+w), so that

u = −( I + CeP22)−1CeP21︸︷︷︸

H21(s)

v−( I + CeP22)−1Ce︸︷︷︸

H22(s)

w. (3.112)

Fromz = P11v+ P12u we obtain

z = P11 − P12( I + CeP22)−1CeP21︸︷︷︸

H11(s)

v−P12( I + CeP)−1Ce︸︷︷︸H12(s)

w. (3.113)

A more compact notation is[zu

]=[

H11(s) H12(s)H21(s) H22(s)

]︸︷︷︸

H(s)

[v

w

]. (3.114)

125


From this we find for the steady-state mean square error

limt→∞ E

(zT(t)z(t)+ uT(t)u(t)

) = limt→∞ E

([z(t)u(t)

]T [z(t)u(t)

])(3.115)

= tr∫ ∞

−∞H(j2π f )H∼(j2π f ) d f (3.116)

= ‖H‖22. (3.117)

Hence, solving the LQG problem amounts to minimizing theH2 norm of the closed-loop systemof Fig. 3.10 with(v,w) as input and(z,u) as output.

The configuration of Fig. 3.10 is a special case of the configuration of Fig. 3.11. In the latterdiagramw is theexternal input(v andw in Fig. 3.10). The signalz is theerror signal,whichideally should be zero (z and u in Fig. 3.10). Furthermore,u is the control input,and y theobserved output.The blockG is thegeneralized plant, andCe the compensator. Note that thesign reversion at the compensator input in Fig. 3.10 has been absorbed into the compensatorCe.

y

z

u

w

G

Ce

Figure 3.11.: The standardH2 problem

Exercise 3.4.2 (Generalized plant for the LQG problem). Show that for the LQG problem thegeneralized plant in state space form may be represented as

x(t) = Ax(t)+ [G 0

][v(t)w(t)

]+ Bu(t), (3.118)

z(t)u(t)y(t)

=

D

0C

x(t)+

0 0

0 00 I

[v(t)

w(t)

]+0

I0

u(t). (3.119)

�

3.4.3. The standard H2 problem and its solution

The standardH2 optimization problem is the problem of choosing the compensatorK in theblock diagram of Fig. 3.11 such that it

1. stabilizes the closed-loop system, and

2. minimizes theH2-norm of the closed-loop system (withw as input andz as output).

We represent the generalized plantG of Fig. 3.11 in state space form as

x(t) = Ax(t)+ B1w(t)+ B2u(t), (3.120)

z(t) = C1x(t)+ D11w(t)+ D12u(t), (3.121)

y(t) = C2x(t)+ D21w(t)+ D22u(t). (3.122)

126

3.5. Feedback system design byH2 optimization

The H2 problem may be solved by reducing it to an LQG problem. This is done in§ 3.7.8(p. 149). The derivation necessitates the introduction of some assumptions, which are listed inthe summary that follows. They are natural assumptions for LQG problems.

Summary 3.4.3 (Solution of the H2 problem). Consider the standardH2 optimization problemfor the generalized plant

x(t) = Ax(t)+ B1w(t)+ B2u(t), (3.123)

z(t) = C1x(t) + D12u(t), (3.124)

y(t) = C2x(t)+ D21w(t)+ D22u(t). (3.125)

Assumptions:

• The systemx(t) = Ax(t)+ B2u(t), y(t) = C2x(t) is stabilizable and detectable.

• The matrix[ A−sI B1

C2 D21

]has full row rank for everys= jω, andD21 has full row rank.

• The matrix[ A−sI B2

C1 D12

]has full column rank for everys= jω, andD12 has full column rank.

Under these assumptions the optimal output feedback controller is

˙x(t) = Ax(t)+ B2u(t)+ K[y(t)− C2x(t)− D22u(t)] (3.126)

u(t) = −Fx(t). (3.127)

The observer and state feedback gain matrices are

F = (DT12D12)

−1(BT2 X + DT

12C1), K = (YCT2 + B1DT

21)(D21DT21)

−1. (3.128)

The symmetric matricesX andY are the unique positive-definite solutions of the algebraic Ric-cati equations

AT X + X A+ CT1 C1 − (X B2 + CT

1 D12)(DT12D12)

−1(BT2 X + DT

12C1) = 0,

AY+ AYT + B1BT1 − (YCT

2 + B1DT21)(D21DT

21)−1(C2Y + D21BT

1 ) = 0.(3.129)

�

The condition thatD12 has full column rank means that there is “direct feedthrough” fromthe inputu to the error signalz. Likewise, the condition thatD21 has full row rank means that thenoisew is directly fed through to the observed outputy.

TheH2 optimization problem and its solution are discussed at length in Saberi, Sannuti, andChen (1995). In§§ 3.5 (p. 127) and 3.6 (p. 133) we discuss the application ofH2 optimization tocontrol system design.

3.5. Feedback system design by H2 optimization

3.5.1. Introduction

In this section we review how LQG andH2 optimization may be used to design SISO and MIMOlinear beedback systems.

127


3.5.2. Parameter selection for the LQG problem

We discuss how to select the design parameters for the LQG problem without a cross term in theperformance index and without cross correlation between the noises. The LQG problem consistsof the minimization of

limt→∞ E[zT(t)Qz(t)+ uT(t)Ru(t)] (3.130)

for the system

x(t) = Ax(t)+ Bu(t)+ Gv(t)z(t) = Dx(t)y(t) = Cx(t)+w(t)

t ∈ R. (3.131)

Important design parameters are the weighting matricesQ and R and the intensitiesV andW.In the absence of specific information about the nature of the disturbances also the noise inputmatrix G may be viewed as a design parameter. Finally there usually is some freedom in theselection of the control outputz; this means that also the matrixD may be considered a designparameter.

We discuss some rules of thumb for selecting the design parameters. They are based on theassumption that we operate in the asymptotic domain where the weighting matricesR (the weighton the input) andW (the measurement noise intensity) are small.

1. First the parametersD, Q and R for the regulator part are selected. These quantitiesdetermine the dominant characteristics of the closed-loop system.

a) D determines the controlled outputz. Often the controlled outputz is also the mea-sured outputy. The case wherez is not y is calledinferential control. There may becompelling engineering reasons for selectingz different fromy.

b) In the SISO caseQ may be chosen equal to 1.

In the MIMO caseQ is best chosen to be diagonal according to the rules of§ 3.2.3(p. 106).

c) In the SISO caseR is a scalar design parameter. It is adjusted by trial and error untilthe desired bandwidth is achieved (see also§ 3.2.6, p. 109)).

In the MIMO case one may letR = ρR0, where the fixed matrixR0 is selectedaccording to the rules of§ 3.2.3 (p. 106) andρ is selected to achieve the desiredbandwidth.

2. Next, the design parameters for theobserver partare determined. They are chosen toachieve loop transfer recovery, as described in§ 3.3.6 (p. 122).

a) To take advantage of loop transfer recovery we need to takeG = B. LTR is onlyeffective if the open-loop transfer functionP(s) = C(sI − A)−1B has no right-halfplane zeros, or only has right-half plane zeros whose magnitudes are sufficientlymuch greater than the desired bandwidth.

b) In the SISO case we letV = 1.

In the MIMO case we may selectV to be diagonal by the “dual” of the rules of§ 3.2.3 (p. 106). This amounts to choosing each diagonal entry ofV proportionalto the inverse of the square root of the amplitude of the largest disturbance that mayoccur at the corresponding input channel.

128


c) In the SISO caseW is a scalar parameter that is chosen small enough to achieveloop transfer recovery. Asymptotically the finite observer poles are the zeros ofdetC(sI − A)−1B. These closed-loop poles correspond to canceling pole-zero pairsbetween the plant and the compensator. The far-away observer poles determine thebandwidth of the compensator and the high-frequency roll-off frequency for the com-plementary sensitivity. The magnitude of the dominant observer poles should be per-haps a decade larger than the magnitude of the dominant regulator poles.In the MIMO case we letW = σW0. W0 is chosen diagonally with each diagonal en-try proportional to the inverse of the square root of the largest expected measurementerror at the corresponding output channel. The scalarσ is chosen small enough toachieve LTR.

3.5.3. H2 Interpretation

In this subsection we discuss the interpretation of the LQG problem as anH2 optimization prob-lem.

If we chooseG = B to take advantage of loop transfer recovery then the open-loop equations(3.110–3.111) may be rewritten as

z = R(s)(u+ v), (3.132)

y = P(s)(u+ v)+w, (3.133)

where

P(s) = C(sI − A)−1B, R(s) = D(sI − A)−1B. (3.134)

The corresponding block diagram is represented in Fig. 3.12. IfP is invertible then by block

Ce P

R

−+

+

+++

y

zu

v

w

Figure 3.12.: Block diagram for loop recovery design

diagram substitution Fig. 3.12 may be redrawn as in Fig. 3.13, whereW0 = RP−1. In the case ofnon-inferential controlW0 = I .

We consider the frequency domain interpretation of the arrangement of Fig. 3.13. By settingup two appropriate signal balances it is easy to find that

z = W0SPv− W0Tw, (3.135)

u = −T′v− Uw. (3.136)

Here

S= ( I + PCe)−1, T = ( I + PCe)

−1PCe, (3.137)

T′ = ( I + CeP)−1CeP, U = Ce( I + PCe)−1. (3.138)

129


Ce P Wo−+

+

+++

y

u

v

w

z

Figure 3.13.: Equivalent block diagram

S is the sensitivity matrix,T the complementary sensitivity matrix, andU the input sensitivitymatrix. T′ is the complementary sensitivity function if the loop is broken at the plant input ratherthan at the plant output. In the SISO caseT′ = T.

Exercise 3.5.1 (Transfer matrices). Derive (3.135–3.138). �

From (3.135–3.136) we find for the performance index

‖H‖22 = tr

∫ ∞

−∞(W0SPP∼S∼W∼

0 + W0TT∼W∼0 + T′T′∼ + UU∼) d f. (3.139)

The argument j2π f is omitted from each term in the integrand. Inspection reveals that the per-formance index involves a trade-off of the sensitivityS, the complementary sensitivitiesT andT′, and the input sensitivityU. The importance given to each of the system functions depends onW0 andP, which act as frequency dependent weighting functions.

The weighting functions in (3.139) arise from the LQG problem and are not very flexible.For more freedom we generalize the block diagram of Fig. 3.13 to that of Fig. 3.14.V1 andV2

are shaping filters andW1 andW2 weighting filters that may be used to modify the design. It isnot difficult to establish that

z1 = W1SPV1v− W1TV2w, (3.140)

z2 = −W2T′V1v− W2UV2w. (3.141)

As a result, the performance index now takes the form

‖H‖22 = tr

∫ ∞

−∞(W1SPV1V∼

1 P∼S∼W∼1 + W1TV2V∼

2 T∼W∼1

+ W2T′V1V′∼1 T′∼W∼

2 + W2UV2V∼2 U∼W∼

2 ) d f. (3.142)

In the next subsections some applications of this generalized problem are discussed.

3.5.4. Design for integral control

There are various ways to obtain integrating action in the LQG framework. We discuss a solu-tion that follows logically from the frequency domain interpretation (3.142). For simplicity weonly consider the SISO case. For the MIMO case the idea may be carried through similarly byintroducing integrating action in each input channel as for the SISO case.

Integral control aims at suppressing constant disturbances, which requires makingS(0)= 0.If the system has no natural integrating action then integrating action needs to be introduced in

130


Ce P

W2

W1 z1

z2

V1

V2

−

+

++

+

y

u

v

w

Figure 3.14.: Generalized configuration

the compensator. Inspection of (3.142) shows that takingV1(0) = ∞ forcesS(0) to be zero —otherwise the integral cannot be finite. Hence, we takeV1 as a rational function with a pole at 0.In particular, we let

V1(s) = s+ α

s. (3.143)

V1 may be seen as a filter that shapes the frequency distribution of the disturbance. The positiveconstantα models the width of the band over which the low-frequency disturbance extends.

Further inspection of (3.142) reveals that the functionV1 also enters the third term of theintegrand. In the SISO case the factorT′ in this term reduces toT. If S(0) = 0 then by com-plementarityT(0) = 1. This means that this third term is infinite at frequency 0,unless W2 hasa factors that cancels the corresponding factors in the numerator ofV1. This has a clear inter-pretation: If the closed-loop system is to suppress constant disturbances then we need to allowconstant inputs — hence we needW2(0) = 0.

More in particular we could take

W2(s) = ss+ α

W2o(s), (3.144)

whereW2o remains to be chosen but usually is taken constant. This choice ofW2 reduces theweight on the input over the frequency band where the disturbances are large. This allows thegain to be large in this frequency band.

A practical disadvantage of chooosingV1 as in (3.143) is that it makes the open-loop systemunstabilizable, because of the integrator outside the loop. This violates one of the assumptions of§ 3.4.3 (p. 126) required for the solution of theH2 problem. The difficulty may be circumventedby a suitable partitioning of the state space and the algebraic Riccati equations (Kwakernaak andSivan 1972). We prefer to eliminate the problem by the block diagram substitutions (a)→ (b)→ (c) of Fig. 3.15. The end result is that an extra factors+α

s is included in both the plant transferfunction and the weighting function for the input. The extra factor in the weighting function onthe input cancels against the factorss+α that we include according to (3.144).W2o remains.

Additionally an extra factor ss+α is included in the compensator. If the modified problem leads

to an optimal compensatorC0 then the optimal compensator for the original problem is

C(s) = s+ α

sC0(s). (3.145)

131


W2

W2

W2

W2 o

W1 z1

z2

z2

z2V2

−

+

+ +

++

+

+ +

y

u

v

v

v

w

s

s s

s

s s

s+α

s+α s+α

s+α

s+α s+α

Ce

Ce Ce

Co

P

P

(a)

(b) (c)

P

Po

Figure 3.15.: Block diagram substitutions for integral control (a)→(b)→(c)

This compensator explicitly includes integrating action.Note that by the block diagram substitutions this method of obtaining integral control comes

down to including an integrator in the plant. After doing the design for the modified plant theextra factor is moved over to the compensator. This way of ensuring integrating action is calledthe integrator in the loopmethod. We apply it in the example of§ 3.6.3 (p. 135). The method isexplained in greater generality in§ 6.7 (p. 279) in the context ofH∞ optimization.

3.5.5. High-frequency roll-off

Solving the LQG problems leads to compensators with a strictly proper transfer matrix. Thismeans that the high-frequency roll-off of the compensator and of the input sensitivity is 1decade/decade (20 dB/decade). Correspondingly the high-frequency roll-off of the complemen-tary sensitivity is at least 1 decade/decade.

Exercise 3.5.2 (High-frequency roll-off). Prove that LQG optimal compensators are strictlyproper. Check for the SISO case what the resulting roll-off is for the input sensitivity function andthe complementary sensitivity function, dependent on the high-frequency roll-off of the plant.

For some applications it may be desirable to have a steeper high-frequency roll-off. Inspectionof (3.142) shows that extra roll-off may be imposed by letting the weighting functionW2 increasewith frequency. Consider the SISO case and suppose thatV2(s) = 1. Let

W2(s) = ρ(1+ rs), (3.146)

with r a positive constant. Then by inspecting the fourth term in the integrand of (3.142) weconclude that the integral can only converge if at high frequencies the input sensitivityU, and,hence, also the compensator transfer functionCe, rolls off at at least 2 decades/decade.

132

3.6. Examples and applications

The difficulty that there is no state space realization for the blockW2 with the transfer function(3.146) may be avoided by the block diagram substitution of Fig. 3.16. If the modified problem

W1 z1

z2

V2

−

+

++

+

y

u

v

w

W2 V1

Ce P

z2

+

+

v

v

W2−1W2

V1

Ce

Co

P

(a)

(b)

Figure 3.16.: Block diagram substitution for high-frequency roll-off

is solved by the compensatorC0 then the optimal compensator for the original problem is

Ce(s) = C0(s)W2(s)

= C0(s)ρ(1+ rs)

. (3.147)

The extra roll-off is apparent. Even more roll-off may be obtained by lettingW2(s) = O(sm) as|s| → ∞, with m≥ 2.

For a more general exposition of the block diagram substitution method see§ 6.7 (p. 279).


3.6.1. Introduction

In this section we present two design applications ofH2 theory: A simple SISO system and a notvery complicated MIMO system.

3.6.2. LQG design of a double integrator plant

We consider the double integrator plant

P(s) = 1s2. (3.148)

133


The design target is a closed-loop system with a bandwidth of 1 rad/s. Because the plant has anatural double integration there is no need to design for integral control and we expect excel-lent low-frequency characteristics. The plant has no right-half zeros that impair the achievableperformance. Neither do the poles.

In state space form the system may be represented as

x =[0 10 0

]︸︷︷︸

A

x+[01

]︸︷︷︸

B

u, y = [1 0

]︸︷︷︸C

x. (3.149)

Completing the system equations we have

x = Ax+ Bu+ Gv, (3.150)

y = Cx+w, (3.151)

z = Dx. (3.152)

We choose the controlled variablez equal to the measured variabley. so thatD = C. To profitfrom loop transfer recovery we letG = B. In the SISO case we may chooseQ = V = 1 withoutloss of generality. Finally we writeR= ρ andW = σ, with the constantsρ andσ to be determined.

We first consider the regulator design. In the notation of§ 3.2.6 (p. 109) we havek = 1,ψ(s) = 1 andχol(s) = s2. It follows from (3.34) that the closed-loop characteristic polynomialχcl for state feedback satisfies

χcl(−s)χcl(s) = χol(−s)χol(s)+ k2

ρψ(−s)ψ(s) = s4 + 1

ρ. (3.153)

The roots of the polynomial on the right-hand side are12

√2(±1 ± j)/ρ

14 . To determine the

closed-loop poles we select those two roots that have negative real parts. They are given by

12

√2(−1± j)/ρ

14 . (3.154)

Figure 3.17 shows the loci of the closed-loop poles asρ varies. As the magnitude of the closed-

-2 -1.5 -1 -0.5 0-2

-1

0

1

2

Re

Im

1/ρ

1/ρ

Figure 3.17.: Loci of the closed-loop poles

loop pole pair is 1/ρ14 the desired bandwidth of 1 rad/s is achieved forρ = 1.

We next turn to the observer design. By Exercise 3.3.2(c) (p. 119) the observer characteristicpolynomialχ f satisfies

χ f (−s)χ f (s) = χol(s)χol(−s)[1 + 1σ

M(s)M(−s)] = s4 + 1σ4, (3.155)

134


whereM(s) = C(sI − A)−1G = 1/s2 andχol(s) = s2. This expression is completely similar tothat for the regulator characteristic polynomial, and we conclude that the observer characteristicvalues are

12

√2(−1± j)/σ

14 . (3.156)

By the rule of thumb of§ 3.5.2 (p. 128) we choose the magnitude 1/σ14 of the observer pole pair

10 times greater than the bandwidth, that is, 10 rad/s. It follows thatσ = 0.0001.By numerical computation8 it is found that the optimal compensator transfer function is

Ce(s) = 155.6(s+ 0.6428)s2 + 15.56s+ 121.0

. (3.157)

This is a lead compensator.

10-2 1010 10 10100-1 -1 11 102

-80

-60

-40

-20

20 20

0


[dB

]

|S | |T |

10-2 100 102-80

-60

-40

-20

0


[dB

]

σ = .01

σ = .01

σ = .0001

σ = .0001

σ = .000001

σ = .000001

Figure 3.18.: Sensitivity function and complementary sensitivity function fortheH2 design

Figure 3.18 shows the magnitude plots of the sensitivity and complementary sensitivity func-tions for σ = .01, σ = .0001 andσ = .000001. The smallerσ is the better loop transfer isrecovered. Since the high-frequency roll-off of the complementary sensitivity of 40 dB/decadesets in at the angular frequency 1/σ

14 it is advantageous not to chooseσ too small. Takingσ

large, though, results in extra peaking at crossover. The valueσ = .0001 seems a reasonablecompromise. Figure 3.19 gives the closed-loop step response. The valueσ = .000001 gives thebest response but that forσ = .0001 is very close.

3.6.3. A MIMO system

As a second example we consider the two-input two-output plant with transfer matrix (Kwaker-naak 1986)

P(s) = 1

s21

s+ 20 1

s+ 2

. (3.158)

8This may be done very conveniently with MATLAB using the Control Toolbox (Control Toolbox 1990).

135


σ = .01σ = .0001

σ = .000001

0 5 100

0.5

1

1.5

time [s]

am

plit

ud

e

step response

Figure 3.19.: Closed-loop step response for theH2 design

Figure 3.20 shows the block diagram. The plant is triangularly coupled. It is easy to see that itmay be represented in state space form as

x =0 1 0

0 0 00 0 −2

︸︷︷︸A

x +0 0

1 00 1

︸︷︷︸B

u, y =[1 0 10 0 1

]︸︷︷︸

C

x. (3.159)

The first two components of the state represent the block 1/s2 in Fig. 3.20, and the third the block1/(s+ 2).

s+2

u1

u2 y2

y1+

+

1

1

s2

Figure 3.20.: MIMO system

The plant has no right-half plane poles or zeros, so that there are no fundamental limitationsto its performance.

Exercise 3.6.1 (Poles and zeros). What are the open-loop poles and zeros?Hint: For thedefinition of the zeros see§ 3.2.6 (p. 109). �

We aim at a closed-loop bandwidth of 1 rad/s on both channels, with good low- and high-frequency characteristics.

We complete the system description to

x = Ax+ Bu+ Gv, (3.160)

y = Cx+w, (3.161)

z = Dx. (3.162)

To take advantage of loop transfer recovery we letG = B. As the controlled outputz is availablefor feedback we haveD = C. Assuming that the inputs and outputs are properly scaled we choose

Q = I, R= ρ I, V = I, W = σ I, (3.163)

136


with the positive scalarsρ andσ to be determined. First we consider the design of the regulator

-4 -3 -2 -1 0-2

-1

0Im

Re

1

2

1/ρ1/ρ

1/ρ

Figure 3.21.: Loci of the regulator poles

part, which determines the dominant characteristics. By numerical solution of the appropriatealgebraic Riccati equation for a range of values ofρ we obtain the loci of closed-loop poles ofFig. 3.21. Asρ decreases the closed-loop poles move away from the double open-loop pole at0 and the open-loop pole at−2. Forρ = 0.8 the closed-loop poles are−2.5424 and−.7162±j .7034. The latter pole pair is dominant with magnitude 1.0038, which is the correct value for aclosed-loop bandwidth of 1 rad/s.

Next we consider the loci of the optimal observer poles as a function ofσ. Like in the doubleintegrator example, they are identical to those of the regulator poles.

Exercise 3.6.2 (Identical loci). Check that this is a consequence of choosingQ= V = I , R= ρ I ,W = σ I , D = C andG = B. �

Again following the rule that the dominant observer poles have magnitude 10 times that ofthe dominant regulator poles we letσ = 5 × 10−5. This results in the optimal observer poles−200.01 and−7.076± j 7.067. The latter pole pair has magnitude 10. Using standard softwaretheH2 solution may now be found. Figure 3.22 shows the magnitudes of the four entries of theresulting 2× 2 sensitivity matrixS.

The attenuation of disturbances that enter the system at the first output correponds to theentriesS11 and S21 and is quite adequate, thanks to the double integrator in the correspondinginput channel. The attenuation of disturbances that affect the second output (represented byS12

andS22) is disappointing, however. The reason is that the low-frequency disturbances generatedby the double integrator completely dominate the disturbances generated in in the other channel.

We improve the performance of the second channel by introducing integrating action. Ap-plication of the integrator-in-the-loop method of§ 3.5.4 (p. 130) amounts to including an extrablock

s+ α

s(3.164)

in the second channel, as indicated in Fig. 3.23. After completion of the design the extra blockis absorbed into the compensator. Representing the extra block by the state space realization

137


-100

-50

[dB]


0

10 -2 1010 10 100-1 1 3102

|S11|

[dB]


-100

-50

0

10-2

1010 10 100-1 1 3

102

|S21|

[dB]

angular frequency [rad/s]10

-21010 10 10

0-1 1 310

2-100

-50

0

|S22|

[dB]

angular frequency [rad/s]10

-21010 10 10

0-1 1 310

2-100

-50

0

|S12|

Figure 3.22.: Magnitudes of the entries of the sensitivity matrixS

s+2s+αs

u1

u2u2' y2

y1+

+

1

1

s2

Figure 3.23.: Expanded plant for integrating action in the second channel

138


x4 = u′2, u2 = αx4 + u′

2 we obtain the modified plant

x =

0 1 0 00 0 0 00 0 −2 α

0 0 0 0

︸︷︷︸A

x+

0 01 00 10 1

︸︷︷︸B

u, y =[1 0 1 00 0 1 0

]︸︷︷︸

C

x. (3.165)

The inputu now has the componentsu1 andu′2, andx has the componentsx1, x2, x3, andx4.

Again we letD = C, G = B, Q = V = I , R= ρ I andW = σ I , with ρ andσ to be determined.We chooseα = 1 so that the low-frequency disturbance at the second input channel extends upto the desired bandwidth. Figure 3.24 shows the loci of the regulator poles withρ as parameter.

-4 -3 -2 -1 0-2

-1

0Im

1/ρ

1/ρ

1/ρ

Re

1

2

Figure 3.24.: Loci of the regulator poles of the extended system

Three of the loci move out to∞ from the open-loop poles 0, 0, and−2. The fourth moves outfrom the open-loop pole at 0 to the open-loop zero at−α = −1. Forρ = 0.5 the regulator polesare−2.7200,−0.8141± j 0.7394 and−0.6079. The latter pole turns out to be nondominant.The pole pair−0.8141± j 0.7394, which has magnitude 1.0998, determines the bandwidth.

The loci of the optimal observer poles are again identical to those for the regulator poles.For σ = 5× 10−5 the observer poles are−200.01,−7.076± j 7.067 and−1. The latter pole isnondominant and the pole pair−7.076± j 7.067 has magnitude 10.

Figure 3.25 shows the magnitudes of the four entries of the sensitivity and complementarysensitivity matricesSandT that follow forρ = .5 andσ = 5× 10−5. The results are now muchmore acceptable.

Note that the off-diagonal entries ofS andT are small (though less so in the crossover re-gion). This means that the feedback compensator to an extent achieves decoupling. This is aconsequence of the high feedback gain at low frequencies. Infinite gain at all frequencies withunit feedback would make the closed-loop transfer matrix equal to the unit matrix, and, hence,completely decouple the system. The decoupling effect is also visible in Fig. 3.26, which showsthe entriessi j , i, j = 1,2 of the closed-loop response to unit steps on the two inputs.


In this appendix we provide sketches of several of the proofs for this chapter.

139


-100





-50

0

[dB]

|S11|

|S22|

|T11| |T12|

|T22|

[dB]

[dB]

[dB]

-100

-50

0

10-2 1010 10 100-1 1 310 2


|S12|

[dB]

-100

-50

0

10-2 1010 10 100-1 1 310210-2 1010 10 100-1 1 3102


|S21|

[dB]

-100

-50

0

10-2

1010 10 100-1 1 3

102

10-2

1010 10 100-1 1 3

102

10-2 1010 10 100-1 1 3102

-100

-50

0


|T21|

[dB]

10-2 1010 10 100-1 1 3102-100

-50

0

angular frequency [rad/s]angular frequency [rad/s]

[dB]

10-2 1010 10 100-1 1 3102-100

-50

0

-100

-50

0

Figure 3.25.: Magnitudes of the entries of the sensitivity matrixS and thecomplementary sensitivity matrixT

140


0 5

time [s]

time [s]

time [s]

time [s]

10-0.5

0

0.5

1

s11

s21

s12

s22

1.5

0 5 10-0.5

0

0.5

1

1.5

0 5 10-0.5

0

0.5

1

1.5

0 5 10-0.5

0

0.5

1

1.5

Figure 3.26.: Entries of the closed-loop unit step response matrix

141


3.7.1. Outline of the solution of the regulator problem

We consider the problem of minimizing∫ ∞

0[zT(t)Qz(t)+ uT(t)Ru(t)] dt (3.166)

for the stabilizable and detectable system

x(t) = Ax(t)+ Bu(t), z(t) = Dx(t). (3.167)

Lyapunov equation. We first study how to compute the quantity∫ t1

0xT(s)Q(s)x(s) dt + xT(t1)X1x(t1) (3.168)

for solutions of the time-varying state differential equationx(t) = A(t)x(t). Q(t) is a time-dependentnonnegative-definite symmetric matrix andX1 is a nonnegative-definite symmetric matrix. Substitution ofx(s)= 8(s,0)x(0), with8 the state transition matrix of the system, yields∫ t1

0xT(s)Q(s)x(s) ds+ xT(t1)X1x(t1)

= xT(0)(∫ t1

08T(s,0)Q(s)8(s,0) ds+8T(t1,0)X18(t1,0)

)x(0).

(3.169)

Define the time-dependent matrix

X(t)=∫ t1

t8T(s, t)Q(s)8(s, t) ds+8T(t1, t)X18(t1, t). (3.170)

Then ∫ t1

0xT(s)Q(s)x(s) ds+ xT(t1)X1x(t1) = xT(0)X(0)x(0). (3.171)

Differentiation ofX(t) with respect tot using ∂∂t8(s, t)= 8(s, t)A(t) shows thatX(t) satisfies the matrix

differential equation and terminal condition

−X(t)= AT(t)X(t)+ X(t)A(t)+ Q(t), X(t1) = X1. (3.172)

This equation is aLyapunov matrix differential equation. If A is constant and stable andR is constant thenast1 → ∞ the matrixX(t) approaches a constant nonnegative-definite matrixX that is independent ofX1

and is the unique solution of thealgebraic Lyapunov equation

0 = AT X + X A+ Q. (3.173)

Solution of the regulator problem. We consider how to determine the time-dependent gainF(t) suchthat the state feedbacku(t) = −F(t)x(t), 0 ≤ t ≤ t1, minimizes the performance criterion∫ t1

0[zT(t)Qz(t)+ uT(t)Ru(t)] dt + xT(t1)X1x(t1)

=∫ t1

0xT(t)

(DTQD+ FT(t)RF(t)

)x(t) dt + xT(t1)X1x(t1)

(3.174)

for the systemx(t)= Ax(t)+ Bu(t)= [ A− BF(t)]x(t). X1 is a nonnegative-definite symmetric matrix. Itfollows that∫ ∞

0[zT(t)Qz(t)+ uT(t)Ru(t)] dt + xT(t1)X1x(t1) = xT(0)X(0)x(0), (3.175)

142


whereX(t) is the solution of

−X(t)= [ A− BF(t)]TX(t)+ X(t)[ A− BF(t)] + DTQD+ FT(t)RF(t), (3.176)

X(t1) = X1. Completion of the square on the right-hand side (with respect toF(t) results in

−X(t) = [F(t)− R−1BT X(t)]TR[F(t)− R−1BT X(t)]

+ AT X(t)+ X(t)A+ DT QD− X(t)BR−1BT X(t), X(t1) = X1. (3.177)

Tracing the solutionX(t) backwards fromt1 to 0 we see that at each timet the least increase ofX(t) resultsif the gainF(t) is selected as

F(t)= −R−1BT X(t). (3.178)

Correspondingly, (3.177) reduces to thematrix differential Riccati equation

−X(t)= AT X(t)+ X(t)A+ DT QD− X(t)BR−1BT X(t), X(t1) = X1. (3.179)

If the system is stabilizable then the minimum of the performance criterion on the left-hand side of (3.174)is a nonincreasing function of the terminal timet1. Because it is bounded from below (by 0) the criterion hasa well-defined limit ast1 → ∞. Hence, alsoX(t) has a well-defined limit ast1 → ∞. Because of the timeindependence of the system this limitX is independent oft. The limit is obviously nonnegative-definite,and satisfies the algebraic Riccati equation

0 = AT X + X A+ DT QD− XBR−1BT X. (3.180)

Correspondingly the optimal feedback gain is time-invariant and equal to

F = −R−1BT X. (3.181)

The left-hand side of (3.174) can only converge to a finite limit ifz(t) = Dx(t) → 0 as t → ∞. Bydetectability this implies thatx(t)→ 0, that is, the closed-loop system is stable. If the system is observablethen X is positive-definite; otherwise there exist nonzero initial states such thatz(t)= Dx(t)= 0 for t ≥ 0.

3.7.2. Kalman-Jakubovi c-Popov equality

The Kalman-Jakuboviˇc-Popov equality is a generalization of the return difference equality that we use in§ 3.2.5 (p. 108). The KJP equality establishes the connection between factorizations and algebraic Riccatiequations.

Summary 3.7.1 (Kalman-Jakubovi c-Popov equality). Consider the linear time-invariant systemx(t) =Ax(t)+ Bu(t), y(t) = Cx(t)+ Du(t), with transfer matrixG(s) = C(sI − A)−1B+ D, and letQ and Rbe given symmetric constant matrices. Suppose that the algebraic matrix Riccati equation

0 = AT X + X A+ CTQC− (X B+ CTQD)(DTQD+ R)−1(BT X + DTQC) (3.182)

has a symmetric solutionX. Then

R+ G∼(s)QG(s)= J∼(s)LJ(s). (3.183)

The constant symmetric matrixL and the rational matrix functionJ are given by

L = R+ DT QD, J(s) = I + F(sI − A)−1B, (3.184)

with F = L−1(BT X + DTQC). The zeros of the numerator of detJ are the eigenvalues of the matrixA− BF. �

143


We use the notationG∼(s) = GT(−s).The KJP equality arises in the study of the regulator problem for the systemx(t) = Ax(t)+ Bu(t),

y(t)= Cx(t)+ Du(t), with the criterion∫ ∞

0[yT(t)Qy(t)+ uT(t)Ru(t)] dt. (3.185)

The equation (3.182) is the algebraic Riccati equation associated with this problem, andu(t) = −Fx(t) isthe corresponding optimal state feedback law.

The KJP equality is best known for the caseD = 0 (see for instance Kwakernaak and Sivan (1972)). Itthen reduces to thereturn difference equality

J∼(s)RJ(s)= R+ G∼(s)QG(s). (3.186)

Kalman-Jakuboviˇc-Popov equality.The proof starts from the algebraic Riccati equation

0 = AT X + X A+ CTQC− (X B+ CTQD)L−1(BT X + DTQC), (3.187)

with L = R+ DTQD. From the relationF = L−1(BT X + DT QC) we haveBT X + DT QC = LF, so thatthe Riccati equation may be written as

0 = AT X + X A+ CTQC− FTLF. (3.188)

This in turn we rewrite as

0 = −(−sI − AT)X − X(sI − A)+ CTQC− FTLF. (3.189)

Premultiplication byBT(−sI − AT)−1 and postmultiplication by(sI − A)−1B results in

0 = − BT X(sI − A)−1B− BT(−sI − AT)−1X B

+ BT(−sI − AT)−1(CTQC− FTLF)(sI − A)−1B.(3.190)

SubstitutingBT X = LF − DTQC we find

0 = (DTQC− LF)(sI − A)−1B+ BT(−sI − AT)−1(CTQD− FTL)

+ BT(−sI − AT)−1(CTQC− FTLF)(sI − A)−1B. (3.191)

Expansion of this expression, substitution ofC(sI − A)−1B = G(s)− D and F(sI − A)−1B = J(s)− Iand simplification lead to the desired result

R+ G∼(s)QG(s)= J∼(s)LJ(s). (3.192)

3.7.3. Robustness under state feedback

We consider an open-loop stable system with loop gain matrixL that satisfies the return difference inequality

( I + L)∼ R( I + L) ≥ R on the imaginary axis. (3.193)

We prove that if the loop gain is perturbed toW L, with W stable rational, then the closed-loop systemremains stable as long as

RW+ W∼ R> R on the imaginary axis. (3.194)

The proof follows Anderson and Moore (1990). First consider the case thatR = I . It follows from thereturn difference inequality thatL∼ + L + L∼L ≥ 0 on the imaginary axis, or

L−1 + (L−1)∼ + I ≥ 0 on the imaginary axis. (3.195)

144


The perturbed system is stable ifI + W L has no zeros in the right-half complex plane. Equivalently, theperturbed system is stable if for 0≤ ε ≤ 1 no zeros of

I + [(1− ε)I + εW]L (3.196)

cross the imaginary axis. Hence, the perturbed system is stable if only if

L−1 + (1− ε)I + εW = Mε (3.197)

is nonsingular on the imaginary axis for all 0≤ ε ≤ 1. Substitution ofL−1 = Mε − (1 − ε)I − εW into(3.195) yields

Mε + M∼ε ≥ 2(1− ε)I + ε(W+ W∼) = (2− ε)I + ε(W+ W∼ − I ). (3.198)

Inspection shows that if

W + W∼ > I (3.199)

on the imaginary axis thenMε + M∼ε > 0 on the imaginary axis for all 0≤ ε ≤ 1, which means thatMε is

nonsingular on the imaginary axis. Hence, ifW+ W∼ > I on the imaginary axis then the perturbed systemis stable.

Exercise 3.7.2 (The case R 6= I ). Show that the case thatR is not necessarily the unit matrix may bereduced to the previous case by factoringR— for instance by Choleski factorization — asR= RT

0 R0. Workthis out to prove that the closed-loop system remains stable under perturbation satisfyingRW+ W∼ R> Ron the imaginary axis. �

3.7.4. Riccati equations and the Hamiltonian matrix

We consider the algebraic Riccati equation

AT X + X A+ Q− (X B+ S)R−1(BT X + ST) = 0 (3.200)

and the associated Hamiltonian matrix

H =[

A− BR−1ST −BR−1BT

−Q+ SR−1ST −(A− BR−1ST)T

]. (3.201)

Summary 3.7.3 (Riccati equation and the Hamiltonian matrix).

1. If λ is an eigenvalue ofH then also−λ is an eigenvalue ofH .

2. Given a solutionX of the Riccati equation, defineF = R−1(BT X + ST). Then

H[

IX

]=[

IX

](A− BF). (3.202)

3. If λ is an eigenvalue ofA − BF corresponding to the eigenvectorx thenλ is also an eigenvalue ofH , corresponding to the eigenvector[

IX

]x. (3.203)

Hence, if then × n matrix A − BF hasn eigenvalues with negative real parts — such as in thesolution of the LQ problem of Summary 3.2.1 (p. 105) — then the eigenvalues ofH consist of thesen eigenvalues ofA− BF and their negatives.

145


4. Assume thatH has no eigenvalues with zero real part. Then there is a similarity transformationUthat bringsH into upper triangular formT such that

H = UTU−1 = U

[T11 T12

0 T22

]U, (3.204)

where the eigenvalues of then × n diagonal blockT11 all have negative real parts and those ofT22

have positive real parts. Write

U =[U11 U12

U21 U22

], (3.205)

where each of the subblocks has dimensionsn × n. Then there is a solutionX of the Riccati equa-tion such that the eigenvalues ofA − BF all have strictly negative real part, if and only ifU11 isnonsingular. In that case

X = U21U−111 (3.206)

is the unique solution of the Riccati equation such that the eigenvalues ofA − BF all have strictlynegative real part. For the LQ problem of Summary 3.2.1 (p. 105)U11 is nonsingular.

�

For the transformation under 4 there are several possibilities. One is to bringH into Jordan normalform. For numerical computation it is to great advantage to use the Schur transformation.

Riccati equation and the Hamiltonian matrix (sketch).

1. The Hamiltonian matrix is of the form

H =[

A QR −AT

], (3.207)

with all blocks square, andQ andR symmetric. We have for the characteristic polynomial ofH

det(λ I − H ) = det

[λ I − A −Q

−R λ I + AT

](1)= det

[−λ I + A QR −λ I − AT

](2)= (−1)n det

[R −λ I − AT

−λ I + A Q

](3)= det

[−λ I − AT RQ −λ I + A

](4)= det

[−λ I + A QR −λ I − AT

](5)= det

[−λ I + A −Q−R −λ I − AT

](3.208)

In step (1) we multiply the matrix by−1. In step (2) we interchange the first and second rows ofblocks and in step (3) the firt and second columns of blocks. In step (4) we transpose the matrix. Instep (5) we multiply the second row and the second column of blocks by−1.

Inspection shows that the characteristic polynomial does not change ifλ is replaced withλ. Hence,if λ is an eigenvalue, so is−λ.

2. Using the Riccati equation we obtain from (3.201)

H[

IX

]=

[A− BF

−Q + SR−1ST − (A− BR−1ST)T X

](3.209)

=[

A− BFX A− X BF

]=[

IX

](A− BF). (3.210)

3. If (A− BF)x = λx then

H

[IX

]x =

[IX

](A− BF)x = λ

[IX

]x. (3.211)

146


4. FromH U = UT we obtain

H[U11

U21

]=[U11

U21

]T11. (3.212)

After multiplying on the right byU11 it follows that

H[

IU21U−1

11

]=[

IU21U−1

11

]U11T11U

−111 . (3.213)

We identify X = U21U−111 and A− BF = U11T11U−1

11 . For the LQ problem the nonsingularity ofU11

follows by the existence ofX such thatA− BF is stable.

3.7.5. The Kalman filter

In this subsection we outline the derivation of the Kalman filter.

Linear system driven by white noise Consider the stable linear systemx(t) = Ax(t)+ v(t), drivenby white noise with intensityV, that is,Ev(t)vT(s) = Vδ(t − s). The state of the system

x(t) =∫ t

−∞eA(t−s) v(s) ds, t ∈ R, (3.214)

is a stationary stochastic process with covariance matrix

Y = Ex(t)xT(t)=∫ t

−∞

∫ t

−∞eA(t−s1 )

(Ev(s1)v

T(s2))eAT (t−s2 ) ds1ds2

=∫ t

−∞eA(t−s)VeAT (t−s) ds=

∫ ∞

0eAτVeATτ dτ. (3.215)

It follows that

AY+ Y AT =∫ ∞

0

(AeAτVeATτ + eAτVeATτ AT

)dτ

=∫ ∞

0

ddτ

(eAτVeATτ

)dτ = eAτVeATτ

∣∣∣∞0

= −V. (3.216)

Hence, the covariance matrixY is the unique solution of the Lyapunov equation

AY+ Y AT + V = 0. (3.217)

Observer error covariance matrix. We consider the systemx(t) = Ax(t)+ Bu(t)+ v(t), y(t) =Cx(t)+w(t), wherev is white noise with intensityV andw white noise with intensityW. The estimationerrore(t)= x(t)− x(t) of the observer

˙x(t) = Ax(t)+ Bu(t)+ K[y(t)− Cx(t)] (3.218)

satisfies the differential equation

e(t) = (A− KC)e(t)− Gv(t)+ Kw(t). (3.219)

The noise process−Gv(t)+ Kw(t) is white noise with intensityGTVG+ KTW K. Hence, if the errorsystem is stable then the error covariance matrixY = Ee(t)eT(t) is the unique solution of the Lyapunovequation

(A− KC)Y+ Y(A− KC)T + GTVG+ KTW K = 0. (3.220)

147


The Kalman filter. We discuss how to choose the observer gainK to minimize the error covariancematrixY. To this end we complete the square (inK) and rewrite the Lyapunov equation (3.220) as

(K − YCTW−1)W(K − YCTW−1)T + AY+ Y AT + GVGT − YCTW−1CY= 0. (3.221)

Suppose that there exists a gainK that stabilizes the error system and minimizes the error variance matrixY.Then changing the gain toK + εK, with ε a small scalar andK an arbitrary matrix of the same dimensionsasK, should only affectY quadratically inε. Inspection of (3.221) shows that this implies

K = YCTW−1. (3.222)

With this gain the observer reduces to the Kalman filter. The minimal error variance matrixY satisfies theRiccati equation

AY+ Y AT + GVGT − YCTW−1CY= 0. (3.223)

3.7.6. Minimization of the steady-state mean square error under statefeedback

We consider the problem of choosing the gainF of the state feedback lawu(t) = −Fx(t) to minimize thesteady state mean square error

E(zT(t)Qz(t)+ uT(t)Ru(t)

)(3.224)

for the systemx(t)= Ax(t)+ Bu(t)+ v(t). The white noisev has intensityV.If the feedback law stabilizes the systemx(t) = (A− BF)x(t)+ v(t) then the steady-state covariance

matrixY of the state is given by

Y = Ex(t)xT(t) =∫ ∞

0e(A−BF)sVe(A−BF)T s ds. (3.225)

Hence we have for the steady-state mean square error


)= E(xT(t)DTQDx(t)+ xT(t)FTRFx(t)

)= E tr

(x(t)xT(t)DTQD+ x(t)xT(t)FT RF

)= tr Y(DTQD+ FT RF

). (3.226)

We rewrite this in the form


)= tr Y(DT QD+ FT RF

)= tr

∫ ∞

0e(A−BF)sVe(A−BF)Ts ds

(DT QD+ FT RF

)= tr V

∫ ∞

0e(A−BF)T s

(DTQD+ FT RF

)e(A−BF)s ds︸︷︷︸

X

= tr V X. (3.227)

X is the solution of the Lyapunov equation

(A− BF)TX + X(A− BF)+ DT QD+ FT RF = 0. (3.228)

X and, hence, trV X, is minimized by choosingF = R−1BT X, with X the solution of the Riccati equationAT X + X A+ DT QD− X BR−1BT X = 0.

148


3.7.7. Loop transfer recovery

We study an LQG optimal system with measurement noise intensityW = σW0 asσ ↓ 0 under the assump-tions thatG = B and that the plant transfer matrixG(s) = C(sI − A)−1B is square with stable inverseG−1(s).

Under the assumptionG = B we have in the absence of any measurement noisew

y = C(sI − A)−1B(u+ v) = G(s)(u+ v). (3.229)

Because by assumptionG−1 is stable the input noisevmay be recovered with arbitrary precision by approx-imating the inverse relation

v = G−1(s)y− u (3.230)

with sufficient accuracy. From the noisev and the known inputu the statex may in turn be reconstructedwith arbitrary precision. Hence, we expect that asσ decreases to 0 the covarianceYσ of the estimation errordecreases to the zero matrix.

Under the assumptionG = B the Riccati equation for the optimal observer is

AYσ + YσAT + BV BT − YσCTW−1CYσ = 0. (3.231)

We rewrite this as

AYσ + YσAT + BV BT − σKσW0KTσ = 0, (3.232)

with Kσ = YσCTW−1 the gain. Inspection shows that ifYσ ↓ 0 then necessarilyKσ → ∞. In fact we maywrite

Kσ ≈ 1√σ

BUσ asσ ↓ 0, (3.233)

whereUσ is a square nonsingular matrix (which may depend onσ) such thatUσW0UTσ = V.

We study the asymptotic behavior of the loop gain

Lσ (s) = F(sI − A+ BF + KσC)−1KσC(sI − A)−1B

≈ F(sI − A+ BF + 1√σ

BUσC)−1 1√σ

BUσC(sI − A)−1B

≈ F(sI − A+ 1√σ

BUσC)−1 1√

σBUσC(sI − A)−1B

= F(sI − A)−1

(I + 1√

σBUσC(sI − A)−1

)−1 1√σ

BUσC(sI − A)−1B

(1)= F(sI − A)−1B1√σ

Uσ

(I + 1√

σC(sI − A)−1BUσ

)−1

C(sI − A)−1B

= F(sI − A)−1BUσ(I√σ + C(sI − A)−1BUσ

)−1C(sI − A)−1B. (3.234)

In step (1) we use the well-known matrix identity( I + AB)−1A = A( I + BA)−1. Inspection of the finalequality shows that

Lσ (s)σ↓0−→ F(sI − A)−1B, (3.235)

which is the loop gain under full state feedback.

3.7.8. Solution of the H2 optimization problem

We solve the standardH2 optimization problem of§ 3.4.3 (p. 126) as if it is an LQG problem, that is, weset out to minimize the steady-state value of

EzT(t)z(t) (3.236)

under the assumption thatw is a white noise input with intensity matrixI .

149


State feedback. We first consider the solution with state feedback. For this it is enough to study theequations

x(t) = Ax(t)+ B1w(t)+ B2u(t), (3.237)

z(t) = C1x(t)+ D11w(t)+ D12u(t). (3.238)

If D11 6= 0 then the outputz has a white noise component that may well make the mean square output(3.236) infinite. We therefore assume thatD11 = 0. Under this assumption we have

z(t) = C1x(t)+ D12u(t) = [I D12

][C1x(t)u(t)

]= [

I D12

] [z0(t)u(t)

], (3.239)

wherez0(t) = C1x(t). As a result,

EzT(t)z(t) = E[zT

0(t) uT(t)][ I

DT12

][I D12

] [ z0

u(t)

]

= E[zT

0(t) uT(t)][ I D12

DT12 DT

12D12

][z0

u(t)

]. (3.240)

This defines a linear regulator problem with a cross term in the output and input. It has a solution if thesystemx(t)= Ax(t)+ B2u(t), z0(t) = C1x(t) is stabilizable and detectable, and the weighting matrix[

I D12

DT12 DT

12D12

](3.241)

is positive-definite. A necessary and sufficient condition for the latter is thatDT12D12 be nonsingular. The

solution to the regulator problem is a state feedback law of the form

u(t) = −Fx(t). (3.242)

The gain matrixF may easily be found from the results of§ 3.2.8 (p. 113) and is given in Summary 3.4.3(p. 118).

Output feedback. If the state is not available for feedback then it needs to be estimated with a Kalmanfilter. To this end we consider the equations

x(t) = Ax(t)+ B1w(t)+ B2u(t), (3.243)

y(t) = C2x(t)+ D21w(t)+ D22u(t). (3.244)

The second equation may be put into the standard form for the Kalman filter if we considery(t)− D22u(t)as the observed variable rather thany(t). If we denote the observation noise asv(t)= D21w(t) then

x(t) = Ax(t)+ B1w(t)+ B2u(t), (3.245)

y(t)− D22u(t) = C2x(t)+ v(t) (3.246)

defines a stochastic system with cross correlated noise terms. We have

E

[w(t)v(t)

][wT(s) vT(s)

] = E

[I

D21

]w(t)wT(s)

[I DT

21

](3.247)

=[

I DT21

D21 D21DT21

]δ(t − s). (3.248)

Suppose that the systemx(t) = Ax(t)+ B1w(t), y(t) = C2x(t) is stabilizable and detectable, and theintensity matrix[

I DT21

D21 D21DT21

](3.249)

150


is positive-definite. A necessary and sufficient condition for the latter is thatD21DT21 be nonsingular. Then

there exists a well-defined Kalman filter of the form

˙x(t) = Ax(t)+ B2u(t)+ K[y(t)− C2x(t)− D22u(t)]. (3.250)

The gain matrixK may be solved from the formulas of§ 3.3.4 (p. 120). Once the Kalman filter (3.250) isin place the optimal input for the output feedback problem is obtained as

u(t) = −Fx(t). (3.251)

F is the same state feedback gain as in (3.242).

151


152

4. Multivariable Control System Design

Overview– Design of controllers for multivariable systems requires anassessment of structural properties of transfer matrices. The zeros andgains in multivariable systems have directions.

With norms of multivariable signals and systems it is possible to ob-tain bounds for gains, bandwidth and other system properties.

A possible approach to multivariable controller design is to reduce theproblem to a series of single loop controller design problems. Examplesare decentralized control and decoupling control.

The internal model principle applies to multivariable systems and, forexample, may be used to design for multivariable integral action.

4.1. Introduction

Many complex engineering systems are equipped with several actuators that may influence theirstatic and dynamic behavior. Commonly, in cases where some form of automatic control is re-quired over the system, also several sensors are available to provide measurement informationabout important system variables that may be used for feedback control purposes. Systems withmore than one actuating control input and more than one sensor output may be considered asmultivariablesystems ormulti-input-multi-output (MIMO)systems. The control objective formultivariable systems is to obtain a desirable behavior of several output variables by simultane-ously manipulating several input channels.

4.1.1. Examples of multivariable feedback systems

The following two examples discuss various phenomena that specifically occur in MIMO feed-back systems and not in SISO systems, such as interaction between loops and multivariablenon-minimum phase behavior.

Example 4.1.1 (Two-tank liquid flow process). Consider the flow process of Fig. 4.1. Theincoming flowφ1 and recycle flowφ4 act as manipulable input variables to the system, and theoutgoing flowφ3 acts as a disturbance. The control objective is to keep the liquid levelsh1 andh2

between acceptable limits by applying feedback from measurements of these liquid levels, while

153


φ1

φ2

φ3φ4

h1

h2

Figure 4.1.: Two-tank liquid flow process with recycle

h1,ref

h2,ref

h1

h2

k1

k2

Pφ4

φ1

Figure 4.2.: Two-loop feedback control of the two-tank liquid flow process

154

4.1. Introduction

accommodating variations in the output flowφ3. As derived in Appendix 4.6, a dynamic model,linearized about a given steady state, is[

h1(t)h2(t)

]=[−1 0

1 0

][h1(t)h2(t)

]+[1 10 −1

][φ1(t)φ4(t)

]+[

0−1

]φ3(t). (4.1)

Here thehi andφi denote the deviations from the steady state levels and flows. Laplace transfor-mation yields[

H1(s)H2(s)

]=[s+ 1 0−1 s

]−1([1 10 −1

][81(s)84(s)

]+[

0−1

]83(s)

),

which results in the transfer matrix relationship[H1(s)H2(s)

]=[

1s+1

1s+1

1s(s+1) − 1

s+1

][81(s)84(s)

]+[

0− 1

s

]83(s). (4.2)

�

The example demonstrates the following typical MIMO system phenomena:

• Each of the manipulable inputsφ1 andφ4 affects each of the outputs to be controlledh1

andh2, as a consequence the transfer matrix

P(s) =[

1s+1

1s+1

1s(s+1) − 1

s+1

](4.3)

has nonzero entries both at the diagonal and at the off-diagonal entries.

• Consequently, if we control the two output variablesh1 andh2 using two separate controlloops, it is not immediately clear which input to use to controlh1, and which one to controlh2. This issue is called theinput/output pairingproblem.

• A possible feedback scheme using two separate loops is shown in Fig. 4.2. Note that inthis control scheme, there exists a coupling between both loops due to the non-diagonalterms in the transfer matrix (4.3). As a result, the plant transfer function in the open upperloop fromφ1 to h1, with the lower loop closed with proportional gaink2, depends onk2,

P11(s)∣∣llc = 1

s+ 1

(1− k2

s(s+ 1− k2)

). (4.4)

Herellc indicates that the lower loop is closed. Owing to the negative steady state gain inthe lower loop, negative values ofk2 stabilize the lower loop. The dynamics ofP11(s)

∣∣llc

changes withk2:

k2 = 0 : P11(s)∣∣llc = 1

s+1,

k2 = −1 : P11(s)∣∣llc = s+1

s(s+2) ,

k2 = −∞ : P11(s)∣∣llc = 1

s .

The phenomenon that the loop gain in one loop also depends on the loop gain in anotherloop is calledinteraction. Themultiple loopcontrol structure used here does not explicitlyacknowledge interaction phenomena. A control structure using individual loops, such asmultiple loop control, is an example of adecentralizedcontrol structure, as opposed to acentralizedcontrol structure where all measured information is available for feedback inall feedback channels.

155


h1,ref

h2,ref

h1

h2

k1

k2

Pφ4

φ1

Kpre

controllerK

Figure 4.3.: Decoupling feedback control of two-tank liquid flow process

• A different approach to multivariable control is to try to compensate for the plant interac-tion. In Example 4.1.1 a precompensatorKpre in series with the plant can be found whichremoves the interaction completely. The precompensatorKpre is to be designed such thatthe transfer matrixPKpre is diagonal. Figure 4.3 shows the feedback structure. Supposethat the precompensatorKpre is given the structure

Kpre(s) =[

1 Kpre12 (s)

Kpre21 (s) 1

](4.5)

thenPKpre is diagonal by selecting

Kpre12 (s) = −1, Kpre

21 (s) = 1s. (4.6)

The compensated plantPKpre for which a multi-loop feedback is to be designed as a nextstep, reads

P(s)Kpre(s) =[

1s 00 − 1

s

]. (4.7)

Closing both loops with proportional gainsk1 > 0 andk2 < 0 leads to a stable system. Indoing so, a multivariable controllerK has been designed having transfer matrix

K(s) = Kpre(s)

[k1 00 k2

]=[

k1 −k2

k1/s k2

]. (4.8)

This approach of removing interaction before actually closing multiple individual loops iscalleddecoupling control.

The next example shows that a multivariable system may show non-minimum phase behavior.

Example 4.1.2 (Multivariable non-minimum phase behavior). Consider the two input, twooutput linear system[

y1(s)y2(s)

]=[ 1

s+12

s+31

s+11

s+1

][u1(s)u2(s)

]. (4.9)

If the lower loop is closed with constant gain,u2(s) = −k2y2(s), then for high gain values(k2 → ∞) the lower feedback loop is stable but has a zero in the right-half plane,

y1(s) =(

1s+ 1

− 2s+ 3

)u1(s) = − s− 1

(s+ 1)(s+ 3)u1(s). (4.10)

156

4.1. Introduction

Thus under high gain feedback of the lower loop, the upper part of the system exhibits non-minimum phase behavior. Conversely if the upper loop is closed under high gain feedback,u1 = −k1 y1 with k1 → ∞, then

y2(s) = − s− 1(s+ 1)(s+ 3)

u2(s). (4.11)

Apparently, the non-minimum phase behavior of the system is not connected to one particularinput-output relation, but shows up in both relations. One loop can be closed with high gainsunder stability of the loop, and the other loop is then restricted to have limited gain due to thenon-minimum phase behavior. Analysis of the transfer matrix

P(s) =[ 1

s+12

s+31

s+11

s+1

]= 1(s+ 1)(s+ 3)

[s+ 3 2s+ 2s+ 3 s+ 3

](4.12)

shows that it loses rank ats = 1. In the next section it will be shown thats = 1 is an unstabletransmission zeroof the multivariable system and this limits the closed loop behavior irrespec-tive of the controller used. Consider the method of decoupling precompensation. Express theprecompensator as

Kpre(s) =[

K11(s) K12(s)K21(s) K22(s)

]. (4.13)

Decoupling means thatPKpre is diagonal. Inserting the matrix (4.12) intoPKpre, a solution fora decouplingKpre is

K11(s) = 1, K22(s) = 1, K12(s) = −2s+ 1s+ 3

, K21(s) = −1. (4.14)

Then the compensated system is

P(s)Kpre(s) =[ 1

s+12

s+31

s+11

s+1

][1 −2s+1

s+3−1 1

]=[− s−1(s+1)(s+3) 0

0 − s−1(s+1)(s+3)

]. (4.15)

The requirement of decoupling has introduced a right-half plane zero in each of the loops, so thateither loop now only can have a restricted gain in the feedback loop. �

The example suggests that the occurrence of non-minimum phase phenomena is a property ofthe transfer matrix and exhibits itself in a matrix sense, not connected to a particular input-outputpair. It shows that a zero of a multivariable system has no relationship with the possible zeros ofindividual entries of the transfer matrix.

In the example the input-output pairing has been the natural one: outputi is connected bya feedback loop to inputi. This is however quite an arbitrary choice, as it is the result of ourmodel formulation that determines which inputs and which outputs are ordered as one, two andso on. Thus the selection of the most useful input-output pairs is anon-trivial issue inmultiloopcontrolor decentralized control, i.e. in control configurations where one has individual loops asin the example. A classical approach towards dealing with multivariable systems is to bring amultivariable system to a structure that is a collection of one input, one output control problems.This approach ofdecoupling controlmay have some advantages in certain practical situations,and was thought to lead to a simpler design approach. However, as the above example showed,decoupling may introduce additional restrictions regarding the feedback properties of the system.

157


4.1.2. Survey of developments in multivariable control

The first papers on multivariable control systems appeared in the fifties and considered aspectsof noninteracting control. In the sixties, the work of Rosenbrock (1970) considered matrix tech-niques to study questions of rational and polynomial representation of multivariable systems. Thepolynomial representation was also studied by Wolovich (1974). The books by Kailath (1980)and Vardulakis (1991) provide a broad overview. The use of Nyquist techniques for multivari-able control design was developed by Rosenbrock (1974a). The generalization of the Nyquistcriterion and of root locus techniques to the multivariable case can be found in the work ofPostlethwaite and MacFarlane (1979). The geometric approach to multivariable state-space con-trol design is contained in the classical book by Wonham (1979) and in the book by Basile andMarro (1992). A survey of classical design methods for multivariable control systems can befound in (Korn and Wilfert 1982), (Lunze 1988) and in the two books by Tolle (1983), (1985).Modern approaches to frequency domain methods can be found in (Raisch 1993), (Maciejowski1989), and (Skogestad and Postlethwaite 1995). Interaction phenomena in multivariable processcontrol systems are discussed in terms of a process control formulation in (McAvoy 1983). Amodern, process-control oriented approach to multivariable control is presented in (Morari andZafiriou 1989). The numerical properties of several computational algorithms relevant to the areaof multivariable control design are discussed in (Svaricek 1995).

4.2. Poles and zeros of multivariable systems

In this section, a number of structural properties of multivariable systems are discussed. Theseproperties are important for the understanding of the behavior of the system under feedback. Thesystems are analyzed both in the time domain and frequency domain.

4.2.1. Polynomial and rational matrices

Let P be a proper real-rational transfer matrix.Real-rationalmeans that every entryPij of P isa ratio of two polynomials having real coefficients. Any rational matrixP may be written as afraction

P = 1d

N

whereN is apolynomial matrix1 andd is the monic least common multiple of all denominatorpolynomials of the entries ofP. Polynomial matrices will be studied first. Many results aboutpolynomial matrices may be found in, e.g., the books by MacDuffee (1956), Gohberg, Lancaster,and Rodman (1982) and Kailath (1980).

In the scalar case a polynomial has no zeros if and only if it is a nonzero constant, or, toput it differently, if and only if its inverse is polynomial as well. The matrix generalization is asfollows.

Definition 4.2.1 (Unimodular polynomial matrix). A polynomial matrix isunimodular if itsquare and its inverse exists and is a polynomial matrix. �

It may be shown that a square polynomial matrixU is unimodular if and only if detU is anonzero constant. Unimodular matrices are considered having no zeros and, hence, multiplicationby unimodular matrices does not affect the zeros.

1That is, a matrix whose entries are polynomials.

158


Summary 4.2.2 (Smith form of a polynomial matrix). For every polynomial matrixN thereexist unimodularU andV such that

U NV =

ε′1 0 0 0 0 0

0 ε′2 0 0 0 0

0 0... 0 0 0

0 0 0 ε′r 0 0

0 0 0 0 0 00 0 0 0 0 0

︸︷︷︸S

(4.16)

whereε′i, (i = 1, . . . , r) are monic polynomials with the property thatεi/εi−1 is polynomial. The

matrix S is known as theSmith formof N and the polynomialsε′i the invariant polynomialsof

N.In this caser is thenormal rankof the polynomial matrixN, and thezerosof N are defined

as the zeros of one or more of its invariant polynomials. �

The Smith formSof N may be obtained by applying toN a sequence ofelementary row andcolumn operations, which are:

• multiply a row/column by a nonzero constant;

• interchange two rows or two columns;

• add a row/column multiplied by a polynomial to another row/column.

Example 4.2.3 (Elementary row/column operations). The following sequence of elementaryoperations brings the polynomial matrix

N(s) =[

s+ 1 s− 1s+ 2 s− 2

]

to diagonal Smith-form:[s+ 1 s− 1s+ 2 s− 2

](1)⇒

[s+ 1 s− 1

1 −1

](2)⇒

[s+ 1 2s

1 0

](3)⇒

[0 2s1 0

](4)⇒

[1 00 s

].

Here, in step (1) we subtracted row 1 from row 2; in step (2) we added the first column to thesecond column; in step (3),s+ 1 times the second row was subtracted from row 1. Finally, instep (4), we interchanged the rows and then divided the (new) second row by a factor 2.

Elementaryrow operations correspond thepremultiplication by unimodular matrices, andelementarycolumnoperations correspond thepostmultiplication by unimodular matrices. Forthe above four elementary operations these are

(1) premultiply byUL1(s) :=[

1 0−1 1

];

(2) postmultiply byVR2(s) :=[1 10 1

];

(3) premultiply byUL3(s) :=[1 −(s+ 1)0 1

];

159


(4) premultiply byUL4(s) :=[

0 11/2 0

].

Instead of applying sequentially multiplication onN we may also first combine the sequenceof unimodular matrices into two unimodular matricesU = UL4UL3UL1 andV = VR2, and thenapply theU andV,[ −1 1

1+ s/2 −1/2− s/2

]︸︷︷︸

U(s) = UL4(s)UL3(s)UL1(s)

[s+ 1 s− 1s+ 2 s− 2

]︸︷︷︸

N(s)

[1 10 1

]︸︷︷︸

V(s) = VR2(s)

=[1 00 s

]︸︷︷︸

S(s)

.

This clearly shows thatS is the Smith-form ofN. The polynomial matrixN has rank 2 and hasone zero ats= 0. �

Now consider a rational matrixP. We write P as

P = 1d

N (4.17)

whereN is a polynomial matrix andd a scalar polynomial. We immediately get a generalizationof the Smith form.

Summary 4.2.4 (Smith-McMillan form of a rational matrix). For every rational matrixP thereexist unimodular polynomial matricesU andV such that

U PV =

ε1ψ1

0 0 0 0 00 ε2

ψ20 0 0 0

0 0... 0 0 0

0 0 0 εrψr

0 00 0 0 0 0 00 0 0 0 0 0

︸︷︷︸M

(4.18)

where theεi andψi are coprime such that

εi

ψi= ε′

i

d(4.19)

and theε′i, (i = 1, . . . , r) are the invariant polynomials ofN = dP. The matrixM is known as

theSmith-McMillan formof P.In this caser is thenormal rankof the rational matrixP. The transmission zerosof P are

defined as the zeros of∏r

i=1 εi and thepolesare the zeros of∏r

i=1ψi . �

Definitions for other notions of zeros may be found in (Rosenbrock 1970), (Rosenbrock 1973)and (Rosenbrock 1974b), and in (Schrader and Sain 1989).

Example 4.2.5 (Smith-McMillan form). Consider the transfer matrix

P(s) =[

1s

1s(s+1)

1s+1

1s+1

]= 1

s(s+ 1)

[s+ 1 1

s s

]= 1

d(s)N(s).

The Smith form ofN is obtained by the following sequence of elementary operations:[s+ 1 1

s s

]⇒[s+ 1 1−s2 0

]⇒[1 s+ 10 s2

]⇒[1 00 s2

]

160


Division of each diagonal entry byd yields the Smith-McMillan form[ 1s(s+1) 0

0 ss+1

].

The set of transmission zeros is{0}, the set of poles is{−1,−1,0}. This shows that transmissionzeros of a multivariable transfer matrix may coincide with poles without being canceled if theyoccur in different diagonal entries of the Smith-McMillan form. In the determinant they docancel. Therefore from the determinant

detP(s) = 1(s+ 1)2

we may not always uncoverall poles and transmission zeros ofP. (GenerallyP need not besquare so its determinant may not even exist.) �

An s0 ∈ C is apoleof P if and only if it is a pole of one or more entriesPij of P. An s0 ∈ C

that is not a pole ofP is a transmission zero ofP if and only if the rank ofP(s0) is strictlyless than the normal rankr as defined by the Smith-McMillan form ofP. For square invertiblematricesP there further holds thats0 is a transmission zero ofP if and only it is pole ifP−1. Forexample

P(s) =[1 1/s0 1

]

has a transmission zero ats= 0 — even though detP(s) = 1 — becauseP−1(s) = [1 −1/s0 1

]has

a pole ats= 0.If the normal rank of anny × nu plant P is r then at mostr entries of the outputy = Pu can

be given independent values by manipulating the inputu. This generally means thatr entries ofthe output are enough for feedback control. LetK be anynu × ny controller transfer matrix, thenthe sensitivity matrixSand complementary sensitivity matrixT as previously defined are for themultivariable case (see Exercise 1.5.4)

S= ( I + PK)−1, T = ( I + PK)−1PK. (4.20)

If r < ny, then theny × ny loop gainPK is singular, so thatShas one or more eigenvaluesλ = 1for any s in the complex plane, in particular everywhere on the imaginary axis. In such cases‖S(jω)‖ in whatever norm does not converge to zero asω → 0. This indicates an undesiredsituation which is to be prevented by ascertaining the rank ofP to equal the number of the becontroller outputs. This must be realized by proper selection and application of actuators andsensors in the feedback control system.

Feedback around a nonsquareP can only occur in conjunction with a compensatorK whichmakes the series connectionPK square, as required by unity feedback. Such controllersK aresaid tosquare downthe plantP. We investigate down squaring.

Example 4.2.6 (Squaring down). Consider

P(s) = [1s

1s2

].

Its Smith-McMillan form isM(s) = [1/s2 0

]. The plant has no transmission zeros and has a

double pole ats= 0. As pre-compensator we propose

K0(s) =[1a

].

161


This results in the squared-down system

P(s)K0(s) = s+ as2

.

Apparently squaring down may introduces transmission zeros. In this example the choicea >0 creates a zero in the left-half plane, so that subsequent high gain feedback can be applied,allowing a stable closed-loop system with high loop gain. �

In the example,P does not have transmission zeros. A general 1× 2 plant P(s) =[a(s) b(s)

]has transmission zeros if and only ifa and b have common zeros. Ifa and b

are in some sense randomly chosen then it is very unlikely thata andb have common zeros.Generally it holds that nonsquare plants have no transmission zeros assuming the entries ofPij

are in some sense uncorrelated with other entries. However, many physical systems bear structurein the entries of their transfer matrix, so that these cannot be considered as belonging to the classof generic systems. Many physically existing nonsquare systems actually turn out to possesstransmission zeros.

The theory of transmission zero placement by squaring down is presently not complete, al-though a number of results are available in the literature. Sain and Schrader (1990) describe thegeneral problem area, and results regarding the squaring problem are described in (Horowitz andGera 1979), (Karcanias and Giannakopoulos 1989), (Le and Safonov 1992), (Sebakhy, ElSingaby,and ElArabawy 1986), (Stoorvogel and Ludlage 1994), and (Shaked 1976).

Example 4.2.7 (Squaring down). Consider the transfer matrixP and its Smith-McMillan form

P(s) = 1

s1

s(s2−1)

1 1s2−1

0 ss2−1

= 1

s(s2 − 1)

s2 − 1 1

s(s2 − 1) s0 s2

(4.21)

=1 0 0

s 0 1s2 −1 0

︸︷︷︸U−1(s)

1

s(s2−1) 00 s0 0

︸︷︷︸M(s)

[s2 − 1 1

1 0

]︸︷︷︸

V−1(s)

. (4.22)

There is one transmission zero{0} and the set of poles is{−1,1,0}. The postcompensatorK0 isnow 2× 3, which we parameterize as

K0 =[

a b cd e f

].

We obtain the newly formed set of transmission zeros as the set of zeros of

det

[a b cd e f

]1 0s 0s2 −1

= (cd− af )+ (ce− bf )s,

where the second matrix in this expression are the first two columns ofU−1. Thus one singletransmission zero can be assigned to a desired location. Choose this zero ats = −1. This leadsfor example to the choicea = 0, b = 0, c = 1, d = 1, e= 1, f = 0,

K0(s) :=[0 0 11 1 0

]

162


and the squared-down system

K0(s)P(s) =[

0 ss2−1

s+1s

1s(s−1)

]= 1

s(s2 − 1)

[0 s2

(s+ 1)(s2 − 1) s+ 1

].

This squared-down matrix may be shown to have Smith-McMillan form[ 1s(s2−1) 0

0 s(s+ 1)

]and consequently the set of transmission zeros is{0,−1}, the set of poles remain unchanged as{−1,1,0}. These values are as expected, i.e., the zero ats = 0 has been retained, and the newzero at−1 has been formed. Note thats= 0 is both a zero and a pole. �

4.2.2. Transmission zeros of state-space realizations

Any properny × nu rational matrixP has astate-space realization

P(s) = C(sIn − A)−1B+ D, A ∈ Rn×n, B ∈ R

n×nu, C ∈ Rny×n, D ∈ R

ny×nu .

Realizations are commonly denoted as a quadruple(A, B,C, D). There are many realizations(A, B,C, D) that define the same transfer matrixP. For one, theorder nis not fixed, A realizationof ordern of P is minimal if no realization ofP exists that has a lower order. Realizations areminimal if and only of the ordern equals the degree of the polynomialψ1ψ2 . . .ψr formed fromthe denominator polynomials in (4.18), see (Rosenbrock 1970). The poles ofP then equal theeigenvalues ofA, which, incidentally, shows that computation of poles is a standard eigenvalueproblem. Numerical computation of the transmission zeros may be more difficult. In some specialcases, computation can be done by standard operations, in other cases specialized numericalalgorithms have to be used.

Lemma 4.2.8 (Transmission zeros of a minimal state-space system). Let (A, B,C, D) be aminimal state-space realization of ordern of a proper real-rationalny × nu transfer matrixP ofrankr . Then the transmission zeross0 ∈ C of P as defined in Summary 4.2.4 are the zeross0 ofthe polynomial matrix[

A− s0 I BC D

]. (4.23)

That is,s0 is transmission zero if and only if rank[

A−s0 I BC D

]< n+ r . �

The proof of this result may be found in Appendix 4.6. Several further properties may bederived from this result.

Summary 4.2.9 (Invariance of transmission zeros). The zeross0 of (4.23) are invariant underthe following operations:

• nonsingularinput spacetransformations(T2) andoutput spacetransformations(T3):[A− sI B

C D

]⇒[

A− sI BT2

T3C T3DT2

]

• nonsingularstate spacetransformations(T1):[A− sI B

C D

]⇒[

T1AT−11 − sI T1B

CT−11 D

]

163


• Static output feedbackoperations, i.e. for the systemx = Ax+ Bu, y = Cx+ Du, we applythe feedback lawu = −Ky+ v wherev acts as new input. Assuming that det( I + DK) 6=0, this involves the transformation:[

A− sI BC D

]⇒[

A− BK( I + DK)−1C − sI B( I + K D)−1

( I + DK)−1C ( I + DK)−1D

]

• State feedback(F) andoutput injection(L) operations,[A− sI B

C D

]⇒[

A− BF − sI BC − DF D

]⇒[

A− BF − LC + LDF − sI B− LDC − DF D

]�

The transmission zeros have a clear interpretation as that ofblocking certain exponentialinputs, (Desoer and Schulman 1974), (MacFarlane and Karcanias 1976).

Example 4.2.10 (Blocking property of transmission zeros). Suppose that the system realiza-tion P(s)= C(sI − A)−1B+ D is minimal and thatP is left-invertible, that is rankP = nu. Thens0 is a transmission zero if and only if

[A−s0 I B

C D

]has less than full column rank, in other words if

and only if there exist vectorsxs0, us0—not both2 zero—that satisfy[A− s0 I B

C D

][xs0

us0

]= 0. (4.24)

Then the state and input defined as

x(t) = xs0es0t, u(t) = us0e

s0t

satisfy the system equations

x(t) = Ax(t)+ Bu(t), Cx(t)+ Du(t) = 0. (4.25)

We see thaty = Cx+ du is identically zero. Transmission zeross0 hence correspond to certainexponential inputs whose output is zero for all time. �

Under constant high gain output feedbacku = Ky the finite closed loop poles generallyapproach the transmission zeros of the transfer matrixP. In this respect the transmission zerosof a MIMO system play a similar role in determining performance limitations as do the zerosin the SISO case. See e.g. (Francis and Wonham 1975). IfP is square and invertible, then thetransmission zeros are the zeros of the polynomial

det

[A− sI B

C D

](4.26)

If in addition D is invertible, then

det

[A− sI B

C D

][I 0

−D−1C I

]= det

[(A− BD−1C)− sI B

0 D

]. (4.27)

So then the transmission zeros are the eigenvalues ofA − BD−1C. If D is not invertible thencomputation of transmission zeros is less straightforward. We may have to resort to computationof the Smith-McMillan form, but preferably we use methods based on state-space realizations.

2In fact here neitherxs0 nor us0 is zero.

164


Example 4.2.11 (Transmission zeros of the liquid flow system). Consider the Example 4.1.1with plant transfer matrix

P(s) =[ 1

s+11

s+11

s(s+1) − 1s+1

]= 1

s(s+ 1)

[s s1 −s

].

Elementary row and column operations successively applied to the polynomial part lead to theSmith form[

s s1 −s

](1)⇒[s+ 1 0

1 −s

](2)⇒[s+ 1 s(s+ 1)

1 0

](3)⇒[1 00 s(s+ 1)

].

The Smith-McMillan form hence is

M(s) = 1s(s+ 1)

[1 00 s(s+ 1)

]=[ 1

s(s+1) 00 1

].

The liquid flow system has no zeros but has two poles{0,−1}.We may compute the poles and zeros also from the state space representation ofP,

P(s) = C(sI − A)−1B+ D,

[A− sI B

C D

]=

−1− s 0 1 11 −s 0 −1

−1 0 0 00 −1 0 0

.

The realization is minimal because bothB and C are square and nonsingular. The poles aretherefore the eigenvalues ofA, which indeed are{0,−1} as before. There are no transmissionzeros because det

[A−sI B

C D

]is a nonzero constant (verify this). �

Example 4.2.12 (Transmission zeros via state space and transfer matrix representation).Consider the system with state space realization(A, B,C, D), with

A =0 0 0

1 0 00 1 0

, B =

0 1

1 00 0

, C =

[0 1 00 0 1

], D =

[0 00 0

].

The system’s transfer matrixP equals

P(s) = C(sI − A)−1B+ D =[

1/s 1/s2

1/s2 1/s3

]= 1

s3

[s2 ss 1

].

Elementary operations on the numerator polynomial matrix results in[s2 ss 1

]⇒[0 s0 1

]⇒[1 00 0

]

so the Smith-McMillan form ofP is

M(s) = 1s3

[1 00 0

]=[1/s3 0

0 0

].

The system therefore has three poles, all at zero:{0,0,0}. As the matrixA ∈ R3×3 has an

equal number of eigenvalues, it must be that the realization is minimal and that its eigenvalues

165


coincide with the poles of the system. From the Smith-McMillan form we see that there are notransmission zeros. This may also be verified from the matrix pencil

[A− sI B

C D

]=

−s 0 0 0 11 −s 0 1 00 1 −s 0 00 −1 0 0 00 0 −1 0 0

.

Elementary row and column operations do not affect the rank of this matrix pencil. Thereforethe rank equals that of

−s 0 0 0 11 −s 0 1 00 1 −s 0 00 −1 0 0 00 0 −1 0 0

⇒

−s 0 0 0 11 0 0 1 00 0 0 0 00 −1 0 0 00 0 −1 0 0

⇒

0 0 0 0 10 0 0 1 00 0 0 0 00 −1 0 0 00 0 −1 0 0

.

As shas disappeared from the matrix pencil it is direct that the rank of the matrix pencil does notdepend ons. The system hence has no transmission zeros. �

4.2.3. Numerical computation of transmission zeros

If a transfer matrixP is given, the numerical computation of transmission zeros is in general basedon a minimal state space realization ofP because reliable numerical algorithms for rational andpolynomial matrix operations are not yet numerically as reliable as state space methods. Webriefly discuss two approaches.

Algorithm 4.2.13 (Transmission zeros via high gain output feedback). The approach has beenproposed by Davison and Wang (1974), (1978). Let(A, B,C, D) be a minimal realization ofPand assume thatP is either left-invertible or right-invertible. Then:

1. Determine an arbitrary full rank output feedback matrixK ∈ Rnu×ny, e.g. using a random

number generator.

2. Determine the eigenvalues of the matrix

Zρ := A+ BK(1ρ

I − DK)−1C

for various large real values ofρ, e.g.ρ = 1010, . . . ,1020 at double precision computation.Now asρ goes to∞ we observe that some eigenvalues go to infinity will others converge(the so calledfiniteeigenvalues).

3. For square systems, the transmission zeros equal the finite eigenvalues ofZρ. For non-square systems, the transmission zeros equal the finite eigenvalues ofZρ for almostallchoices of the output feedback matrixK.

4. In cases of doubt, vary the values ofρ andK and repeat the calculations.�

See Section 4.6 for a sketch of the proof.

166

4.3. Norms of signals and systems

Algorithm 4.2.14 (Transmission zeros via generalized eigenvalues). The approach has beenproposed in (Laub and Moore 1978) and makes use of theQZ algorithm for solving the gen-eralized eigenvalue problem. Let(A, B,C, D) be a minimal realization ofP of ordern havingthe additional property that it is left-invertible (if the system is right-invertible, then use the dualsystem(AT,CT, BT, DT)). Then:

1. Define the matricesM andL as

M =[

In 00 0

], L =

[A B

−C D

].

2. Compute a solution to thegeneralized eigenvalueproblem i.e. determine all valuess ∈ C

andr ∈ Cn+nu satisfying

[sM− L]r = 0

3. The set offinitevalues fors are the transmission zeros of the system.

Although reliable numerical algorithms exist for the solution of generalized eigenvalue prob-lems, a major problem is to decide which values ofs belong to the finite transmission zeros andwhich values are to be considered as infinite. However, this problem is inherent in all numericalapproaches to transmission zero computation. �


For a SISO system with transfer functionH the interpretation of|H(jω)| as a “gain” from inputto output is clear, but how may we define “gain” for a MIMO system with a transfermatrix H?One way to do this is to use norms‖H(jω)‖ instead of absolute values. There are many differentnorms and in this section we review the most common norms for both signals and systems. Thetheory of norms leads to a re-assessment of stability.

4.3.1. Norms of vector-valued signals

We consider continuous-time signalsz defined on the time axisR. These signals may be scalar-valued (with values inR or C) or vector-valued (with values inRn or Cn). They may be addedand multiplied by real or complex numbers and, hence, are elements of what is known as avectorspace.

Given such a signalz, its norm is a mathematically well-defined notion that is a measure forthe “size” of the signal.

Definition 4.3.1 (Norms). Let X be a vector space over the real or complex numbers. Then afunction

‖ · ‖ : X → R (4.28)

that mapsX into the real numbersR is anorm if it satisfies the following properties:

1. ‖x‖ ≥ 0 for all x ∈ X (nonnegativity),

2. ‖x‖ = 0 if and only if x = 0 (positive-definiteness),

3. ‖λx‖ = |λ| · ‖x‖ for every scalarλ and allx ∈ X (homogeneity with respect to scaling),

167


4. ‖x + y‖ ≤ ‖x‖ + ‖y‖ for all x ∈ X andy ∈ X (triangle inequality).

The pair(X, ‖ · ‖) is called anormedvector space. If it is clear which norm is used,X by itselfis often called a normed vector space. �

A well-known norm is thep-norm of vectors inCn.

Example 4.3.2 (Norms of vectors in Cn). Suppose thatx = (x1, x2, · · · , xn) is ann-dimensional

complex-valued vector, that is,x is an element ofCn. Then for 1≤ p ≤ ∞ the p-normof x isdefined as

‖x‖p ={ (∑n

i=1 |xi|p)1/p

for 1 ≤ p<∞,

maxi=1,2,··· ,n |xi| for p = ∞.(4.29)

Well-known special cases are the norms

‖x‖1 =n∑

i=1

|xi|, ‖x‖2 =(

n∑i=1

|xi|2)1/2

, ‖x‖∞ = maxi=1,2,··· ,n

|xi|. (4.30)

‖x‖2 is the familiarEuclidean norm. �

The p-norm for constant vectors may easily be generalized to vector-valued signals.

Definition 4.3.3 ( Lp-norm of a signal). For any 1≤ p ≤ ∞ the p-normor Lp-norm‖z‖Lp of acontinuous-time scalar-valued signalz, is defined by

‖z‖Lp ={ (∫∞

−∞ |z(t)|p dt)1/p

for 1 ≤ p< ∞,

supt∈R |z(t)| for p = ∞.(4.31)

If z(t) is vector-valued with values inRn or Cn, this definition is generalized to

‖z‖Lp ={ (∫∞

−∞ ‖z(t)‖p dt)1/p

for 1 ≤ p<∞,

supt∈R ‖z(t)‖ for p = ∞,(4.32)

where‖ · ‖ is any norm on then-dimensional spaceRn or Cn. �

The signal norms that we mostly need are theL2-norm, defined as

‖z‖L2 =(∫ ∞

−∞‖z(t)‖2

2 dt

)1/2

(4.33)

and theL∞-norm, defined as

‖z‖L∞ = supt∈R

‖z(t)‖∞. (4.34)

The square‖z‖2L2

of the L2-norm is often called theenergyof the signalz, and theL∞-norm‖z‖L∞ its amplitudeor peak value.

Example 4.3.4 ( L2 and L∞-norm). Consider the signalz with two entries

z(t) =[

e−t1(t)

2e−3t1(t)

].

168


Here1(t) denotes the unit step, soz(t) is zero for negative time. Fromt = 0 onwards both entriesof z(t) decay to zero ast increases. Therefore

‖z‖L∞ =∥∥∥∥[12

]∥∥∥∥∞

= 2.

The square of theL2-norm follows as

‖z‖2L2

=∫ ∞

0e−2t + 4e−6t dt = e−2t

−2+ 4

e−6t

−6

∣∣∣∣t=∞

t=0= 1

2+ 4

6= 7

6

The energy ofz equals 7/6, itsL2-norm is√

7/6. �

4.3.2. Singular values of vectors and matrices

We review the notion of singular values of a matrix3. If A is ann× mcomplex-valued matrix thenthe matricesAH A andAAH are both nonnegative-definite Hermitian. As a result, the eigenvaluesof both AH A andAAH are all real and nonnegative. The min(n,m) largest eigenvalues

λi (AH A), λi (AAH), i = 1, 2, . . . , min(n,m), (4.35)

of AH A and AAH, respectively, ordered in decreasing magnitude, are equal. The remainingeigenvalues, if any, are zero. The square roots of these min(n,m) eigenvalues are called thesingular valuesof the matrixA, and are denoted

σi (A) = λ1/2i (AH A) = λ

1/2i (AAH), i = 1, 2, . . . , min(n,m). (4.36)

Obviously, they are nonnegative real numbers. The number ofnonzerosingular values equals therank of A.

Summary 4.3.5 (Singular value decomposition). Givenn × m matrix A let6 be the diagonaln × m matrix whose diagonal elements areσi(A), i = 1, 2, . . ., min(n,m). Then there existsquare unitary matricesU andV such that

A = U6VH. (4.37)

A (complex-valued) matrixU is unitary if UHU = UUH = I , with I a unit matrix of correctdimensions. The representation (4.37) is known as thesingular value decomposition (SVD)ofthe matrixA.

The largest and smallest singular valueσ1(A) andσmin(n,m)(A) respectively are commonlydenoted by an overbar and an underbar

σ(A) = σ1(A), σ(A) = σmin(n,m)(A).

Moreover, the largest singular valueσ(A) is a norm ofA. It is known as thespectral norm. �

There exist numerically reliable methods to compute the matricesU andV and the singularvalues. These methods are numerically more stable than any available for the computation ofeigenvalues.

3See for instance Section 10.8 of Noble (1969).

169


Example 4.3.6 (Singular value decomposition). The singular value decomposition of the 3× 1matrix A given by

A =0

34

(4.38)

is A = U6VH, where

U = 0 −0.6 −0.8

0.6 0.64 −0.480.8 −0.48 0.36

, 6 =

5

00

, V = 1. (4.39)

The matrixA has a single nonzero singular value (because it has rank 1), equal to 5. Hence, thespectral norm ofA is σ1(A) = 5. �

Exercise 4.3.7 (Singular value decomposition). Let A = U6VH be the singular value decom-position of then × m matrix A, with singular valuesσi , i = 1, 2, · · · , min(n,m). Denote thecolumns of then × n unitary matrixU asui, i = 1, 2, · · · , n, and those of them× m unitarymatrix V asvi , i = 1, 2,· · · , m. Prove the following statements:

1. For i = 1, 2, · · · , min(n,m) the column vectorui is an eigenvector ofAAH correspond-ing to the eigenvalueσ2

i . Any remaining columns are eigenvectors corresponding to theeigenvalue 0.

2. Similarly, for i = 1, 2, · · · , min(n,m) the column vectorvi is an eigenvector ofAH A cor-responding to the eigenvalueσ2

i . Any remaining columns are eigenvectors correspondingto the eigenvalue 0.

3. Fori = 1, 2,· · · , min(n,m) the vectorsui andvi satisfy

Avi = σiui, AHui = σivi . (4.40)�

Exercise 4.3.8 (Singular values). Given is a squaren × n matrix A. Prove the following prop-erties (compare p. R-5 of (Chiang and Safonov 1988)):

1. σ(A) = maxx∈Cn, x6=0‖Ax‖2‖x‖2

.

2. σ(A) = minx∈Cn, x6=0‖Ax‖2‖x‖2

.

3. σ(A) ≤ |λi(A)| ≤ σ(A), with λi(A) the ith eigenvalue ofA.

4. If A−1 exists thenσ(A) = 1/σ(A−1) andσ(A) = 1/σ(A−1).

5. σ(αA) = |α|σ(A), with α any complex number.

6. σ(A+ B) ≤ σ(A)+ σ(B).

7. σ(AB) ≤ σ(A)σ(B).

8. σ(A)− σ(B) ≤ σ(A+ B) ≤ σ(A)+ σ(B).

9. max(σ(A), σ(B)) ≤ σ([ A B]) ≤ √2max(σ(A), σ(B)).

10. maxi, j |Aij | ≤ σ(A) ≤ nmaxi, j |Aij |, with Aij the(i, j) element ofA.

11.∑n

i=1σ2i (A) = tr(AH A).

�

170


4.3.3. Norms of linear operators and systems

On the basis of normed signal space we may define norms of operators that act on these signals.

Definition 4.3.9 (Induced norm of a linear operator). Suppose thatφ is a linear mapφ : U →Y from a normed spaceU with norm‖ · ‖U to a normed spaceY with norm‖ · ‖Y . Then thenorm of the operatorφ induced by the norms‖ · ‖U and‖ · ‖Y is defined by

‖φ‖ = sup‖u‖U 6=0

‖φu‖Y

‖u‖U. (4.41)

�

Exercise 4.3.10 (Induced norms of linear operators). The space of linear operators from avector spaceU to a vector spaceY is itself a vector space.

1. Show that the induced norm‖ · ‖ as defined by (4.41) is indeed a norm on this spacesatisfying the properties of Definition 4.3.1.

Prove that this norm has the following additional properties:

2. If y = φu, then‖y‖Y ≤ ‖φ‖ · ‖u‖U for anyu ∈ U .

3. Submultiplicative property:Let φ1 : U → V andφ2 : V → Y be two linear operators,with U , V andY normed spaces. Then‖φ1φ2‖ ≤ ‖φ1‖ · ‖φ2‖, with all the norms induced.

�

Constant matrices represent linear maps, so that (4.41) may be used to define norms of ma-trices. The spectral normσ(M) is the norm induced by the 2-norm (see Exercise 4.3.8(1)).

Exercise 4.3.11 (Matrix norms induced by vector norms). Consider them× k complex-valuedmatrix M as a linear mapCk → Cm defined byy = Mu. Then depending on the norms definedonCk andCm we obtain different matrix norms.

1. Matrix norm induced by the 1-norm. Prove that the norm of the matrixM induced by the1-norm (both forU andY ) is themaximum absolute column sum

‖M‖ = maxj

∑i

|Mij |, (4.42)

with Mij the(i, j) entry of M.

2. Matrix norm induced by the∞-norm. Prove that the norm of the matrixM induced by the∞-norm (both forU andY ) is themaximum absolute row sum

‖M‖ = maxi

∑j

|Mij |, (4.43)

with Mij the(i, j) entry of M.

Prove these statements. �

171


u φ y

Figure 4.4.: Input-output mapping system

4.3.4. Norms of linear systems

We next turn to a discussion of the norm of a system. Consider a system as in Fig. 4.4, whichmaps the input signalu into an output signaly. Given an inputu, we denote the output of thesystem asy = φu. If φ is a linear operator the system is said to be linear. The norm of the systemis now defined as the norm of this operator.

We establish formulas for the norms of a linear time-invariant system induced by theL2- andL∞-norms of the input and output signals.

Summary 4.3.12 (Norms of linear time-invariant systems). Consider a MIMO convolutionsystem with impulse response matrixh, so that its IO mapy = φu is determined as

y(t) =∫ ∞

−∞h(τ)u(t − τ) dτ, t ∈ R. (4.44)

Moreover letH denote the transfer matrix, i.e.,H is the Laplace transform ofh.

1. L∞-induced norm.The norm of the system induced by theL2-norm (4.33) if it exists, isgiven by

maxi=1,2,··· ,m

∫ ∞

−∞

k∑j=1

|hi j (t)| dt, (4.45)

wherehi j is the(i, j) entry of them× k impulse response matrixh.

2. L2-induced norm.SupposeH is a rational matrix. The norm of the system induced by theL2-norm exists if and only ifH is proper and has no poles in the closed right-half plane.In that case theL2-induced norm (4.34) equals theH∞-normof the transfer matrix definedas

‖H‖H∞ = supω∈R

σ(H(jω)). (4.46)

�

A sketch of the proof is found in§ 4.6. For SISO systems the expressions for the two normsobtained in Summary 4.3.12 simplify considerably.

Summary 4.3.13 (Norms for SISO systems). The norms of a SISO system with (scalar) impulseresponseh and transfer functionH induced by theL∞- andL2-norms are successively given byactionof the impulse response

‖h‖L1 =∫ ∞

−∞|h(t)| dt, (4.47)

and the peak value on the Bode plotif H has no unstable poles


|H(jω)|. (4.48)

If H has unstable poles then the induced norms do not exist. �

172


Example 4.3.14 (Norms of a simple system). As a simple example, consider a SISO first-ordersystem with transfer function

H(s) = 11+ sθ

, (4.49)

with θ a positive constant. The corresponding impulse response is

h(t) ={

1θe−t/θ for t ≥ 0,

0 for t < 0.(4.50)

It is easily found that the norm of the system induced by theL∞-norm is

‖h‖1 =∫ ∞

0

1θ

e−t/θ dt = 1. (4.51)

The norm induced by theL2-norm follows as


1|1+ jωθ| = sup

ω∈R

1√1+ ω2θ2

= 1. (4.52)

For this example the two system norms are equal. Usually they are not. �

Exercise 4.3.15 ( H2-norm and Hankel norm of MIMO systems). Two further norms of lin-ear time-invariant systems are commonly encountered. Consider a stable MIMO system withtransfer matrixH and impulse response matrixh.

1. H2-norm.Show that theH2-norm defined as

‖H‖H2=√

tr

(∫ ∞

−∞HT(−j2π f )H(j2π f )d f

)=√

tr

(∫ ∞

−∞hT(t)h(t)dt

)(4.53)

is a norm. The notation tr indicates the trace of a matrix.

2. Hankel norm.The impulse response matrixh defines an operator

y(t) =∫ 0

−∞h(t − τ)u(τ) dτ, t ≥ 0. (4.54)

which maps continuous-time signalsu defined on the time axis(−∞,0] to continuous-time signalsy defined on the time axis [0,∞). The Hankel norm‖H‖H of the system withimpulse response matrixh is defined as the norm of the map given by (4.54) induced bythe L2-norms ofu : (−∞,0] → R

nu and y : [0,∞) → Rny. Prove that this is indeed a

norm.

3. Compute theH2-norm and the Hankel norm of the SISO system of Example 4.3.14.Hint:To compute the Hankel norm first show that ify satisfies (4.54) andh is given by (4.50)then‖y‖2

2 = 12θ (∫ 0−∞ eτ/θu(τ) dτ)2. From this, prove that‖H‖H = 1

2.�

Summary 4.3.16 (State-space computation of common system norms). Suppose a systemhas proper transfer matrixH and letH(s)= C(sIn − A)−1B+ D be a minimal realization ofH.Then

173


1. ‖H‖H2is finite if and only if D = 0 andA is astability matrix4. In that case

‖H‖H2=

√tr BTY B=

√tr CXCT

whereX andY are the solutions of the linear equations

AX+ X AT = −BBT, ATY + Y A= −CTC.

The solutionsX andY are unique and are respectively known as thecontrollability gramianandobservability gramian.

2. The Hankel norm‖H‖H is finite only if A is a stability matrix. In that case

‖H‖H =√λmax(XY)

whereX andY are the controllability gramian and observability gramian. The matrixXYhas real nonnegative eigenvalues only, andλmax(XY) denotes the largest of them.

3. TheH∞-norm‖H‖H∞ is finite if and only if A is a stability matrix. Thenγ ∈ R is an upperbound

‖H‖H∞ < γ

if and only if σ(D) < γ and the 2n× 2n matrix[A 0

−CTC −AT

]−[ −BCT D

](γ2 I − DT D)−1 [DTC BT

](4.55)

has no imaginary eigenvalues. Computation of theH∞-norm involves iteration onγ.

Proofs are listed in Appendix 4.6. �

4.4. BIBO and internal stability

Norms shed a new light on the notions of stability and internal stability. Consider the MIMOinput-output mapping system of Fig. 4.4 with linear input-output mapφ. In § 1.3.2 we definedthe system to be BIBO stable if any bounded inputu results in a bounded outputy = φu. Nowbounded means having finite norm, and so different norms may yield different versions of BIBOstability. Normally theL∞- or L2-norms ofu andy are used.

We review a few fairly obvious facts.

Summary 4.4.1 (BIBO stability).

1. If ‖φ‖ is finite, with‖ · ‖ the norm induced by the norms of the input and output signals,then the system is BIBO stable.

2. Suppose that the system is linear time-invariant with rational transfer matrixH. Then thesystem is BIBO stable (both when theL∞- and theL2-norms ofu andy are used) if andonly if H is proper and has all its poles in the open left-half complex plane.

�4A constant matrixA is astability matrixif it is square and all its eigenvalues have negative real part.

174

4.4. BIBO and internal stability

Exercise 4.4.2 (Proof).

1. Prove the statements 1 and 2.

2. Show by a counterexample that the converse of 1 is not true, that is, for certain systems andnorms the system may be BIBO stable in the sense as defined while‖φ‖ is not necessarilyfinite.

3. In the literature the following better though more complicated definition of BIBO stabilityis found: The system is BIBO stable if for every positive real constantN there exists apositive real constantM such that‖u‖ ≤ N implies ‖y‖ ≤ M. With this definition thesystem is BIBO stable if and only if‖φ‖ <∞. Prove this.

�

Given the above it will be no surprise that we say that a transfer matrix isstable if it isproper and has all its poles in the open left-half complex plane. In§ 1.3.2 we already definedthe notion ofinternal stability for MIMO interconnected systems: An interconnected systemis internally stable if the system obtained by adding external input and output signals in everyexposed interconnection is BIBO stable. Here we specialize this for the MIMO system of Fig. 4.5.

r e u yPK

Figure 4.5.: Multivariable feedback structure

r e u yPK

w

Figure 4.6.: Inputsr , w and outputse, u defining internal stability

Assume that feedback system shown in Fig. 4.5 ismultivariablei.e., u or y have more thanone entry. The closed loop of Fig. 4.5 is by definitioninternally stableif in the extended closedloop of Fig. 4.6 the maps from(w, r ) to (e,u) are BIBO stable. In terms of the transfer matrices,these maps are[

eu

]=

[I P

−K I

]−1[rw

](4.56)

=[

I − P( I + K P)−1K −P( I + K P)−1

( I + K P)−1K ( I + K P)−1

][rw

](4.57)

Necessary for internal stability is that

det( I + P(∞)K(∞)) 6= 0.

175


Feedback systems that satisfy this nonsingularity condition are said to bewell-posed, (Hsu andChen 1968). The following is a variation of Eqn. (1.44).

Lemma 4.4.3 (Internal stability). SupposeP and K are proper, having minimal realizations(AP, BP,CP, DP) and (AK, BK,CK, DK). Then the feedback system of Fig. 4.5 is internallystable if and only if det( I + DPDK) 6= 0 and the polynomial

det( I + P(s)K(s))det(sI − AP)det(sI − AK )

has all its zeros in the open left-half plane. �

If the ny × nu systemP is not square, then the dimensionny × ny of I + PK and dimensionnu × nu of I + K P differ. The fact that det( I + PK) = det( I + K P) allows to evaluate thestability of the system at the loop location having the lowest dimension.

Example 4.4.4 (Internal stability of closed-loop system). Consider the loop gain

L(s) =[

1s+1

1s−1

0 1s+1

].

The unity feedback aroundL gives as closed-loop characteristic polynomial:

det( I + L(s))det(sI − AL) =(

s+ 2s+ 1

)2

(s+ 1)2(s− 1) = (s+ 2)2(s− 1). (4.58)

The closed loop system hence is not internally stable. Evaluation of the sensitivity matrix

( I + L(s))−1 =[ s+1

s+2 − s+1(s+2)2

0 s+1s+2

]

shows that this transfer matrix in the closed-loop system is stable, but the complementary sensi-tivity matrix

( I + L(s))−1L(s) =[

1s+2

s2+2s+3(s−1)(s+2)2

0 1s+2

]

is unstable, confirming that the closed-loop system indeed is not internally stable. �

Example 4.4.5 (Internal stability and pole-zero cancellation). Consider

P(s) =[

1s

1s

2s+1

1s

], K(s) =

[1 2s

s−1

0 − 2ss−1

].

The unstable poles= 1 of the controller does not appear in the productPK,

P(s)K(s) =[ 1

s 02

s+12

s+1

].

This may suggest as in the SISO case that the map fromr to u will be unstable. Indeed thelower-left entry of this map is unstable,

K( I + PK)−1 =[ ∗ ∗− 4s2

(s−1)(s+1)(s+3) ∗].

176

4.5. MIMO structural requirements and design methods

Instability may also be verified using Lemma 4.4.3; it is readily verified that

det( I + P(s)K(s))det(sI − AP)det(sI − AK)

= det

[ s+1s 02

s+1s+3s+1

]s2(s+ 1)(s− 1) = s(s− 1)(s+ 1)(s+ 3)

and this has a zero ats= 1. �

Exercise 4.4.6 (Non-minimum phase loop gain). In Example 4.1.2onedecoupling precompen-sator is found that diagonalizes the loop gainPK with the property that both diagonal elementsof PK have a right half-plane zero ats = 1. Show thateveryinternally stabilizing decouplingcontrollerK has this property. �

K Pe u

v

zr

Figure 4.7.: Disturbance rejection for MIMO systems

Exercise 4.4.7 (Internal model control & MIMO disturbance rejection). Consider the MIMOsystem shown in Fig. 4.7 and suppose that the plantP is stable.

1. Show that a controller with transfer matrixK stabilizes if and only ifQ := K( I + PK)−1

is a stable transfer matrix.

2. ExpressK andS := ( I + PK)−1 in terms ofP andQ. Why is it useful to expressK andS in terms ofP andQ?

3. Let P be the 2× 2 systemP(s) = 1s+1

[1 s

−s 1

]and suppose thatv is a persistent harmonic

disturbancev(t) = [12

]eiω0t for some knownω0.

a) Assumeω0 6= ±1. Find a stabilizingK (using the parameterization by stableQ) suchthat in closed loop the effect ofv(t) on z(t) is asymptotically rejected (i.e.,e(t)→ 0ast → ∞ for r (t) = 0).

b) Does there exist such a stabilizingK if ω0 = ±1? (Explain.)�


During both the construction and the design of a control system it is mandatory that structuralrequirements of plant and controller and the control configuration are taken into account. Forexample, the choice of placement of the actuators and sensors may affect the number of right-half plane zeros of the plant and thereby may limit the closed loop performance, irrespectivewhich controller is used. In this section a number of such requirements will be identified. Theserequirements show up in the MIMO design methods that are discussed in this section.

177


4.5.1. Output controllability and functional reproducibility

Various concepts ofcontrollability are relevant for the design of control systems. A minimalrequirement for almost any control system, to be included in any control objective, is the re-quirement that the output of the system can be steered to any desired position in output space bymanipulating the input variables. This property is formalized in the concept ofoutput controlla-bility.

Summary 4.5.1 (Output controllability). A time-invariant linear system with inputu(t) andoutput y(t) is said to beoutput controllableif for any y1, y2 ∈ Rp there exists an inputu(t),t ∈ [t1, t2] with t1 < t2 that brings the output fromy(t1) = y1 to y(t2) = y2. In case the systemhas a strictly proper transfer matrix and is described by a state-space realization(A, B,C), thesystem isoutput controllableif and only if the constant matrix[

CB C AB C A2B . . . C An−1B]

(4.59)

has full row rank �

If in the above definitionC = In then we obtain the definition of state controllability. Theconcept of output controllability is generally weaker: a system(A, B,C)may be output control-lable and yet not be (state) controllable5. The property of output controllability is aninput-outputpropertyof the system while (state) controllability is not.

Note that the concept of output controllability only requires that the output can be given adesired value at each instant of time. A stronger requirement is to demand the output to be able tofollow any preassigned trajectory in time over a given time interval. A system capable of satisfy-ing this requirement is said to beoutput functional reproducibleor functional controllable. Func-tional controllability is a necessary requirement for output regulator and servo/tracking problems.Brockett and Mesarovic (1965) introduced the notion ofreproducibility, termedoutput control-lability by Rosenbrock (1970).

Summary 4.5.2 (Output functional reproducibility). A system having proper real-rationalny × nu transfer matrix is said to befunctionally reproducibleif rank P = ny. In particular forfunctionally reproducible it is necessary thatny ≤ nu. �

Example 4.5.3 (Output controllability and Functional reproducibility). Consider the lineartime-invariant state-space system of Example 4.2.12,

A =0 0 0

1 0 00 1 0

, B =

0 1

1 00 0

, C =

[0 1 00 0 1

]. (4.60)

This forms a minimal realization of the transfer matrix

P(s) = C(sI − A)−1B =[

1s

1s2

1s2

1s3

]. (4.61)

The system(A, B,C) is controllable and observable, andC has full rank. Thus the systemalso is output controllable. However, the rank ofP is 1. Thus the system isnot functionallyreproducible. �

5Here we assume thatC has full row rank, which is a natural assumption because otherwise the entries ofy are linearlydependent.

178


4.5.2. Decoupling control

A classical approach to multivariable control design consists of the design of a precompensatorthat brings the system transfer matrix to diagonal form, with subsequent design of the actual feed-back loops for the various single-input, single-output channels separately. This allows the tuningof individual controllers in separate feedback loops, and it is thought to provide an acceptablecontrol structure providing ease of survey for process operators and maintenance personnel. Thesubject of noninteracting or decoupling control as discussed in this section is based on the worksof Silverman (1970), Williams and Antsaklis (1986). The presentation follows that of Williamsand Antsaklis (1996).

In this section we investigate whether and how a squareP can be brought to diagonal form byapplying feedback and/or static precompensation. Suppose thatP has state-space representation

x(t) = Ax(t)+ Bu(t),

y(t) = Cx(t)+ Du(t)

with u andy having equally many entries,ny = nu = m, i.e., P(s)= C(sI − A)−1B+ D square.Assume in what follows thatP is invertible. The inverse transfer matrixP−1—necessarily arational matrix—may be viewed as a precompensator ofP that diagonalizes the series connectionPP−1. If the direct feedthrough termD = P(∞) of P is invertible, then the inverseP−1 is properand has realization

P−1(s) = −D−1C(sI − A+ BD−1C)−1BD−1 + D−1.

Exercise 4.5.4 (Realization of inverse). Verify this. �

If D is singular then the inverseP−1—assuming it exists—is not proper. In this case weproceed as follows. Define the indicesfi ≥ 0 (i = 1, . . .m) such thatD f defined as

D f (s) = diag(sf1, sf2, . . . , sfm) (4.62)

is such that

D := lim|s|→∞

D f (s)P(s) (4.63)

is defined and every row ofD has at least one nonzero entry. This identifies the indicesfi

uniquely. The fi equal the maximum of the relative degrees of the entries in theith row of P.Indeed, then by construction the largest relative degree in each row ofD f P is zero, and as aconsequence lim|s|→∞ D f (s)P(s) has a nonzero value at precisely the entries of relative degreezero. The indicesfi may also be determined using the realization ofP: If the ith row of D is notidentical to zero thenfi = 0, otherwise

fi = min{ k> 0 | row i of C Ak−1B is not identical to zero}.

It may be shown thatfi ≤ n, wheren is the state dimension. The indicesfi so defined are knownas thedecoupling indices.

Summary 4.5.5 (Decoupling state feedback). If D as defined in (4.63) is nonsingular, then theregular state feedback

u(t) = D−1(−C x(t)+ v(t))

179


where

C =

C1∗ Af1

C2∗ Af2

...Cm∗ Afm

renders the system with outputy and new inputv decoupled with diagonal transfer matrixD−1

f (s) = diag(s− f1, s− f2, . . . , s− fm). The compensated plant is shown in Fig. 4.8(b). �

HereCi∗ is denotes theith row of the matrixC. A proof is given in Appendix 4.6. Thedecoupled plantD−1

f has all its poles ats = 0 and it has no transmission zeros. We also know

that regular state-feedback does not change the zeros of the matrix[

A−sI BC D

]. This implies that

after state feedback all non-zero transmission zeros of the system are canceled by poles. In otherwords, after state feedback the realization has unobservable modes at the open loop transmissionzeros. This is undesirable ifP has unstable transmission zeros as it precludes closed loop stabilityusing output feedback. IfP has no unstable transmission zeros, then we may proceed with thedecoupled plantD−1

f and use (diagonal)K to further the design, see Fig. 4.8. A decoupled plant

(a) (b)

ee uu yy v

xC

D−1KK PP

Decoupled plantD−1f

Figure 4.8.: (a) closed loop with original plant, (b) with decoupled plant

that is stable may offer better perspectives for further performance enhancement by multiloopfeedback in individual channels. To obtain a stable decoupled plant we take instead of (4.62) aD f of the form

D f = diag(p1, . . . , pm)

where thepi are strictly Hurwitz6 if monic7 polynomials of degreefi . The formulae are nowmore messy, but the main results holds true also for this choice ofD f :

Summary 4.5.6 (Decoupling, stabilizing state feedback). Suppose the transfer matrixP ofthe plant is proper, squarem× m and invertible, and suppose it has no unstable transmissionzeros. Let(A, B,C, D) be a minimal realization ofP and let fi be the decoupling indices andsuppose thatpi are Hurwitz polynomials of degreefi . Then there is a regular state feedbacku(t) = Fx(t) + Wv(t) for which the loop fromv(t) to y(t) is decoupled. Its realization iscontrollable and detectable and has a stable transfer matrix diag( 1

p1, . . . , 1

pm). �

Example 4.5.7. (Diagonal decoupling by state feedback precompensation).Consider a plantwith transfer matrix

P(s) =[

1/s 1/s2

−1/s2 −1/s2

].

6A strictly Hurwitzpolynomial is a polynomial whose zeros have strictly negative real part.7A monicpolynomial is a polynomial whose highest degree coefficient equals 1.

180


A minimal realization ofP is

[A BC D

]=

0 0 0 0 0 11 0 0 0 1 00 0 0 0 1 −10 0 1 0 0 00 1 0 0 0 00 0 0 1 0 0

.

Its transmission zeros follow after a sequence of elementary row and column operations

P(s) = 1s2

[s 11 −1

]⇒ 1

s2

[s+ 1 1

0 1

]⇒ 1

s2

[1 00 s+ 1

]=[1/s2 0

0 (s+ 1)/s2

].

Thus P has one transmission zeros ats = −1. It is a stable zero hence we may proceed. Thedirect feedthrough termD is the zero matrix so the decoupling indicesfi are all greater thanzero. We need to computeCi∗ Ak−1B.

C1∗ B = [0 1 0 0

]

0 11 01 −10 0

= [

1 0]

it is not identically zero so thatf1 = 1. Likewise

C2∗ B = [0 0 0 1

]

0 11 01 −10 0

= [

0 0].

It is zero hence we must computeC2∗ AB,

C2∗ AB= [1 −1

].

As it is nonzero we havef2 = 2. This gives us

D =[

C1∗ BC2∗ AB

]=[1 01 −1

].

The matrixD is nonsingular so a decoupling state feedback exists. Take

D f (s) =[s+ 1 0

0 (s+ 1)2

].

Its diagonal entries have the degreesf1 = 1 and f2 = 2 as required. A realization of

D f (s)P(s) =[(s+ 1)/s (s+ 1)/s2

(s+ 1)2/s2 −(s+ 1)2/s2

]

is (A, B,C ,D ) with

C =[1 1 0 00 0 2 1

], D =

[1 01 −1

].

181


Now the regular state feedback

u(t) =[1 01 −1

]−1(−[1 1 0 00 0 2 1

]x(t)+ v(t)

)renders the system with inputv and outputy, decoupled. After applying this state feedback wearrive at the state-space system

x(t) = A+ BD−1(−C x(t)+ v(t)) =

−1 −1 −2 10 −1 0 00 0 −2 −10 0 1 0

x(t)+

1 −11 00 10 0

v(t)

y(t) = Cx(t)+ Dv(t) =[0 1 0 00 0 0 1

]x(t)

(4.64)

and its transfer matrix by construction is

D−1f (s) =

[1/(s+ 1) 0

0 1/(s+ 1)2

].

Note that a minimal realization ofD−1f has order 3 whereas the realization (4.64) has order 4.

Therefore (4.64) is not minimal. This is typical for decoupling procedures. �

Exercise 4.5.8 (An invertible system that is not decouplable by state feedback). Considerthe plant with transfer matrix

P(s) =[1/(s+ 1) 1/s1/(s− 1) 1/s

].

1. Find the transmission zeros.

2. Show that the methods of Summary 4.5.5 and Summary 4.5.6 are not applicable here.

3. Find a stable precompensatorK0 that renders the productPK0 diagonal.�

4.5.3. Directional properties of gains

In contrast to a SISO system, a MIMO system generally does not have a unique gain. A trivialexample is the 2× 2 system with constant transfer matrix

P(s) =[1 00 2

].

The gain is in between 1 and 2 depending on the direction of the input.There are various ways to define gains for MIMO systems. A natural generalization of the

SISO gain|y(jω)|/|u(jω)| from inputu(jω) to outputy(jω) is to use norms instead of merelyabsolute values. We take the 2-norm.

Summary 4.5.9 (Bounds on gains). For any given inputu and fixed frequencyω there holds

σ(P(jω)) ≤ ‖P(jω)u(jω)‖2

‖u(jω)‖2≤ σ(P(jω)) (4.65)

and the lower boundσ(P(jω)) and upper boundσ(P(jω)) are achieved for certain inputsu. �

182


10-2

10-1

100

101

102

10-4

10-3

10-2

10-1

100

101

Figure 4.9.: Lower and upper bound on the gain ofT(jω)

Example 4.5.10 (Upper and lower bounds on gains). Consider the plant with transfer matrix

P(s) =[

1s

1s2+√

2s+1

− 12(s+1)

1s

].

In unity feedback the complementary sensitivity matrixT = P( I + P)−1 has dimension 2× 2.So at each frequencyω the matrixT(jω) has two singular values. Figure 4.9 shows the twosingular valuesσ(jω) andσ(jω) as a function of frequency. Near the crossover frequency thesingular values differ considerably. �

Let

P(jω) = Y(jω)6(jω)U∗(jω) (4.66)

be an SVD (at each frequency) ofP(jω), that is,Y(jω) andU(jω) are unitary and

6(jω) = diag(σ1(jω), σ2(jω), . . . , σmin(ny,nu)(jω))

with

σ1(jω) ≥ σ2(jω) ≥ . . . ≥ σmin(ny,nu)(jω) ≥ 0.

The columnsuk(jω) of U(jω) theinput principal directions, (Postlethwaite, Edmunds, and Mac-Farlane 1981) and they have the special property that their gains are precisely the correspondingsingular values

‖P(jω)uk(jω)‖2

‖uk(jω)‖2= σk(jω)

and the responseyk(jω) = P(jω)uk(jω) in fact equalsσk(jω) times thekth column ofY(jω).The columns ofY(jω) are theoutput principal directionsand theσk(jω) theprincipal gains.

Thecondition numberκ defined as

κ(P(jω)) = σ(P(jω))σ(P(jω))

(4.67)

183


is a possible measure of the difficulty of controlling the system. If for a plant all principal gainsare the same, such as whenP(jω) is unitary, thenκ(jω) = 1. If κ(jω) � 1 then the loop gainL = PK around the crossover frequencymaybe difficult to shape. Note that the condition numberis not invariant under scaling. Describing the same system in terms of a different set of physicalunits leads to a different condition number. A high condition number indicates that the system is“close” to losing its full rank, i.e. close to not satisfying the property of functional controllability.

Example 4.5.11 (Multivariable control system—design goals). The performance of a multi-variable feedback system can be assessed using the notion of principal gains as follows. Thebasic idea follows the ideas previously discussed for single input, single output systems.

The disturbance rejecting properties of the loop transfer require the sensitivity matrix to besmall:

σ(S(jω))� 1.

At the same time the restriction of the propagation of measurement noise requires

σ(T(jω))� 1.

As in the SISO case these two requirements are conflicting. Indeed, using the triangle inequalitywe have the bounds

|1− σ(S(jω))| ≤ σ(T(jω)) ≤ 1+ σ(S(jω)) (4.68)

and

|1− σ(T(jω))| ≤ σ(S(jω)) ≤ 1+ σ(T(jω)). (4.69)

It is useful to try to make the difference betweenσ(T) andσ(T), and betweenσ(S) andσ(S),not too large in the cross over region. Making them equal would imply that the whole plantbehaves identical in all directions and this is in general not possible.

We must also be concerned to make the control inputs not too large. Thus the transfer matrix( I + K P)−1K from r to u must not be too large.

‖u(jω)‖2

‖r (jω)‖2= ‖( I + K(jω)P(jω))−1K(jω)r (jω)‖2

‖r (jω)‖2

≤ σ(( I + K(jω)P(jω))−1K(jω))

≤ σ(( I + K(jω)P(jω))−1)σ(K(jω))

≤ σ(K(jω))σ( I + K(jω)P(jω))

≤ σ(K(jω))1− σ(K(jω)P(jω))

≤ σ(K(jω))1− σ(K(jω))σ(P(jω))

. (4.70)

Here several properties of Exercise 4.3.8 are used and in the last two inequalities it is assumedthatσ(K(jω))σ(P(jω)) < 1. The upperbound shows that whereσ(P(jω)) is not too large thatσ(K(jω))� 1 guarantees a small gain fromr (jω) to u(jω). The upper bound (4.70) of this gainis easily determined numerically.

184


The direction of an important disturbance can be taken into consideration to advantage. Letv(jω) be a disturbance acting additively on the output of the system. Complete rejection of thedisturbance would require an input

u(jω) = −P−1(jω)v(jω). (4.71)

Thus

‖u(jω)‖2

‖v(jω)‖2= ‖P−1(jω)v(jω)‖2

‖v(jω)‖2(4.72)

measures the magnitude ofu needed to reject a unit magnitude disturbance acting in the directionv. The most and least favorable disturbance directions are those for whichv is into the directionof the principal output directions corresponding toσ(P(jω)) andσ(P(jω)), respectively. Thisleads to thedisturbance condition numberof the plant,

κv(P(jω)) = ‖P−1(jω)v(jω)‖2

‖v(jω)‖2σ(P(jω)) (4.73)

which measures the input needed to reject the disturbancev relative to the input needed to rejectthe disturbance acting in the most favorable direction. It follows that

1 ≤ κv(P) ≤ κ(P) (4.74)

andκv(P) generalizes the notion of condition number of the plant defined in (4.67). �

4.5.4. Decentralized control structures

Dual to decoupling control where the aim is make the plantP diagonal, isdecentralized controlwhere it is the controllerK that is restricted to be diagonal or block diagonal,

K = diag{Ki} =

K1 0K2

...0 Km

. (4.75)

This control structure is calledmultiloop controland it assumes that there as many control inputsas control outputs. In a multiloop control structure it is in general desirable to make an orderingof the input and output variables such that the interaction between the various control loops is assmall as possible. This is theinput-output pairing problem.

Example 4.5.12 (Relative gain of a two-input-two-output system). Consider a two-input-two-output system[

y1

y2

]= P

[u1

u2

]

and suppose we decide to close the loop by a diagonal controlleruj = Kj(r j − yj ), that is, thefirst component ofu is only controlled by the first component ofy, and similarlyu2 is controlledonly by y2. Suppose we leave the second control loop open for the moment and that we vary thefirst control loop. If the open loop gain fromu2 to y2 does not change a lot as we vary the firstloop u1 = K1(r1 − y1), it is save to say that we can design the second control loop independent

185


from the first, or to put it differently, that the second control loop is insensitive to tuning of thefirst. This is desirable. In summary, we want the ratio

λ := gain fromu2 to y2 if first loop is opengain fromu2 to y2 if first loop is closed

(4.76)

preferably close to 1. To make thisλ more explicit we assume now that the reference signalris constant and that all signals have settled to their constant steady state values. Now if the firstloop is open then the first input entryu1 is zero (or constant). However if the first loop isclosedthen—assuming perfect control—it is the outputy1 that is constant (equal tor1). This allows usto express (4.76) as

λ =dy2/du2

∣∣u1

dy2/du2

∣∣y1

where∣∣u1

expresses thatu1 is considered constant in the differentiation. This expression forλ

exists and may be computed ifP(0) is invertible:

dy2

du2

∣∣∣∣u1

= d P21(0)u1 + P22(0)u2

du2

∣∣∣∣u1

= P22(0)

and becauseu = P−1(0)y we also have

dy2

du2

∣∣∣∣y1

= 1du2/dy2

∣∣∣∣y1

= 1d P−1

21 (0)y1+P−122 (0)y2

dy2

∣∣∣∣y1

= 1

P−122 (0)

= P−122 (0).

The relative gainλ hence equalsP22(0)P−122 (0). �

For general square MIMO systemsP we want to consider therelative gain array (RGA)which is the (rational matrix)3 defined element-wise as

3i j = Pij P−1ji

or, equivalently as a matrix, as

3 = P◦ (P−1)T (4.77)

where◦ denotes theHadamard productwhich is the entry-wise product of two matrices of thesame dimensions.

Summary 4.5.13 (Properties of the RGA).

1. In constant steady state there holds that

3i j (0) =dyi

duj

∣∣all loops(uk, yk), k 6= j open

dyi

duj

∣∣all loops(uk, yk), k 6= j closed

2. The sum of the elements of each row or each column of3 is 1

3. Any permutation of rows and columns inP results in the same permutations of rows andcolumns in3

186


4. 3 is invariant under diagonal input and output scaling

5. If P is diagonal or upper or lower triangular then3 = Im

6. The normσ(3(jω)) of the RGA is believed to be closely related to the minimized con-dition number ofP(jω) under diagonal scaling, and thus serves as an indicator for badlyconditioned systems. See (Nett and Manousiouthakis 1987), (Grosdidier, Morari, and Holt1985), and (Skogestad and Morari 1987).

7. The RGA measures the sensitivity ofP to relative element-by-element uncertainty. Itindicates thatP becomes singular by an element perturbation fromPij to Pij [1 − λ−1

i j ].

If 3 deviates a lot fromIm then this is an indication that interaction is present in the system.�

Example 4.5.14 (Relative Gain Array for 2× 2 systems). For the two-input, two-output case

P =[

P11 P12

P21 P22

], (P−1)T = 1

P11P22 − P21P12

[P22 −P21

−P12 P11

],

the relative gain array3 = P ◦ P−T equals

3 =[λ 1− λ

1− λ λ

], λ := P11P22

P11P22 − P21P12.

We immediate see that rows and columns add up to one, as claimed. Some interesting specialcases are

P =[

P11 00 P22

]⇒ 3 =

[1 00 1

]

P =[

P11 0P21 P22

]⇒ 3 =

[1 00 1

]

P =[

0 P12

P21 0

]⇒ 3 =

[0 11 0

]

P =[

p p−p p

]⇒ 3 =

[0.5 0.50.5 0.5

]In the first and second example, the relative gain suggests to pairu1 with y1, andu2 with y1. Inthe first example this pairing is also direct from the fact thatP is diagonal. In the third examplethe RGA isantidiagonal, so we want to pairu1 with y2 andu2 with y1. In the fourth example allentries of the RGA are the same, hence no pairing can be deduced from the RGA.

If P is close to singular then several entries of3 are large. The corresponding pairings are tobe avoided, and possibly no sensible pairing exists. For example

P =[a aa a(1+ δ)

]⇒ 3 =

[1+δδ

−1δ−1

δ1+δδ

]

with δ ≈ 0. �

Although the RGA has been derived originally by Bristol (1966) for the evaluation ofP(s)at steady-states = 0 assuming stable steady-state behavior, the RGA may be useful over thecomplete frequency range. The followingi/o pairing ruleshave been in use on the basis of anheuristic understanding of the properties of the RGA:

187


1. Prefer those pairing selections where3(jω) is close to unity around the crossover fre-quency region. This prevents undesirable stability interaction with other loops.

2. Avoid pairings with negative steady-state or low-frequency values ofλii (jω) on the diago-nal.

The following result by Hovd and Skogestad (1992) makes more precise the statement in (Bris-tol 1966) relating negative entries in the RGA, and nonminimum-phase behavior. See also thecounterexample against Bristol’s claim in (Grosdidier and Morari 1987).

Summary 4.5.15 (Relative Gain Array and RHP transmission zeros). Suppose thatP hasstable elements having no poles nor zeros ats= 0. Assume that the entries of3 are nonzero andfinite for s→ ∞. If 3i j shows different signs when evaluated ats = 0 ands = ∞, then at leastone of the following statements holds:

1. The entryPij has a zero in the right half plane

2. P has a transmission zero in the right half plane

3. The subsystem ofP with input j and outputi removed has a transmission zero in the righthalf plane

�

As decentralized or multiloop control structures are widely applied in industrial practice, arequirement arises regarding the ability to take one loop out of operation without destabilizingthe other loops. In a problem setting employing integral feedback, this leads to the notion ofdecentralized integral controllability (Campo and Morari 1994).

Definition 4.5.16 (Decentralized integral controllability (DIC)). The systemP is DIC if thereexists a decentralized (multiloop) controller having integral action in each loop such that thefeedback system is stable and remains stable when each loopi is individually detuned by a factorεi for 0 ≤ εi ≤ 1. �

The definition of DIC implies that the systemP must be open-loop stable. It is also assumedthat the integrator in the control loop is put out of order whenεi = 0 in loop i. The steady-stateRGA provides a tool for testing on DIC. The following result has been shown by Grosdidier,Morari, and Holt (1985).

Summary 4.5.17 (Steady-state RGA and stability). Let the plantP be stable and square, andconsider a multiloop controllerK with diagonal transfer matrix having integral action in eachdiagonal element. Assume further that the loop gainPK is strictly proper. If the RGA3(s)of the plant contains a negative diagonal value fors= 0 then the closed-loop system satisfies atleast one of the following properties:

1. The overall closed-loop system is unstable

2. The loop with the negative steady-state relative gain is unstable by itself

3. The closed-loop system is unstable if the loop corresponding to the negative steady-staterelative gain is opened

�

A further result regarding stability under multivariable integral feedback has been providedby Lunze (1985); see also (Grosdidier, Morari, and Holt 1985), and (Morari 1985).

188


Summary 4.5.18 (Steady-state gain and DIC). Consider the closed loop system consisting ofthe open-loop stable plantP and the controllerK having transfer matrix

K(s) := α

sKss (4.78)

Assume unity feedback around the loop and assume that the summing junction in the loop in-troduces a minus sign in the loop. Then a necessary condition for the closed loop system to bestable for allα in an interval 0≤ α ≤ α with α > 0 is that the matrixP(0)Kss must have all itseigenvalues in the open right half plane. �

Further results on decentralized i/o pairing can be found in (Skogestad and Morari 1992),and in (Hovd and Skogestad 1994). The inherent control limitations of decentralized controlhave been studied by Skogestad, Hovd, and Lundström (1991). Further results on decentralizedintegral controllability can be found in (Le, Nwokah, and Frazho 1991) and in (Nwokah, Frazho,and Le 1993).

4.5.5. Internal model principle and the servomechanism problem

Theservomechanism problemis to design a controller – calledservocompensator– that rendersthe plant output as insensitive as possible to disturbances while at the same asymptotically trackcertain reference inputsr . We discuss a state space solution to this problem; a solution that worksfor SISO as well as MIMO systems. Consider a plant with state space realization

x(t) = Ax(t)+ Bu(t) u(t) ∈ Rnu, y ∈ R

ny, x(0) = x0 ∈ Rn

y(t) = Cx(t)+ v(t) (4.79)

and assume thatv is a disturbance that is generated by a dynamic system having all of its poleson the imaginary axis

xv(t) = Avxv(t), xv(0) = xv0v(t) = Cvxv(t). (4.80)

The disturbance ispersistentbecause the eigenvalues fofAv are assumed to lie on the imaginaryaxis. Without loss of generality we further assume that(Av,Cv) is an observable pair. Then apossible choice of theservocompensatoris the compensator (controller) with realization

xs(t) = Asxs(t)+ Bse(t), xs(0) = xs0

e(t) = r (t)− y(t) (4.81)

whereAs is block-diagonal withny blocks

As = diag(Av, Av, . . . , Av) (4.82)

and whereBs is of compatible dimensions and such that(As, Bs) is controllable, but otherwisearbitrary. The composite system, consisting of the plant (4.79) and the servocompensator (4.81)yields[

x(t)xs(t)

]=

[A 0

−BsC As

][x(t)xs(t)

]+[

B0

]u(t)+

[0Bs

](r (t)− v(t))

y(t) = [C 0

] [ x(t)xs(t)

]+ v(t).

189


As its stands the closed loop is not stable, but it may be stabilized by an appropriate state feedback

u(t) = Fx(t)+ Fsxs(t) (4.83)

whereF and Fs are chosen by any method for designing stabilizing state feedback, such as LQtheory. The resulting closed loop is shown in Fig. 4.10.

e xs

x

K P

F

Fsu y

v

r

Figure 4.10.: Servomechanism configuration

Summary 4.5.19 (Servocompensator). Suppose that the following conditions are satisfied.

1. (A, B) is stabilizable and(A,C) is detectable,

2. nu ≥ ny, i.e. there are at least as many inputs as there are outputs

3. rank[

A−λi BC D

] = n+ ny for every eigenvalueλi of Av.

ThenF, Fs can be selected such that the closed loop system matrix[A+ BF BFs

−BsC As

](4.84)

has all its eigenvalues in the open left-half plane. In that case the controller (4.81),(4.83) is aservocompensator in thate(t)→ 0 ast → ∞ for any of the persistentv that satisfy (4.80). �

The system matrixAs of the servocompensator contains several copies of the system matricesAv that definesv. This means that the servocompensator contains the mechanism that generatesv; it is an example of the more generalinternal model principle. The conditions as mentionedimply the following, see (Davison 1996) and (Desoer and Wang 1980).

• No transmission zero of the system should coincide with one of the imaginary axis polesof the disturbance model

• The system should be functionally controllable

• The variablesr andv occur in the equations in a completely interchangeable role, so thatthe disturbance model used forv can also be utilized forr . That is, for anyr that satisfies

xv(t) = Avxv(t), r (t) = Cr xv(t)

the error signale(t) converges to zero. Hence the outputy(t) approachesr (t) ast → ∞.

• Asymptotically, whene(t) is zero, the disturbancev(t) is still present and the servocom-pensator acts in an open loop fashion as the autonomous generator of a compensatingdisturbance at the outputy(t), of equal form asv(t) but of opposite sign.

190


• If the state of the system is not available for feedback, then the use of a state estimator ispossible without altering the essentials of the preceding results.

Example 4.5.20 (Integral action). If the servocompensator is taken to be an integratorxs = ethen constant disturbancesv are asymptotically rejected and constant reference inputsr asymp-totically tracked. We shall verify the conditions of Summary 4.5.19 for this case. As before theplant P is assumed strictly proper with state space realization and corrupted with noisev,

x(t) = Ax(t)+ Bu(t), y(t) = Cx(t)+ v(t).

Combined with the integrating action of the servocompensatorxs(t) = e(t), with e(t) = r (t)−y(t), we obtain[

x(t)xs(t)

]=

[A 00 0

][x(t)xs(t)

]+[

B0

]u(t)+

[0I

]e(t)

=[

A 0−C 0

][xxs

]+[

B0

]u(t)+

[0I

](r (t)− v(t)).

Closing the loop with state feedbacku(t) = Fx(t)+ Fsxs(t) renders the closed loop state spacerepresentation[

x(t)xs(t)

]=

[A+ BF BFs

−C 0

][x(t)xs(t)

]+[0I

](r (t)− v(t))

y(t) = [C 0

] [ x(t)xs(t)

]+ v(t).

All closed-loop poles can be arbitrarily assigned by state feedbacku(t)= Fx(t)+ Fsxs(t) if andonly if

([A 0C 0

],[

B0

])is controllable. This is the case if and only if[

B AB A2B · · ·0 CB C AB · · ·

]

has full row rank. The above matrix may be decomposed as[A BC 0

][0 B AB · · ·Im 0 0 · · ·

]

and therefore has full row rank if and only if

1. (A, B) is controllable,

2. nu ≥ ny,

3. rank

[A BC 0

]= n+ ny.

(In fact the third condition implies the second.) The three conditions are exactly what Sum-mary 4.5.19 states. If the full statex is not available for feedback then we may replace thefeedback lawu(t) = Fx(t)+ Fsxs(t) by an approximating observer,

˙x(t) = (A− KC)x(t)+ Bu(t)+ Ky(t)

u(t) = Fx(t)+ Fsxs(t).

�

191


4.6. Appendix: Proofs and Derivations

Derivation of process dynamic model of the two-tank liquid flow process.A dynamic processmodel for the liquid flow process of figure 4.1 is derived under the following assumptions:

• The liquid is incompressible;

• The pressure in both vessels is atmospheric;

• The flowφ2 depends instantaneously on the static pressure defined by the liquid levelH1;

• The flowφ4 is the instantaneous result of a flow-controlled recycle pump;

• φ3 is determined by a valve or pump outside the system boundary of the process underconsideration, independent of the variables under consideration in the model.

Let ρ denote density of the liquid expressed in kg/m3, and let A1 and A2 denote the cross-sectional areas (in m2) of each vessel, then the mass balances over each vessel and a relation foroutflow are:

A1ρh1(t) = ρ[φ1(t)+ φ4(t)− φ2(t)] (4.85)

A2ρh2(t) = ρ[φ2(t)− φ4(t)− φ3(t)] (4.86)

The flow rateφ2(t) of the first tank is assumed to be a function of the levelh1 of this tank. Thecommon model is that

φ2(t) = k√

h1(t). (4.87)

wherek is a valve dependent constant. Next we linearize these equations around an assumedequilibrium state(hk,0, φi,0). Define the deviations from the equilibrium values with tildes, thatis,

φi(t) = φi,0 + φi (t), i = 1 . . .4

hk(t) = hk,0 + hk(t), k = 1,2. (4.88)

Linearization of equation (4.87) around(hk,0, φi,0) gives

φ2(t) = k√h1,0

12

˙h1(t) = φ2,0

2h1,0

˙h1(t). (4.89)

Inserting this relation into equations (4.85), (4.86) yields the linearized equations[ ˙h1(t)˙h2(t)

]=[− φ2,0

2A1h1,00

φ2,0

2A1h1,00

][h1(t)h2(t)

]+[ 1

A1

1A1

0 − 1A2

][φ1(t)φ4(t)

]+[

0− 1

A2

]φ3(t) (4.90)

Assuming thatφ4,0 = φ1,0, and thusφ2,0 = 2φ1,0, and taking numerical valuesφ1,0 = 1, A1 =A2 = 1, andh1,0 = 1 leads to Equation (4.1).

Proof of Algorithm 4.2.13.We only prove the square case, i.e., whereD andK are sqaure. Asρgoes to infinity, the zeross of

A− sI B 0C D I0 1

ρI K

192


converge to the transmission zeros. Now apply the following two elementary operations to theabove matrix: subtractρK times the second column from the third column and then interchangethese columns. Then we get the matrix

A− sI −ρBK BC I − ρDK D0 0 1

ρI

.

In this form it is clear that its zeros are the zeros of[

A−sI −rhoBC I−ρDK

]. However, since[

A− sI −ρBKC I − ρDK

][I 0

−( I − ρDK)−1C I

]︸︷︷︸

nonsingular

=[

A+ BK( 1ρ

I − DK)−1C − sI ρB0 I − ρDK

]

we see that these zeros are simply the eigenvalues ofA+ BK( 1ρ

I − DK)−1C.

Proof of Summary 4.3.16.

1. If D is not zero then the impulse response matrixh(t) = CeAtB1(t) + δ(t)D containsDirac delta functions and as a result theH2-norm can not be finite. SoD = 0 and

h(t) = CeAtB1(t).

If A is not a stability matrix thenh(t) is unbounded, hence theH2-norm is infinite. SoD = 0 andA is a stability matrix. Then

‖H‖2H2

= tr∫ ∞

−∞h(t)Th(t)dt

= tr∫ ∞

0BTeAT tCTCeAtB dt

= tr(BT

∫ ∞

0eAT tCTCeAt dt︸︷︷︸

Y

B).

The so defined matrixY satisfies

ATY+ Y A=∫ ∞

0ATeAT tCTCeAt + eAT tCTCeAt A dt= eAT tCTCeAt

∣∣t=∞t=0 = −CTC. (4.91)

That is, Y satisfies the Lyapunov equationATY + Y A= −CTC, and asA is a stabilitymatrix the solutionY of (4.91) is well known to be unique.

2. Let X andY be the controllability and observability gramians. By the property of state, theoutputy for positive time is a function of the statex0 at t = 0. Then∫ ∞

0y(t)T y(t)dt =

∫ ∞

0xT

0 eAT tCTCeAtx0 dt = xT0Yx0.

If X satisfiesAX + X AT = −BBT then its inverseZ := X−1 satisfiesZ A + AT Z =−Z BBT Z. By completion of the square we may write

ddt

xT Zx = 2xT Zx = 2xT(Z Ax+ Z Bu) = xT(Z A+ AT Z)x + 2xT Z Bu

= xT(−Z BBTZ)xT + 2xT Z Bu

= uTu− (u− BT Zx)T(u− BT Zx).

193


Therefore, assumingx(−∞) = 0,∫ 0

−∞‖u(t)‖2

2 − ‖u(t)− BT Zx(t)‖22 dt = xT(t)Zx(t)

∣∣t=0t=−∞ = xT

0 Zx0 = xT0 X−1x0.

From this expression it follows that the smallestu (in norm) that steersx from x(−∞)= 0to x(0) = x0 is u(t) = BT Zx(t) (verify this). This then shows that

supu

∫∞0 yT(t)y(t)dt∫ 0−∞ uT(t)u(t)dt

= xT0Yx0

xT0 X−1x0

.

Now this supremum is less then someγ2 for somex0 if and only if

xT0Yx0 < γ

2 · xT0 X−1x0.

There exist suchx0 iff Y < γ2X−1, which in turn is equivalent to thatλi(Y X) =λi (X1/2Y X1/2) < γ2. This proves the result.

3. ‖H‖H∞ < γ holds if and only ifH is stable andγ2 I − H∼ H is positive definite on jR∪∞.This is the case iff it is positive definite at infinity (i.e.,σ(D) < γ, i.e.γ2 I − DT D > 0) andnowhere on the imaginary axisγ2 I − H∼ H is singular. We will show thatγ2 I − H∼ H hasimaginary zeros iff (4.55) has imaginary eigenvalues. It is readily verified that a realizationof γ2 I − H∼ H is

A− sI 0 −B−CTC −AT − sI CT DDTC BT γ2 I − DT D

By the Schur complement results applied to this matrix we see that

det

[A− sI 0−CTC −AT − sI

]· det(γ2I − H∼ H) = det(γ2 I − DT D) · det(Aham− sI).

whereAham is the Hamiltonian matrix (4.55). AsA has no imaginary eigenvalues we havethatγ2 I − H∼ H has no zeros on the imaginary axis iffAhamhas no imaginary eigenvalues.

Proof of Summary 4.5.5.We make use of the fact that if

x(t) = Ax(t)+ Bu(t), x(0) = 0,

y(t) = Cx(t)

then the derivativey(t) satisfies

y(t) = Cx(t) = C Ax(t)+ CBu(t)

The matrixD f (s) = diag(sf1, . . . , sfm) is a diagonal matrix ofdifferentiators. Since the planty = Pusatisfies

x(t) = Ax(t)+ Bu(t), x(0) = 0,

y(t) = Cx(t)+ Du(t) (4.92)

194


it follows that the component-wise differentiatedv defined asv := D f y satisfies

x(t) = Ax(t)+ Bu(t), x(0) = 0,

v(t) =

d f1

dt f1y1(t)

d f2

dt f2y2(t)...

d fm

dt fm ym(t)

=

C1∗ Af1

C2∗ Af2

...CmAfm

︸︷︷︸C

x(t)+ Du(t)

By assumptionD is nonsingular, sou(t) = D−1(−C x(t) + v(t)). Inserting this in the staterealization (4.92) yields

x(t) = (A− BD−1C )x(t)+ BD−1v(t), x(0) = 0,

y(t) = (C − DD−1C )x(t)+ DD−1v(t)

Sincev = D f y it must be that the above is a realization of theD−1f .

Proof of Lemma 4.2.8 (sketch).We only prove it for the case that none of the transmission zeross0 are poles. Then(A− s0 In)

−1 exists for any transmission zero, ands0 is a transmission zero ifand only if P(s0) drops rank. Since[

A− s0 In BC D

][In (s0 In − A)−1B0 I

]=[

A− s0 In 0C P(s0)

]

we see that the rank of[

A−s0 I BC D

]equalsn plus the rank ofP(s0). Hences0 is a transmission zero

of P if and only if[

A−s0 In BC D

]drops rank.

Proof of Norms of linear time-invariant systems.We indicate how the formulas for the systemnorms as given in Summary 4.3.12 are obtained.

1. L∞-induced norm.In terms of explicit sums, theith componentyi of the outputy = h ∗ uof the convolution system may be bounded as

|yi(t)| =∣∣∣∣∣∫ ∞

−∞

∑j

hi j (τ)uj(t − τ) dτ

∣∣∣∣∣ ≤∫ ∞

−∞

∑j

|hi j (τ)| · |uj(t − τ)| dτ

≤(∫ ∞

−∞

∑j

|hi j (τ)| dτ

)· ‖u‖L∞ , t ∈ R, (4.93)

so that

‖y‖L∞ ≤ maxi

(∫ ∞

−∞

∑j

|hi j (τ)| dτ

)· ‖u‖L∞ . (4.94)

This shows that

‖φ‖ ≤ maxi

∫ ∞

−∞

∑j

|hi j (τ)| dτ. (4.95)

The proof that the inequality may be replaced by equality follows by showing that thereexists an inputu such that (4.94) is achieved withequality.

195


2. L2-induced norm.By Parseval’s theorem (see e.g. (Kwakernaak and Sivan 1991)) we havethat

‖y‖2L2

‖u‖2L2

=∫∞−∞ yH(t)y(t)dt∫∞−∞ uH(t)u(t)dt

=1

2π

∫∞−∞ yH(jω)y(jω)dω

12π

∫∞−∞ uH(jω)u(jω)dω

(4.96)

=∫∞−∞ uH(jω)HH(jω)H(jω)u(jω)dω∫∞

−∞ uH(jω)u(jω)dω(4.97)

wherey is the Laplace transform ofy andu that ofu. For any fixed frequencyω we havethat

supu( jω)

uH(jω)HH(jω)H(jω)u(jω)uH(jω)u(jω)

= σ2(H(jω)).

Therefore

supu

‖y‖2L2

‖u‖2L2

≤ supω

σ2(H(jω)). (4.98)

The right hand side of this inequality is by definition‖H‖2H∞

. The proof that the inequalitymay be replaced by equality follows by showing that there exists an inputu for which(4.98) is achieved with equality within an arbitrarily small positive marginε.

196

5. Uncertainty Models and Robustness

Overview– Various techniques exist to examine the stability robustnessof control systems subject to parametric uncertainty.

Parametric and nonparametric uncertainty with varying degree ofstructure may be captured by thebasic perturbation model.The sizeof the perturbation is characterized by bounds on the norm of the pertur-bation. The small gain theorem provides the tool to analyze the stabilityrobustness of this model.

These methods allow to generalize the various stability robustnessresults for SISO systems of Chapter 2 in several ways.

5.1. Introduction

In this chapter we discuss various paradigms for representing uncertainty about thedynamicpropertiesof a plant. Moreover, we present methods to analyze the effect of uncertainty onclosed-loop stability and performance.

Section 5.2 is devoted toparametricuncertainty models. The idea is to assume that theequations that describe the dynamics of the plant and compensator (in particular, their transferfunctions or matrices) are known, but that there is uncertainty about the precise values of variousparameters in these equations. The uncertainty is characterized by anintervalof possible values.We discuss some methods to analyze closed-loop stability under this type of uncertainty. Themost famous result isKharitonov’s theorem.

In § 5.3 we introduce the so-calledbasic perturbation modelfor linear systems, which admitsa much wider class of perturbations, includingunstructured perturbations.Unstructured pertur-bations may involve changes in the order of the dynamics and are characterized bynorm bounds.Norm bounds are bounds on the norms of operators corresponding to systems.

In § 5.4 we review the fixed point theorem of functional analysis to the small gain theorem ofsystem theory. With the help of that we establish in§ 5.5 sufficient and necessary conditions forthe robust stability of the basic perturbation model. Next, in§ 5.6 the basic stability robustnessresult is applied to proportional, proportional inverse perturbations and fractional perturbationsof feedback loops.

The perturbation models of§ 5.6 are relatively crude. Doyle’sstructured singular valueallows much finer structuring. It is introduced in§ 5.7. An appealing feature of structured

197


singular value analysis is that it allows studyingcombinedstability and performance robustnessin a unified framework. This is explained in§ 5.8.

In § 5.9 a number of proofs for this chapter are presented.

5.2. Parametric robustness analysis

5.2.1. Introduction

In this section we review theparametricapproach to robustness analysis. In this view of uncer-tainty the plant and compensator transfer functions are assumed to be given, but contain severalparameters whose values are not precisely known. We illustrate this by an example.

prefilter

errorsignal

plantinput

plantoutput

compensatorr

F

e

C

u

Pz

+ −

referenceinput

Figure 5.1.: Feedback system

Example 5.2.1 (Third-order uncertain plant). Consider a feedback configuration as in Fig. 5.1where the plant transfer function is given by

P(s) = gs2(1+ θs)

. (5.1)

The gaing is not precisely known but is nominally equal tog0 = 1 [s−2]. The numberθ is aparasitic time constant and nominally 0 [s]1. A system of this type can successfully be controlledby a PD controller with transfer functionC(s) = k+ Tds. The latter transfer function is not properso we modify it to


, (5.2)

where the time constantT0 is small so that the PD action is not affected at low frequencies. Withthese plant and compensator transfer functions, the return differenceJ = 1+ L = 1+ PC is givenby

J(s) = 1+ gs2(1+ θs)

k + Tds1+ T0s

= θT0s4 + (θ+ T0)s3 + s2 + gTds+ gks2(1+ θs)(1+ T0s)

. (5.3)

Hence, the stability of the feedback system is determined by the locations of the roots of theclosed-loop characteristic polynomial

χ(s) = θT0s4 + (θ+ T0)s3 + s2 + gTds+ gk. (5.4)

Presumably, the compensator can be constructed with sufficient precision so that the values ofthe parametersk, Td, and T0 are accurately known, and such that the closed-loop system isstable at the nominal plant parameter valuesg = g0 = 1 andθ = 0. The question is whether thesystem remains stable under variations ofg andθ, that is, whether the roots of the closed-loopcharacteristic polynomial remain in the open left-half complex plane. �

1All physical units are SI. From this point on they are usually omitted.

198


Because we pursue this example at some length we use the opportunity to demonstrate howthe compensator may be designed by using the classicalroot locusdesign tool. We assume thatthe ground rules for the construction of root loci are known.

k =

k =

8

8−k = 0

Im

Re

Figure 5.2.: Root locus for compensator consisting of a simple gain

Example 5.2.2 (Compensator design by root locus method). Given the nominal plant withtransfer function

P0(s) = g0

s2, (5.5)

the simplest choice for a compensator would be a simple gainC(s) = k. Varyingk from 0 to∞or from 0 to−∞ results in the root locus of Fig. 5.2. For no value ofk stability is achieved.

Modification to a PD controller with transfer functionC(s) = k + sTd amounts to addition ofa zero at−k/Td. Accordingly, keeping−k/Td fixed while varyingk the root locus of Fig. 5.2 isaltered to that of Fig. 5.3 (only the part fork ≥ 0 is shown). We assume that a closed-loopbandwidth of 1 is required. This may be achieved by placing the two closed-loop poles at12

√2(−1± j). The distance of this pole pair from the origin is 1, resulting in the desired closed-

loop bandwidth. Setting the ratio of the imaginary to the real part of this pole pair equal to 1ensures an adequate time response with a good compromise between rise time and overshoot.

doublepole at 0−k/Td

Im

Re

zero at

Figure 5.3.: Root locus for a PD compensator

The closed-loop characteristic polynomial corresponding to the PD compensator transferfunctionC(s) = k + sTd is easily found to be given by

s2 + g0Tds+ g0k. (5.6)

Choosingg0Td =√

2 andg0k = 1 (i.e.,Td =√

2 andk = 1) places the closed-loop poles at thedesired locations. The zero in Fig. 5.3 may now be found at−k/Td = − 1

2

√2 = −0.7071.

The final step in the design is to make the compensator transfer function proper by changingit to


. (5.7)

199


−1/To −k/Td

Im

Repole

atzero

atdoublepole at 0

Figure 5.4.: Root locus for the modified PD compensator

This amounts to adding a pole at the location−1/T0. Assuming thatT0 is small the root lo-cus now takes the appearance shown in Fig. 5.4. The corresponding closed-loop characteristicpolynomial is

T0s3 + s2 + g0Tds+ g0k. (5.8)

Keeping the valuesTd =√

2 andk = 1 and choosingT0 somewhat arbitrarily equal to 1/10 (whichplaces the additional pole in Fig. 5.4 at−10) results in the closed-loop poles

−0.7652± j0.7715, −8.4697. (5.9)

The dominant pole pair at12√

2(−1± j) has been slightly shifted and an additional non-dominantpole at−8.4697 appears. �

5.2.2. Routh-Hurwitz Criterion

In the parametric approach, stability robustness analysis comes down to investigating the rootsof a characteristic polynomial of the form

χ(s) = χn(p)sn + χn−1(p)s

n−1 + · · · + χ0(p), (5.10)

whose coefficientsχn(p), χn−1(p), · · · , χ0(p) depend on the parameter vectorp. Usually it isnot possible to determine the dependence of the roots onp explicitly.

Sometimes, if the problem is not very complicated, the Routh-Hurwitz criterion may beinvoked for testing stability. For completeness we summarize this celebrated result (see forinstance Chen (1970)). Recall that a polynomial isHurwitz if all its roots have zero or negativereal part. It isstrictly Hurwitz if all its roots have strictly negative real part.

Summary 5.2.3 (Routh-Hurwitz criterion). Consider the polynomial

χ(s) = a0sn + a1sn−1 + · · · + an−1s+ an (5.11)

with real coefficientsa0, a1, · · · , an. From these coefficients we form theHurwitz tableau

a0 a2 a4 a6 · · ·a1 a3 a5 a7 · · ·b0 b2 b4 · · ·b1 b3 b5 · · ·· · ·

(5.12)

200


The first two rows of the tableau are directly taken from the “even” and “odd” coefficients of thepolynomialχ, respectively. The third row is constructed from the two preceding rows by letting

[b0 b2 b4 · · ·] = [

a2 a4 a6 · · ·]− a0

a1

[a3 a5 a7 · · ·] . (5.13)

Note that the numerator ofb0 is obtained as minus the determinant of the 2× 2 matrix formedby the first column of the two rows above and the column to the right of and aboveb0. Similarly,the numerator ofb2 is obtained as minus the determinant of the 2× 2 matrix formed by the firstcolumn of the two rows above and the column to the right of and aboveb2. All further entries ofthe third row are formed this way. Missing entries at the end of the rows are replaced with zeros.We stop adding entries to the third row when they all become zero.

The fourth rowb1, b3, · · · is formed from the two rows above it in the same way the thirdrow is formed from the first and second rows. All further rows are constructed in this manner.We stop aftern+ 1 rows.

Given this tableau, a necessary and sufficient condition for the polynomialχ to be strictlyHurwitz is that all the entriesa0, a1, b0, b1, · · · of the first column of the tableau are nonzero andhave the same sign.

A usefulnecessarycondition for stability (known as Descartes’rule of signs) is that all thecoefficientsa0, a1, · · · , an of the polynomialχ are nonzero and all have the same sign. �

In principle, the Routh-Hurwitz criterion allows establishing thestability regionof the pa-rameter dependent polynomialχ as given by (5.10), that is, the set of all parameter valuesp forwhich the roots ofχ all have strictly negative real part. In practice, this is often not simple.

Example 5.2.4 (Stability region for third-order plant). By way of example we consider thethird-order plant of Examples 5.2.1 and 5.2.2. Using the numerical values established in Example5.2.2 we have from Example 5.2.1 that the closed-loop characteristic polynomial is given by

χ(s) = θ

10s4 + (θ+ 1

10)s3 + s2 + g

√2s+ g. (5.14)

The parameter vectorp has componentsg andθ. The Hurwitz tableau may easily be found to begiven by

θ10 1 g

θ+ 110 g

√2

b0 g

b1

g

(5.15)

where

b0 = θ+ 110 −

√2

10 gθ

θ+ 110

, (5.16)

b1 = (θ+ 110 −

√2

10 gθ)√

2− (θ+ 110)

2

θ+ 110 −

√2

10 gθg. (5.17)

201


Inspection of the coefficients of the closed-loop characteristic polynomial (5.14) shows that anecessary condition for closed-loop stability is that bothg and θ be positive. This conditionensures the first, second and fifth entries of the first column of the tableau to be positive. Thethird entryb0 is positive ifg< g3(θ), whereg3 is the function

g3(θ) = 5√

2(θ+ 110)

θ. (5.18)

The fourth entryb1 is positive ifg< g4(θ), with g4 the function

g4(θ) = 5(θ+ 110)(

√2− 1

10 − θ)

θ. (5.19)

Figure 5.5 shows the graphs ofg3 andg4 and the resulting stability region.The caseθ = 0 needs to be considered separately. Inspection of the root locus of Fig. 5.4

(which applies ifθ = 0) shows that forθ = 0 closed-loop stability is obtained for allg> 0. �

g

g3

g4

θ

40

20

00 1 2

Figure 5.5.: Stability region

Exercise 5.2.5 (Stability margins). The stability of the feedback system of Example 5.2.4 isquite robust with respect to variations in the parametersθ andg. Inspect the Nyquist plot of thenominal loop gain to determine the various stability margins of§ 1.4.2 and Exercise 1.4.9(b) ofthe closed-loop system. �

5.2.3. Gridding

If the number of parameters is larger than two or three then it seldom is possible to establish thestability region analytically as in Example 5.2.4. A simple but laborious alternative method isknown asgridding. It consists of covering the relevant part of the parameter space with a grid,and testing for stability in each grid point. The grid does not necessarily need to be rectangular.

Clearly this is a task that easily can be coded for a computer. The number of grid points in-creases exponentially with the dimension of the parameter space, however, and the computationalload for a reasonably fine grid may well be enormous.

Example 5.2.6 (Gridding the stability region for the third-order plant). Figure 5.6 shows theresults of gridding the parameter space in the region defined by 0.5 ≤ g ≤ 4 and 0≤ θ ≤ 1.The small circles indicate points where the closed-loop system is stable. Plus signs correspondto unstable points. Each point may be obtained by applying the Routh-Hurwitz criterion or anyother stability test. �

202


++

+++

++++

4

2

00 1

θ

g

Figure 5.6.: Stability region obtained by gridding

5.2.4. Kharitonov’s theorem

Gridding is straightforward but may involve a tremendous amount of repetitious computation.In 1978, Kharitonov (1978a) (see also Kharitonov (1978b)) published a result that subsequentlyattracted much attention in the control literature2 because it may save a great deal of work.Kharitonov’s theorem deals with the stability of a system with closed-loop characteristic polyno-mial

χ(s) = χ0 + χ1s+ · · · + χnsn. (5.20)

Each of the coefficientsχi is known to be bounded in the form

χi≤ χi ≤ χi, (5.21)

with χiandχi given numbers fori = 0, 1,· · · , n. Kharitonov’s theorem allows verifying whether

all these characteristic polynomials are strictly Hurwitz by checking onlyfour special polynomi-als out of the infinite family.

Summary 5.2.7 (Kharitonov’s theorem). Each member of the infinite family of polynomials

χ(s) = χ0 + χ1s+ · · · + χnsn, (5.22)

with

χi≤ χi ≤ χi, i = 0, 1, 2, · · · , n, (5.23)

is strictly Hurwitz if and only if each of the four Kharitonov polynomials

k1(s) = χ0+ χ

1s+ χ2s2 + χ3s3 + χ

4s4 + χ

5s5 + χ6s6 + · · · , (5.24)

k2(s) = χ0 + χ1s+ χ2s2 + χ

3s3 + χ4s4 + χ5s5 + χ

6s6 + · · · , (5.25)

k3(s) = χ0 + χ1s+ χ

2s2 + χ3s3 + χ4s4 + χ

5s5 + χ

6s6 + · · · , (5.26)

k4(s) = χ0+ χ1s+ χ2s2 + χ

3s3 + χ

4s4 + χ5s5 + χ6s6 + · · · (5.27)

is strictly Hurwitz. �

Note the repeated patterns of under- and overbars. A simple proof of Kharitonov’s theoremis given by Minnichelli, Anagnost, and Desoer (1989). We see what we can do with this resultfor our example.

2For a survey see Barmish and Kang (1993).

203


Example 5.2.8 (Application of Kharitonov’s theorem). From Example 5.2.1 we know thatthe stability of the third-order uncertain feedback system is determined by the characteristicpolynomial

χ(s) = θT0s4 + (θ+ T0)s3 + s2 + gTds+ gk, (5.28)

where in Example 5.2.2 we tookk = 1, Td =√

2 andT0 = 1/10. Suppose that the variations ofthe plant parametersg andθ are known to be bounded by

g ≤ g ≤ g, 0 ≤ θ ≤ θ. (5.29)

Inspection of (5.28) shows that the coefficientsχ0, χ1, χ2, χ3, andχ4 are correspondinglybounded by

g ≤ χ0 ≤ g,

g√

2 ≤ χ1 ≤ g√

2,

1 ≤ χ2 ≤ 1,

T0 ≤ χ3 ≤ T0 + θ,

0 ≤ χ4 ≤ T0θ.

(5.30)

We assume that

0 ≤ θ ≤ 0.2, (5.31)

so thatθ = 0.2, but consider two cases for the gaing:

1. 0.5 ≤ g ≤ 5, so thatg = 0.5 andg = 5. This corresponds to a variation in the gain by afactor of ten. Inspection of Fig. 5.5 shows that this region of variation is well within thestability region. It is easily found that the four Kharitonov polynomials are

k1(s) = 0.5+ 0.5√

2s+ s2 + 0.3s3, (5.32)

k2(s) = 5+ 5√

2s+ s2 + 0.1s3 + 0.02s4, (5.33)

k3(s) = 5+ 0.5√

2s+ s2 + 0.3s3 + 0.02s4, (5.34)

k4(s) = 0.5+ 5√

2+ s2 + 0.1s3. (5.35)

By the Routh-Hurwitz test or by numerical computation of the roots using MATLAB oranother computer tool it may be found that onlyk1 is strictly Hurwitz, whilek2, k3, andk4 are not Hurwitz. This shows that the polynomialχ0 + χ1s+ χ2s2 + χ3s3 + χ4s4 isnot strictly Hurwitz for all variations of the coefficients within the bounds (5.30). Thisdoesnot prove that the closed-loop system is not stable for all variations ofg andθ withinthe bounds (5.29), however, because the coefficientsχi of the polynomial do not varyindependentlywithin the bounds (5.30).

2. 0.5 ≤ g ≤ 2, so thatg = 0.5 andg = 2. The gain now only varies by a factor of four.Repeating the calculations we find that each of the four Kharitonov polynomials is strictlyHurwitz, so that the closed-loop system is stable for all parameter variations.

�

204


5.2.5. The edge theorem

Example 5.2.8 shows that Kharitonov’s theorem usually only yields sufficient but not necessaryconditions for robust stability in problems where the coefficients of the characteristic polynomialdo not vary independently. We therefore consider characteristic polynomials of the form

χ(s) = χ0(p)sn + χ1(p)s

n−1 + · · · + χn(p), (5.36)

where the parameter vectorp of uncertain coefficientspi , i = 1, 2,· · · , N, enterslinearly into thecoefficientsχi(p), i = 0, 1,· · · , n. Such problems are quite common; indeed, the example we arepursuing is of this type. We may rearrange the characteristic polynomial in the form

χ(s) = φ0(s)+N∑

i=1

piφi(s), (5.37)

where the polynomialsφi , i = 0, 1, · · · , N are fixed and given. Assuming that each of theparameterspi lies in a bounded interval of the formp

i≤ pi ≤ pi , i = 1, 2,· · · , N, the family of

polynomials (5.37) forms apolytopeof polynomials. Then theedge theorem(Bartlett, Hollot, andLin 1988) states that to check whether each polynomial in the polytope is Hurwitz it is sufficientto verify whether the polynomials on each of theexposed edgesof the polytope are Hurwitz. Theexposed edges are obtained by fixingN − 1 of the parameterspi at their minimal or maximalvalue, and varying the remaining parameter over its interval.

The edge theorem actually is stronger than this.

Summary 5.2.9 (Edge theorem). Let D be a simply connected domain in the complex plane.Then all the roots of each polynomial (5.37) are contained inD if and only if the roots of eachpolynomial on the exposed edges of the polytope are contained inD . �

A simply connected domain is a domain such that every simple closed contour (i.e., a contourthat does not intersect itself) inside the domain encloses only points of the domain. Althoughobviously application of the edge theorem involves much more work than that of Kharitonov’stheorem it produces more results.

Example 5.2.10 (Application of the edge theorem). We apply the edge theorem to the third-order uncertain plant. From Example 5.2.1 we know that the closed-loop characteristic polyno-mial is given by

χ(s) = θT0s4 + (θ+ T0)s

3 + s2 + gTds+ gk (5.38)

= (T0s3 + s2)+ θ(T0s4 + s3)+ g(Tds+ k). (5.39)

Assuming that 0≤ θ ≤ θ andg ≤ g ≤ g this forms a polytope. By the edge theorem, we need tocheck the locations of the roots of the four “exposed edges”

θ = 0 : T0s3 + s2 + gTds+ gk, g ≤ g ≤ g,

θ = θ : θT0s4 + (θ+ T0)s3 + s2 + gTds+ gk, g ≤ g ≤ g,

g = g : θT0s4 + (θ+ T0)s3 + s2 + gTds+ gk, 0 ≤ θ ≤ θ,

g = g : θT0s4 + (θ+ T0)s3 + s2 + gTds+ gk, 0 ≤ θ ≤ θ.

(5.40)

Figure 5.7 shows the patterns traced by the various root loci so defined withT0, Td and k asdetermined in Example 5.2.2, andθ = 0.2, g = 0.5, g = 5. This is the case where in Example

205


-400 -300 -200 -100 0-10

-5

0

5

10Fourth edge

-10 -5 0-10

-5

0

5

10First edge

-15 -10 -5 0-5

0

5Second edge

-150 -100 -50 0-1

-0.5

0

0.5

1Third edge

Figure 5.7.: Loci of the roots of the four exposed edges

5.2.10 application of Kharitonov’s theorem was not successful in demonstrating robust stability.Figure 5.7 shows that the four root loci are all contained within the left-half complex plane.By the edge theorem, the closed-loop system is stable for all parameter perturbations that areconsidered. �

5.2.6. Testing the edges

Actually, to apply the edge theorem it is not necessary to “grid” the edges, as we did in Example5.2.10. By a result of Białas (1985) the stability of convex combinations of stable polynomials3

p1 and p2 of the form

p = λp1 + (1− λ)p2, λ ∈ [0, 1], (5.41)

may be established by a single test. Given a polynomial

q(s) = q0sn + q1sn−1 + q2sn−2 + · · · + qn, (5.42)

define itsHurwitz matrix Q(Gantmacher 1964) as then× n matrix

Q =

q1 q3 q5 · · · · · · · · ·q0 q2 q4 · · · · · · · · ·0 q1 q3 q5 · · · · · ·0 q0 q2 q4 · · · · · ·0 0 q1 q3 q5 · · ·· · · · · · · · · · · · · · · · · ·

. (5.43)

Białas’ result may be rendered as follows.

3Stable polynomialis the same as strictly Hurwitz polynomial.

206


Summary 5.2.11 (Białas’ test). Suppose that the polynomialsp1 and p2 are strictly Hurwitzwith their leading coefficient nonnegative and the remaining coefficients positive. LetP1 andP2

be their Hurwitz matrices, and define the matrix

W = −P1P−12 . (5.44)

Then each of the polynomials

p = λp1 + (1− λ)p2, λ ∈ [0,1], (5.45)

is strictly Hurwitz iff the real eigenvalues ofW all are strictly negative. �

Note that no restrictions are imposed on the non-real eigenvalues ofW.

Example 5.2.12 (Application of Białas’ test). We apply Białas’ test to the third-order plant. InExample 5.2.10 we found that by the edge theorem we need to check the locations of the rootsof the four “exposed edges”

T0s3 + s2 + gTds+ gk, g ≤ g ≤ g,

θT0s4 + (θ+ T0)s3 + s2 + gTds+ gk, g ≤ g ≤ g,

θT0s4 + (θ+ T0)s3 + s2 + gTds+ gk, 0 ≤ θ ≤ θ,

θT0s4 + (θ+ T0)s3 + s2 + gTds+ gk, 0 ≤ θ ≤ θ.

(5.46)

The first of these families of polynomials is the convex combination of the two polynomials

p1(s) = T0s3 + s2 + gTds+ gk, (5.47)

p2(s) = T0s3 + s2 + gTds+ gk, (5.48)

which both are strictly Hurwitz for the given numerical values. The Hurwitz matrices of thesetwo polynomials are

P1 =

1 gk 0

T0 gTd 0

0 1 gk

, P2 =

1 gk 0

T0 gTd 0

0 1 gk

, (5.49)

respectively. Numerical evaluation, withT0 = 1/10,Td = √2, k = 1, g = 0.5, andg = 5, yields

W = −P1P−12 =

−1.068 0.6848 0

−0.09685 −0.03152 00.01370 −0.1370 −0.1

. (5.50)

The eigenvalues ofW are−1,−0.1, and−0.1. They are all real and negative. Hence, by Białas’test, the system is stable on the edge under investigation.

Similarly, it may be checked that the system is stable on the other edges. By the edge theorem,the system is stable for the parameter perturbations studied. �

Exercise 5.2.13 (Application of Białas’ test). Test the stability of the system on the remainingthree edges as given by (5.46). �

207


5.3. The basic perturbation model

5.3.1. Introduction

This section presents a quite general uncertainty model. It deals with both parametric — orstructured— andunstructuredperturbations, although it is more suited to the latter. Unstructuredperturbations are perturbations that may involve essential changes in the dynamics of the system,characterized bynorm bounds.We first consider the fundamental perturbation paradigm, andthen apply it to a number of special cases.

5.3.2. The basic perturbation model

Figure 5.8 shows the block diagram of the basic perturbation model. The block marked “H” isthe system whose stability robustness under investigation. Its transfer matrixH is sometimescalled theinterconnection matrix. The block “1H” represents a perturbation of the dynamics ofthe system. This perturbation model is simple but the same time powerful4. We illustrate it bysome applications.

p qH

∆H

Figure 5.8.: Basic perturbation model

5.3.3. Application: Perturbations of feedback systems

The stability robustness of a unit feedback loop is of course of fundamental interest.

− − −+

++ +

(a) (b) (c)

L L

q p

L

q p∆L ∆L

H H

Figure 5.9.: Perturbation models of a unit feedback loop.(a) Nominal system. (b) Additive perturbation. (c) Multiplicative perturbation

Example 5.3.1 (The additive perturbation model). Figure 5.9(a) shows a feedback loop withloop gainL. After perturbing the loop gain fromL to L +1L, the feedback system may be rep-

4It may be attributed to Doyle (1984).

208

5.3. The basic perturbation model

resented as in Fig. 5.9(b). The block within the dashed lines is the unperturbed system, denotedH, and1L is the perturbation1H in the basic model.

To computeH, denote the input to the perturbation block1L asq and its output asp, asin Fig. 5.8. Inspection of the diagram of Fig. 5.9(b) shows that with the perturbation block1L

taken away, the system satisfies the signal balance equationq = −p− Lq, so that

q = −( I + L)−1 p. (5.51)

It follows that the interconnection matrixH is

H = −( I + L)−1 = −S, (5.52)

with S the sensitivity matrix of the feedback system. The model of Fig. 5.9(b) is known as theadditiveperturbation model. �

Example 5.3.2 (The multiplicative perturbation model). An alternative perturbation model,calledmultiplicative or proportional perturbation model, is shown in Fig. 5.9(c). The transfermatrix of the perturbed loop gain is

(1+1L)L, (5.53)

which is equivalent to an additive perturbation1L L. The quantity1L may be viewed as therelative sizeof the perturbation. From the signal balanceq = L(−p− q) we obtainq = −( I +L)−1Lp, so that the interconnection matrix is

H = −( I + L)−1L = −T. (5.54)

T is the complementary sensitivity matrix of the feedback system. �

5.3.4. Application: Parameter perturbations

The basic model may also be used for perturbations that are much more structured than those inthe previous application.

Example 5.3.3 (Parameter perturbation). Consider the third-order plant of Example 5.2.1 withtransfer function

P(s) = gs2(1+ sθ)

. (5.55)

The parametersg andθ are uncertain. It is not difficult to represent this transfer function by ablock diagram where the parametersg andθ are found in distinct blocks. Figure 5.10 shows oneway of doing this. It does not matter that one of the blocks is a pure differentiator. Note the way

+

−u

s

y

θ

1/s2g

Figure 5.10.: Block diagram for third-order plant

the factor 1+ sθ in the denominator is handled by incorporating a feedback loop.

209


−

++

+

−

+ +

C s( )

q1 p1 p2 q2

g

s θ

y

H

∆g ∆θ

1/s2

u

Figure 5.11.: Application of the the basic perturbation model to the third-order plant

After perturbingg to g +1g andθ to θ +1θ and including a feedback compensator withtransfer functionC the block diagram may be arranged as in Fig. 5.11. The large block insidethe dashed lines is the interconnection matrixH of the basic perturbation model.

To compute the interconnection matrixH we consider the block diagram of Fig. 5.11. In-spection shows that with the perturbation blocks1g and1θ omitted, the system satisfies thesignal balance equation

q2 = −s(p2 + θq2)+ 1s2(p1 − gC(s)q2) . (5.56)

Solution forq2 yields

q2 = 1/s2

1+ sθ+ C(s)g/s2 p1 − s1+ sθ+ C(s)g/s2 p2. (5.57)

Further inspection reveals thatq1 = −C(s)q2, so that

q1 = − C(s)/s2

1+ sθ+ C(s)g/s2p1 + sC(s)

1+ sθ+ C(s)g/s2p2. (5.58)

Consequently, the interconnection matrix is

H(s) = 1

1+ sθ+ C(s)gs2

[− C(s)s2 sC(s)

1s2 −s

](5.59)

=1

1+sθ

1+ L(s)

[C(s)−1

][− 1s2 s

], (5.60)

whereL = PC is the (unperturbed) loop gain. �

5.4. The small gain theorem

5.4.1. Introduction

In § 5.5 we analyze the stability of the basic perturbation model using what is known as thesmallgain theorem.The small gain theorem is an application of thefixed point theorem, one of the

210

5.4. The small gain theorem

most important results of functional analysis5.

Summary 5.4.1 (Fixed point theorem). Suppose thatU is a complete normed space6. LetL : U → U be a map such that

‖L(u)− L(v)‖ ≤ k ‖u− v‖ for all u, v ∈ U , (5.61)

for some real constant 0≤ k< 1. Such a mapL is called acontraction. Then:

1. There exists auniquesolutionu∗ ∈ U of the equation

u = L(u). (5.62)

The pointu∗ is called afixed pointof the equation.

2. The sequenceun, n = 0, 1, 2,· · · , defined by

un+1 = L(un), n = 0, 1, 2, · · · , (5.63)

converges tou∗ for every initial pointu0 ∈ U . This method of approaching the fixed pointis calledsuccessive approximation.

�

LL q

(a) (b)

p

++

Figure 5.12.: Left: feedback loop. Right: feedback loop with internal inputand output.

5.4.2. The small gain theorem

With the fixed point theorem at hand, we are now ready to establish the small gain theorem (seefor instance Desoer and Vidyasagar (1975)).

Summary 5.4.2 (Small gain theorem). In the feedback loop of Fig. 5.12(a), suppose thatL :U → U is a BIBO stable system for some complete normed spaceU . Then a sufficient conditionfor the feedback loop of Fig. 5.12(a) to be internally stable is that the input-output mapL is acontraction. IfL is linear then it is a contraction if and only if

‖L‖ < 1, (5.64)5For an easy introduction to this material see for instance the well-known though no longer new book by Luenberger

(1969).6The normed spaceU is complete if every sequenceui ∈ U , i = 1, 2,· · · such that limn,m→∞ ‖un − um‖ = 0 has a limit

in U .

211


where‖ · ‖ is the norm induced by the signal norm,

‖L‖ := supu∈U

‖Lu‖U

‖u‖U.

�

The proof may be found in§ 5.9. The small gain theorem gives asufficientcondition forinternal stability that is very simple but often also very conservative. The power of the smallgain theorem is that it does not only apply to linear time-invariant systems but also to nonlineartime-varying systems. The signal spaces of signals of finite energy‖u‖L2 and the signals of finiteamplitude‖u‖L∞ are two examples of complete normed spaces, hence, for these norms the smallgain theorem applies.

Example 5.4.3 (Small gain theorem). Consider a feedback system as in Fig. 5.12(a), whereLis a linear time-invariant system with transfer function

L(s) = − k1+ sθ

. (5.65)

θ is a positive time constant andk a positive or negative gain. We investigate the stability of thefeedback system with the help of the small gain theorem.

1. Norm induced by theL∞-norm. First consider BIBO stability in the sense of theL∞-norm on the input and output signals. By inverse Laplace transformation it follows that theimpulse response corresponding toL is given by

l (t) ={

− kθe−t/θ for t ≥ 0,

0 otherwise.(5.66)

The norm‖L‖ of the system induced by theL∞-norm is (see Summary 4.3.13)

‖L‖ = ‖l‖L1 =∫ ∞

−∞|l (t)| dt = |k|. (5.67)

Hence, by the small gain theorem a sufficient condition for internal stability of the feedbacksystem is that

−1< k < 1. (5.68)

2. Norm induced by theL2-norm. For positiveθ the systemL is stable so according toSummary 4.3.13 theL2-induced norm exists and equals

‖L‖H∞ = supω∈R

|L(jω)| = supω∈R

| k1+ jωθ

| = |k|. (5.69)

Again we find from the small gain theorem that (5.68) is a sufficient condition for closed-loop stability.

In conclusion, we determine the exact stability region. The feedback equation of the system ofFig. 5.12(b) isw = v+ Lw. Interpreting this in terms of Laplace transforms we may solve forw

to obtain

w = 11− L

v, (5.70)

212

5.5. Stability robustness of the basic perturbation model

so that the closed-loop transfer function is

11− L(s)

= 1

1+ k1+sθ

= 1+ sθ(1+ k)+ sθ

. (5.71)

Inspection shows that the closed-loop transfer function has a single pole at−(1+ k)/θ. The cor-responding system is BIBO stable if and only if 1+ k > 0. The closed-loop system is internallystable if and only if

k > −1. (5.72)

This stability region is much larger than that indicated by (5.68). �

Remark. In the robust control literature theH∞-norm of a transfer matrixH is commonlydenoted by‖H‖∞ and not by‖H‖H∞ as we have done so far. In the remainder of the lecturenotes we will adopt the notation predominantly used and hence write‖H‖∞ to mean theH∞-norm, and we refer to this norm simply as the∞-norm. This will not lead to confusion as thep-norm for p = ∞ defined for vectors in Chapter 4 will not return in the remainder of the lecturenotes.


5.5.1. Introduction

We study the internal stability of the basic perturbation model of Fig. 5.8, which is repeated inFig. 5.13(a).

H H

w1

w2

v1

v2

+

+

∆H ∆H+

+

(a) (b)

Figure 5.13.: (a) Basic perturbation model. (b) Arrangement for internal sta-bility

5.5.2. Sufficient conditions

We assume that the model isnominally stable, that is, internally stable if1H = 0. This isequivalent to the assumption that the interconnection systemH by itself is BIBO stable. Similarly,we assume that also1H by itself is BIBO stable.

213


Introducing internal inputs and outputs as in Fig. 5.1(b), we obtain the signal balance equa-tions

w1 = v1 +1H(v2 + Hw1), w2 = v2 + H(v1 +1Hw1). (5.73)

Rearrangement yields

w1 = 1H Hw1 + v1 +1Hv2, (5.74)

w2 = H1Hw2 + Hv1 + v2. (5.75)

Inspection shows that by the fixed point theorem the equations have bounded solutionsw1 andw2 for any boundedv1 andv2 if the inequalities

‖1H H‖ < 1 and ‖H1H‖ < 1 (5.76)

both hold.Following the work of Doyle we use the system norm induced by theL2-norm7. This is the

∞-norm of the system’s transfer matrix as introduced in Summary 4.3.12.We reformulate the conditions‖1H H‖∞ < 1 and‖H1H‖∞ < 1 in two ways. First, since

‖1H H‖∞ = supω σ(H(jω)1H(jω)) and by property 7 of 4.3.8σ(H(jω)1H(jω)) ≤ σ(H(jω)) ·σ(1H(jω)) it follows that‖1H H‖∞ < 1 is implied by

σ(H(jω)) · σ(1H(jω)) < 1 for all ω ∈ R. (5.77)

The same condition implies‖H1H‖∞ < 1. Hence, (5.77) is a sufficient condition for stabilityof the perturbed system. It provides a frequency-dependent bound of the perturbations that areguaranteed not to destabilize the system.

Another way of reformulating the conditions‖1H H‖∞ < 1 and‖H1H‖∞ < 1 is to replacethem with the sufficient condition

‖1H‖∞ · ‖H‖∞ < 1. (5.78)

This follows from the submultiplicative property of the induced norm of Exercise 4.3.10(3).

5.5.3. Necessity

The inequality (5.77) is asufficientcondition for internal stability. It is easy to find examples thatadmit perturbations which violate (5.77) but at the same time donot destabilize the system. Itmay be proved, though8, that if robust stability is desired forall perturbations satisfying (5.77)then the condition is also necessary. This means that it is always possible to find a perturbationthat violates (5.77) within an arbitrarily small margin but destabilizes the system.

The condition (5.78) is equivalent to

‖1H‖∞ <1

‖H‖∞. (5.79)

Also this condition is necessary for robust stability if stability is required for all perturbationssatisfying the bound.

The latter condition is particularly elegant if the perturbations arescaledsuch that they satisfy‖1H‖∞ ≤ 1. Then a necessary and sufficient condition for robust stability is that‖H‖∞ < 1.

7Use of the norm induced by theL∞-norm has also been considered in the literature (see e.g. Dahleh and Ohda (1988)).8See Vidysagar (1985) for a proof for the rational case.

214


Summary 5.5.1 (Stability of the basic perturbation model). Suppose that in the basic pertur-bation model of Fig. 5.13(a) bothH and1H are BIBO stable.

1. Sufficient for internal stability is that

σ(1H(jω)) <1

σ(H(jω))for all ω ∈ R, (5.80)

with σ denoting the largest singular value. If stability is required forall perturbationssatisfying this bound then the condition is also necessary.

2. Another sufficient condition for internal stability is that

‖1H‖∞ <1

‖H‖∞. (5.81)

If stability is required forall perturbations satisfying this bound then the condition is alsonecessary.

3. In particular, a sufficient and necessary condition for robust stability under all perturbationssatisfying‖1H‖∞ ≤ 1 is that‖H‖∞ < 1.

�

5.5.4. Examples

We consider two applications of the basic stability robustness result.

Example 5.5.2 (Stability under perturbation). By way of example, we study which perturba-tions of the real parameterθ preserve the stability of the system with transfer function

P(s) = 11+ sθ

. (5.82)

Arranging the system as in Fig. 5.14(a) we obtain the perturbation model of Fig. 5.14(b), withθ0

u

θ s

y u

p q

s

H

y

∆θ

(a) (b)

θο

+

−

+

−

+ +

Figure 5.14.: (a) Block diagram. (b) Perturbation model

the nominal value ofθ and1θ its perturbation. Because the system should be nominally stablewe needθ0 to be positive.

The interconnection matrixH of the standard perturbation model is obtained from the signalbalance equationq = −s(p+ θ0q) (omittingu andy). Solution forq yields

q = −s1+ sθ0

p. (5.83)

215


It follows that

H(s) = −s1+ sθ0

= −sθ0

1+ sθ0

1θ0, (5.84)

so that

σ(H(jω)) = |H(jω)| = 1θ0

√ω2θ2

0

1+ ω2θ20

(5.85)

and

‖H‖∞ = supω

σ(H(jω)) = 1θ0. (5.86)

Since σ(1H(jω)) = |1θ| = ‖1H‖∞, both (1) and (2) of Summary 5.5.1 imply that internalstability is guaranteed for all perturbations such that|1θ| < θ0, or

−θ0 < 1θ < θ0. (5.87)

Obviously, though, the system is stable for allθ > 0, that is, for all perturbations1θ such that

−θ0 < 1θ <∞. (5.88)

Hence, the estimate for the stability region is quite conservative. On the other hand, it is easyto find a perturbation that violates (5.87) and destabilizes the system. The perturbation1θ =−θ0(1+ ε), for instance, withε positive but arbitrarily small, violates (5.87) with an arbitrarilysmall margin, anddestabilizesthe system (because it makesθ negative).

Note that the class of perturbations such that‖1θ‖∞ ≤ θ0 is much larger than just real per-turbations satisfying (5.87). For instance,1θ could be a transfer function, such as

1θ(s) = θ0α

1+ sτ, (5.89)

with τ any positive time constant, andα a real number with magnitude less than 1. This “para-sitic” perturbation leaves the system stable. �

Example 5.5.3 (Two-parameter perturbation). A more complicated example is the feedbacksystem discussed in Examples 5.2.1, 5.2.2 and 5.3.3. It consists of a plant and a compensatorwith transfer functions

P(s) = gs2(1+ sθ)

, C(s) = k + Tds1+ T0s

, (5.90)

respectively. In Example 5.3.3 we found that the interconnection matrixH with respect to per-turbations in the parametersg andθ is

H(s) =1

1+sθ0

1+ L0(s)

[C(s)−1

] [− 1s2 s

], (5.91)

with g0 andθ0 denoting the nominal values of the two uncertain parametersg andθ. Beforecontinuing the analysis we modify the perturbation model of Fig. 5.11 to that of Fig. 5.15. Thismodel includes scaling factorsα1 andβ1 for the perturbation1g and scaling factorsα2 andβ2 for the perturbation1θ. The scaled perturbations are denotedδg andδθ, respectively. The

216


−

++

+

−

++

C s( )u

g

q1 q2p1 p2

β1 α1

1/s2

s θy

H

δg δθ

α2 β2

Figure 5.15.: Scaled perturbation model for the third-order plant

productα1β1 = ε1 is the largest possible perturbation ing, whileα2β2 = ε2 is the largest possibleperturbation inθ. It is easily verified that the interconnection matrix corresponding to Fig. 5.15is

H(s) =1

1+sθ0

1+ L0(s)

[β1C(s)−β2

][− α1s2 α2s

]. (5.92)

It may also easily be worked out that the largest eigenvalue ofHT(−jω)H(jω) is

σ2(ω) =1

1+ω2θ20

|1+ L0(jω)|2(β2

1|C(jω)|2 + β22

)( α21

ω4+ α2

2ω2

), ω ∈ R. (5.93)

The other eigenvalue is 0. The nonnegative quantityσ(ω) is the largest singular value ofH(jω).By substitution ofL0 andC into (5.93) it follows that

σ2(ω) =(β2

1(k2 + ω2T2

d )+ β22(1+ ω2T2

0 ))(α2

1 + α22ω

6)

|χ(jω)|2 , (5.94)

with χ the closed-loop characteristic polynomial

χ(s) = θ0T0s4 + (θ0 + T0)s3 + s2 + g0Tds+ g0k. (5.95)

First we note that if we choose the nominal valueθ0 of the parasitic time constantθ equal tozero — which is a natural choice — andε2 = α2β2 6= 0 thenσ(∞) = ∞, so that stability is notguaranteed. Indeed, this situation admits negative values ofθ, which destabilize the closed-loopsystem. Hence, we need to chooseθ0 positive but such that the nominal closed-loop system isstable, of course.

Second, inspection of (5.94) shows that for fixed uncertaintiesε1 = α1β1 andε2 = α2β2 thefunction σ depends on the way the individual values of the scaling constantsα1, β1, α2, andβ2 are chosen. On first sight this seems surprising. Reflection reveals that this phenomenonis caused by the fact that the stability robustness test is based onfull perturbations1H . Fullperturbations are perturbations such that all entries of the perturbation transfer matrix1H arefilled with dynamical perturbations.

217


We choose the numerical valuesk = 1, Td = √2, andT0 = 1/10 as in Example 5.2.2. In

Example 5.2.8 we consider variations in the time constantθ between 0 and 0.2. Correspondingly,we letθ0 = 0.1 andε2 = 0.1. For the variations in the gaing we study two cases as in Example5.2.8:

1. 0.5 ≤ g ≤ 5. Correspondingly, we letg0 = 2.75 andε1 = 2.25.

2. 0.5 ≤ g ≤ 2. Correspondingly, we takeg0 = 1.25 andε1 = 0.75.

For lack of a better choice we select for the time being

α1 = β1 = √ε1, α2 = β2 = √

ε2. (5.96)

Figure 5.16 shows the resulting plots ofσ(ω) for the cases (1) and (2). In case (1) the peak valueis about 67.1, while in case (2) it is approximately 38.7. Both cases fail the stability robustnesstest by a very wide margin, although from Example 5.2.4 we know that in both situations stabilityis ensured. The reasons that the test fails are that (i) the test is based on full perturbations ratherthan the structured perturbations implied by the variations in the two parameters, and (ii) the testalso allowsdynamicalperturbations rather than the real perturbations implied by the parametervariations.

10−2

10−1

100

101

102

103

80

40

0

ω

(a)

(b)

σ( ( ω))H j

Figure 5.16.: Plots ofσ(H(jω)) for the two-parameter plant

Less conservative results may be obtained by allowing the scaling factorsα1, β1, α2, andβ2

to be frequency-dependent, and to choose them so as to minimizeσ(ω) for eachω. Substitutingβ1 = ε1/α1 andβ2 = ε2/α2 we obtain

σ2(ω) =ε2

1(k2 + ω2T2

d )+ ε21(k

2+ω2T2d )ω

6

ρ+ ε2

2(1+ ω2T20 )ρ+ ε2

2(1+ ω2T20 )ω

6

|χ(jω)|2 , (5.97)

whereρ = α21/α

22. It is easy to establish that for fixedω the quantityσ2(ω) is minimized for

ρ = ω3 ε1

ε2

√k2 + ω2T2

d

1+ ω2T20

, (5.98)

and that for this value ofρ

σ(ω) =ε1

√k2 + ω2T2

d + ε2ω3√

1+ ω2T20

|χ(jω)| , ω ≥ 0. (5.99)

218


10−2

10−1

100

101

102

103

ω

0

1

2

(a)

(b)

σ( ( ω))H j

Figure 5.17.: Plots ofσ(H(jω)) with optimal scaling for the two-parameterplant

Figure 5.17 shows plots ofσ for the same two cases (a) and (b) as before. In case (a) the peakvalue of σ is about 1.83, while in case (b) it is 1. Hence, only in case (b) robust stability isestablished. The reason that in case (a) robust stability is not proved is that the perturbationmodel allows dynamic perturbations. Only if the size of the perturbations is downsized by afactor of almost 2 robust stability is guaranteed.

In Example 5.7.11 in the section on the structured singular value this example is furtherdiscussed. �

+

+

+

x

A

qp∆

sI1

Figure 5.18.: Perturbation of the state space system

Exercise 5.5.4 (Stability radius). Hinrichsen and Pritchard (1986) investigate what perturba-tions of the matrixA make the system described by the state differential equation

x(t) = Ax(t), t ∈ R, (5.100)

unstable. Assume that the nominal systemx(t) = Ax(t) is stable. The number

r (A) = infA+1 ∈ Rn×n has at least one

eigenvalue in the closed right-half plane

σ(1) (5.101)

is called thestability radiusof the matrixA. The matrix norm used is the spectral norm. If onlyreal-valued perturbations1 are considered thenr (A) is thereal stability radius, denotedrR(A).If the elements of1 may also assume complex values thenr (A) is thecomplex stability radius,denotedrC(A).

219


1. Prove that

rR(A) ≥ rC(A) ≥ 1‖F‖∞

, (5.102)

with F the rational matrix given byF(s) = (sI − A)−1. Hint: Represent the system bythe block diagram of Fig. 5.18.

2. A more structured perturbation model results by assuming thatA is perturbed as

A −→ A+ B1C, (5.103)

with B andC matrices of suitable dimensions, and1 the perturbation. Adapt the blockdiagram of Fig. 5.18 and prove that the associated complex stability radiusrC(A, B,C)satisfiesrC(A, B,C) ≥ 1/‖H‖∞, with H(s) = C(sI − A)−1B.

Hinrichsen and Pritchard (1986) prove that actuallyrC(A) = 1/‖F‖∞ and rC(A, B,C) =1/‖H‖∞. Thus, explicit formulas are available for the complex stability radius. The real sta-bility radius is more difficult to determine. Only very recently results have become available(Qiu, Bernhardsson, Rantzer, Davison, Young, and Doyle 1995). �

5.5.5. Nonlinear perturbations

The basic stability robustness result also applies fornonlinearperturbations. Suppose that theperturbation1H in the block diagram of Fig. 5.13(a) is a nonlinear operator that maps the signale(t), t ∈ R, into the signal(1He)(t), t ∈ R. Assume that there exists a positive constantk suchthat

‖1He1 −1He2‖ ≤ k‖e1 − e2‖ (5.104)

for every two inpute1,e2 to the nonlinearity. We use theL2-norm for signals. By application ofthe fixed point theorem it follows that the perturbed system of Fig. 5.13(a) is stable if

k‖H‖∞ < 1. (5.105)

Suppose for instance that the signals are scalar, and that1H is a static nonlinearity described by

(1He)(t) = f (e(t)), (5.106)

with f : R → R. Let f satisfy the inequality∣∣∣∣ f (e)e

∣∣∣∣ ≤ c, e 6= 0, e∈ R, (5.107)

with c a nonnegative constant. Figure 5.19 illustrates the way the functionf is bounded. We callf asector bounded function. It is not difficult to see that (5.104) holds withk = c. It follows thatthe perturbed system is stable for any static nonlinear perturbation satisfying (5.107) as long asc ≤ 1/‖H‖∞.

Similarly, the basic stability robustness result holds fortime-varyingperturbations and per-turbations that are both nonlinear and time-varying, provided that these perturbations are suitablybounded.

220

5.6. Stability robustness of feedback systems

e

f (e)

ce

−ce

0

Figure 5.19.: Sector bounded function


5.6.1. Introduction

In this section we apply the results of§ 5.5 to various perturbation models for single-degree-of-freedom feedback systems. We successively discuss proportional, proportional inverse andfractional perturbations for MIMO systems. The results of§ 1.4 emerge as special cases.

5.6.2. Proportional perturbations

In § 5.3 we consider additive and multiplicative (orproportional) perturbation models for single-degree-of-freedom feedback systems. The proportional model is preferred because it representstherelativesize of the perturbation. The corresponding block diagram is repeated in Fig. 5.20(a).It represents a perturbation

−

++

−

++

L

q p

H

L

W

q p

V

H

∆L

δL

(a) (b)

Figure 5.20.: Proportional and scaled perturbations of a unit feedback loop

L −→ ( I +1L)L (5.108)

Sinceq = −L(p+ q), so thatq = −( I + L)−1Lp, the interconnection matrix is

H = −( I + L)−1L = −T, (5.109)

221


with T = ( I + L)−1L = L( I + L)−1 the complementary sensitivity matrixof the closed-loopsystem.

To scalethe proportional perturbation we modify the block diagram of Fig. 5.20(a) to that ofFig. 5.20(b). This diagram represents a perturbation

L −→ ( I + VδLW)L, (5.110)

whereV andW are available to scale such that‖δL‖∞ ≤ 1. For this configuration the intercon-nection matrix isH = −WTV.

Application of the results of Summary 5.5.1 leads to the following conclusions.

Summary 5.6.1 (Robust stability of feedback systems for proportional perturbations). As-sume that the feedback system of Fig. 5.20(a) is nominally stable.

1. A sufficient condition for the closed-loop system to be stable under proportional perturba-tions of the form

L −→ (1+1L)L (5.111)

is that1L be stable with

σ(1L(jω)) <1

σ(T(jω))for all ω ∈ R. (5.112)

with T = ( I + L)−1L = L( I + L)−1 the complementary sensitivity matrix of the feedbackloop. If stability is required forall perturbations satisfying the bound then the condition isalso necessary.

2. Underscaledperturbations

L −→ (1+ VδLW)L (5.113)

the system is stable for all‖δL‖∞ ≤ 1 if and only if

‖WTV‖∞ < 1. (5.114)�

For SISO systems part (1) of this result reduces to Doyle’s robustness criterion of§ 1.4. Infact, Summary 5.6.1 is the original MIMO version of Doyle’s robustness criterion (Doyle 1979).

5.6.3. Proportional inverse perturbations

The result of the preceding subsection confirms the importance of the complementary sensitivitymatrix (or function)T for robustness. We complement it with a “dual” result involving thesensitivity functionS. To this end, consider the perturbation model of Fig. 5.21(a), where theperturbation1L−1 is included in a feedback loop. The model represents a perturbation

L −→ ( I +1L−1)−1L. (5.115)

Assuming thatL has an inverse, this may be rewritten as the perturbation

L−1 −→ L−1( I +1L−1). (5.116)

222


−+

−

−

+−

(a) (b)

L

p q

L

H H

V W

p q

∆L−1

δL−1

Figure 5.21.: Proportional inverse perturbation of unit feedback loop

Hence, the perturbation represents a proportional perturbation of theinverseloop gain. Theinterconnection matrixH follows from the signal balanceq = −p− Lqso thatq = −( I + L)−1 p.Hence, the interconnection matrix is

H = −( I + L)−1 = −S. (5.117)

S= ( I + L)−1 is the sensitivity matrix of the feedback loop.We allow for scaling by modifying the model to that of Fig. 5.21(b). This model represents

perturbations of the form

L−1 −→ L−1( I + VδL−1W), (5.118)

whereV andW provide freedom to scale such that‖δL−1‖∞ ≤ 1. The interconnection matrixnow is H = −W SV.

Application of the results of Summary 5.5.1 yields the following conclusions.

Summary 5.6.2 (Robust stability of feedback systems for proportional inverse perturba-tions). Assume that the feedback system of Fig. 5.21(a) and (b) is nominally stable.

1. Underproportional inverse perturbationsof the form

L−1 −→ L−1( I +1L−1) (5.119)

a sufficient condition for stability is that1L−1 be stable with

σ(1L−1(jω)) <1

σ(S(jω))for all ω ∈ R, (5.120)

with S= ( I + L)−1 the sensitivity matrix of the feedback loop. If stability is required forall perturbations satisfying the bound then the condition is also necessary.

2. Underscaledinverse perturbations of the form

L−1 −→ L−1( I + VδL−1W) (5.121)

the closed-loop system is stable for all‖δL−1‖∞ ≤ 1 if and only if

‖W SV‖∞ < 1. (5.122)�

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5.6.4. Example

We illustrate the results of Summaries 5.6.1 and 5.6.2 by application to an example.

Example 5.6.3 (Robustness of a SISO closed-loop system). In Example 5.2.1 we considereda SISO single-degree-of-freedomfeedback system with plant and compensator transfer functions

P(s) = gs2(1+ sθ)

, C(s) = k + Tds1+ T0s

, (5.123)

respectively. Nominally the gaing equalsg0 = 1, while the parasitic time constantθ is nominally0. In Example 5.2.2 we chose the compensator parameters ask = 1, Td = √

2 andT0 = 1/10.We use the results of Summaries 5.6.1 and 5.6.2 to study what perturbations of the parameters

g andθ leave the closed-loop system stable.

1. Loop gain perturbation model.Starting with the expression

L(s) = P(s)C(s) = gs2(1+ sθ)

k + Tds1+ T0s

(5.124)

it is easy to find that the proportional loop gain perturbations are

1L(s) = L(s)− L0(s)L0(s)

=g−g0

g0− sθ

1+ sθ. (5.125)

Figures 5.22(a) and (b) show the magnitude plot of the inverse 1/T0 of the nominal com-plementary sensitivity function. Inspection shows that 1/T0 assumes relatively small val-ues in the low-frequency region. This is where the proportional perturbations of the loopgain need to be the smallest. Forθ = 0 the proportional perturbation (5.125) of the loopgain reduces to

1L(s) = g− g0

g0, (5.126)

and, hence, is constant. The minimal value of the function 1/|T0| is about 0.75. Therefore,for θ = 0 the robustness criterion 1 of Summary 5.6.1 allows relative perturbations ofg upto 0.75, so that 0.25< g< 1.75.

For g = g0, on the other hand, the proportional perturbation (5.125) of the loop gain re-duces to

1L(s) = −sθ1+ sθ

. (5.127)

Figure 5.22(a) shows magnitude plots of this perturbation for several values ofθ. Therobustness criterion (a) of Summary 5.6.1 permits values ofθ up to about 1.15. For thisvalue ofθ the gaing can be neither increased nor decreased without violating the criterion.

For smaller values of the parasitic time constantθ a wider range of gains is permitted. Theplots of of|1L| of Fig. 5.22(b) demonstrate that forθ = 0.2 the gaing may vary betweenabout 0.255 and 1.745.

The stability bounds ong andθ are conservative. The reason is of course that the perturba-tion model allows a much wider class of perturbations than just those caused by changesin the parametersg andθ.

224


.01 .011 1100 100

1000 1000

1 1

.001 .001

(a) (b)

ω ω

1/|T |o 1/|T |o

θ = 0.2θ = 0.6

θ = 1.15

g = 1.745

g = 1.2

g = 0.2

Figure 5.22.: Robust stability test for the loop gain perturbation model:(a) 1/|T0| and the relative perturbations forg = 1. (b) 1/|T0| and the relative perturbations forθ = 0.2.

2. Inverse loop gain perturbation model.The proportional inverse loop gain perturbation isgiven by

1L−1(s) = L−1(s)− L−10 (s)

L−10 (s)

= g0 − gg

+ sθg0

g. (5.128)

We apply the results of Summary 5.6.2. Figure 5.23 gives the magnitude plot of the in-verse 1/S0 of the nominal sensitivity function. Inspection shows that for the inverse loopgain perturbation model the high-frequency region is the most critical. By inspection of(5.128) we see that ifθ 6= 0 then|1L−1(∞)| = ∞, so that robust stability is not ensured.Apparently, this model cannot handle high-frequency perturbations caused by parasiticdynamics.

For θ = 0 the proportional inverse loop gain perturbation reduces to

1L−1(s) = g0 − gg

, (5.129)

and, hence, is constant. The magnitude of 1/S0 has a minimum of about 0.867, so thatfor θ = 0 stability is ensured for| g0−g

g | < 0.867, or 0.536< g < 7.523. This range ofvariation is larger than that found by application of the proportional loop gain perturbationmodel.

Again, the result is conservative. It is disturbing that the model does not handle parasiticperturbations.

�

The derivation of the unstructured robust stability tests of Summary 5.6.1 is based on the smallgain theorem, and presumes the perturbations1L to beBIBO stable. This is highly restrictiveif the loop gainL by itself represents an unstable system, which may easily occur (in particular,when the plant by itself is unstable).

It is well-known (Vidysagar 1985) that the stability assumption on the perturbation may berelaxed to the assumption that both the nominal loop gain and the perturbed loop gain have thesame number of right-half plane poles.

Likewise, the proofs of the inverse perturbation tests of Summary 5.6.2 require the perturba-tion of theinverseof the loop gain to be stable. This is highly restrictive if the loop gain by itself

225


10−2

10−1

100

101

102

102

104

1

1/| |So

ω

Figure 5.23.: Magnitude plot of 1/S0

has right-half plane zeros. This occurs, in particular, when the plant has right-half plane zeros.The requirement, however, may be relaxed to the assumption that the nominal loop gain and theperturbed loop gain have thesame number of right-half plane zeros.

5.6.5. Fractional representation

The stability robustness analysis of feedback systems based on perturbations of the loop gain orits inverse is simple, but often overly conservative.

Another model that is encountered in the literature9 relies on what we term herefractionalperturbations. It combines, in a way, loop gain perturbations and inverse loop gain perturbations.In this analysis, the loop gainL is represented as

L = N D−1, (5.130)

where thedenominator Dis a square nonsingular rational or polynomial matrix, andN a rationalor polynomial matrix. Any rational transfer matrixL may be represented like this in many ways.If D and N are polynomial then the representation is known as a (right)10 polynomial matrixfraction representation. If D and N are rational and proper with all their poles in the openleft-half complex plane then the representation is known as a (right)rational matrix fractionrepresentation.

Example 5.6.4 (Fractional representations). For a SISO system the fractional representationis obvious. Suppose that

L(s) = gs2(1+ sθ)

. (5.131)

Clearly we have the polynomial fractional representationL = N D−1 with N(s)= g andD(s) =s2(1+ sθ). The fractional representation may be made rational by letting

D(s) = s2(1+ sθ)d(s)

, N(s) = gd(s)

, (5.132)

with d any strictly Hurwitz polynomial. �9The idea originates from Vidyasagar (Vidyasagar, Schneider, and Francis 1982; Vidysagar 1985). It is elaborated in

McFarlane and Glover (1990).10Because the denominator is on the right.

226


Right fractional representation may also be obtained for MIMO systems (see for instanceVidysagar (1985)).

−

+−

++

p1 p2q2q1

L

H

∆D ∆N

Figure 5.24.: Fractional perturbation model

5.6.6. Fractional perturbations

Figure 5.24 shows the fractional perturbation model. Inspection shows that the perturbation isgiven by

L −→ ( I +1N)L( I +1D)−1, (5.133)

or

N D−1 −→ ( I +1N)N D−1( I +1D)−1. (5.134)

Hence, the numerator and denominator are perturbed as

N −→ ( I +1N)N, D −→ ( I +1D)D. (5.135)

Thus,1D and1N representproportional perturbations of the denominatorandof the numerator,respectively.

By block diagram substitution it is easily seen that the configuration of Fig. 5.24 is equivalentto that of Fig. 5.25(a). The latter, in turn, may be rearranged as in Fig. 5.25(b). Here

p = −1Dq1 +1Nq2 = 1L

[q1

q2

], (5.136)

with

1L = [−1D 1N] . (5.137)

To establish the interconnection matrixH of the configuration of Fig. 5.25(b) we consider thesignal balance equationq1 = p− Lq1. It follows that

q1 = ( I + L)−1 p = Sp, (5.138)

with S= ( I + L)−1 the sensitivity matrix. Sinceq2 = −Lq1 we have

q2 = −L( I + L)−1 p = −Tp, (5.139)

227


−

−

+

+

+

−

q2

q2

q1

q1

p1p2∆N

∆L

∆D

L

L +

+

H

H

p

(a)

(b)

Figure 5.25.: Equivalent configurations

with T = L( I + L)−1 the complementary sensitivity matrix. Inspection of (5.138–5.139) showsthat the interconnection matrix is

H =[

S−T

]. (5.140)

Investigation of the frequency dependence of the greatest singular value ofH(jω) yields infor-mation about the largest possible perturbations1L that leave the loop stable.

It is useful to allow for scaling by modifying the configuration of Fig. 5.25(b) to that ofFig. 5.26. This modification corresponds to representing the perturbations as

1D = VδDW1, 1N = VδNW2, (5.141)

whereV, W1, andW2 are suitably chosen (stable) rational matrices such that

‖δL‖∞ ≤ 1, with δL = [−δD δN]. (5.142)

Accordingly, the interconnection matrix changes to

H =[

W1SV−W2TV

]. (5.143)

Application of the results of Summary 5.5.1 yields the following conclusions.

Summary 5.6.5 (Numerator-denominator perturbations). In the stable feedback configurationof Fig. 5.27, suppose that the loop gain is perturbed as

L = N D−1 −→ [( I + VδNW2)N][ ( I + VδDW1)D]−1 (5.144)

228


−

+

+L

W2

q2 q1

W1V

p

δL

H

Figure 5.26.: Perturbation model with scaling

with V, W1 andW2 stable transfer matrices. Then the closed-loop system is stable for all stableperturbationsδP = [−δD δN] such that‖δP‖∞ ≤ 1 if and only if

‖H‖∞ < 1. (5.145)

Here we have

H =[

W1SV−W2TV

], (5.146)

with S= ( I + L)−1 the sensitivity matrix andT = L( I + L)−1 the complementary sensitivitymatrix. �

−L


The largest singular value ofH(jω), with H given by (5.146), equals the square root of thelargest eigenvalue of

HT(−jω)H(jω) = VT(−jω)[ST(−jω)WT1 (−jω)W1(jω)S(jω)

+ TT(−jω)WT2 (−jω)W2(jω)T(jω)]V(jω). (5.147)

For SISO systems this is the scalar function

HT(−jω)H(jω) = |V(jω)|2[|S(jω)|2|W1(jω)|2 + |T(jω)|2|W2(jω)|2]. (5.148)

5.6.7. Discussion

We consider how to arrange the fractional perturbation model. In the SISO case, without loss ofgenerality we may take the scaling functionV equal to 1. ThenW1 represents the scaling factor

229


for the denominator perturbations andW2 that for the numerator perturbations. We accordinglyhave

‖H‖2∞ = sup

ω∈R

(|S(jω)|2|W1(jω)|2 + |T(jω)|2|W2(jω)|2). (5.149)

For well-designed control systems the sensitivity functionS is small at low frequencies while thecomplementary sensitivity functionT is small at high frequencies. Figure 5.28 illustrates this.Hence,W1 may be large at low frequencies andW2 large at high frequencies without violating

.01 .011 1100 100

1 1

.01 .01

.001 .001

|S| |T|

ω ω

Figure 5.28.: Sensitivity function and complementary sensitivity function fora well-designed feedback system

the robustness condition‖H‖∞ < 1. This means that at low frequencies we may allow largedenominator perturbations, and at high frequencies large numerator perturbations.

The most extreme point of view is to structure the perturbation model such that all low fre-quency perturbations are denominator perturbations, and all high frequency perturbations arenumerator perturbations. Since we may trivially write

L = 11L

, (5.150)

modeling low frequency perturbations as pure denominator perturbations implies modeling lowfrequency perturbations asinverseloop gain perturbations. Likewise, modeling high frequencyperturbations as pure numerator perturbations implies modeling high frequency perturbations asloop gain perturbations. This amounts to taking

|W1(jω)| ={

WL−1(ω) at low frequencies,0 at high frequencies,

(5.151)

|W2(jω)| ={

0 at low frequencies,WL(ω) at high frequencies,

(5.152)

with WL−1 a bound on the size of the inverse loop perturbations, andWL a bound on the size ofthe loop perturbations.

Obviously, the boundary between “low” and “high” frequencies lies in the “crossover” region,that is, near the frequency where the loop gain crosses over the zero dB line. In this frequencyregion neitherSnor T is small.

230


Another way of dealing with this perturbation model is to modify the stability robustness testto checking whether for eachω ∈ R

|1L−1(jω)| < 1|S(jω)| or |1L(jω)| < 1

|T(jω)| . (5.153)

This test amounts to verifying whether either the proportional loop gain perturbation test succeedsor the proportional inverse loop gain test. Obviously, its results are less conservative than theindividual tests.

Example 5.6.6 (Numerator-denominator perturbations). Again consider the feedback systemthat we studied before, most recently in Example 5.6.3, with nominal plantP0(s) = g0/s2, per-turbed plant

P(s) = gs2(1+ sθ)

, (5.154)

and compensatorC(s)= (k+ sTd)/(1+ sT0). We use the design values of Example 5.2.2. Plotsof the magnitudes of the nominal sensitivity and complementary sensitivity functionsS andTare given in Fig. 5.28.S is small up to a frequency of almost 1, whileT is small starting from aslightly higher frequency. The crossover region is located about the frequency 1.

Figure 5.29 illustrates that the perturbation from the nominal parameter valuesg0 = 1 andθ0 = 0 to g = 5, θ = 0.2, for instance, fails the two separate tests, but passes the test (5.153).Hence, the perturbation leaves the closed-loop system stable. �

1/|S| 1/|S|1/|T| 1/|T|

(a) (b) (c)10000

100

1

.01 .01.011 1 1100 100100ω ω ω

∆L−1 ∆L−1∆L ∆L

Figure 5.29.: (a), (b) Separate stability tests. (c) Combined test.

Feedback systems are robustly stable for perturbations in the frequency regions where eitherthe sensitivity is small (at low frequencies) or the complementary sensitivity is small (at highfrequencies). In thecrossover regionneither sensitivity is small. Hence, the feedback system isnot robust for perturbations that strongly affect the crossover region.

In the crossover region the uncertainty therefore should be limited. On the one hand thislimitation restrictsstructureduncertainty — caused by load variations and environmental changes— that the system can handle. On the other hand,unstructureduncertainty — deriving fromneglected dynamics and parasitic effects — should be kept within bounds by adequate modeling.

5.6.8. Plant perturbation models

In the previous subsections we modeled the uncertainty as an uncertainty in the loop gainL, whichresults in interconnection matrices in terms of this loop gain, such asSandT. It is important to

231


realize that we have a choice in how to model the uncertainty and that we need not necessarilydo that in terms of the loop gain. In particular as the uncertainty is usually present in the plantP only, and not in controllerK, it makes sense to model the uncertainty as such. Table 5.1 listsseveral ways to model plant uncertainty.

plant perturbed plant interconnection matrixH perturbation matrix1

P P+ V1PW −W KSV 1P

P ( I + V1PW)P −WTV 1P

P P( I + V1PW) −W KSPV 1P

P ( I + V1PW)−1 P −W SV 1P

P P( I + V1PW)−1 −W( I + K P)−1V 1P

D−1N (D + V1DW1)−1(N + V1N W2) −[ W1S

W2KS

]D−1V

[1D 1N

]ND−1 (N + W11N V)( D + W21DV)−1 −VD−1

[KSW1 ( I+KP)−1W2

] [1N1D

]Table 5.1.: A list of perturbation models with plantP and controllerK

5.7. Structured singular value robustness analysis

5.7.1. Introduction

We return to the basic perturbation model of Fig. 5.8, which we repeat in Fig. 5.30(a). Asdemonstrated in§ 5.3, the model is very flexible in representing both structured perturbations(i.e., variations of well-defined parameters) and unstructured perturbations.

The stability robustness theorem of Summary 5.5.1 (p. 215) guarantees robustness under allperturbations1 such thatσ(1(jω)) · σ(H(jω)) < 1 for all ω ∈ R. Perturbations1(jω) whosenormσ(1(jω)) does not exceed the number 1/σ(H(jω)), however, are completelyunstructured.As a result, often quite conservative estimates are obtained of the stability region forstructuredperturbations. Several examples that we considered in the preceding sections confirm this.

Doyle (1982) proposed another measure for stability robustness based on the notion of whathe callsstructured singular value.In this approach, the perturbation structure is detailed as inFig. 5.30(b) (Safonov and Athans 1981). The overall perturbation1 has the block diagonal form

1 =

11 0 · · · 00 12 0 · · ·· · · · · · · · · · · ·0 · · · 0 1K

. (5.155)

Each diagonal block entry1i has fixed dimensions, and has one of the following forms:

• 1i = δ I , with δ a real number. If the unit matrixI has dimension 1 this represents a realparameter variation. Otherwise, this is arepeated real scalar perturbation.

232


(a) (b)

H H

∆

∆2

∆K

∆1

Figure 5.30.: (a) Basic perturbation model. (b) Structured perturbationmodel

• 1i = δ I , with δ a stable11 scalar transfer matrix. This represents ascalar or repeatedscalar dynamic perturbation.

• 1i is a (not necessarily square) stable transfer matrix. This represents amultivariabledynamic perturbation.

A wide variety of perturbations may be modeled this way.

ω

Im

Re10

Figure 5.31.: Nyquist plot of det( I − H1) for a destabilizing perturbation

We study which are the largest perturbations1with the given structure that do not destabilizethe system of Fig. 5.30(a). Suppose that a given perturbation1 destabilizes the system. Then bythe generalized Nyquist criterion of Summary 1.3.13 the Nyquist plot of det( I − H1) encirclesthe origin at least once, as illustrated in Fig. 5.31. Consider the Nyquist plot of det( I − εH1),with ε a real number that varies between 0 and 1. Forε = 1 the modified Nyquist plot coincideswith that of Fig. 5.31, while forε = 0 the plot reduces to the point 1. Since obviously the plotdepends continuously onε, there must exist a value ofε in the interval(0, 1] such that theNyquist plot of det( I − εH1) passes through the origin. Hence, if1 destabilizes the perturbed

11A transfer function or matrix is “stable” if all its poles are in the open left-half plane.

233


system, there existε ∈ (0, 1] andω ∈ R such that det( I − εH(jω)1(jω)) = 0. Therefore,1doesnot destabilize the perturbed system if and only if there do not existε ∈ (0, 1] andω ∈ R

such that det( I − εH(jω1(jω)) = 0.Fix ω, and letκ(H(jω)) be thelargestreal number such that det( I − H(jω)1(jω)) 6= 0 for

all 1(jω) (with the prescribed structure) such thatσ(1(jω)) < κ(H(jω)). Then, obviously, iffor a given perturbation1(jω), ω ∈ R,

σ(1(jω)) < κ(H(jω)) for all ω ∈ R (5.156)

then det( I − εH(jω)1(jω)) 6= 0 for ε ∈ (0,1] andω ∈ R. Therefore, the Nyquist plot of det( I −H1) does not encircle the origin, and, hence,1 does not destabilize the system. Conversely, itis always possible to find a perturbation1 that violates (5.156) within an arbitrarily small marginthat destabilizes the perturbed system. The numberκ(H(jω)) is known as themultivariablerobustness marginof the systemH at the frequencyω (Safonov and Athans 1981).

We note that for fixedω

κ(H(jω)) = sup{γ | σ(1(jω)) < γ ⇒ det( I − H(jω)1(jω)) 6= 0}= inf{σ(1(jω)) | det( I − H(jω)1(jω)) = 0}. (5.157)

If no structure is imposed on the perturbation1 thenκ(H(jω)) = 1/σ(H(jω)). This led Doyleto terming the numberµ(H(jω)) = 1/κ(H(jω)) thestructured singular valueof the complex-valued matrixH(jω).

Besides being determined byH(jω) the structured singular value is of course also dependenton the perturbation structure. Given the perturbation structure of the perturbations1 of thestructured perturbation model, define byD the class ofconstantcomplex-valued matrices of theform

1 =

11 0 · · · 00 12 0 · · ·· · · · · · · · · · · ·0 · · · 0 1K

. (5.158)

The diagonal block1i has the same dimensions as the corresponding block of the dynamicperturbation, and has the following form:

• 1i = δ I , with δ a real number, if the dynamic perturbation is a scalar or repeated real scalarperturbation.

• 1i = δ I , with δ a complex number, if the dynamic perturbation is a scalar dynamic or arepeated scalar dynamic perturbation.

• 1i is a complex-valued matrix if the dynamic perturbation is a multivariable dynamicperturbation.

5.7.2. The structured singular value

We are now ready to define thestructured singular valueof a complex-valued matrixM:

Definition 5.7.1 (Structured singular value). Let M be ann × m complex-valued matrix, andD a set ofm× n structured perturbation matrices. Then thestructured singular valueof M giventhe setD is the numberµ(M) defined by

1µ(M)

= inf1∈D , det( I−M1)=0

σ(1). (5.159)

If det( I − M1) 6= 0 for all1 ∈ D thenµ(M) = 0. �

234


The structured singular valueµ(M) is the inverse of the norm of the smallest perturbation1

(within the given classD) that makesI + M1 singular. Thus, the largerµ(M), the smaller theperturbation1 is that is needed to makeI + M1 singular.

Example 5.7.2 (Structured singular value). By way of example we consider the computationof the structured singular value of a 2× 2 complex-valued matrix

M =[m11 m12

m21 m22

], (5.160)

under the real perturbation structure

1 =[11 00 12

], (5.161)

with 11 ∈ R and12 ∈ R. It is easily found that

σ(1) = max(|11|, |12|), det( I + M1) = 1+ m1111 + m2212 + m1112, (5.162)

with m = det(M) = m11m22 − m12m21. Hence, the structured singular value ofM is the inverseof

κ(M) = inf{11∈R, 12∈R: 1+m1111+m2212+m1112=0}

max(|11|, |12|). (5.163)

Elimination of, say,12 results in the equivalent expression

κ(M) = inf11∈R

max(|11|,∣∣∣∣ 1+ m1111

m22 + m11

∣∣∣∣). (5.164)

Suppose thatM is numerically given by

M =[1 23 4

]. (5.165)

Then it may be found (the details are left to the reader) that

κ(M) = −5+ √33

4, (5.166)

so that

µ(M) = 4

−5+ √33

= 5+ √33

2= 5.3723. (5.167)

�

Exercise 5.7.3 (Structured singular value). Fill in the details of the calculation. �

Exercise 5.7.4 (Alternative characterization of the structured singular value). Prove (Doyle1982) that if all perturbations are complex then

µ(M) = max1∈ D : σ(1)≤1

ρ(M1), (5.168)

with ρ denoting the spectral radius — that is, the magnitude of the largest eigenvalue (in magni-tude). Show that if on the other hand some of the perturbations are real then

µ(M) = max1∈ D : σ(1)≤1

ρR(M1). (5.169)

H1ereρR(A) is largest of the magnitudes of the real eigenvalues of the complex matrixA. If Ahas no real eigenvalues thenρR(A) = 0. �

235


5.7.3. Structured singular value and robustness

We discuss the structured singular value in more detail in§ 5.7.4, but first summarize its applica-tion to robustness analysis.

Summary 5.7.5 (Structured robust stability). Given the stable unperturbed systemH the per-turbed system of Fig. 5.30(b) is stable for all perturbations1 such that1(jω) ∈ D for all ω ∈ R

if

σ(1(jω)) <1

µ(H(jω))for all ω ∈ R, (5.170)

with µ the structured singular value with respect to the perturbation structureD . If robust sta-bility is required with respect toall perturbations within the perturbation class that satisfy thebound (5.170) then the condition is besides sufficient also necessary. �

Given a structured perturbation1 such that1(jω) ∈ D for everyω ∈ R, we have

‖1‖∞ = supω∈R

σ(1(jω)). (5.171)

Suppose that the perturbations arescaled, so that‖1‖∞ ≤ 1. Then Summary 5.7.5 implies thatthe perturbed system is stable ifµH < 1, where

µH = supω∈R

µ(H(jω)). (5.172)

With some abuse of terminology, we callµH thestructured singular valueof H. Even more canbe said:

Summary 5.7.6 (Structured robust stability for scaled perturbations). The perturbed systemof Fig. 5.30(b) is stable for all stable perturbations1 such that1(jω) ∈ D for all ω ∈ R and‖1‖∞ ≤ 1 if and only ifµH < 1. �

Clearly, if the perturbations are scaled to a maximum norm of 1 then a necessary and sufficientcondition for robust stability is thatµH < 1.

5.7.4. Properties of the structured singular value

Before discussing the numerical computation of the structured singular value we list some of theprincipal properties of the structured singular value (Doyle 1982; Packard and Doyle 1993).

Summary 5.7.7 (Principal properties of the structured singular value). The structured singu-lar valueµ(M) of a matrixM under a structured perturbation setD has the following properties:

1. Scaling property:

µ(αM) = |α| µ(M) (5.173)

for everyα ∈ R. If none of the perturbations are real then this holds as well for everycomplexα.

2. Upper and lower bounds:Suppose thatM is square. Then

ρR(M) ≤ µ(M) ≤ σ(M), (5.174)

236


with ρR defined as in Exercise 5.7.4.

If none of the perturbations is real then

ρ(M) ≤ µ(M) ≤ σ(M), (5.175)

with ρ denoting the spectral radius. The upper bounds also apply ifM is not square.

3. Preservation of the structured singular value under diagonal scaling:Suppose that theith diagonal block of the perturbation structure has dimensionsmi × ni . Form two blockdiagonal matricesD andD whoseith diagonal blocks are given by

Di = di Ini , Di = di Imi , (5.176)

respectively, withdi a positive real number.Ik denotes ak × k unit matrix. Then

µ(M) = µ(DMD−1). (5.177)

4. Preservation of the structured singular value under unitary transformation:Suppose thatthe ith diagonal block of the perturbation structure has dimensionsmi × ni . Form theblock diagonal matricesQ and Q whoseith diagonal blocks are given by themi × mi

unitary matrixQi and theni × ni unitary matrixQi , respectively. Then

µ(M) = µ(QM) = µ(MQ). (5.178)�

The scaling property is obvious. The upper bound in 2 follows by considering unrestrictedperturbations of the form1 ∈ Cm×n (that is,1 is a full m× n complex matrix). The lowerbound in 2 is obtained by considering restricted perturbations of the form1 = δ I , with δ a realor complex number. The properties 3 and 4 are easily checked.

Exercise 5.7.8 (Proof of the principal properties of the structured singular value). Fill in thedetails of the proof of Summary 5.7.7. �

5.7.5. A useful formula

The following formula for the structured singular value of a 2× 2 dyadic matrix has usefulapplications.

Summary 5.7.9 (Structured singular value of a dyadic matrix). The structured singular valueof the 2× 2 dyadic matrix

M =[a1b1 a1b2

a2b1 a2b2

]=[a1

a2

] [b1 b2

], (5.179)

with a1, a2, b1, andb2 complex numbers, with respect to the perturbation structure

1 =[11 00 12

], 11 ∈ C, 12 ∈ C, (5.180)

is

µ(M) = |a1b1| + |a2b2|. (5.181)

�

The proof is given in§ 5.9.

237


5.7.6. Numerical approximation of the structured singular value

Exact calculation of the structured singular value is often not possible and in any case compu-tationally intensive. The numerical methods that are presently available for approximating thestructured singular value are based on calculating upper and lower bounds for the structuredsingular value.

Summary 5.7.10 (Upper and lower bounds for the structured singular value).

1. D-scaling upper bound.Let the diagonal matricesD andD be chosen as in (c) of Summary5.7.7. Then with property (b) we have

µ(M) = µ(DMD−1) ≤ σ(DMD−1). (5.182)

The rightmost side may be numerically minimized with respect to the free positive num-bersdi that determineD andD.

Suppose that the perturbation structure consists ofm repeated scalar dynamic perturbationblocks andM full multivariable perturbation blocks. Then if 2m+ M ≤ 3 the minimizedupper bound actuallyequalsthe structured singular value12

2. Lower bound.Let Q be a unitary matrix as constructed in (d) of Summary 5.7.7. Withproperty (b) we have (for complex perturbations only)

µ(M) = µ(MQ) ≥ ρ(MQ). (5.183)

Actually (Doyle 1982),

µ(M) = maxQ

ρ(MQ), (5.184)

with Q varying over the set of matrices as constructed in (d) of Summary 5.7.7.�

Practical algorithms for computing lower and upper bounds on the structured singular valuefor complex perturbations have been implemented in the MATLAB µ-Analysis and SynthesisToolbox(Balas, Doyle, Glover, Packard, and Smith 1991). The closeness of the bounds is ameasure of the reliability of the calculation. Real perturbations are not handled by this versionof theµ-toolbox. Algorithms for real and mixed complex-real perturbations are the subject ofcurrent research (see for instance Young, Newlin, and Doyle (1995b) and Young, Newlin, andDoyle (1995a)).

The MATLAB Robust Control Toolbox(Chiang and Safonov 1992) provides routines for cal-culating upper bounds on the structured singular value for both complex and real perturbations.

5.7.7. Example

We apply the singular value method for the analysis of the stability robustness to an example.

Example 5.7.11 (SISO system with two parameters). By way of example, consider the SISOfeedback system that was studied in Example 5.2.1 and several other places. A plant with transferfunction

P(s) = gs2(1+ sθ)

(5.185)

12If there arereal parametric perturbations then this result remains valid, provided that we use a generalization ofD-scaling called(D,G)-scaling. This is a generalization that allows to exploit realness of perturbations.

238


is connected in feedback with a compensator with transfer function

C(s) = k + sTd

1+ sT0. (5.186)

We use the design values of the parameters established in Example 5.2.2. In Example 5.3.3 itwas found that the interconnection matrix for scaled perturbations in the gaing and time constantθ is given by

H(s) =1

1+sθ0

1+ L0(s)

[β1C(s)

−β2

][− α1s2 α2s

], (5.187)

where the subscripto indicates nominal values, andL0 = P0C is the nominal loop gain. Thenumbersα1, α2, β1, andβ2 are scaling factors such that|g − g0| ≤ ε1 with ε1 = α1β1, and|θ− θ0| ≤ ε2 with ε2 = α2β2. The interconnection matrixH has a dyadic structure. Application

10−2

10−1

100

101

102

103

ω

0

1

2

(a)

(b)

µ( ( ω))H j

Figure 5.32.: Structured singular values for the two-parameter plant for twocases

of the result of Summary 5.7.9 shows that its structured singular value with respect tocomplexperturbations in the parameters is

µ(H(jω)) =√

11+ω2θ2

0

|1+ L0(ω)|(α1β1

|C(jω)|ω2 + α2β2ω

)

=ε1

√k2 +ω2T2

d + ε2ω3√

1+ ω2T20

|χ0(jω)| , (5.188)

with χ0(s) = θ0T0s4 + (θ0 + T0)s3 + s2 + g0Tds+ g0k the nominal closed-loop characteristicpolynomial.

Inspection shows that the right-hand side of this expression is identical to that of (5.99) inExample 5.5.3, which was obtained by a singular value analysis based on optimal scaling. Inview of the statement in Summary 5.7.10(1) this is no coincidence.

Figure 5.32 repeats the plots of of Fig. 5.17 of the structured singular value for the two casesof Example 5.5.3. For case (a) the peak value of the structured singular value is greater than1 so that robust stability is not guaranteed. Only in case (b) robust stability is certain. Sincethe perturbation analysis is based on dynamic rather than parameter perturbations the results areconservative. �Exercise 5.7.12 (Computation of structured singular values with M ATLAB ). Reproduce theplots of Fig. 5.32 using the appropriate numerical routines from theµ-Toolbox or the RobustControl Toolbox for the computation of structured singular values. �

239


5.8. Combined performance and stability robustness

5.8.1. Introduction

In the preceding section we introduced the structured singular value to study the stability robust-ness of the basic perturbation model. We continue in this section by consideringperformanceand its robustness. Figure 5.33 represents a control system withexternalinputw, such as dis-turbances, measurement noise, and reference signals, andexternaloutputz. The output signalz represents anerror signal,and ideally should be zero. The transfer matrixH is the transfermatrix from the external inputw to the error signalz.

w H z

Figure 5.33.: Control system

Example 5.8.1 (Two-degree-of-freedom feedback system). To illustrate this model, considerthe two-degree-of-freedom feedback configuration of Fig. 5.34.P is the plant,C the compen-sator, andF a precompensator. The signalr is the reference signal,v the disturbance,m themeasurement noise,u the plant input, andy the control system output. It is easy to find that

y = ( I + PC)−1PCFr+ ( I + PC)−1v− ( I + PC)−1PCm

= T Fr + Sv− Tm, (5.189)

whereS is the sensitivity matrix of the feedback loop andT the complementary sensitivity ma-trix. The tracking errorz = y− r is given by

z= y− r = (T F − I )r + Sv− Tm. (5.190)

Considering the combined signal

w = rv

m

(5.191)

as the external input, it follows that the transfer matrix of the control system is

H = [T F − I S −T

]. (5.192)

�

The performance of the control system of Fig. 5.33 is ideal ifH = 0, or, equivalently,‖H‖∞ =0. Ideals cannot always be obtained, so we settle for the norm‖H‖∞ to be “small,” rather thanzero. By introducing suitable frequency dependent scaling functions (that is, by modifyingH toW HV) we may arrange that “satisfactory” performance is obtained if and only if

‖H‖∞ < 1. (5.193)

Example 5.8.2 (Performance specification). To demonstrate how performance may be speci-fied this way, again consider the two-degree-of-freedom feedback system of Fig. 5.34. For sim-plicity we assume that it is a SISO system, and that the reference signal and measurement noise

240


+

−

++

++

r F Cu

P

v

y

m

Figure 5.34.: Two-degree-of-freedom feedback control system.

are absent. It follows from Example 5.8.1 that the control system transfer function reduces to thesensitivity functionSof the feedback loop:

H = 11+ PC

= S. (5.194)

Performance is deemed to be adequate if the sensitivity function satisfies the requirement

.01 .1 1 10 100

10

1

.1

.01

.001

.0001

ω

| |Sspec

Figure 5.35.: Magnitude plot ofSspec.

|S(jω)| < |Sspec(jω)|, ω ∈ R, (5.195)

with Sspeca specified rational function, such as

Sspec(s) = s2

s2 + 2ζω0s+ ω20

. (5.196)

The parameterω0 determines the minimal bandwidth, while the relative damping coefficientζ specifies the allowable amount of peaking. A doubly logarithmic magnitude plot ofSspec isshown in Fig. 5.35 forω0 = 1 andζ = 1

2.The inequality (5.195) is equivalent to|S(jω)V(jω)| < 1 for ω ∈ R, with V = 1/Sspec. Re-

definingH asH = SV this reduces to

‖H‖∞ < 1. (5.197)

�

241


By suitable scaling we thus may arrange that the performance of the system of Fig. 5.33 isconsidered satisfactory if

‖H‖∞ < 1. (5.198)

In what follows we exploit the observation that by (1) of Summary 5.5.1 this specification isequivalent to the requirement that the artificially perturbed system of Fig. 5.36 remains stableunder all stable perturbations10 such that‖10‖∞ ≤ 1.

w H

∆0

z

Figure 5.36.: Artificially perturbed system

5.8.2. Performance robustness

Performance is said to berobustif ‖H‖∞ remains less than 1 under perturbations. To make thisstatement more specific we consider the configuration of Fig. 5.37(a). The perturbation1 maybe structured, and is scaled such that‖1‖∞ ≤ 1. We describe the system by the interconnectionequations[

zq

]= H

[w

p

]=[

H11 H12

H21 H22

][w

p

]. (5.199)

H11 is the nominal control system transfer matrix. We define performance to be robust if

1. the perturbed system remains stable under all perturbations, and

2. the∞-norm of the transfer matrix of the perturbed system remains less than 1 under allperturbations.

wH

z

qp

∆ ∆

∆0

w z

p qH

(a) (b)

Figure 5.37.: (a) Perturbed control system. (b) Doubly perturbed system.

242


Necessary and sufficient for robust performance is that the norm of the perturbed transfer matrixfromw to z in the perturbation model of Fig. 5.37(a) is less than 1 for every perturbation1 withnorm less than or equal to 1. This, in turn, is equivalent to the condition that the augmentedperturbation model of Fig. 5.37(b) is stable for every “full” perturbation10 and every structuredperturbation1, both with norm less than or equal to 1. Necessary and sufficient for this is that

µH < 1, (5.200)

with µ the structured singular value ofH with respect to the perturbation structure defined by[10 00 1

]. (5.201)

Summary 5.8.3 (Robust performance and stability). Robust performance of the system ofFig. 5.37(a) is achieved if and only if

µH < 1, (5.202)

whereµH is the structured singular value ofH under perturbations of the form[10 00 1

]. (5.203)

10 is a “full” perturbation, and1 structured as specified. �

−

+ +

− +

+

+

+

Cu

P

w

z

Cu

P

pq

w

yz

W2

W1

δP

(a)

(b)

Figure 5.38.: SISO feedback system. (a) Nominal. (b) Perturbed.

5.8.3. SISO design for robust stability and performance

We describe an elementary application of robust performance analysis using the structured singu-lar value (compare Doyle, Francis, and Tannenbaum (1992)). Consider the feedback control sys-tem configuration of Fig. 5.38(a). A SISO plantP is connected in feedback with a compensatorC.

243


• Performance is measured by the closed-loop transfer function from the disturbancew to thecontrol system outputz, that is, by the sensitivity functionS= 1/(1+ PC). Performanceis considered satisfactory if

|S(jω)W1(jω)| < 1, ω ∈ R, (5.204)

with W1 a suitable weighting function.

• Plant uncertainty is modeled by the scaled uncertainty model

P −→ P(1+ δPW2), (5.205)

with W2 a stable function representing the maximal uncertainty, andδP a scaled stableperturbation such that‖δP‖∞ ≤ 1.

The block diagram of Fig. 5.38(b) includes the weighting filterW1 and the plant perturbationmodel. By inspection we obtain the signal balance equationy = w+ p− PCy, so that

y = 11+ PC

(w+ p). (5.206)

By further inspection of the block diagram it follows that

z = W1y = W1S(w+ p), (5.207)

q = −W2PCy = −W2T(w+ p). (5.208)

Here

S= 11+ PC

, T = PC1+ PC

(5.209)

are the sensitivity function and the complementary sensitivity function of the feedback system,respectively. Thus, the transfer matrixH in the configuration of Fig. 5.37(b) follows from[

qz

]=[−W2T −W2T

W1S W1S

]︸︷︷︸

H

[pw

]. (5.210)

H has the dyadic structure

H =[

W2TW1S

] [−1 1]. (5.211)

With the result of Summary 5.7.9 we obtain the structured singular value of the interconnectionmatrix H as

µH = supω∈R

µ(H(jω)) = supω∈R

(|W1(jω)S(jω)| + |W2(jω)T(jω)|) (5.212)

= ‖ |W1S| + |W2T| ‖∞. (5.213)

By Summary 5.8.3, robust performance and stability are achieved if and only ifµH < 1.

244


.01.01 11 100100

11

1010

.1.1

.01.01

.001.001

ω ω

| |S | |T

Figure 5.39.: Nominal sensitivity and complementary sensitivity functions

Example 5.8.4 (Robust performance of SISO feedback system). We consider the SISO feed-back system we studied in Example 5.2.1 (p. 198) and on several other occasions, with nominalplant and compensator transfer functions

P0(s) = g0

s2, C(s) = k + sTd

1+ sT0, (5.214)

respectively. We use the design values of Example 5.2.2 (p. 199). In Fig. 5.39 magnitude plots aregiven of the nominal sensitivity and complementary sensitivity functions of this feedback system.The nominal sensitivity function is completely acceptable, so we impose as design specificationthat under perturbation∣∣∣∣ S(jω)

S0(jω)

∣∣∣∣ ≤ 1+ ε, ω ∈ R, (5.215)

with S0 the nominal sensitivity function and the positive numberε a tolerance. This comes downto choosing the weighting functionW1 in (5.204) as

W1 = 1(1+ ε)S0

. (5.216)

We consider how to select the weighting functionW2, which specifies the allowable perturba-tions. With W1 chosen as in (5.216) the robust performance criterionµH < 1 according to(5.212) reduces to

11+ ε

+ |W2(jω)T0(jω)| < 1, ω ∈ R, (5.217)

with T0 the nominal complementary sensitivity function. This is equivalent to

|W2(jω)| <ε

1+ε|T0(jω)| , ω ∈ R. (5.218)

Figure 5.40(a) shows a plot of the right-hand side of (5.218) forε = 0.25. This right-hand side isa frequency dependent bound for the maximally allowable scaled perturbationδP. The plot showsthat for low frequencies the admissible proportional uncertainty is limited toε/(1 + ε) = 0.2,

245


.01 1 100

1

10

.1

.01

.001

bound

ω

| |∆P

(a) Uncertainty bound and admissibleperturbation

.01 1 100

1

10

.1

.01

.001

ω

| |S

|( ) |1+ ε So

(b) Effect of perturbation onS

Figure 5.40.: The bound and its effect onS

but that the system is much more robust with respect to high-frequency perturbations. In thecrossover frequency region the allowable perturbation is even less than 0.2.

Suppose, as we did before, that the plant is perturbed to

P(s) = gs2(1+ sθ)

. (5.219)

The corresponding proportional plant perturbation is

1P(s) = P(s)− P0(s)P0(s)

=g−g0

g0− sθ

1+ sθ. (5.220)

At low frequencies the perturbation has approximate magnitude|g − g0|/g0. For performancerobustness this number definitely needs to be less than the minimum of the bound on the right-hand side of (5.218), which is aboutε1+ε 0.75.

If ε = 0.25 then the latter number is about 0.15. Figure 5.40(a) includes a magnitude plotof 1P for |g− g0|/g0 = 0.1 (that is,g = 0.9 or g = 1.1) andθ = 0.1. For this perturbation theperformance robustness test is just satisfied. Ifg is made smaller than 0.9 or larger than 1.1 orθ

is increased beyond 0.1 then the perturbation fails the test.Figure 5.40(b) shows the magnitude plot of the sensitivity functionSfor g = 1.1 andθ = 0.1.

Comparison with the bound|(1+ ε)S0| shows that performance is robust, as expected. �

Exercise 5.8.5 (Bound on performance robustness). In Example 5.8.4 we used the perfor-mance specification (in the form ofW1) to find bounds on the maximally allowable scaled pertur-bationsδP. Conversely, we may use the uncertainty specification (in the form ofW2) to estimatethe largest possible performance variations.

1. Suppose thatg varies between 0.5 and 1.5 andθ between 0 and 0.2. Determine a functionW2 that tightly bounds the uncertainty1P of (5.220).

2. Use (5.218) to determine the smallest value ofε for which robust performance is guaran-teed.

246


3. Compute the sensitivity functionSfor a number of values of the parametersg andθ withinthe uncertainty region. Check how tight the bound on performance is that follows from thevalue ofε obtained in 2.

�


In this Appendix to Chapter 5 the proofs of certain results in are sketched.

5.9.1. Small gain theorem

We prove Summary 5.4.2.

Proof of Small gain theorem.The proof of the small gain theorem follows by applying the fixed point the-orem to the feedback equationw = v + Lw that follows from Fig. 5.12(b). Denote the normed signalspace thatL operates on asU . For any fixed inputv ∈ U , if L is a contraction, so is the map defined byw 7→ v+ Lw. Hence, ifL is a contraction, for everyv ∈ U the feedback equation has a unique solutionw ∈ U . It follows that the system with inputv and outputw is BIBO stable.

If L is a contraction with contraction constantk then it follows from‖Lu‖U ≤ k‖u‖U that ‖L‖ ≤ k.Hence,‖L‖ < 1. Conversely, if‖L‖ < 1, for anyu andv in U we have‖Lu − Lv‖U = ‖L(u − v)‖U ≤‖L‖ · ‖u − v‖U , so thatL is a contraction. This proves thatL is a contraction if and only if its norm isstrictly less than 1.

5.9.2. Structured singular value of a dyadic matrix

We prove Summary 5.7.9. In fact we prove the generalization that for rank 1 matrices

M =

a1

a2

...

[b1 b2 · · ·] , ai ,bj ∈ C (5.221)

and perturbation structure

1 =11 0 · · ·

0 12 · · ·· · · · · · · · ·

, 1i ∈ C (5.222)

there holds thatµ(M) =∑i |ai bi |.

Proof of structured singular value of dyadic matrix.Given M as the product of a column and row vectorM = abwe have

det( I − M1) = det( I − ab1) = det(1− b1a) = 1−∑

i

ai bi1i . (5.223)

This gives a lower bound for the real part of det( I − M1),

det( I − M1) = 1−∑

i

ai bi1i ≥ 1−∑

i

|ai bi |maxi

|1i | = 1−∑

i

|ai bi |σ(1). (5.224)

For det( I − M1) to be zero we need at least that the lower bound is not positive, that is, thatσ(1) ≥1/∑

i |ai bi |. Now take1 equal to

1 = 1∑i |ai bi |

sgn(a1b1) 0 · · ·0 sgn(a2b2) · · ·· · · · · · ...

. (5.225)

247


Thenσ(1) = 1/∑

i |ai bi | and it is such that

det( I − M1) = 1−∑

j

aj bj1 j = 1−∑

j

aj bjsgnaj bj∑

i |ai bi | = 0. (5.226)

Hence a smallest1 (in norm) that makesI − M1 singular has norm 1/∑

i |ai bi |. Thereforeµ(M) =∑i |ai bi |.

It may be shown that theD-scaling upper bound for rank 1 matrices equalsµ(M) in this case.

248

6. H∞-Optimization and µ-Synthesis

Overview– Design byH∞-optimization involves the minimization of thepeak magnitude of a suitable closed-loop system function. It is very wellsuited to frequency response shaping. Moreover, robustness against plantuncertainty may be handled more directly than withH2 optimization.

Design byµ-synthesis aims at reducing the peak value of the struc-tured singular value. It accomplishes joint robustness and performanceoptimization.

6.1. Introduction

In this chapter we introduce what is known asH∞-optimization as a design tool for linear mul-tivariable control systems.H∞-optimization amounts to the minimization of the∞-norm of arelevant frequency response function. The name derives from the fact that mathematically theproblem may be set in the spaceH∞ (named after the British mathematician G. H. Hardy), whichconsists of all bounded functions that are analytic in the right-half complex plane. We do not goto this length, however.

H∞-optimization resemblesH2-optimization, where the criterion is the 2-norm. Becausethe 2- and∞-norms have different properties the results naturally are not quite the same. Animportant aspect ofH∞ optimization is that it allows to include robustness constraints explicitlyin the criterion.

In § 6.2 (p. 250) we discuss themixed sensitivity problem.This specialH∞ problem is animportant design tool. We show how this problem may be used to achieve the frequency responseshaping targets enumerated in§ 1.5 (p. 27).

In § 6.3 (p. 258) we introduce thestandard problemof H∞-optimization, which is the mostgeneral version. The mixed sensitivity problem is a special case. Several other special cases ofthe standard problem are exhibited. Closed-loop stability is defined and a number of assumptionsand conditions needed for the solution of theH∞-optimization problem are reviewed.

The next few sections are devoted to the solution of the standardH∞-optimization problem.In § 6.4 (p. 264) we derive a formula for so-calledsuboptimalsolutions and establish a lowerbound for the∞-norm. We also establish how allstabilizingsuboptimal solutions, if any exist,may be obtained.

The key tool in solvingH∞-optimization problems isJ-spectral factorization. In § 6.5(p. 272) we show that by using state space techniques theJ-spectral factorization that is re-

249

6. H∞-Optimization andµ-Synthesis

quired for the solution of the standard problem may be reduced to the solution of two algebraicRiccati equations.

In § 6.6 (p. 275) we discuss optimal (as opposed tosuboptimal) solutions and highlight someof their peculiarities. Section 6.7 (p. 279) explains how integral control and high-frequency roll-off may be handled.

Section 6.8 (p. 288) is devoted to an introductory exposition ofµ-synthesis. This approximatetechnique for joint robustness and performance optimization usesH∞ optimization to reduce thepeak value of the structured singular value. Section 6.9 (p. 292) is given over to a rather elaboratedescription of an application ofµ-synthesis.

Section 6.10 (p. 304), finally, contains a number of proofs.

6.2. The mixed sensitivity problem

6.2.1. Introduction

In § 5.6.6 (p. 227) we studied the stability robustness of the feedback configuration of Fig. 6.1.We considered fractional perturbations of the type

C P−


L = N D−1 −→ ( I + VδNW2)N D−1( I + VδDW1)−1. (6.1)

The frequency dependent matricesV, W1, and W2 are so chosen that the scaled perturbationδp = [−δD δN] satisfies‖δP‖∞ ≤ 1. Stability robustness is guaranteed if

‖H‖∞ < 1, (6.2)

where

H =[

W1SV−W2TV

]. (6.3)

S= ( I + L)−1 andT = L( I + L)−1 are the sensitivity matrix and the complementary sensitivitymatrix of the closed-loop system, respectively, withL the loop gainL = PC.

Given a feedback system with compensatorC and corresponding loop gainL = PC thatdoes not satisfy the inequality (6.2) one may look for a different compensator that does achieveinequality. An effective way of doing this is to consider the problem ofminimizing‖H‖∞ withrespect to all compensatorsC that stabilize the system. If the minimal valueγ of ‖H‖∞ isgreater than 1 then no compensator exists that stabilizes the systems for all perturbations suchthat‖δP‖∞ ≤ 1. In this case, stability robustness is only obtained for perturbations satisfying‖δP‖∞ ≤ 1/γ.

The problem of minimizing∥∥∥∥[

W1SV−W2TV

]∥∥∥∥∞

(6.4)

250


(Kwakernaak 1983; Kwakernaak 1985) is a version of what is known as themixed sensitivityproblem(Verma and Jonckheere 1984). The name derives from the fact that the optimizationinvolves both the sensitivity and the complementary sensitivity function.

In what follows we explain that the mixed sensitivity problem cannot only be used to verifystability robustness for a class of perturbations, but also to achieve a number of other importantdesign targetsfor the one-degree-of-freedom feedback configuration of Fig. 6.1.

Before starting on this, however, we introduce a useful modification of the problem. We maywrite W2TV = W2L( I + L)−1V = W2PC( I + L)−1V = W2PUV, where

U = C( I + PC)−1 (6.5)

is the input sensitivity matrix introduced in§ 1.5 (p. 27) . For a fixed plant we may absorbthe plant transfer matrixP (and the minus sign) into the weighting matrixW2, and consider themodified problem of minimizing∥∥∥∥

[W1SVW2UV

]∥∥∥∥∞

(6.6)

with respect to all stabilizing compensators. We redefine the problem of minimizing (6.6) as themixed sensitivity problem.

We mainly consider the SISO mixed sensitivity problem. The criterion (6.6) then reduces tothe square root of the scalar quantity

supω∈R

(|W1(jω)S(jω)V(jω)|2 + |W2(jω)U(jω)V(jω)|2). (6.7)

Many of the conclusions also hold for the MIMO case, although their application may be moreinvolved.

6.2.2. Frequency response shaping

The mixed sensitivity problem may be used for simultaneously shaping the sensitivity and inputsensitivity functions. The reason is that the solution of the mixed sensitivity sensitivity problemoften has theequalizing property(see§ 6.6.1, p. 275). This property implies that the frequencydependent function

|W1(jω)S(jω)V(jω)|2 + |W2(jω)U(jω)V(jω)|2, (6.8)

whose peak value is minimized, actually is aconstant(Kwakernaak 1985). If we denote theconstant asγ2, with γ nonnegative, then it immediately follows from

|W1(jω)S(jω)V(jω)|2 + |W2(jω)U(jω)V(jω)|2 = γ2 (6.9)

that for the optimal solution

|W1(jω)S(jω)V(jω)|2 ≤ γ2, ω ∈ R,

|W2(jω)U(jω)V(jω)|2 ≤ γ2, ω ∈ R.(6.10)

Hence,

|S(jω)| ≤ γ

|W1(jω)V(jω)| , ω ∈ R, (6.11)

|U(jω)| ≤ γ

|W2(jω)V(jω)| , ω ∈ R. (6.12)

251


By choosing the functionsW1, W2, andV correctly the functionsS andU may be made smallin appropriate frequency regions. This is also true if the optimal solution does not have theequalizing property.

If the weighting functions are suitably chosen (in particular, withW1V large at low frequen-cies andW2V large at high frequencies), then often the solution of the mixed sensitivity problemhas the property that the first term of the criterion dominates at low frequencies and the secondat high frequencies:

|W1(jω)S(jω)V(jω)|2︸︷︷︸dominates at low frequencies

+ |W2(jω)U(jω)V(jω)|2︸︷︷︸dominates at high frequencies

= γ2. (6.13)

As a result,

|S(jω)| ≈ γ

|W1(jω)V(jω)| for ω small, (6.14)

|U(jω)| ≈ γ

|W2(jω)V(jω)| for ω large. (6.15)

This result allows quite effective control over the shape of the sensitivity and input sensitivityfunctions, and, hence, over the performance of the feedback system.

Because the∞-norm involves the supremum frequency response shaping based on minimiza-tion of the∞-norm is more direct than for theH2 optimization methods of§ 3.5 (p. 127).

Note, however, that the limits of performance discussed in§ 1.6 (p. 40) can never be violated.Hence, the weighting functions must be chosen with the respect due to these limits.

6.2.3. Type k control and high-frequency roll-off

In (6.14–6.15), equality may often be achievedasymptotically.

Type k control. Suppose that|W1(jω)V(jω)| behaves as 1/ωk asω→ 0, with k a nonnegativeinteger. This is the case ifW1(s)V(s) includes a factorsk in the denominator. Then|S(jω)|behaves asωk asω → 0, which implies a typek control system, with excellent low-frequencydisturbance attenuation ifk ≥ 1. If k = 1 then the system has integrating action.

High-frequency roll-off. Likewise, suppose that|W2(jω)V(jω)| behaves asωm asω → ∞.This is the case ifW2V is nonproper, that is, if the degree of the numerator ofW2V exceeds thatof the denominator (bym). Then|U(jω)| behaves asω−m asω→ ∞. FromU = C/(1+ PC) itfollows thatC = U/(1+ U P). Hence, ifP is strictly proper andm ≥ 0 then alsoC behaves asω−m, andT = PC/(1+ PC) behaves asω−(m+e), with e the pole excess1 of P.

Hence, by choosingm we pre-assign thehigh-frequency roll-offof the compensator transferfunction, and the roll-offs of the complementary and input sensitivity functions. This is importantfor robustness against high-frequency unstructured plant perturbations.

Similar techniques to obtain typek control and high-frequency roll-off are used in§ 3.5.4(p. 130) and§ 3.5.5 (p. 132) , respectively, forH2 optimization.

1The pole excess of a rational transfer functionP is the difference between the number of poles and the number ofzeros. This number is also known as therelative degreeof P

252


6.2.4. Partial pole placement

There is a further important property of the solution of the mixed sensitivity problem that needsto be discussed before considering an example. This involves a pole cancellation phenomenonthat is sometimes misunderstood. The equalizing property of§ 6.2.2 (p. 251) implies that

W1(s)W1(−s)S(s)S(−s)V(s)V(−s)+ W2(s)W2(−s)U(s)U(−s)V(s)V(−s)= γ2 (6.16)

for all s in the complex plane. We write the transfer functionP and the weighting functionsW1,W2, andV in rational form as

P = ND, W1 = A1

B1, W2 = A2

B2, V = M

E, (6.17)

with all numerators and denominators polynomials. If also the compensator transfer function isrepresented in rational form as

C = YX

(6.18)

then it easily follows that

S= DXDX + NY

, U = DYDX + NY

. (6.19)

The denominator

Dcl = DX + NY (6.20)

is the closed-loop characteristic polynomial of the feedback system. SubstitutingS andU asgiven by (6.19) into (6.16) we easily obtain

D∼ D · M∼ M · (A∼1 A1B∼

2 B2X∼ X + A∼2 A2B∼

1 B1Y∼Y)

E∼ E · B∼1 B1 · B∼

2 B2 · D∼cl Dcl

= γ2. (6.21)

If A is any rational or polynomial function thenA∼ is defined byA∼(s) = A(−s).Since the right-hand side of (6.21) is a constant, all polynomial factors in the numerator of the

rational function on the left cancel against corresponding factors in the denominator. In particular,the factorD∼ D cancels. If there are no cancellations betweenD∼ D and E∼EB∼

1 B1B∼2 B2 then

the closed-loop characteristic polynomialDcl (which by stability has left-half plane roots only)necessarily has among its roots those roots ofD that lie in the left-half plane, and the mirrorimages with respect to the imaginary axis of those roots ofD that lie in the right-half plane. Thismeans that the open-loop poles (the roots ofD), possibly after having been mirrored into theleft-half plane,reappearas closed-loop poles.

This phenomenon, which is not propitious for good feedback system design, may be pre-vented by choosing the denominator polynomialE of V equal to the plant denominator polyno-mial D, so thatV = M/D. With this special choice of the denominator ofV, the polynomialEcancels againstD in the left-hand side of (6.21), so that the open-loop poles donot reappear asclosed-loop poles.

Further inspection of (6.21) shows that if there are no cancellations betweenM∼ M andE∼ EB∼

1 B1B∼2 B2, and we assume without loss of generality thatM has left-half plane roots only,

then the polynomialM cancels against a corresponding factor inDcl. If we takeV proper(whichensuresV(jω) to be finite at high frequencies) then the polynomialM has the same degree asD,and, hence, has the same number of roots asD.

253


All this means that by letting

V = MD, (6.22)

where the polynomialM has the same degree as the denominator polynomialD of the plant, theopen-loop poles (the roots ofD) are reassigned to the locations of the roots ofM. By suitablychoosing the remaining weighting functionsW1 andW2 these roots may often be arranged to bethedominantpoles.

This technique, known aspartial pole placement(Kwakernaak 1986; Postlethwaite, Tsai,and Gu 1990) allows further control over the design. It is very useful in designing for a specifiedbandwidth and good time response properties.

In the design examples in this chapter it is illustrated how the ideas of partial pole placementand frequency shaping are combined.

A discussion of the root loci properties of the mixed sensitivity problem may be found inChoi and Johnson (1996).

6.2.5. Example: Double integrator

We apply the mixed sensitivity problem to the same example as in§ 3.6.2 (p. 133), whereH2

optimization is illustrated2. Consider a SISO plant with nominal transfer function

P0(s) = 1s2. (6.23)

The actual, perturbed plant has the transfer function

P(s) = gs2(1+ sθ)

, (6.24)

whereg is nominally 1 and the nonnegative parasitic time constantθ is nominally 0.

Perturbation analysis. We start with a preliminary robustness analysis. The variations in theparasitic time constantθ mainly cause high-frequency perturbations, while the low-frequencyperturbations are primarily the effect of the variations in the gaing. Accordingly, we modelthe effect of the parasitic time constant as anumeratorperturbation, and the gain variations asdenominatorperturbations, and write

P(s) = N(s)D(s)

=1

1+sθs2

g

. (6.25)

Correspondingly, the relative perturbations of the denominator and the numerator are

D(s)− D0(s)D0(s)

= 1g

− 1,N(s)− N0(s)

N0(s)= −sθ

1+ sθ. (6.26)

The relative perturbation of the denominator is constant over all frequencies, also in the crossoverregion. Because the plant is minimum-phase trouble-free crossover may be achieved (that is,without undue peaking of the sensitivity and complementary sensitivity functions). Hence, weexpect that — in the absence of other perturbations — values of|1/g− 1| up to almost 1 may betolerated.

2Much of the text of this subsection has been taken from Kwakernaak (1993).

254


The size of the relative perturbation of the numerator is less than 1 for frequencies below1/θ, and equal to 1 for high frequencies. To prevent destabilization it is advisable to makethe complementary sensitivity small for frequencies greater than 1/θ. As the complementarysensitivity starts to decrease at the closed-loop bandwidth, the largest possible value ofθ dictatesthe bandwidth. Assuming that performance requirements specify the system to have a closed-loop bandwidth of 1, we expect that — in the absence of other perturbations — values of theparasitic time constantθ up to 1 do not destabilize the system.

Thus, both for robustness and for performance, we aim at a closed-loop bandwidth of 1with small sensitivity at low frequencies and a sufficiently fast decrease of the complementarysensitivity at high frequencies with a smooth transition in the crossover region.

Choice of the weighting functions. To accomplish this with a mixed sensitivity design, wesuccessively consider the choice of the functionsV = M/D (that is, of the polynomialM), W1

andW2.To obtain a good time response corresponding to the bandwidth 1, which does not suffer from

sluggishness or excessive overshoot, we assign two dominant poles to the locations12

√2(−1± j).

This is achieved by choosing the polynomialM as

M(s) = [s− 12

√2(−1+ j)][s− 1

2

√2(−1− j)] = s2 + s

√2+ 1, (6.27)

so that

V(s) = s2 + s√

2+ 1s2

. (6.28)

We tentatively choose the weighting functionW1 equal to 1. Then if the first of the two terms ofthe mixed sensitivity criterion dominates at low frequencies from we have from (6.14) that

|S(jω)| ≈ γ

|V(jω)| = γ

∣∣∣∣ (jω)2

(jω)2 + jω√

2+ 1

∣∣∣∣ at low frequencies. (6.29)

Figure 6.2 shows the magnitude plot of the factor 1/V. The plot implies a very good low-

1/| |V

ω

.01 .1 1 10 100

1

.1

.01

.001

10

.0001

Figure 6.2.: Bode magnitude plot of 1/V

frequency behavior of the sensitivity function. Owing to the presence of the double open-loop

255


pole at the origin the feedback system is of type 2. There is no need to correct this low frequencybehavior by choosingW1 different from 1.

Next contemplate the high-frequency behavior. For high frequenciesV is constant and equalto 1. Consider choosingW2 as

W2(s) = c(1+ rs), (6.30)

with c andr nonnegative constants such thatc 6= 0. Then for high frequencies the magnitude ofW2(jω) asymptotically behaves asc if r = 0, and ascrω if r 6= 0.

Hence, if r = 0 then the high-frequency roll-off of the input sensitivity functionU andthe compensator transfer functionC is 0 and that of the complementary sensitivityT is 2decades/decade (40 dB/decade).

If r 6= 0 thenU and C roll off at 1 decade/decade (20 dB/decade) andT rolls off at 3decades/decade (60 dB/decade).

c=10

c=1

c=1/10

c=1/100

c=1/10

c=1

c=10

c=1/100

.01 .1 1 10 100 .01 .1 1 10 100

10

1

.1

.01

.001

10

1

.1

.01

.001

sensitivity | |Scomplementarysensitivity | |T

ω ω

Figure 6.3.: Bode magnitude plots ofSandT for r = 0

Solution of the mixed sensitivity problem. We first study the caser = 0, which results in aproper but not strictly proper compensator transfer functionC, and a high-frequency roll-off ofTof 2 decades/decade. Figure 6.3 shows3 the optimal sensitivity functionSand the correspondingcomplementary sensitivity functionT for c = 1/100,c = 1/10, c = 1, andc = 10. Inspectionshows that asc increases,|T| decreases and|S| increases, which conforms to expectation. Thesmallerc is, the closer the shape of|S| is to that of the plot of Fig. 6.2.

We choosec = 1/10. This makes the sensitivity small with little peaking at the cut-offfrequency. The corresponding optimal compensator has the transfer function

C(s) = 1.2586s+ 0.619671+ 0.15563s

, (6.31)

and results in the closed-loop poles12

√2(−1± j) and−5.0114. The two former poles dominate

the latter pole, as planned. The minimal∞-norm is‖H‖∞ = 1.2861.

3The actual computation of the compensator is discussed in Example 6.6.1 (p. 276).

256


Robustness against high-frequency perturbations may be improved by making the comple-mentary sensitivity functionT decrease faster at high frequencies. This is accomplished by takingthe constantr nonzero. Inspection ofW2 as given by (6.30) shows that by choosingr = 1 theresulting extra roll-off ofU, C, andT sets in at the frequency 1. Forr = 1/10 the break pointis shifted to the frequency 10. Figure 6.4 shows the resulting magnitude plots. Forr = 1/10

r = 1

r = 1/10

r = 0

r = 1

r = 1/10

r = 0

.01 .1 1 10 100 .01 .1 1 10 100

10

1

.1

.01

.001

10

1

.1

.01

.001

sensitivity| |Scomplementarysensitivity | |T

ω ω

Figure 6.4.: Bode magnitude plots ofSandT for c = 1/10

the sensitivity function has little extra peaking while starting at the frequency 10 the comple-mentary sensitivity function rolls off at a rate of 3 decades/decade. The corresponding optimalcompensator transfer function is

C(s) = 1.2107s+ 0.5987

1+ 0.20355s+ 0.01267s2, (6.32)

which results in the closed-loop poles12

√2(−1± j) and−7.3281± j1.8765. Again the former

two poles dominate the latter. The minimal∞-norm is‖H‖∞ = 1.3833.

Inspection of the two compensators (6.31) and (6.32) shows that both basically are lead com-pensators with high-frequency roll-off.

Robustness analysis. We conclude this example with a brief analysis to check whether ourexpectations about robustness have come true. Given the compensatorC = Y/X the closed-loop characteristic polynomial of the perturbed plant isDcl(s) = D(s)X(s) + N(s)Y(s) =(1+ sθ)s2X(s)+ gY(s). By straightforward computation, which involves fixing one of the twoparametersg andθ and varying the other, the stability region of Fig. 6.5 may be establishedfor the compensator (6.31). That for the other compensator is similar. The diagram shows thatfor θ = 0 the closed-loop system is stable for allg > 0, that is, for all−1< 1

g − 1< ∞. Thisstability interval is larger than predicted. Forg = 1 the system is stable for 0≤ θ < 1.179, whichalso is a somewhat larger interval than expected.

The compensator (6.31) is similar to the modified PD compensator that is obtained in Example5.2.2 (p. 199) by root locus design. Likewise, the stability region of Fig. 6.5 is similar to that ofFig. 5.5.

257


0 .5 1 1.50

2

4

6

g

θ

Figure 6.5.: Stability region

6.3. The standard H∞ optimal regulation problem

6.3.1. Introduction

The mixed sensitivity problem is a special case of the so-calledstandardH∞ optimal regulationproblem (Doyle 1984). This standard problem is defined by the configuration of Fig. 6.6. The

plant

G

compensator

K

w z

yu

Figure 6.6.: The standardH∞ optimal regulation problem.

“plant” is a given system with two sets of inputs and two sets of outputs. It is often referred toas thegeneralized plant. The signalw is anexternal input, and represents driving signals thatgenerate disturbances, measurement noise, and reference inputs. The signalu is thecontrol input.The outputz has the meaning ofcontrol error, and ideally should be zero. The outputy, finally,is theobserved output, and is available for feedback. The plant has an open-loop transfer matrixG such that[

zy

]= G

[w

u

]=[

G11 G12

G21 G22

][w

u

]. (6.33)

By connecting the feedback compensator

u = Ky (6.34)

we obtain fromy = G21w+ G22u the closed-loop signal balance equationy = G21w+ G22Ky,so thaty = ( I − G22K)−1G21w. Fromz= G11w+ G12u = G11x+ G12Ky we then have

z= [G11 + G12K( I − G22K)−1G21]︸︷︷︸H

w. (6.35)

258

6.3. The standardH∞ optimal regulation problem

Hence, the closed-loop transfer matrixH is

H = G11 + G12K( I − G22K)−1G21. (6.36)

The standardH∞-optimal regulation problem is the problem of determining a compensator withtransfer matrixK that

1. stabilizes the closed-loop system, and

2. minimizes the∞-norm‖H‖∞ of the closed-loop transfer matrixH from the external inputw to the control errorz.

W V

K P W

z w

yu

z

2

2

1 1

v

−+

+

Figure 6.7.: The mixed sensitivity problem.

To explain why the mixed sensitivity problem is a special case of the standard problem we con-sider the block diagram of Fig. 6.7, where the compensator transfer function now is denotedKrather thanC. In this diagram, the external signalw generates the disturbancev after passingthrough a shaping filter with transfer matrixV. The “control error”zhas two components,z1 andz2. The first componentz1 is the control system output after passing through a weighting filterwith transfer matrixW1. The second componentz2 is the plant inputu after passing through aweighting filter with transfer matrixW2.

It is easy to verify that for the closed-loop system

z=[

z1

z2

]=[

W1SV−W2UV

]︸︷︷︸

H

w, (6.37)

so that minimization of the∞-norm of the closed-loop transfer matrixH amounts to minimiza-tion of∥∥∥∥

[W1SVW2UV

]∥∥∥∥∞. (6.38)

In the block diagram of Fig. 6.7 we denote the input to the compensator asy, as in the standardproblem of Fig. 6.6. We read off from the block diagram that

z1 = W1Vw + W1Pu,

z2 = W2u,

y = −Vw − Pu.

(6.39)

259


Hence, the open-loop transfer matrixG for the standard problem is

G = W1V W1P

0 W2

−V −P

(6.40)

Other well-known special cases of the standard problemH∞ problem are themodel matchingproblemand theminimum sensitivity problem(see Exercises 6.3.2 and 6.3.3, respectively.)

TheH∞-optimal regulation problem is treated in detail by Green and Limebeer (1995) andZhou, Doyle, and Glover (1996).

Exercise 6.3.1 (Closed-loop transfer matrix). Verify (6.37). �

Exercise 6.3.2 (The model matching problem). The model matching problem consists of find-ing a stable transfer matrixK that minimizes

‖P− K‖L∞ := supω

σ(P(jω)− K(jω)) (6.41)

with P a given (unstable) transfer matrix. Find a block diagram that represents this problem andshow that it is a standard problem with generalized plant

G =[

P − II 0

]. (6.42)

The problem is also known as the (generalized)Nehariproblem. �

Exercise 6.3.3 (Minimum sensitivity problem). A special case of the mixed sensitivity prob-lem is the minimization of the infinity norm‖W SV‖∞ of the weighted sensitivity matrixS forthe closed-loop system of Fig. 6.1. Show that this minimum sensitivity problem is a standardproblem with open-loop plant

G =[

WV W P−V −P

]. (6.43)

The SISO version of this problem historically is the firstH∞ optimization problem that wasstudied. It was solved in a famous paper by Zames (1981). �

Exercise 6.3.4 (Two-degree-of-freedom feedback system). As a further application considerthe two-degree-of-freedom configuration of Fig. 6.8. In Fig. 6.9 the diagram is expanded with

F C Pr u

v

z

m

o+

++

− +

+

Figure 6.8.: Two-degree- of-freedom feedback configuration

a shaping filterV1 that generates the disturbancev from the driving signalw1, a shaping filter

260


V2 that generates the measurement noisem = V2w2 from the driving signalw2, a shaping filterV0 that generates the reference signalr from the driving signalw3, a weighting filterW1 thatproduces the weighted tracking errorz1 = W1(z0 − r ), and a weighting filterW2 that produces theweighted plant inputz2 = W2u. Moreover, the compensatorC and the prefilterF are combinedinto a single blockK.

V0

V1

V2

W2

W1

r

y2

z2

z1z0

w2

w1

w3

y1u

P

v

K +

+

−

+

+

+

Figure 6.9.: Block diagram for a standard problem

Define the “control error”z with componentsz1 andz2, the observed outputy with compo-nentsy1 andy2 as indicated in the block diagram, the external inputw with componentsw1, w2,andw3, and the control inputu. Determine the open-loop generalized plant transfer matrixGand the closed-loop transfer matrixH. �

6.3.2. Closed-loop stability

We need to define what we mean by the requirement that the compensatorK “stabilizes theclosed-loop system” of Fig. 6.6. First, we take stability in the sense of internal stability, asdefined in§ 1.3.2 (p. 12). Second, inspection of the block diagram of Fig. 6.10 shows that wecertainly may require the compensator to stabilize the loop around the blockG22, but that wecannot expect that it is always possible to stabilize the entire closed-loop system. “The looparoundG22 is stable” means that the configuration of Fig. 6.11 is internally stable.

Exercise 6.3.5 (Closed-loop stability). Prove that the configuration of Fig. 6.11 is internallystable if and only if the four following transfer matrices have all their poles in the open left-halfcomplex plane:

( I − KG22)−1, ( I − KG22)

−1K, ( I − G22K)−1, ( I − G22K)−1G22. (6.44)

Prove furthermore that if the loop is SISO with scalar rational transfer functionsK = Y/X, withX andY coprime polynomials4 and G22 = N0/D0, with D0 and N0 coprime, then the loop isinternally stable if and only if the closed-loop characteristic polynomialDcl = D0X − N0Y hasall its roots in the open left-half complex plane. �

4That is,X andY have no common polynomial factors.

261


++

++

w z

u y

K

G11

G12

G21

G22

G

Figure 6.10.: Feedback arrangement for the standard problem

K

G22

Figure 6.11.: The loop aroundG22

262


Since we cannot expect that it is always possible or, indeed, necessary to stabilize the entireclosed-loop system we interpret the requirement that the compensator stabilizes the closed-loopsystem as the more limited requirement that it stabilizes the loop aroundG22.

Example 6.3.6 (Stability for the model matching problem). To illustrate the point that it is notalways possible or necessary to stabilize the entire system consider the model matching problemof Exercise 6.3.2 (p. 260). Since for this problemG22 = 0, internal stability of the loop aroundG22 is achieved iffK has all its poles in the open left-half plane, that is, iffK is stable. This isconsistent with the problem formulation. Note that ifP is unstable then the closed-loop transferfunction H = P− K can never be stable ifK is stable. Hence, for the model matching problemstability of the entire closed-loop system can never be achieved (except in the trivial case thatPby itself is stable). �

Exercise 6.3.7 (Mixed sensitivity problem). Show that for the mixed sensitivity problem “sta-bility of the loop aroundG22” is equivalent to stability of the feedback loop. Discuss what addi-tional requirements are needed to ensure that the entire closed-loop system be stable.�

6.3.3. Assumptions on the generalized plant and well-posedness

Normally, various conditions are imposed on the generalized plantG to ensure its solvability.This is what we assume at this point:

(a) G is rational but not necessarily proper, whereG11 has dimensionsm1 × k1, G12 has di-mensionsm1 × k2, G21 has dimensionsm2 × k1, andG22 has dimensionsm2 × k2.

(b) G12 has full normal column rank5 and G21 has full normal row rank. This implies thatk1 ≥ m2 andm1 ≥ k2.

These assumptions are less restrictive than those usually found in theH∞ literature (Doyle,Glover, Khargonekar, and Francis 1989; Stoorvogel 1992; Trentelman and Stoorvogel 1993).Indeed the assumptions need to be tightened when we present the state space solution of theH∞problem in§ 6.5 (p. 272). The present assumptions allow nonproper weighting matrices in themixed sensitivity problem, which is important for obtaining high frequency roll-off.

Assumption (b) causes little loss of generality. If the rank conditions are not satisfied thenoften the plant inputu or the observed outputy or both may be transformed so that the rankconditions hold.

We conclude with discussingwell-posednessof feedback systems. LetK = Y X−1, with XandY stable rational matrices (possibly polynomial matrices). ThenK( I − G22K)−1 = Y(X −G22Y)−1, so that the closed-loop transfer matrix is

H = G11 + G12Y(X − G22Y)−1G21. (6.45)

The feedback system is said to bewell-posedas long asX − G22Y is nonsingular (except at afinite number of points in the complex plane). A feedback system may be well-posed even ifD0 or X is singular, so thatG and K are not well-defined. This formalism allows consideringinfinite-gain feedback systems, even if such systems cannot be realized.

The compensatorK is stabilizing iff X − G22Y is strictly Hurwitz, that is, has all its zeros6

in the open left-half plane. We accordingly reformulate theH∞ problem as the problem ofminimizing the∞-norm of H = G11 + G12X(X − G22Y)−1G21 with respect to all stable rationalX andY such thatX − G22Y is strictly Hurwitz.

5A rational matrix function has full normal column rank if it has full column rank everywhere in the complex planeexcept at a finite number of points.

6The zeros of a polynomial matrix or a stable rational matrix are those points in the complex plane where it loses rank.

263


Example 6.3.8 (An infinite-gain feedback system may be well-posed). Consider the SISOmixed sensitivity problem defined by Fig. 6.7. From (6.40) we haveG22 = −P. The system iswell-posed ifX + PY is not identical to zero. Hence, the system is well-posed even ifX = 0 (aslong asY 6= 0). ForX = 0 the system has infinite gain. �

6.4. Frequency domain solution of the standard problem

6.4.1. Introduction

In this section we outline the main steps that lead to an explicit formula for so-calledsuboptimalsolutions to the standardH∞ optimal regulation problem. These are solutions such that

‖H‖∞ < γ, (6.46)

with γ a given nonnegative number. The reason for determining suboptimal solutions is thatstrictly optimal solutions cannot be found directly. Once suboptimal solutions may be obtainedoptimal solutions may be approximated by searching on the numberγ.

Suboptimal solutions may be determined by spectral factorization in the frequency domain.This solution method applies to the wide class of problems defined in§ 6.3 (p. 258). In§ 6.5(p. 272) we specialize the results to the well-known state space solution. Some of the proofs ofthe results of this section may be found in the appendix to this chapter (§ 6.10, p. 304). The detailsare described in Kwakernaak (1996). In Subsection 6.4.7 (p. 270) an example is presented.

In what follows the∞-norm notation is to be read as

‖H‖∞ = supω∈R

σ(H(jω)), (6.47)

with σ the largest singular value. This notation is also used ifH has right-half plane poles, sothat the corresponding system is not stable and, hence, has no∞-norm. As long asH has nopoles on the imaginary axis the right-hand side of (6.47) is well-defined.

We first need the following result.

Summary 6.4.1 (Inequality for ∞-norm). Let γ be a nonnegative number andH a rationaltransfer matrix. Then‖H‖∞ ≤ γ is equivalent to either of the following statements:

(a) H∼ H ≤ γ2 I on the imaginary axis;

(b) H H∼ ≤ γ2 I on the imaginary axis.

�

H∼ is defined byH∼(s) = HT(−s). H∼ is called theadjoint of H. If A and B are twocomplex-valued matrices thenA ≤ B means thatB − A is a nonnegative-definite matrix. “Onthe imaginary axis” means for alls= jω, ω ∈ R. The proof of Summary 6.4.1 may be found in§ 6.10 (p. 304).

To motivate the next result we consider the model matching problem of Exercise 6.3.2(p. 260). For this problem we haveH = P − K, with P a given transfer matrix, andK a stabletransfer matrix to be determined. SubstitutingH = P − K into H∼ H ≤ γ2 I we obtain fromSummary 6.4.1(a) that‖H‖∞ ≤ γ is equivalent to

P∼ P− P∼K − K∼ P+ K∼K ≤ γ2 I on the imaginary axis. (6.48)

264


This in turn we may rewrite as

[I K∼][γ2 I − P∼ P P∼

P − I

]︸︷︷︸

5γ

[IK

]≥ 0 on the imaginary axis. (6.49)

This expression defines the rational matrix5γ . Write K = Y X−1, with X andY rational matriceswith all their poles in the left-half complex plane. Then by multiplying (6.49) on the right byXand on the left byX∼ we obtain

[ X∼ Y∼]

[γ2 I − P∼ P P∼

P − I

]︸︷︷︸

5γ

[XY

]≥ 0 on the imaginary axis. (6.50)

The inequality (6.50) characterizes all suboptimal compensatorsK. A similar inequality, lesseasily obtained than for the model matching problem, holds for the general standard problem:

Summary 6.4.2 (Basic inequality). The compensatorK = Y X−1 achieves‖H‖∞ ≤ γ for thestandard problem, withγ a given nonnegative number, if and only if

[ X∼ Y∼]5γ

[XY

]≥ 0 on the imaginary axis, (6.51)

where5γ is the rational matrix

5γ =[

0 I

−G∼12 −G∼

22

][G11G∼

11 − γ2 I G11G∼21

G21G∼11 G21G∼

21

]−1[0 −G12

I −G22

]. (6.52)

�

An outline of the proof is found in§ 6.10 (p. 304).

6.4.2. Spectral factorization

A much more explicit characterization of all suboptimal compensators (stabilizing or not) thanthat of Summary 6.4.2 follows by finding aspectral factorizationof the rational matrix5γ , ifone exists.

Note that5γ is para-Hermitian, that is,5∼γ =5γ. A para-Hermitian rational matrix5 has a

spectral factorization if there exists a nonsingular square rational matrixZ such that bothZ andits inverseZ−1 have all their poles in the closed left-half complex plane, and

5 = Z∼ J Z, (6.53)

with J a signature matrix. Zis called aspectral factor. J is a signature matrix if it is a constantmatrix of the form

J =[

In 00 − Im

], (6.54)

265


with In and Im unit matrices of dimensionsn × n andm× m, respectively. On the imaginaryaxis5 is Hermitian. If5 has no poles or zeros7 on the imaginary axis thenn is the number ofpositive eigenvalues of5 on the imaginary axis andm the number of negative eigenvalues.

The factorization5 = Z∼ J Z, if it exists, is not at all unique. IfZ is a spectral factor, then sois U Z, whereU is any rational or constant matrix such thatU∼ JU = J. U is said to beJ-unitary.There are many such matrices. If

J =[1 00 −1

], (6.55)

for instance, then all constant matricesU, with

U =[

c1 coshα c2 sinhαc1 sinhα c2 coshα

], (6.56)

α ∈ R, c1 = ±1, c2 = ±1, satisfyUT JU = J.

6.4.3. All suboptimal solutions

Suppose that5γ has a spectral factorization

5γ = Z∼γ J Zγ, (6.57)

with

J =[

Im2 00 − Ik2

]. (6.58)

It may be proved that such a spectral factorization generally exists providedγ is sufficiently large.Write [

XY

]= Z−1

γ

[AB

]. (6.59)

with A anm2 × m2 rational matrix andB a k2 × m2 rational matrix, both with all their poles inthe closed left-half plane. Then (6.51) reduces to

A∼ A ≥ B∼ B on the imaginary axis. (6.60)

Hence, (6.59), withA and B chosen so that they satisfy (6.60), characterizes all suboptimalcompensators. It may be proved that the feedback system defined this way is well-posed iffA isnonsingular.

This characterization of all suboptimal solutions has been obtained under the assumption that5γ has a spectral factorization. Conversely, it may be shown that if suboptimal solutions existthen5γ has a spectral factorization (Kwakernaak 1996).

Summary 6.4.3 (Suboptimal solutions). Givenγ, a suboptimal solution of the standard prob-lem exists iff5γ has a spectral factorization5γ = Z∼

γ J Zγ, with

J =[

Im2 00 − Ik2

]. (6.61)

7The zeros of5 are the poles of5−1.

266


If this factorization exists then all suboptimal compensators are given byK = Y X−1, with[XY

]= Z−1

γ

[AB

], (6.62)

with A nonsingular square rational andB rational, both with all their poles in the closed left-halfplane, and such thatA∼ A ≥ B∼ B on the imaginary axis. �

There are many matrices of stable rational functionsA and B that satisfyA∼ A ≥ B∼ B onthe imaginary axis. An obvious choice isA = I , B = 0. We call this thecentral solution Thecentral solution as defined here is not at all unique, because as seen in§ 6.4.2 (p. 265) the spectralfactorization (6.57) is not unique.

6.4.4. Lower bound

Suboptimal solutions exist for those values ofγ for which5γ is spectrally factorizable. Thereexists a lower boundγ0 so that5γ is factorizable for allγ > γ0. It follows thatγ0 is also a lowerbound for the∞-norm, that is,

‖H‖∞ ≥ γ0 (6.63)

for every compensator.

Summary 6.4.4 (Lower bound). Defineγ1 as the first value ofγ asγ decreases from∞ forwhich the determinant of[

G11G∼11 − γ2 I G11G∼

21

G21G∼11 G21G∼

21

](6.64)

has aγ-dependent8 zero on the imaginary axis. Similarly, letγ2 be the first value ofγ asγdecreases from∞ for which the determinant of[

G∼11G11 − γ2 I G∼

11G12

G∼12G11 G∼

12G12

](6.65)

has aγ-dependent zero on the imaginary axis. Then5γ is spectrally factorizable forγ > γ0,with

γ0 = max(γ1, γ2). (6.66)

Also,

‖H‖∞ ≥ γ0. (6.67)

The lower boundγ0 may be achieved. �

The proof may be found in Kwakernaak (1996). Note that the zeros of the determinant (6.64)are the poles of5γ , while the zeros of the determinant of (6.65) are the poles of5−1

γ .

8That is, a zero that varies withγ.

267


6.4.5. Stabilizing suboptimal compensators

In § 6.4.4 a representation is established of all suboptimal solutions of the standard problem fora given performance levelγ. Actually, we are not really interested in all solutions, but in allstabilizingsuboptimal solutions, that is, all suboptimal compensators that make the closed-loopsystem stable (as defined in§ 6.3.2, p. 261).

Before discussing how to find such compensators we introduce the notion of acanonicalspectral factorization. First, a square rational matrixM is said to bebiproper if both M and itsinverseM−1 are proper.M is biproper iff lim|s|→∞ M(s) is finite and nonsingular.

The spectral factorization5= Z∼ J Z of a biproper rational matrix5 is said to be canonical ifthe spectral factorZ is biproper. Under the general assumptions of§ 6.4 (p. 264) the matrix5γ isnot necessarily biproper. If it is not biproper then the definition of what a canonical factorization isneeds to be adjusted (Kwakernaak 1996). The assumptions on the state space version of standardproblem that are introduced in§ 6.5 (p. 272) ensure that5γ is biproper.

We consider the stability of suboptimal solutions. According to§ 6.3.3 (p. 263) closed-loopstability depends on whether the zeros ofX − G22Y all lie in the open left-half plane. Denote

Z−1γ = Mγ =

[Mγ,11 Mγ,12

Mγ,21 Mγ,22

](6.68)

Then by (6.62) all suboptimal compensators are characterized by

X = Mγ,11A+ Mγ,12B, Y = Mγ,21A+ Mγ,22B. (6.69)

Correspondingly,

X − G22Y = (Mγ,11 − G22Mγ,21)A+ (Mγ,12A− G22Mγ,22)B

= (Mγ,11 − G22Mγ,21)( I + TU)A. (6.70)

HereT = (Mγ,11 − G22Mγ,21)−1(Mγ,12 − G22Mγ,22) andU = BA−1.

Suppose that the factorization of5γ is canonical.Then it may be proved (Kwakernaak 1996)that as the real numberε varies from 0 to 1, none of the zeros of

(Mγ,11 − G22Mγ,21)( I + εTU)A (6.71)

crosses the imaginary axis. This means that all the zeros ofX − G22Y are in the open left-halfcomplex plane iff all the zeros of(Mγ,11 − G22Mγ,21)A are in the open left-half complex plane.It follows that a suboptimal compensator is stabilizing iff bothA andMγ,11 − G22Mγ,21 have alltheir roots in the open left-half plane. The latter condition corresponds to the condition that thecentralcompensator, that is, the compensator obtained forA = I andB = 0, be stabilizing.

Thus, we are led to the following conclusion:

Summary 6.4.5 (All stabilizing suboptimal compensators). Suppose that the factorization5γ = Z∼

γ J Zγ is canonical.

1. A stabilizing suboptimal compensator exists if and only if the central compensator is sta-bilizing.

2. If a stabilizing suboptimal compensator exists thenall stabilizing suboptimal compensatorsK = Y X−1 are given by[

XY

]= Z−1

γ

[AB

], (6.72)

268


with A nonsingular square rational andB rational, both with all their poles in the openleft-half complex plane, such thatA has all its zeros in the open left-half complex planeandA∼ A ≥ B∼ B on the imaginary axis.

�

Exercise 6.4.6 (All stabilizing suboptimal solutions). Prove that if the factorization is canon-ical then all stabilizing suboptimal compensatorsK = Y X−1, if any exist, may be characterizedby

[XY

]= Z−1

γ

[IU

], (6.73)

with U rational stable such that‖U‖∞ ≤ 1. �

6.4.6. Search procedure for near-optimal solutions

In the next section we discuss how to perform the spectral factorization of5γ . Assuming that weknow how to do this numerically — analytically it can be done in rare cases only — the searchprocedure of Fig. 6.12 serves to obtain near-optimal solutions.Optimalsolutions are discussedin § 6.6 (p. 275).

No

ΠγDoes a canonical spectral factorization of exist?

Compute the lower bound , or guess a lower bound

Compute the central compensator

Is the compensator close enough to optimal?

Is the compensator stabilizing?

No

Yes

Yes

No

Yes

γ γ> o

γ o

Choose a valueof

Increaseγ

Exit

Decreaseγ

Figure 6.12.: Search procedure

269


6.4.7. Example: Model matching problem

By way of example, we consider the solution of the model matching problem of Exercise 6.3.2(p. 260) with

P(s) = 11− s

. (6.74)

Lower bound. By (6.42) we have

G(s) =[

1s−1 −1

1 0

]. (6.75)

It may easily be found that the determinants of the matrices (6.64) and (6.65) are both equal to−γ2. It follows thatγ1 = γ2 = 0 so that alsoγ0 = 0. Indeed, the “compensator”K(s) = P(s) =1/(1− s) achieves‖H‖∞ = ‖P− K‖∞ = 0. Note that this compensator is not stable.

Spectral factorization. It is straightforward to find from (6.52) and (6.75) that for this problem5γ is given by

5γ(s) =

1−

1γ2

1−s2

1γ2

1+s1γ2

1−s − 1γ2

. (6.76)

We needγ > 0. It may be verified that forγ 6= 12 the matrix5γ has the spectral factor and

signature matrix

Zγ (s) =

γ2− 34

γ2− 14

+s

1+s

1γ2− 1

41+s

−14

γ(γ2− 14 )

1+s

1γ

(γ2+ 1

4γ2− 1

4+s

)1+s

, J =

[1 00 −1

]. (6.77)

The inverse ofZγ is

Z−1γ (s) =

γ2+ 14

γ2− 14

+s

1+s −γ

γ2− 14

1+s14

γ2− 14

1+s

γ(γ2− 3

4γ2− 1

4+s)

1+s

. (6.78)

5γ is biproper. Forγ 6= 12 both Zγ andZ−1

γ are proper, so that the factorization is canonical. Forγ = 1

2 the spectral factorZγ is not defined. It may be proved that forγ = 12 actually no canonical

factorization exists. Noncanonical factorizationdoexist for this value ofγ, though.

Suboptimal compensators. It follows by application of Summary 6.4.3 (p. 266) after somealgebra that forγ 6= 1

2 all suboptimal “compensators” have the transfer function

K(s) =14

γ2− 14

+ γ(γ2− 3

4

γ2− 14

+ s)

U(s)(γ2+ 1

4

γ2− 14

+ s)

− γ

γ2− 14U(s)

, (6.79)

with U = B/A such thatA∼ A ≥ B∼ B on the imaginary axis. We consider two special cases.

270


“Central” compensator. Let A = 1, B = 0, so thatU = 0. We then have the “central” com-pensator

K(s) =14

γ2− 14

γ2+ 14

γ2− 14

+ s. (6.80)

Forγ > 12 the compensator is stable and for 0≤ γ < 1

2 it is unstable. The resulting “closed-looptransfer function” may be found to be

H(s) = P(s)− K(s) =γ2

γ2− 14(1+ s)

(γ2+ 1

4

γ2− 14

+ s)(1− s). (6.81)

The magnitude of the corresponding frequency response function takes its peak value at thefrequency 0, and we find

‖H‖∞ = |H(0)| = γ2

γ2 + 14

. (6.82)

It may be checked that‖H‖∞ ≤ γ, with equality only forγ = 0 andγ = 12.

Equalizing compensator. ChooseA = 1, B = 1, so thatU = 1. After simplification we obtainfrom (6.79) the compensator

K(s) =γ

(s+ γ+ 1

2 − 12γ

γ+ 12

)s+ γ− 1

2

γ+ 12

. (6.83)

For this compensator we have the closed-loop transfer function

H(s) = γ(1+ s)(s− γ− 1

2

γ+ 12)

(1− s)(s+ γ− 12

γ+ 12). (6.84)

The magnitude of the corresponding frequency response functionH(jω) is constant and equal toγ. Hence, the compensator (6.83) is equalizing with‖H‖∞ = γ.

Stabilizing compensators. The spectral factorization of5γ exists forγ > 0, and is canonicalfor γ 6= 1

2. The central compensator (6.80) is stabilizing iff the compensator itself is stable,which is the case forγ > 1

2. Hence, according to Summary 6.4.5 (p. 268) forγ > 12 stabilizing

suboptimal compensators exist. They are given by

K(s) =14

γ2− 14

+ γ(γ2− 3

4

γ2− 14

+ s)

U(s)(γ2+ 1

4

γ2− 14

+ s)

− γ

γ2− 14U(s)

, (6.85)

with U scalar stable rational such that‖U‖∞ ≤ 1. In particular, the equalizing compensator(6.83), obtained by lettingU(s) = 1, is stabilizing.

Since the central compensator is not stabilizing forγ < 12, we conclude that no stabilizing

compensators exist such that‖H‖∞ < 12.

271


The optimal compensator. Investigation of (6.79) shows that forγ → 12 the transfer function

of all suboptimal compensators approachesK0(s)= 12. This compensator is (trivially) stable, and

the closed-loop transfer function is

H0(s) = P(s)− K0(s) = 12

1+ s1− s

, (6.86)

so that‖H0‖∞ = 12. Since forγ < 1

2 no stabilizing compensators exist we conclude that thecompensatorK0 = 1

2 is optimal,that is, minimizes‖H‖∞ with respect to all stable compensators.

6.5. State space solution

6.5.1. Introduction

In this section we describe how to perform the spectral factorization needed for the solution ofthe standardH∞ optimal regulator problem. To this end, we first introduce a special version ofthe problem instate space form.In § 6.5.5 (p. 274) a more general version is considered. Thestate space approach leads to a factorization algorithm that relies on the solution of two algebraicRiccati equations.

6.5.2. State space formulation

Consider a standard plant that is described by the state differential equation

x = Ax+ Bu+ Fw1, (6.87)

with A, B, and F constant matrices, and the vector-valued signalw1 one component of theexternal inputw. The observed outputy is

y = Cx+w2, (6.88)

with w2 the second component ofw, andC another constant matrix. The error signalz, finally,is assumed to be given by

z=[

Hxu

], (6.89)

with alsoH a constant matrix. It is easy to see that the transfer matrix of the standard plant is

G(s) =

H(sI − A)−1F 0 H(sI − A)−1B

0 0 IC(sI − A)−1F I C(sI − A)−1B

. (6.90)

This problem is a stylized version of the standard problem that is reminiscent of the LQG regu-lator problem.

In this and other state space versions of the standardH∞ problem it is usual to require thatthe compensator stabilize the entire closed-loop system, and not just the loop aroundG22. Thisnecessitates the assumption that the systemx = Ax+ Bu, y = Hx be stabilizable and detectable.Otherwise, closed-loop stabilization is not possible.

272

6.5. State space solution

6.5.3. Factorization of 5γ

In § 6.10 (p. 304) it is proved that for the state space standard problem described in§ 6.5.2 therational matrix5γ as defined in Summary 6.4.2 (p. 265) may be spectrally factored as5γ =Z∼γ J Zγ, where the inverseMγ = Z−1

γ of the spectral factor is given by

Mγ (s) =[

I 00 γ I

]+[

C

−BT X2

](sI − A2)

−1( I − 1γ2

X1X2)−1[ X1CT γB]. (6.91)

The constant matrixA2 is defined as

A2 = A+ (1γ2

FFT − BBT)X2. (6.92)

The matrixX1 is a symmetric solution — if any exists — of the algebraic Riccati equation

0 = AX1 + X1AT + FFT − X1(CTC − 1

γ2HT H)X1 (6.93)

such that

A1 = A+ X1(1γ2

HT H − CTC) (6.94)

is a stability matrix9. The matrixX2 is a symmetric solution — again, if any exists — of thealgebraic Riccati equation

0 = AT X2 + X2A+ HT H − X2(BBT − 1γ2

FFT)X2 (6.95)

such thatA2 is a stability matrix.The numerical solution of algebraic Riccati equations is discussed in§ 3.2.9 (p. 114).

6.5.4. The central suboptimal compensator

Once the spectral factorization has been established, we may obtain the transfer matrix of thecentral compensator asK = Y X−1, with[

XY

]= Z−1

γ

[I0

]= Mγ

[I0

]=[

Mγ,11

Mγ,21

]. (6.96)

After some algebra we obtain from (6.91) the transfer matrixK = Mγ,21M−1γ,11 of the central

compensator in the form

K(s) = −BT X2

(sI − A− 1

γ2FFTX2 + BBT X2

+ ( I − 1γ2

X1X2)−1X1CTC

)−1

( I − 1γ2

X1X2)−1X1CT.

(6.97)

In state space form the compensator may be represented as

˙x = (A+ 1γ2

FFTX2)x + ( I − 1γ2

X1X2)−1X1CT(y− Cx)+ Bu, (6.98)

u = −BT X2x. (6.99)

The compensator consists of an observer interconnected with a state feedback law. The order ofthe compensator is the same as that of the standard plant.

9A matrix is a stability matrix if all its eigenvalues have strictly negative real part.

273


6.5.5. A more general state space standard problem

A more general state space representation of the standard plant is

x = Ax+ B1w+ B2u, (6.100)

z = C1x+ D11w+ D12u, (6.101)

y = C2x+ D21w+ D22u. (6.102)

In the µ-Tools MATLAB toolbox (Balas, Doyle, Glover, Packard, and Smith 1991) and theRobust Control Toolbox (Chiang and Safonov 1992) a solution of the correspondingH∞ problembased on Riccati equations is implemented that requires the following conditions to be satisfied:

1. (A, B2) is stabilizable and(C2, A) is detectable.

2. D12 andD21 have full rank.

3.

[A− jω I B2

C1 D12

]has full column rank for allω ∈ R (hence,D12 is tall10).

4.

[A− jω I B1

C2 D21

]has full row rank for allω ∈ R (hence,D21 is wide).

The solution of this more general problem is similar to that for the simplified problem. It consistsof two algebraic Riccati equations whose solutions define an observercumstate feedback law.The expressions, though, are considerably more complex than those in§ 6.5.3 and§ 6.5.4.

The solution is documented in a paper by Glover and Doyle (1988). More extensive treat-ments may be found in a celebrated paper by Doyle, Glover, Khargonekar, and Francis (1989)and in Glover and Doyle (1989). Stoorvogel (1992) discusses a number of further refinements ofthe problem.

6.5.6. Characteristics of the state space solution

We list a few properties of the state space solution.

1. The assumption of stabilizability and detectability ensures that thewholefeedback systemis stabilized, not just the loop aroundG22. There are interestingH∞ problems that do notsatisfy the stabilizability or detectability assumption, and, hence, cannot be solved withthis algorithm.

2. The eigenvalues of the Hamiltonian corresponding to the Riccati equation (6.93) are thepoles of5γ . Hence, the numberγ1 defined in the discussion on the lower bound in§ 6.4may be determined by finding the first value ofγ asγ decreases from∞ such that any ofthese eigenvalues reaches the imaginary axis.

Likewise, the eigenvalues of the Hamiltonian corresponding to the Riccati equation (6.95)are the poles of the inverse of5γ . Hence, the numberγ2 defined in the discussion on thelower bound in§ 6.4 may be determined by finding the first value ofγ asγ decreases from∞ such that any of these eigenvalues reaches the imaginary axis.

3. The required solutionsX1 andX2 of the Riccati equations (6.93) and (6.95) are nonnega-tive-definite.

10A matrix is tall if it has at least as many rows as columns. It is wide if it has at least as many columns as rows.

274

6.6. Optimal solutions to theH∞ problem

4. The central compensator is stabilizing iff

λmax(X2X1) < γ2, (6.103)

whereλmax denotes the largest eigenvalue. This is a convenient way to test whether stabi-lizing compensators exist.

5. The central compensator is of the same order as the generalized plant (6.100–6.102).

6. The transfer matrixK of the central compensator is strictly proper.

7. The factorization algorithm only applies if the factorization is canonical. If the factoriza-tion is noncanonical then the solution breaks down. This is described in the next section.

6.6. Optimal solutions to the H∞ problem

6.6.1. Optimal solutions

In § 6.4.6 (p. 269) a procedure is outlined for finding optimal solutions that involves a searchon the parameterγ. As the search advances the optimal solution is approached more and moreclosely. There are two possibilities for the optimal solution:

Type A solution. The central suboptimal compensator is stabilizing for allγ ≥ γ0, with γ0 thelower bound discussed in 6.4.4 (p. 267). In this case, the optimal solution is obtained forγ = γ0.

Type B solution. As γ varies, the central compensator becomes destabilizing asγ decreasesbelow some numberγopt with γopt ≥ γ0.

In type B solutions a somewhat disconcerting phenomenon occurs. In the example of§ 6.4.7(p. 270) several of the coefficients of the spectral factor and of the compensator transfer matrixgrow without bound asγ approachesγopt. This is generally true.

In the state space solution the phenomenon is manifested by the fact that asγ approachesγopt

either of the two following eventualities occurs (Glover and Doyle 1989):

(B1.) The matrixI − 1γ2 X2X1 becomes singular.

(B2.) In solving the Riccati equation (6.93) or (6.95) or both the “top coefficient matrix”E1 of§ 3.2.9 (p. 114) becomes singular, so that the solution of the Riccati equation ceases toexist.

In both cases large coefficients occur in the equations that define the central compensator.The reason for these phenomena for type B solutions is that at the optimum the factorization

of5γ is non-canonical.As illustrated in the example of§ 6.4.7, however, the compensator transfer matrix approaches

a well-defined limit asγ → γopt, corresponding to theoptimalcompensator. The type B optimalcompensator is of lower order than the suboptimal central compensator. Also this is generallytrue.

It is possible to characterize and computeall optimalsolutions of type B (Glover, Limebeer,Doyle, Kasenally, and Safonov 1991; Kwakernaak 1991).

An important characteristic of optimal solutions of type B is that the largest singular valueσ(H(jω)) of the optimal closed-loop frequency response matrixH is constantas a function ofthe frequencyω ∈ R. This is known as theequalizing propertyof optimal solutions.

275


Straightforward implementation of the two-Riccati equation algorithm leads to numericaldifficulties for type B problems. As the solution approaches the optimum several or all of thecoefficients of the compensator become very large. Because eventually the numbers become toolarge the optimum cannot be approached too closely.

One way to avoid numerical degeneracy is to represent the compensator in what is calleddescriptorform (Glover, Limebeer, Doyle, Kasenally, and Safonov 1991). This is a modificationof the familiar state representation and has the appearance

Ex = Fx+ Gu, (6.104)

y = Hx+ Lu. (6.105)

If E is square nonsingular then by multiplying (6.104) on the left byE−1 the equations (6.104–6.105) may be reduced to a conventional state representation. The central compensator for theH∞ problem may be represented in descriptor form in such a way thatE becomes singular as theoptimum is attained but all coefficients in (6.104–6.105) remain bounded. The singularity ofEmay be used to reduce the dimension of the optimal compensator (normally by 1).

General descriptor systems (withE singular) are discussed in detail in Aplevich (1991).

6.6.2. Numerical examples

We present the numerical solution of two examples of the standard problem.

Example 6.6.1 (Mixed sensitivity problem for the double integrator). We consider the mixedsensitivity problem discussed in§ 6.2.5 (p. 254). WithP, V, W1 andW2 as in 6.3, by (6.39) thestandard plant transfer matrix is

G(s) =

M(s)s2

1s2

0 c(1+ rs)− M(s)

s2 − 1s2

, (6.106)

whereM(s) = s2 + s√

2 + 1. We consider the caser = 0 andc = 0.1. It is easy to check thatfor r = 0 a state space representation of the plant is

x =[0 01 0

]︸︷︷︸

A

x+[

1√2

]︸︷︷︸

B1

w+[10

]︸︷︷︸B2

u, (6.107)

z=[0 10 0

]︸︷︷︸

C1

x+[10

]︸︷︷︸D11

w+[0c

]︸︷︷︸D12

u, (6.108)

y = [0 −1

]︸︷︷︸C2

x+ [−1]︸︷︷︸

D21

w. (6.109)

Note that we constructed a joint minimal realization of the blocksP andV in the diagram ofFig. 6.7. This is necessary to satisfy the stabilizability condition of§ 6.5.5 (p. 274).

A numerical solution may be obtained with the help of theµ-Tools MATLAB toolbox (Balas,Doyle, Glover, Packard, and Smith 1991). The search procedure forH∞ state space problems

276

6.6. Optimal solutions to theH∞ problem

implemented inµ-Tools terminates atγ = 1.2861 (for a specified tolerance in theγ-search of10−8). The state space representation of the corresponding compensator is

˙x =[−0.3422× 109 −1.7147× 109

1 −1.4142

]x +

[−0.3015−0.4264

]y, (6.110)

u = [−1.1348× 109 − 5.6871× 109] x. (6.111)

Numerical computation of the transfer function of the compensator results in

K(s) = 2.7671× 109s+ 1.7147× 109

s2 + 0.3422× 109s+ 2.1986× 109. (6.112)

The solution is of type B as indicated by the large coefficients. By discarding the terms2 in thedenominator we may reduce the compensator to

K(s) = 1.2586s+ 0.61971+ 0.1556s

, (6.113)

which is the result stated in (6.31)(p. 256). It may be checked that the optimal compensatorpossesses the equalizing property: the mixed sensitivity criterion does not depend on frequency.

The caser 6= 0 is dealt with in Exercise 6.7.2 (p. 288). �

Example 6.6.2 (Disturbance feedforward). Hagander and Bernhardsson (1992) discuss the ex-ample of Fig. 6.13. A disturbancew acts on a plant with transfer function 1/(s+ 1). The plant

w

yK s( )

u

z2

z1x1 x2s+1s+111 1

ρ

actuator plant

++

Figure 6.13.: Feedforward configuration

is controlled by an inputu through an actuator that also has the transfer function 1/(s+ 1). Thedisturbancew may be measured directly, and it is desired to control the plant output by feedfor-ward control from the disturbance. Thus, the observed outputy is w. The “control error” has ascomponents the weighted control system outputz1 = x2/ρ, with ρ a positive coefficient, and thecontrol inputz2 = u. The latter is included to prevent overly large inputs.

In state space form the standard plant is given by

x =[−1 0

1 −1

]︸︷︷︸

A

x+[01

]︸︷︷︸B1

w+[10

]︸︷︷︸B2

u, (6.114)

277


z=[

0 1ρ

0 0

]︸︷︷︸

C1

x+[00

]︸︷︷︸D11

w+[01

]︸︷︷︸D12

u, (6.115)

y = [0 0

]︸︷︷︸C2

x + [1]︸︷︷︸

D21

w. (6.116)

The lower bound of§ 6.4.4 (p. 267) may be found to be given by

γ0 = 1√1+ ρ2

. (6.117)

Define the number

ρc =√√

5+ 12

= 1.270. (6.118)

Hagander and Bernhardsson prove this (see Exercise 6.6.3):

1. Forρ > ρc the optimal solution is of type A.

2. Forρ < ρc the optimal solution is of type B.

Forρ = 2, for instance, numerical computation usingµ-Tools with a tolerance 10−8 results in

γopt = γ0 = 1√5

= 0.4472, (6.119)

with the central compensator

˙x =[−1.6229 −0.4056

1 −1

]x+

[0

0.7071

]y, (6.120)

u = [−0.8809 − 0.5736] x. (6.121)

The coefficients of the state representation of the compensator are of the order of unity, and thecompensator transfer function is

K(s) = −0.4056s− 0.4056ss2 + 2.6229s+ 2.0285

. (6.122)

There is no question of order reduction. The optimal solution is of type A.Forρ = 1 numerical computation leads to

γopt = 0.7167> γ0 = 0.7071, (6.123)

with the central compensator

˙x =[−6.9574× 107 −4.9862× 107

1 −1

]x +

[01

]y, (6.124)

u = [−6.9574× 107 − 4.9862× 107] x. (6.125)

278

6.7. Integral control and high-frequency roll-off

The compensator transfer function is

K(s) = −4.9862× 107s− 4.9862× 107

s2 + 0.6957× 108s+ 1.1944× 108. (6.126)

By discarding the terms2 in the denominator this reduces to the first-order compensator

K(s) = −0.7167s+ 1

s+ 1.7167. (6.127)

The optimal solution now is of type B. �

Exercise 6.6.3 (Disturbance feedforward).

1. Verify that the lower boundγ0 is given by (6.117).

2. Prove thatρc as given by (6.118) separates the solutions of type A and type B.

3. Prove that ifρ < ρc then the minimal∞-normγopt is the positive solution of the equationγ4 + 2γ3 = 1/ρ2 (Hagander and Bernhardsson 1992).

4. Check the frequency dependence of the largest singular value of the closed-loop frequencyresponse matrix for the two types of solutions.

5. What do you have to say about the optimal solution forρ = ρc?�


6.7.1. Introduction

In this section we discuss the application ofH∞ optimization, in particular the mixed sensitivityproblem, to achieve two specific design targets: integral control and high-frequency roll-off.Integral control is dealt with in§ 6.7.2. In§ 6.7.3 we explain how to design for high-frequencyroll-off. Subsection 6.7.4 is devoted to an example.

The methods to obtain integral control and high frequency roll-off discussed in this sectionfor H∞ design may also be used withH2 optimization. This is illustrated for SISO systems in§ 3.5.4 (p. 130) and§ 3.5.5 (p. 132).

6.7.2. Integral control

In § 2.3 (p. 64) it is explained that integral control is a powerful and important technique. Bymaking sure that the control loop contains “integrating action” robust rejection of constant dis-turbances may be obtained, as well as excellent low-frequency disturbance attenuation and com-mand signal tracking. In§ 6.2.3 (p. 252) it is claimed that the mixed sensitivity problem allows to

C P−

Figure 6.14.: One-degree-of-freedom feedback system

279


design for integrating action. Consider the SISO mixed sensitivity problem for the one-degree-of-freedom configuration of Fig. 6.14. The mixed sensitivity problem amounts to the minimizationof the peak value of the function

|V(jω)W1(jω)S(jω)|2 + |V(jω)W2(jω)U(jω)|2, ω ∈ R. (6.128)

SandU are the sensitivity function and input sensitivity function

S= 11+ PC

, U = C1+ PC

, (6.129)

respectively, andV, W1 andW2 suitable frequency dependent weighting functions.If the plant P has “natural” integrating action, that is,P has a pole at 0, then no special

provisions are needed. In the absence of natural integrating action we may introduce integratingaction by letting the productVW1 have a pole at 0. This forces the sensitivity functionS to havea zero at 0, because otherwise (6.128) is unbounded atω = 0.

There are two ways to introduce such a pole intoVW1:

1. Let V have a pole at 0, that is, take

V(s) = V0(s)s

, (6.130)

with V0(0) 6= 0. We call this theconstant disturbance modelmethod.

2. Let W1 have a pole at 0, that is, take

W1(s) = W1o(s)s

, (6.131)

with W1o(0) 6= 0. This is theconstant error suppressionmethod.

We discuss these two possibilities.

Constant disturbance model. We first consider lettingV have a pole at 0. AlthoughVW1

needs to have a pole at 0, the weighting functionVW2 cannot have such a pole. IfVW2 has a poleat 0 thenU would be forced to be zero at 0.SandU cannot vanish simultaneously at 0. Hence,if V has a pole at 0 thenW2 should have a zero at 0.

This makes sense. Including a pole at 0 inV means thatV contains a model for constantdisturbances. Constant disturbances can only be rejected by constant inputs to the plant. Hence,zero frequency inputs should not be penalized byW2. Therefore, ifV is of the form (6.130) thenwe need

W2(s) = sW2o(s), (6.132)

with W2o(0) 6= ∞.Figure 6.15(a) defines the interconnection matrixG for the resulting standard problem.

Inspection shows that owing to the pole at 0 in the blockV0(s)/s outside the feedback loopthis standard problem does not satisfy the stabilizability condition of§ 6.5.5 (p. 274).

The first step towards resolving this difficulty is to modify the diagram to that of Fig. 6.15(b),where the plant transfer matrix has been changed tosP(s)/s. The idea is to construct a simulta-neous minimal state realization of the blocksV0(s)/sandsP(s)/s. This brings the unstable poleat 0 “inside the loop.”

280


W s2o( ) V so( )

W s1( )sK s( )

K so( )

P s( )

z w

y uz

2

1

v

−+

+

s

s

(c)

sW s2o( )

sW s2o( )

W s1( )

V so( )

V so( )

K s( ) P s( )

z w

y uz

2

1

v

−+

+

(a)

s

s

ssP s( )

W s1( )K s( )

z w

y uz

2

1

v

−+

+

(b)

Figure 6.15.: Constant disturbance model

281


The difficulty now is, of course, that owing to the cancellation insP(s)/s there is no minimalrealization that is controllable from the plant input. Hence, we remove the factors from the nu-merator ofsP(s)/sby modifying the diagram to that of Fig. 6.15(c). By a minimal joint realiza-tion of the blocksV0(s)/sandP(s)/s the interconnection system may now be made stabilizable.This is illustrated in the example of§ 6.7.4.

The block diagram of Fig. 6.15(c) defines a modified mixed sensitivity problem. Supposethat the compensatorK0 solves this modified problem. Then the original problem is solved bycompensator

K(s) = K0(s)s

. (6.133)

Constant error suppression. We next consider choosingW1(s)= W1o(s)/s. This correspondsto penalizing constant errors in the output with infinite weight. Figure 6.16(a) shows the corre-sponding block diagram. Again straightforward realization results in violation of the stabilizabil-ity condition of § 6.5.5 (p. 274). The offending factor 1/s may be pulled inside the loop as inFig. 6.16(b). Further block diagram substitution yields the arrangement of Fig. 6.16(c).

Integrator in the loop method. Contemplation of the block diagrams of Figs. 6.15(c) and6.16(c) shows that both the constant disturbance model method and the constant error rejectionmethod may be reduced to theintegrator in the loopmethod. This method amounts to connectingan integrator 1/s in series with the plantP as in Fig. 6.17(a), and doing a mixed sensitivity designfor this augmented system.

After a compensatorK0 has been obtained for the augmented plant the actual compensatorthat is implemented consists of the series connectionK(s)= K0(s)/sof the compensatorK0 andthe integrator as in Fig. 6.17(b). To prevent the pole at 0 of the integrator from reappearing in theclosed-loop system it is necessary to letV have a pole at 0. ReplacingV(s) with V0(s)/s in thediagram of Fig. 6.17(a) produces the block diagrams of Figs. 6.15(c) and 6.16(c).

6.7.3. High-frequency roll-off

As explained in§ 6.2.3 (p. 252), high-frequency roll-off in the compensator transfer functionKand the input sensitivity functionU may be pre-assigned by suitably choosing the high-frequencybehavior of the weighting functionW2. If W2(s)V(s) is of ordersm ass approaches∞ thenKandU have a high-frequency roll-off ofmdecades/decade. NormallyV is chosen biproper11 suchthatV(∞)= 1. Hence, to achieve a roll-off ofmdecades/decade it is necessary thatW2 behavesassm for larges.

If the desired roll-offm is greater than 0 then the weighting functionW2 needs to be nonproper.The resulting interconnection matrix

G = W1V W1P

0 W2

−V −P

(6.134)

is also nonproper and, hence, cannot be realized as a state system of the form needed for the statespace solution of theH∞-problem.

The makeshift solution usually seen in the literature is to cut off the roll-on at high frequency.Suppose by way of example that the desired form ofW2 is W2(s) = c(1 + rs), with c and r11That is, bothV and 1/V are proper. This means thatV(∞) is finite and nonzero.

282


W s2( ) V s( )

W s1o( )

W s1o( )

W s1o( )sK s( )

K so( )

P s( )

z w

y uz

2

1

v

−+

+

s

s

(c)

W s2( )

W s1o( )

W s1o( )

W s1o( )

V s( )

K s( ) P s( )

z w

y uz

2

1

v

−+

+

(a)

s

s

s

W s2( ) V s( )

P s( )K s( )

z w

y u z

2

1

v

−+

+

(b)

Figure 6.16.: Constant error suppression

283


W s2( )

W s1( )

V s( )

K so( )

K so( )

K s( )

P s( )

P s( )

P so ( )

z w

y uz

2

1

v

−

−+

+

(a)

(b)

1

1

s

s

Figure 6.17.: Integrator in the loop method

positive constants. ThenW2 may be made proper by modifying it to

W2(s) = c(1+ rs)1+ sτ

, (6.135)

with τ � r a small positive time constant.This contrivance may be avoided by the block diagram substitution of Fig. 6.18 (Krause

1992). If W2 is nonproper then, in the SISO case,W−12 is necessarily proper, and there is no

difficulty in finding a state realization of the equivalent mixed sensitivity problem defined byFig. 6.18 with the modified plantP0 = W−1

2 P.Suppose that the modified problem is solved by the compensatorK0. Then the original

problem is solved by

K = K0W−12 . (6.136)

Inspection suggests that the optimal compensatorK has the zeros ofW2 as its poles. Becausethis is in fact not the case the extra poles introduced intoK cancel against corresponding zeros.This increase in complexity is the price of using the state space solution.

The example of§ 6.7.4 illustrates the procedure.

6.7.4. Example

We present a simple example to illustrate the mechanics of designing for integrating action andhigh-frequency roll-off. Consider the first-order plant

P(s) = ps+ p

, (6.137)

with p = 0.1. These are the design specifications:

1. Constant disturbance rejection.

284


WW

V

K

Ko Po

P W

w

y u

z2

z122 1

v

−+

+−1

Figure 6.18.: Block diagram substitution for nonproperW2

2. A closed-loop bandwidth of 1.

3. A suitable time response to step disturbances without undue sluggishness or overshoot.

4. High-frequency roll-off of the compensator and input sensitivity at 1 decade/decade.

We use the integrator in the loop method of§ 6.7.2 to design for integrating action. Accordingly,the plant is modified to

P0(s) = ps(s+ p)

. (6.138)

Partial pole placement (see§ 6.2.4, p. 253) is achieved by letting

V(s) = s2 + as+ bs(s+ p)

. (6.139)

The choicea = √2 andb = 1 places two closed-loop poles at the roots

12

√2(−1± j) (6.140)

of the numerator ofV. We plan these to be the dominant closed-loop poles in order to achieve asatisfactory time response and the required bandwidth.

To satisfy the high-frequency roll-off specification we let

W2(s) = c(1+ rs), (6.141)

with the positive constantsc andr to be selected12. It does not look as if anything needs to beaccomplished withW1 so we simply letW1(s) = 1.

Figure 6.19 shows the block diagram of the mixed sensitivity problem thus defined. It isobtained from Fig. 6.17(a) with the substitution of Fig. 6.18.

We can now see why in the eventual design the compensatorK has no pole at the zero−1/rof the weighting factorW2. The diagram of Fig. 6.19 defines a standard mixed sensitivity problemwith

P(s) =prc

s(s+ 1r )(s+ p)

, (6.142)

V(s) = s2 + as+ bs(s+ p)

= (s+ 1r )(s

2 + as+ b)

s(s+ 1r )(s+ p)

, (6.143)

12See also Exercise 6.7.1 (p. 287).

285


c +rs(1 ) s s+p( )

s s+p( )s +as+2 1

1

− y uuo

z2

z1

w

++

pK so( )

P so( )

Figure 6.19.: Modified block diagram for the example

andW1(s) = W2(s) = 1.

If V is in the form of the rightmost side of (6.143) then its numerator has a zero at−1/r .By the partial pole assignment argument of§ 6.2.4 (p. 253) this zero reappears as a closed-looppole. Because this zero is also an open-loop pole it is necessarily a zero of the compensatorK0

that solves the mixed sensitivity problem of Fig. 6.19. This zero, in turn, cancels in the transferfunctionK of (6.136).

The system within the larger shaded block of Fig. 6.19 has the transfer matrix representation

z1 =[

s2 + as+ bs(s+ p)

ps(s+ p)

][w

u

]. (6.144)

This system has the minimal state realization

[x1

x2

]=

[0 01 −p

][x1

x2

]+[

ba− p

]w+

[p0

]u, (6.145)

z1 = [0 1

] [x1

x2

]+w. (6.146)

The block

u = 1c(1+ rs)

u0 (6.147)

may be represented in state form as

x3 = −1r

x3 + 1c

u0, (6.148)

u = 1r

x3. (6.149)

Combining the state differential equations (6.145) and (6.148) and arranging the output equationswe obtain the equations for the interconnection system that defines the standard problem as

286


follows:x1

x2

x3

=

0 0 p

r1 −p 00 0 − 1

r

︸︷︷︸A

x1

x2

x3

+

b

a− p0

︸︷︷︸B1

w+0

01c

︸︷︷︸B2

u0, (6.150)

z =[0 1 00 0 0

]︸︷︷︸

C1

x1

x2

x3

+

[10

]︸︷︷︸D11

w+[01

]︸︷︷︸D12

u0 (6.151)

y = [0 −1 0

]︸︷︷︸C2

x1

x2

x3

+ (−1)︸︷︷︸

D21

w. (6.152)

Let c = 1 andr = 0.1. Numerical computation of the optimal compensator yields the type Bsolution

K0(s) = 1.377× 1010s2 + 14.055× 1010s+ 2.820× 1010

s3 + 0.1142× 1010s2 + 1.3207× 1010s+ 1.9195× 1010(6.153)

Neglecting the terms3 in the denominator results in

K0(s) = 1.377s2 + 14.055s+ 2.8200.1142s2 + 1.3207s+ 1.91195

(6.154)

= 12.056(s+ 10)(s+ 0.2047)

(s+ 9.8552)(s+ 1.7049). (6.155)

Hence, the optimal compensator for the original problem is

K(s) = K0(s)sW2(s)

= 12.056(s+ 10)(s+ 0.2047)

(s+ 9.8552)(s+ 1.7049)· 10

s(s+ 10)(6.156)

= 120.56(s+ 0.2047)s(s+ 9.8552)(s+ 1.7049)

. (6.157)

The pole at−10 cancels, as expected, and the compensator has integrating action, as planned.The compensator has a high-frequency roll-off of 2 decades/decade.

Exercise 6.7.1 (Completion of the design). Complete the design of this example.

(a) For the compensator (6.157), compute and plot the sensitivity functionS, the input sensi-tivity function U, and the complementary sensitivity functionT.

(b) Check whether these functions behave satisfactorily, in particular, ifSandT do not peaktoo much. See if these functions may be improved by changing the values ofc andr bytrial and error.

(c) The roll-off of 2 decades/decade is more than called for by the design specifications. Thedesired roll-off of 1 decade/decade may be obtained with the constant weighting functionW2(s) = c. Recompute the optimal compensator for this weighting function withc = 1.

(d) Repeat (a) and (b) for the design of (c).

287


(e) Compute and plot the time responses of the outputz and the plant inputu to a unit stepdisturbancev in the configuration of Fig. 6.20 for the designs selected in (b) and (d).Discuss the results.

�

C Pu

v

z+

+

−

Figure 6.20.: One-degree-of- freedom system with disturbancev

Exercise 6.7.2 (Roll-off in the double integrator example). In Example 6.6.1 (p. 276) thenumerical solution of the double integrator example of§ 6.2.5 (p. 254) is presented forr = 0.Extend this to the caser 6= 0, and verify (6.32). �

6.8. µ-Synthesis

6.8.1. Introduction

In § 5.8 (p. 240) the use of the structured singular value for the analysis of stability and perfor-mance robustness is explained. We briefly review the conclusions.

In Fig. 6.21(a), the interconnection blockH represents a control system, with structuredperturbations represented by the block1. The signalw is the external input to the control systemandz the control error. The input and output signals of the perturbation block are denotedpandq, respectively. The perturbations are assumed to have been scaled such that all possiblestructured perturbations are bounded by‖1‖∞ ≤ 1.

Let H1 denote the transfer function fromw to z (under perturbation1). Again, the transferfunction is assumed to have been scaled in such a way that the control system performance isdeemed satisfactory if‖H1‖∞ ≤ 1.

The central result established in§ 5.8 is that the control system

1. is robustly stable, that is, is stable under all structured perturbations1 such that‖1‖∞ ≤ 1,and

2. has robust performance, that is,‖H1‖∞ ≤ 1 under all structured perturbations1 such that‖1‖∞ ≤ 1,

if and only if

µH < 1. (6.158)

The quantityµH , defined in (5.172), is the structured singular value ofH with respect to pertur-bations as in Fig. 6.21(b), with1 structured and10 unstructured.

Suppose that the control system performance and stability robustness may be modified byfeedback through a compensator with transfer matrixK, as in Fig. 6.22(a). The compensatorfeeds the measured outputy back to the control inputu. Denote byµHK the structured singular

288

6.8. µ-Synthesis

value of the closed-loop transfer matrixHK of the system of Fig. 6.22(b) with respect to struc-tured perturbations1 and unstructured perturbations10. Thenµ-synthesisis any procedure toconstruct a compensatorK (if any exists) that stabilizes the nominal feedback system and makes

µHK < 1. (6.159)

(a) (b)

q pw z w z

q p

∆ ∆

∆0

HH

Figure 6.21.:µ-Analysis for robust stability and performance.

∆ ∆

∆0(a) (b)

q p q p

w z w z

u y u yG

K

G

K

Figure 6.22.:µ-Synthesis for robust stability and performance.

6.8.2. Approximate µ-synthesis for SISO robust performance

In § 5.8 the SISO single-degree-of-freedom configuration of Fig. 6.23 is considered. We analyzeits robustness with respect to proportional plant perturbations of the form

P −→ P(1+ δpW2) (6.160)

with ‖δP‖∞ ≤ 1. The system is deemed to have satisfactory performance robustness if

‖W1S‖∞ ≤ 1 (6.161)

for all perturbations, withS the sensitivity function of the closed-loop system.W1 andW2 aresuitable frequency dependent functions.

289


Ku

P

w

z−

++

Figure 6.23.: Single-degree- of-freedom configuration

By determining the structured singular value it follows in§ 5.8 that the closed-loop system isrobustly stable and has robust performance if and only if

µHK = supω∈R

(|W1(jω)S(jω)| + |W2(jω)T(jω)|) < 1. (6.162)

T is the complementary sensitivity function of the closed-loop system. Hence, the robust perfor-mance and stability problem consists of determining a feedback compensatorK (if any exists)that stabilizes the nominal system and satisfiesµHK < 1.

One way of doing this is tominimizeµHK with respect to all stabilizing compensatorsK.Unfortunately, this is not a standardH∞-optimal control problem. The problem cannot be solvedby the techniques available for the standardH∞ problem. In fact, no solution is known to thisproblem.

By the well-known inequality(a + b)2 ≤ 2(a2 + b2) for any two real numbersa andb itfollows that

(|W1S| + |W2T|)2 ≤ 2(|W1S|2 + |W2T|2). (6.163)

Hence,

supω∈R

(|W1(jω)S(jω)|2 + |W2(jω)T(jω)|2) < 12 (6.164)

impliesµHK < 1. Therefore, if we can find a compensatorK such that

supω∈R

(|W1(jω)S(jω)|2 + |W2(jω)T(jω)|2)1/2 =∥∥∥∥[

W1SW2T

]∥∥∥∥∞< 1

2

√2 (6.165)

then this compensator achieves robust performance and stability. Such a compensator, if anyexists, may be obtained by minimizing∥∥∥∥

[W1SW2T

]∥∥∥∥∞, (6.166)

which is nothing but a mixed sensitivity problem.Thus, the robust design problem has been reduced to a mixed sensitivity problem. This

reduction is not necessarily successful in solving the robust design problem — see Exercise 6.9.1(p. 294).

Exercise 6.8.1 (Proof of inequality). Prove the inequality(a + b)2 ≤ 2(a2 + b2) for any tworeal numbersa andb, for instance by application of the Cauchy-Schwarz inequality to the vectors(a, b) and (1, 1). �

Exercise 6.8.2 (Minimization of upper bound). Show that the upper bound on the right-handside of (6.163) may also be obtained from Summary 5.7.7(2) (p. 236). �

290

6.8. µ-Synthesis

6.8.3. Approximate solution of the µ-synthesis problem

More complicated robust design problems than the simple SISO problem we discussed cannotbe reduced to a tractable problem so easily. Various approximate solution methods to theµ-synthesis problem have been suggested. The best known of these (Doyle 1984) relies on theproperty (see Summary 5.7.7(2–3)

µ(M) ≤ σ[ DMD−1], (6.167)

with D andD suitable diagonal matrices. The problem of minimizing

µHK = supω∈R

µ(HK(jω)) (6.168)

is now replaced with the problem of minimizing the bound

supω∈R

σ[ D(jω)HK(jω)D−1(jω)] = ‖DHK D−1‖∞, (6.169)

where for each frequencyω the diagonal matricesD(jω) andD(jω) are chosen so that the boundis the tightest possible. Minimizing (6.169) with respect toK is a standardH∞ problem, providedD andD are rational stable matrix functions.

Doyle’s method is based onD-K iteration:

Summary 6.8.3 ( D-K iteration).

1. Choose an initial compensatorK that stabilizes the closed-loop system, and compute thecorresponding nominal closed-loop transfer matrixHK.

One way of finding an initial compensator is to minimize‖H0K‖∞ with respect to all sta-

bilizing K, whereH0K is the closed-loop transfer matrix of the configuration of Fig. 6.22

fromw to z with 1 = 10 = 0.

2. Evaluate the upper bound

minD( jω), D( jω)

σ[ D(jω)HK(jω)D−1(jω)], (6.170)

with D andD diagonal matrices as in Summary 5.7.7(c), for a number of values ofω on asuitable frequency grid. The maximum of this upper bound over the frequency grid is anestimate ofµHK .

If µHK is small enough then stop. Otherwise, continue.

3. On the frequency grid, fit stable minimum-phase rational functions to the diagonal entriesof D and D. Because of the scaling property it is sufficient to fit theirmagnitudesonly.The resulting extra freedom (in the phase) is used to improve the fit. Replace the originalmatrix functionsD andD with their rational approximations.

4. Given the rational approximationsD and D, minimize‖DHK D−1‖∞ with respect to allstabilizing compensatorsK. Denote the minimizing compensator asK and the correspond-ing closed-loop transfer matrix asHK . Return to 2.

�

291


Any algorithm that solves the standardH∞-problem exactly or approximately may be usedin step (d). The procedure is continued until a satisfactory solution is obtained. Convergence isnot guaranteed. The method may be implemented with routines provided in theµ-Tools toolbox(Balas, Doyle, Glover, Packard, and Smith 1991). A lengthy example is discussed in§ 6.9.

The method is essentially restricted to “complex” perturbations, that is, perturbations corre-sponding to dynamic uncertainties. “Real” perturbations, caused by uncertain parameters, needto be overbounded by dynamic uncertainties.

6.9. An application of µ-synthesis

6.9.1. Introduction

To illustrate the application ofµ-synthesis we consider the by now familiar SISO plant of Exam-ple 5.2.1 (p. 198) with transfer function

P(s) = gs2(1+ sθ)

. (6.171)

Nominally,g = g0 = 1 andθ = 0, so that the nominal plant transfer function is

P0(s) = g0

s2. (6.172)

In Example 5.2.2 (p. 199) root locus techniques are used to design a modified PD compensatorwith transfer function

C(s) = k + sTd

1+ sT0, k = 1, Td =

√2 = 1.414, T0 = 0.1. (6.173)

The corresponding closed-loop poles are−0.7652± j 0.7715 and−8.4679.In this section we explore howD–K iteration may be used to obtain an improved design that

achieves robust performance.

6.9.2. Performance specification

In Example 5.8.4 (p. 245) the robustness of the feedback system is studied with respect to theperformance specification

|S(jω)| ≤ 1|V(jω)| , ω ∈ R. (6.174)

The functionV is chosen as

1V

= (1+ ε)S0. (6.175)

S0 is the sensitivity function of the nominal closed-loop system and is given by

S0(s) = s2(1+ sT0)

χ0(s), (6.176)

with χ0(s) = T0s3 + s2 + g0Tds+ g0k the nominal closed-loop characteristic polynomial. Thenumberε is a tolerance.

292

6.9. An application ofµ-synthesis

In Example 5.8.4 it is found that robust performance is guaranteed with a toleranceε = 0.25for parameter perturbations that satisfy

0.9 ≤ g ≤ 1.1, 0 ≤ θ < 0.1. (6.177)

The robustness test used in this example is based on a proportional loop perturbation model.In the present example we wish to redesign the system such that performance robustness is

achieved for a modified range of parameter perturbations, namely for

0.5 ≤ g ≤ 1.5, 0 ≤ θ < 0.05. (6.178)

For the purposes of this example we moreover change the performance specification model.Instead of relating performance to the nominal performance of a more or less arbitrary design wechoose the weighting functionV such that

1V(s)

= (1+ ε)s2

s2 + 2ζω0s+ ω20

. (6.179)

Again,ε is a tolerance. The constantω0 specifies the minimum bandwidth andζ determines themaximally allowable peaking. Numerically we chooseε = 0.25,ω0 = 1, andζ = 1

2.The design (6.173) doesnot meet the specification (6.174) withV given by (6.179) for the

range of perturbations (6.178). Figure 6.24 shows plots of the bound 1/|V| together with plotsof the perturbed sensitivity functionsS for the four extreme combinations of parameter values.In particular, the bound is violated when the value of the gaing drops to too small values.

10-1

100

101

102

-60

-40

-20

20

0


bound| |[dB]

S

Figure 6.24.: Magnitude plot ofSfor different parameter value combinationsand the bound 1/V

6.9.3. Setting up the problem

To set up theµ-synthesis problem we first define the perturbation model. As seen in Example5.8.4 the loop gain has proportional perturbations

1P(s) = P(s)− P0(s)P0(s)

=g−g0

g0− sθ

1+ sθ. (6.180)

We bound the perturbations as|1P(jω)| ≤ |W0(jω)| for all ω ∈ R, with

W0(s) = α+ sθ0

1+ sθ0. (6.181)

293


The numberα, with α < 1, is a bound|g − g0|/g0 ≤ α for the relative perturbations ing, andθ0 is the largest possible value for the parasitic time constantθ. Numerically we letα = 0.5and θ0 = 0.05. Figure 6.25 shows plots of the perturbation|δP| for various combinations ofvalues of the uncertain parameters together with a magnitude plot ofW0. Figure 6.26 shows

10-1

100

101

102

-60

-40

-20

0

20


| |[dB]δP

bound

Figure 6.25.: Magnitude plots of the perturbationsδP for different parametervalues and of the boundW0

the corresponding scaled perturbation model. It also includes the scaling functionV used tonormalize performance.

K

Wo δP

Po

V

y

p

uoz

+ +

++

q

w

u

−

Figure 6.26.: Perturbation and performance model

Exercise 6.9.1 (Reduction to mixed sensitivity problem). In § 6.8.2 (p. 289) it is argued thatrobust performance such that‖SV‖∞ < 1 under proportional plant perturbations bounded by‖1P‖∞ ≤ ‖W0‖∞ is achieved iff

supω∈R

(|S(jω)V(jω)| + |T(jω)W0(jω)|) < 1. (6.182)

A sufficient condition for this is that

supω∈R

(|S(jω)V(jω)|2 + |T(jω)W0(jω)|2) < 12

√2. (6.183)

To see whether the latter condition may be satisfied we consider the mixed sensitivity problemconsisting of the minimization of‖H‖∞, with

H =[

SV−TW0

]. (6.184)

294


(a) Show that the mixed sensitivity problem at hand may be solved as a standardH∞ problemwith generalized plant

G =

V P0

0 W0V−1P0

−V −P0

. (6.185)

(b) Find a (minimal) state representation ofG. (Hint: Compare (6.190) and (6.191).) Checkthat condition (b) of§ 6.5.5 (p. 274) needed for the state space solution of theH∞ problemis not satisfied. To remedy this, modify the problem by adding a third componentz1 = ρuto the error signalz, with ρ small.

(c) Solve theH∞-optimization problem for a small value ofρ, say,ρ = 10−6 or even smaller.Use the numerical values introduced earlier:g0 = 1, ε = 0.25,ω0 = 1, ζ = 0.5, α = 0.5,andθ0 = 0.05. Check that‖H‖∞ cannot be made less than about 0.8, and, hence, thecondition (6.183) cannot be met.

(d) Verify that the solution of the mixed sensitivity problem does not satisfy (6.182). Alsocheck that the solution does not achieve robust performance for the real parameter varia-tions (6.178).

The fact that a robust design cannot be obtained this way does not mean that a robust design doesnot exist. �

The next step in preparing forD–K iteration is to compute the interconnection matrixG asin Fig. 6.22 from the interconnection equations

pzy

= G

qw

u

. (6.186)

Inspection of the block diagram of Fig. 6.26 shows that

p = W0u,

z = Vw+ P0(q+ u), (6.187)

y = −Vw− P0(q+ u),

so thatp

zy

=

0 0 W0

P0 W1 P0

−P0 −W1 −P0

︸︷︷︸G

qw

u

. (6.188)

To apply the state space algorithm for the solution of theH∞ problem we need a state spacerepresentation ofG. For the outputz we have from the block diagram of Fig. 6.26

z= Vw+ P0u0 = 11+ ε

s2 + 2ζω0s+ ω20

s2w+ g0

s2u0. (6.189)

295


This transfer function representation may be realized by the two-dimensional state space system

[x1

x2

]=[0 10 0

][x1

x2

]+[ 2ζω0

1+ε 0ω2

01+ε g0

][w

u0

],

z= [1 0

] [x1

x2

]+ 1

1+ εw.

(6.190)

The weighting filterW0 may be realized as

x3 = − 1θ0

x3 + α− 1θ0

u,

p = u+ x3. (6.191)

Exercise 6.9.2 (State space representation). Derive or prove the state space representations(6.190) and (6.191). �

Using the interconnection equationu0 = q+ u we obtain from (6.190–6.191) the overall statedifferential and output equations

x =

0 1 0

0 0 0

0 0 − 1θ0

x+

0 2ζω0

1+ε 0

g0ω2

01+ε g0

0 0 α−1θ0

qw

u

,

p

zy

=

0 0 1

1 0 0−1 0 0

x+

0 0 1

0 11+ε 0

0 − 11+ε 0

qw

u

.

(6.192)

To initialize theD–K iteration we select the compensator that was obtained in Example 5.2.2(p. 199) with transfer function

K0(s) = k + sTd

1+ sT0. (6.193)

This compensator has the state space representation

x = − 1T0

x+ 1T0

y,

u = (k − Td

T0)x + Td

T0y. (6.194)

6.9.4. D–K iteration

TheD–K iteration procedure is initialized by defining a frequency axis, calculating the frequencyresponse of the initial closed-loop systemH0 on this axis, and computing and plotting the lowerand upper bounds of the structured singular value ofH0 on this frequency axis13. The calculationof the upper bound also yields the “D-scales.”

13The calculations were done with the MATLAB µ-tools toolbox. The manual (Balas, Doyle, Glover, Packard, and Smith1991) offers a step-by-step explanation of the iterative process.

296


10-2

10-1

100

101

102

103

0.8

1

1.2

1.4


µ

Figure 6.27.: The structured singular value of the initial design

The plot of Fig. 6.27 shows that the structured singular value peaks to a value of about 1.3.Note that the computed lower and upper bounds coincide to the extent that they are indistin-guishable in the plot. Since the peak value exceeds 1, the closed-loop system does not haverobust performance, as we already know.

The next step is to fit rational functions to theD-scales. A quite good-looking fit may beobtained on the first diagonal entry ofD with a transfer function of order 2. The second diagonalentry is normalized to be equal to 1, and, hence, does not need to be fitted. Figure 6.28 showsthe calculated and fitted scales for the first diagonal entry. Because we have 1× 1 perturbation

10-2

10-1

100

101

102

103

10-4

10-2

100

102


| |D

D-scale

fit

Figure 6.28.: Calculated and fittedD-scales

blocks only, the left and right scales are equal.The following step in theµ-synthesis procedure is to perform anH∞ optimization on the

scaled systemG1 = DGD−1. The result is a sub-optimal solution corresponding to the boundγ1 = 1.01.

In the search procedure for theH∞-optimal solution as described in§ 6.4.6 (p. 269) thesearch is terminated when the optimal value of the search parameterγ has been reached withina prespecified tolerance. This tolerance should be chosen with care. Taking it too small not onlyprolongs the computation but — more importantly — results in a solution that is close to but notequal to the actualH∞ optimal solution. As seen in§ 6.6 (p. 221), solutions that are close tooptimal often have large coefficients. These large coefficients make the subsequent calculations,in particular that of the frequency response of the optimal closed-loop system, ill-conditioned,and easily lead to erroneous results. This difficulty is a weak point of algorithms for the solutionof of theH∞ problem that do not provide the actual optimal solution.

Figure 6.29 shows the structured singular value of the closed-loop system that corresponds tothe compensator that has been obtained after the first iteration. The peak value of the structured

297


10-2

10-1

100

101

102

103

0.8

1

1.2

1.4


µ initial solution

first iteration

second iteration

Figure 6.29.: Structured singular values for the three successive designs

singular value is about 0.97. This means that the design achieves robust stability.To increase the robustness margin anotherD–K iteration may be performed. Again theD-

scale may be fitted with a second-order rational function. TheH∞ optimization results in asuboptimal solution with levelγ2 = 0.91.

Figure 6.29 shows the structured singular value plots of the three designs we now have. Forthe third design the structured value has a peak value of about 0.9, well below the critical value1.

The plot for the structured singular value for the final design is quite flat. This appears to betypical for minimum-µ designs (Lin, Postlethwaite, and Gu 1993).

6.9.5. Assessment of the solution

The compensatorK2 achieves a peak structured singular value of 0.9. It therefore has robustperformance with a good margin. The margin may be improved by furtherD–K iterations. Wepause to analyze the solution that has been obtained.

Figure 6.30 shows plots of the sensitivity function of the closed-loop system with the com-pensatorK2 for the four extreme combinations of the values of the uncertain parametersg andθ.The plots confirm that the system has robust performance.

10-2

10-1

100

101

102

103

-100

-50

0

50


bound| |[dB]

S

Figure 6.30.: Perturbed sensitivity functions forK2 and the bound 1/V

Figure 6.31 gives plots of the nominal system functionsS and U. The input sensitivityfunctionU increases to quite a large value, and has no high-frequency roll-off, at least not inthe frequency region shown. The plot shows that robust performance is obtained at the cost of alarge controller gain-bandwidth product. The reason for this is that the design procedure has noexplicit or implicit provision that restrains the bandwidth.

298


10-2

10-1

100

101

102

103

-100

-50

0

50

100

angular frequency [rad/s]m

ag

nitu

de

[dB

]

U

S

Figure 6.31.: Nominal sensitivitySand input sensitivityU for the compen-satorK2

10-2

10-1

100

101

102

103

-50

0

50

100


ma

gn

itu

de

[dB

]

UV

1/W2

Figure 6.32.: Magnitude plots ofUV and the bound 1/W2

6.9.6. Bandwidth limitation

We revise the problem formulation to limit the bandwidth explicitly. One way of doing this isto impose bounds on the high-frequency behavior of the input sensitivity functionU. This, inturn, may be done by bounding the high-frequency behavior of the weighted input sensitivityUV. Figure 6.32 shows a magnitude plot ofUV for the compensatorK2. We wish to modify thedesign so thatUV is bounded as

|U(jω)V(jω)| ≤ 1|W2(jω)| , ω ∈ R, (6.195)

where

1W2(s)

= β

s+ω1. (6.196)

Numerically, we tentatively chooseβ = 100 andω1 = 1. Figure 6.32 includes the correspondingbound 1/|W2| on |U|.

To include this extra performance specification we modify the block diagram of Fig. 6.26to that of Fig. 6.33. The diagram includes an additional outputz2 while z has been renamedto z1. The closed-loop transfer function fromw to z2 is −W2UV. To handle the performancespecifications‖SV‖∞ ≤ 1 and‖W2UV‖∞ ≤ 1 jointly we impose the performance specification∥∥∥∥

[W1SVW2UV

]∥∥∥∥∞

≤ 1, (6.197)

whereW1 = 1. This is the familiar performance specification of the mixed sensitivity problem.

299


yK

u

z2

W2 W0p δP q

+

+

Po

V

+

+uo z1

w

−

Figure 6.33.: Modified block diagram for extra performance specification

By inspection of Fig. 6.33 we see that the structured perturbation model used to design forrobust performance now is

[qw

]=[δP 0 00 δ1 δ2

]︸︷︷︸

1

p

z1

z2

, (6.198)

with δP, δ1 andδ2 independent scalar complex perturbations such that‖1‖∞ ≤ 1.

BecauseW2 is a nonproper rational function we modify the diagram to the equivalent diagramof Fig. 6.34 (compare§ 6.7, p. 279). The compensatorK and the filterW2 are combined to a

yK W2

K~

z2

~uW2

−1

x4

Wo δPp

q

uoPo

w

V

z1

−+

+

+

+

Figure 6.34.: Revised modified block diagram

new compensatorK = KW2, and any nonproper transfer functions now have been eliminated.It is easy to check that the open-loop interconnection system according to Fig. 6.34 may be

300


represented as

x =

0 1 0 0

0 0 0 g0

0 0 − 1θ0

α−1θ0

0 0 0 −ω1

x+

0 2ζω01+ε 0

g0ω2

01+ε 0

0 0 0

0 0 β

qw

u

,

pz1

z2

y

=

0 0 1 11 0 0 00 0 0 0

−1 0 0 0

x+

0 0 0

0 11+ε 0

0 0 1

0 − 11+ε 0

qw

u

.

(6.199)

To let the structured perturbation1 have square diagonal blocks we rename the external inputw

tow1 and expand the interconnection system with a void additional external inputw2:

x =

0 1 0 0

0 0 0 g0

0 0 − 1θ0

α−1θ0

0 0 0 −ω1

x+

0 2ζω0

1+ε 0 0

g0ω2

01+ε 0 0

0 0 0 0

0 0 0 β

qw1

w2

u

,

pz1

z2

y

=

0 0 1 11 0 0 00 0 0 0

−1 0 0 0

x+

0 0 0 0

0 11+ε 0 0

0 0 0 1

0 − 11+ε 0 0

qw1

w2

u

.

(6.200)

The perturbation model now is[qw

]=[δP 00 10

][pz

]. (6.201)

with δp a scalar block and10 a full 2× 2 block.To obtain an initial design we do anH∞-optimal design on the system withq andp removed,

that is, on the nominal, unperturbed system. This amounts to the solution of a mixed sensitivityproblem. Nine iterations are needed to find a suboptimal solution according to the levelγ0 = 0.97.The latter number is less than 1, which means that nominal performance is achieved. The peakvalue of the structured singular value turns out to be 1.78, however, so that we do not have robustperformance.

One D–K iteration leads to a design with a reduced peak value of the structured singularvalue of 1.32 (see Fig. 6.35). Further reduction does not seem possible. This means that thebandwidth constraint is too severe.

To relax the bandwidth constraint we change the value ofβ in the bound 1/W2 as given by(6.196) from 100 to 1000. Starting withH∞-optimal initial design oneD-K iteration leads to adesign with a peak structured singular value of 1.1. Again, robust performance is not feasible.

Finally, after choosingβ = 10000 the same procedure results after oneD-K iteration in acompensator with a peak structured singular value of 1.02. This compensator very nearly hasrobust performance with the required bandwidth.

Figure 6.35 shows the structured singular value plots for the three compensators that aresuccessively obtained forβ = 100,β = 1000 andβ = 10000.

301


10-2

10-1

100

101

102

103

0.6

0.8

1

1.2

1.4


µβ = 100

β = 1000

β = 10000

Figure 6.35.: Structured singular value plots for three successive designs

6.9.7. Order reduction of the final design

The limited bandwidth robust compensator has order 9. This number may be explained as fol-lows. The generalized plant (6.200) has order 4. In the (single)D–K iteration that yields thecompensator the “D-scale” is fitted by a rational function of order 2. Premultiplying the plant byD and postmultiplying by its inverse increases the plant order to 4+ 2+ 2 = 8. Central subop-timal compensatorsK for this plant hence also have order 8. This means that the correspondingcompensatorsK = KW−1

2 for the configuration of Fig. 6.33 have order 9.It is typical for the D-K algorithm that compensators of high order are obtained. This is

caused by the process of fitting rational scales. Often the order of the compensator may consid-erably be decreased without affecting performance or robustness.

Figure 6.36(a) shows the Bode magnitude and phase plots of the limited bandwidth robust

10

10

-2

-2

10

10

-1

-1

10

10

0

0

10

10

1

1

10

10

2

2

10

10

3

3

10

10

4

4

10

10

5

5-200

-100

0

100

ph

ase

[de

g]


10-4

10-2

100

102

104

ma

gn

itu

de

(a)

(a)

(b),(c)

(b),(c)

Figure 6.36.: Exact (a) and approximate (b, c) frequency responses of thecompensator

compensator. The compensator is minimum-phase and has pole excess 2. Its Bode plot has breakpoints near the frequencies 1, 60 and 1000 rad/s. We construct a simplified compensator in twosteps:

302


1. The break point at 1000 rad/s is caused by a large pole. We remove this pole by omittingthe leading terms9 from the denominator of the compensator. This large pole correspondsto a pole at∞ of the H∞-optimal compensator. Figure 6.36(b) is the Bode plot of theresulting simplified compensator.

2. The Bode plot that is now found shows that the compensator has pole excess 1. It hasbreak points at 1 rad/s (corresponding to a single zero) and at 60 rad/s (corresponding toa double pole). We therefore attempt to approximate the compensator transfer function bya rational function with a single zero and two poles. A suitable numerical routine fromMATLAB yields the approximation

K(s) = 6420s+ 0.6234

(s+ 22.43)2 + 45.312. (6.202)

Figure 6.36(c) shows that the approximation is quite good.Figure 6.37 gives the magnitudes of the perturbed sensitivity functions corresponding to the

four extreme combinations of parameter values that are obtained for this reduced-order compen-sator. The plots confirm that performance is very nearly robust.

Figure 6.38 displays the nominal performance. Comparison of the plot of the input sensitivityU with that of Fig. 6.31 confirms that the bandwidth has been reduced.

10-2

10-1

100

101

102

103

-100

-50

0

50


ma

gn

itu

de

[dB

]

bound

Figure 6.37.: Perturbed sensitivity functions of the reduced-order limited-bandwidth design

10-2

10-1

100

101

102

103

-100

-50

0

50


ma

gn

itu

de

[dB

]

S

U

Figure 6.38.: Nominal sensitivityS and input sensitivityU of the reduced-order limited- bandwidth design

Exercise 6.9.3 (Peak in the structured singular value plot). Figure 6.35 shows that the sin-gular value plot for the limited-bandwidth design has a small peak of magnitude slightly greater

303


than 1 near the frequency 1 rad/s. Inspection of Fig. 6.37, however, reveals no violation of theperformance specification onSnear this frequency. What is the explanation for this peak?�

6.9.8. Conclusion

The example illustrates thatµ-synthesis makes it possible to design for robust performance whileexplicitly and quantitatively accounting for plant uncertainty. The method cannot be used na¨ıvely,though, and considerable insight is needed.

A severe shortcoming is that the design method inherently leads to compensators of highorder, often unnecessarily high. At this time onlyad hocmethods are available to decrease theorder. The usual approach is to apply an order reduction algorithm to the compensator.

Exercise 6.9.4 (Comparison with other designs). The compensator (6.202) is not very dif-ferent from the compensators found in Example 5.2.2 (p. 199) by the root locus method, byH2

optimization in§ 3.6.2 (p. 133). and by solution of a mixed sensitivity problem in§ 6.2.5 (p. 254).

1. Compare the four designs by computing their closed-loop poles and plotting the sensitivityand complementary sensitivity functions in one graph.

2. Compute the stability regions for the four designs and plot them in one graph as in Fig. 6.5.

3. Discuss the comparative robustness and performance of the four designs.�


6.10.1. Introduction

In this Appendix to Chapter 6 we present several proofs for§ 6.4 (p. 264) and§ 6.5 (p. 272).

6.10.2. Proofs for § 6.4

We first prove the result of Summary 6.4.1 (p. 264).

Inequality for the∞-norm. By the definition of the∞-norm the inequality‖H‖∞ ≤ γ is equivalent to thecondition thatσ(H∼(jω)H(jω))≤ γ for ω ∈ R, with σ the largest singular value. This in turn is the sameas the condition thatλi (H∼ H) ≤ γ2 on the imaginary axis for alli , with λi the i th largest eigenvalue. Thisfinally amounts to the condition thatH∼ H ≤ γ2 I on the imaginary axis, which is part (a) of Summary 6.4.1.The proof of (b) is similar.

Preliminary to the proof of the basic inequality forH∞ optimal control of Summary 6.4.2 (p. 250)we establish the following lemma. The lemma presents, in polished form, a result originally obtained byMeinsma (Kwakernaak 1991).

Summary 6.10.1 (Meinsma’s lemma). Let 5 be a nonsingular para-Hermitian rational matrix functionthat admits a factorization5 = Z∼ J Z with

J =[

In 00 −Im

](6.203)

and Z square rational. Suppose thatV is an(n+ m)× n rational matrix of full normal column rank14 andW anm× (n+ m) rational matrix of full normal row rank such thatW∼V = 0. Then

14A rational matrix has full normal column rank if it has full column rank everywhere in the complex plane except at afinite number of points. A similar definition holds for full normal row rank.

304


(a) V∼5V ≥ 0 on the imaginary axis if and only ifW∼5−1W ≤ 0 on the imaginary axis.

(b) V∼5V > 0 on the imaginary axis if and only ifW∼5−1W< 0 on the imaginary axis.

�

Proof of Meinsma’s lemma.(a) V∼5V ≥ 0 on the imaginary axis is equivalent to

V = Z−1

[AB

], (6.204)

with A ann × n nonsingular rational matrix andB ann × m rational matrix such thatA∼ A ≥ B∼ B on theimaginary axis. Define anm× m nonsingular rational matrixA and anm× n rational matrixB such that

[−B A]

[AB

]= 0. (6.205)

It follows that BA = AB, and, hence,A−1B = BA−1. From A∼ A ≥ B∼ B on the imaginary axis it followsthat‖BA−1‖∞ ≤ 1, so that also‖ A−1B‖∞ ≤ 1, and, hence,AA∼ ≥ BB∼ on the imaginary axis. Since

W∼V = W∼ Z−1

[AB

]= 0, (6.206)

it follows from (6.205) that we have

W∼ Z−1 = U[−B A], (6.207)

with U some square nonsingular rational matrix. It follows thatW∼ = U[−B A] Z, and, hence,

W∼5−1W = U( BB∼ − AA∼)U∼. (6.208)

This proves thatW∼5−1W ≤ 0 on the imaginary axis. The proof of (b) is similar.

We next consider the proof of the basic inequality of Summary 6.4.2 (p. 265) for the standardH∞optimal control problem.

Proof of Basic inequality.By Summary 6.4.1(b) (p. 264) the inequality‖H‖∞ ≤ γ is equivalent toH H∼ ≤γ2 I on the imaginary axis. Substitution of

H = G11 + G12K( I − G22K)−1︸︷︷︸L

G21, (6.209)

leads to

G11G∼11 + G11G

∼21L∼ + LG21G

∼11 + LG21G

∼21L∼ ≤ γ2 I (6.210)

on the imaginary axis. This, in turn, is equivalent to

[ I L ]

[G11G∼

11 − γ2 I G11G∼21

G21G∼11 G21G∼

21

]︸︷︷︸

8

[I

L∼

]≤ 0 (6.211)

on the imaginary axis. Consider

8−1 =[811 812

821 822

]−1

(6.212)

=[

I 8128−122

0 I

][811 −8128

−122821 0

0 822

][I 0

8−122821 I

]. (6.213)

305


For the suboptimal solution to exist we need that

8128−122821 −811 = G11G

∼21(G21G

∼21)

−1G21G∼11 − G11G

∼11 + γ2 I ≥ 0 (6.214)

on the imaginary axis. This is certainly the case forγ sufficiently large. Also, obviously822 = G21G∼21 ≥ 0

on the imaginary axis. Then by a well-known theorem (see for instance Wilson (1978)) there exist squarerational matricesV1 andV2 such that

8128−122821 −811 = V∼

1 V1, 822 = V∼2 V2. (6.215)

Hence, we may write

8 =[

I 8128−122

0 I

][V∼

1 00 V∼

2

]︸︷︷︸

Z∼

[−I 00 I

][V1 00 V2

][I 0

8−122821 I

],︸︷︷︸

Z

(6.216)

which shows that8 admits a factorization. It may be proved that the matrix8 is nonsingular. Taking

V =[

IL

], W =

[−LI

], (6.217)

application of Summary 6.10.1(a) (p. 304) to (6.211) (replacing5 with 8) yields

[ −L∼ I ] 8−1

[IL

]≥ 0 (6.218)

on the imaginary axis. Since[−LI

]=[−G12K( I − G22K)−1

I

]=[0 −G12

I −G22

][IK

]( I − G22K)−1 (6.219)

we have

[K∼ I

] [ 0 I−G∼

12 −G∼22

][G11G∼

11 − γ2 I G11G∼21

G21G∼11 G21G∼

21

]−1 [0 −G12

I −G22

]︸︷︷︸

5γ

[IK

]≥ 0. (6.220)

This is what we set out to prove.

6.10.3. Proofs for § 6.5

Our derivation of the factorization of5γ of § 6.5.3 (p. 273) for the state space version of the standardproblem is based on the Kalman-Jakuboviˇc-Popov equality of Summary 3.7.1 (p. 143) in§ 3.7.2. Thisequality establishes the connection between factorizations and algebraic Riccati equations.

Proof (Factorization of5γ ). We are looking for a factorization of

5γ =[

0 I

−G∼12 −G∼

22

][G11G∼

11 − γ2 I G11G∼21

G21G∼11 G21G∼

21

]−1[0 −G12

I −G22

]. (6.221)

First factorization. As a first step, we determine a factorization[G11G∼

11 − γ2 I G11G∼21

G21G∼11 G21G∼

21

]= Z1J1Z∼

1 , (6.222)

such thatZ1 and Z−11 have all their poles in the closed left-half complex plane. Such a factorization is

sometimes called a spectralcofactorization. A spectral cofactor of a rational matrix5 may be obtained by

306


first finding a spectral factorization of5T and then transposing the resulting spectral factor. For the problemat hand, it is easily found that[

G11(s)G∼11(s)− γ2 I G11(s)G∼

21(s)

G21(s)G∼11(s) G21(s)G∼

21(s)

]

=

H(sI − A)−1FFT(−sI − AT)−1HT − γ2 I 0 H(sI − A)−1FFT(−sI − AT)−1CT

0 −γ2 I 0

C(sI − A)−1FFT(−sI − AT)−1HT 0 C(sI − A)−1FFT(−sI − AT)−1CT + I

.

(6.223)

The transpose of the right-hand side may be written asR1 + H∼1 (s)W1H1(s), where

R1 =−γ2 I 0 0

0 −γ2 I 00 0 I

, H1(s) = FT(sI − AT)−1[ HT 0 CT], W1 = I . (6.224)

Appending the distinguishing subscript 1 to all matrices we apply the Kalman-Jakuboviˇc-Popov equality ofSummary 3.7.1 withA1 = AT, B1 = [ HT 0 CT], C1 = FT, D1 = 0, andR1 andW1 as displayed above.This leads to the algebraic Riccati equation

0 = AX1 + X1AT + FFT − X1(CTC − 1

γ2HT H)X1. (6.225)

The corresponding gainF1 and the factorV1 are

F1 =− 1

γ2 H

0C

X1, V1(s) = I +

− 1

γ2 H

0C

X1(sI − AT)−1[ HT 0 CT]. (6.226)

We need a symmetric solutionX1 of the Riccati equation (6.225) such thatAT + ( 1γ2 HT H − CTC)X1, or,

equivalently,

A1 = A+ X1(1γ2

HT H − CTC) (6.227)

is a stability matrix. By transposing the factorV1 we obtain the cofactorization (6.222), with

Z1(s)= VT1 (s) = I +

H

0C

(sI − A)−1X1

[− 1γ2 F 0 C

], (6.228)

J1 = R1 =−γ2 I 0 0

0 −γ2 I 00 0 I

. (6.229)

Second factorization.The next step in completing the factorization of5γ is to show, by invoking somealgebra, that

Z−11

[0 −G12

I −G22

]=

−H(sI − A1)−1CT −H(sI − A1)

−1B

0 −I

I − C(sI − A1)−1X1CT −C(sI − A1)

−1B

=0 0

0 −II 0

+

−H

0−C

(sI − A1)

−1[X1CT B

]. (6.230)

307


To obtain the desired factorization of5γ we again resort to the KJP equality, with (appending the distin-guishing subscript 2)

A2 = A1, B2 = [ X1CT B], R2 = 0, (6.231)

C2 =−H

0−C

, D2 =

0 0

0 −II 0

, W2 = J−1

1 =− 1

γ2 I 0 00 − 1

γ2 I 00 0 I

. (6.232)

This leads to the algebraic Riccati equation (inX2)

0 = AT1 X2 + X2 A1 − 1

γ2HT H + CTC

− ( X2X1 − I )CTC(X1X2 − I )+ γ2X2BBT X2. (6.233)

Substitution ofA1 from (6.227) and rearrangement yields

0 = AT X2 + X2A− ( X2X1 − I )1γ2

HT H(X1X2 − I )

+ X2X1(1γ2

HT H − CTC)X1X2 + γ2X2BBT X2. (6.234)

Substitution ofX1(1γ2 HT H − CTC)X1 from the Riccati equation (6.225) and further rearrangement result

in

0 = ( X2X1 − I )AT X2 + X2A(X1X2 − I )

+ ( X2X1 − I )1γ2

HT H(X1X2 − I )− X2(γ2BBT − FFT)X2. (6.235)

Postmultiplication of this expression byγ(X1X2 − I )−1, assuming that the inverse exists, and premultipli-cation by the transpose yields the algebraic Riccati equation

0 = AT X2 + X2A+ HT H − X2(BBT − 1γ2

FFT)X2, (6.236)

whereX2 = γ2X2(X1X2 − I )−1. After solving (6.236) forX2 we obtainX2 as X2 = X2(X1X2 − γ2 I )−1.The gainF2 and factorV2 for the spectral factorization of5γ are

F2 = L−12 (B

T2 X2 + DT

2 W2C2) =[

C(X1X2 − I )

−γ2BT X2

], (6.237)

V2(s)= I + F2(sI − A2)−1B2. (6.238)

We need to select a solutionX2 of the Riccati equation (6.233) such thatA2 − B2F2 is a stability matrix. Itis proved in the next subitem thatA2 − B2F2 is a stability matrix iff

A2 = A− BF2 = A+ (1γ2

FFT − BBT)X2 (6.239)

is a stability matrix, whereF2 = (BBT − 1γ2 FFT)X2 is the gain corresponding to the algebraic Riccati

equation (6.236).Since

L2 =[

I 00 − 1

γ2 I

], (6.240)

the desired spectral factorZγ of 5γ is

Zγ (s)=[

I 00 1

γI

]V2(s). (6.241)

308


Invoking an amount of algebra it may be found that the inverseMγ of the spectral factorZγ may be ex-pressed as

Mγ (s)=[

I 00 γ I

]+[

C

−BT X2

](sI − A2)

−1( I − 1γ2

X1X2)−1[ X1C

T γB]. (6.242)

Choice of the solution of the algebraic Riccati equation.We require a solution to the algebraic Riccatiequation (6.233) such thatA2 − B2F2 is a stability matrix. Consider

A2 − B2F2 = A1 − X1CTC(X1X2 − I )+ γ2BBT X2. (6.243)

Substitution ofA1 = A+ 1γ2 X1HT H − X1CTC and X = X2(X1X2 − γ2 I )−1 results in

A2 − B2F2 = (AX1X2 − γ2A+ 1γ2

X1HT H X1X2 − X1HT H

−X1CTCX1X2 + γ2BBTX2)(X1X2 − γ2 I )−1. (6.244)

Substitution ofX1(1γ2 HT H − CTC)X1 from (6.225) results in

A2 − B2F2 = (−γ2A− X1HT H + γ2BBT X2 − X1AT X2 − FFTX2)(X1X2 − γ2 I )−1. (6.245)

Further substitution ofHT H + AT X2 from (6.236) and simplification yields

A2 − B2F2 = (X1X2 − γ2 I )(A− BBTX2 + 1γ2

FFT X2)(X1X2 − γ2 I )−1. (6.246)

Inspection shows thatA2 − B2F2 has the same eigenvalues asA2 = A − BBT X2 + 1γ2 FFT X2. Hence,

A2 − B2F2 is a stability matrix iff A2 is a stability matrix.

309


310

A. Matrices

This appendix lists several matrix definitions, formulae and results that are used in the lecturenotes.

In what follows capital letters denote matrices, lower case letters denote column or row vec-tors or scalars. The element in theith row and jth column of a matrixA is denoted byAij .Whenever sumsA+ B and productsABetcetera are used then it is assumed that the dimensionsof the matrices are compatible.

A.1. Basic matrix results

Eigenvalues and eigenvectors

A column vectorv ∈ Cn is aneigenvectorof a square matrixA ∈ Cn×n if v 6= 0 andAv = λv forsomeλ ∈ C. In that caseλ is referred to as aneigenvalueof A. Oftenλi (A) is used to denotethe ith eigenvalue ofA (which assumes an ordering of the eigenvalues, an ordering that shouldbe clear from the context in which it is used). The eigenvalues are the zeros of thecharacteristicpolynomial

χA(λ) = det(λ In − A), (λ ∈ C, )

whereIn denotes then× n identitymatrix orunit matrix.An eigenvalue decompositionof a square matrixA is a decomposition ofA of the form

A = V DV−1, whereV andD are square andD is diagonal.

In this case the diagonal entries ofD are the eigenvalues ofA and the columns ofV are thecorresponding eigenvectors. Not every square matrixA has an eigenvalue decomposition.

The eigenvalues of a squareA and ofT AT−1 are the same for any nonsingularT. In particularχA = χT AT−1.

Rank, trace, determinant, singular and nonsingular matrices

The trace, tr(A) of a square matrixA ∈ Cn×n is defined as tr(A) = ∑ni=1 Aii . It may be shown

that

tr(A) =n∑

i=1

λi (A).

Therank of a (possibly nonsquare) matrixA is the maximal number of linearly independentrows (or, equivalently, columns) inA. It also equals the rank of the square matrixAT A which inturn equals the number of nonzero eigenvalues ofAT A.

311

A. Matrices

Thedeterminantof a square matrixA ∈ Cn×n is usually defined (but not calculated) recur-sively by

det(A) ={ ∑n

j=1(−1) j+1A1 j det(Aminor1 j ) if n> 1

A if n = 1.

Here Aminori j is the (n − 1)× (n − 1)-matrix obtained fromA by removing itsith row and jth

column. The determinant of a matrix equals the product of its eigenvalues, det(A)=∏ni=1λi (A).

A square matrix issingular if det(A) = 0 and isregular or nonsingularif det(A) 6= 0. Forsquare matricesA andB of the same dimension we have

det(AB) = det(A)det(B).

Symmetric, Hermitian and positive definite matrices, the transpose and unitary matrices

A matrix A ∈ Rn×n is (real) symmetricif AT = A. Here AT is the transposeof A is definedelementwise as(AT)i j = Aji , (i, j = 1, . . . ,n).

A matrix A ∈ Cn×n is Hermitian if AH = A. HereAH is thecomplex conjugate transposeofA defined as(AH)i j = Aji (i, j = 1, . . . ,n). Overbarsx+ jy of a complex numberx+ jy denotethe complex conjugate:x+ jy = x− jy.

Every real-symmetric and Hermitian matrixA has an eigenvalue decompositionA = V DV−1

and they have the special property that the matrixV may be chosenunitary which is that thecolumns ofV have unit length and are mutually orthogonal:VHV = I .

A symmetric or Hermitian matrixA is said to benonnegative definiteor positive semi-definiteif xH Ax ≥ 0 for all column vectorsx. We denote this by

A ≥ 0.

A symmetric or Hermitian matrixA is said to bepositive definiteif xH Ax> 0 for all nonzerocolumn vectorsx. We denote this by

A> 0.

For Hermitian matricesA andB the inequalityA ≥ B is defined to mean thatA− B ≥ 0.

Lemma A.1.1 (Nonnegative definite matrices). Let A ∈ Cn×n be a Hermitian matrix. Then

1. All eigenvalues ofA are real valued,

2. A ≥ 0 ⇐⇒ λi (A) ≥ 0 (∀ i = 1, . . . ,n),

3. A> 0 ⇐⇒ λi (A) > 0 (∀ i = 1, . . . ,n),

4. If T is nonsingular thenA ≥ 0 if and onlyTH AT ≥ 0.�

A.2. Three matrix lemmas

Lemma A.2.1 (Eigenvalues of matrix products). SupposeA andBH are matrices of the samedimensionn × m. Then for anyλ ∈ C there holds

det(λ In − AB) = λn−mdet(λ Im − BA). (A.1)

312

A.2. Three matrix lemmas

Proof. One the one hand we have[λ Im BA In

][Im − 1

λB

0 In

]=[λ Im 0A In − 1

λAB

]

and on the other hand[λ Im BA In

][Im 0

−A In

]=[λ Im − BA B

0 In

].

Taking determinants of both of these equations shows that

λmdet( In − 1λ

AB) = det

[λ Im BA In

]= det(λ Im − BA).

So thenonzeroeigenvalues ofAB and BA are the same. This gives the two very usefulidentities:

1. det( In − AB) = det( Im − BA),

2. tr(AB) =∑i λi (AB) =∑

j λ j(BA) = tr(BA).

Lemma A.2.2 (Sherman-Morrison-Woodburry & rank-one update).

(A+ UVH)−1 = A−1 − A−1U( I + VH A−1VH)A−1.

This formula is used mostly ifU = u andV = v are column vectors. ThenUVH = uvH has rankone, and it shows that a rank-one update ofA corresponds to a rank-one update of its inverse,

(A+ uvH)−1 = A−1 − 11+ vH A−1u

(A−1u)(vH A−1)︸︷︷︸rank-one

.

�

Lemma A.2.3 (Schur complement). Suppose a Hermitian matrixA is partitioned as

A =[

P QQH R

]

with P andR square. Then

A> 0 ⇐⇒ P is invertible,P> 0 andR− QH P−1Q> 0.

The matrixR− QH P−1Q is referred to as theSchur complementof P (in A). �

313

A. Matrices

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323

Index

∼, 12411

2-degrees-of-freedom, 502-degrees-of-freedom, 11

acceleration constant, 62action, 172additive perturbation model, 209adjoint

of rational matrix, 264algebraic Riccati equation, 105amplitude, 168ARE, 105

solution, 114asymptotic stability, 12

Bezout equation, 16bandwidth, 28, 80

improvement, 8basic perturbation model, 208BIBO stability, 13biproper, 268Blaschke product, 41Bode

diagram, 69gain-phase relationship, 35magnitude plot, 69phase plot, 69sensitivity integral, 38

Butterworth polynomial, 90

canonical factorization, 268central solution (ofH∞ problem), 267certainty equivalence, 121characteristic polynomial, 14, 311classical control theory, 59closed-loop

characteristic polynomial, 15system matrix, 107transfer matrix, 33

complementarysensitivity function, 23sensitivity matrix, 33

complex conjugate transpose, 312

constant disturbance model, 280constant disturbance rejection, 64constant error suppression, 280contraction, 211control

decentralized-, 185controllability gramian, 174controller

stabilizing-, 14crossover region, 32

decentralizedcontrol, 155, 185integral action, 188

decouplingcontrol, 156indices, 179

delaymargin, 22time, 80

derivative time, 66Descartes’ rule of sign, 201descriptor representation, 276determinant, 311DIC, 188disturbance condition number, 185D-K iteration, 291D-scaling upper bound, 238

edge theorem, 205eigenvalue, 311

decomposition, 311eigenvector, 311elementary column operation, 159elementary row operation, 159energy, 168equalizing property, 275error covariance matrix, 118error signal, 5Euclidean norm, 168

feedback equation, 6fixed point theorem, 211

324

Index

forward compensator, 5Freudenberg-looze equality, 40full perturbation, 217functional

controllability, 178reproducibility, 178

FVE, 80

gain, 87at a frequency, 69crossover frequency, 79margin, 21, 79

generalized Nyquist criterion, 19generalized plant, 126, 258gridding, 202Guillemin-Truxal, 89

H2-norm, 124, 173H2-optimization, 125Hadamard product, 186Hankel norm, 173Hermitian, 312high-gain, 6H∞-norm, 172, 213H∞-problem, 258

central solution, 267equalizing property, 275

Hurwitzmatrix, 206polynomial, 200strictly-, 180tableau, 200

induced norm, 171inferential control, 128∞-norm

of transfer matrix, 213of vector, 168

input-output pairing, 185input sensitivity

function, 30matrix, 33

integral control, 65integrating action, 64integrator in the loop, 132intensity matrix, 116interaction, 155interconnection matrix, 208internal model principle, 65

internal stability, 13, 175ITAE, 90

J-unitary, 266

Kalman-Jakuboviˇc-Popov equality, 108Kalman filter, 118Kharitonov’s theorem, 204KJP

equality, 108inequality, 108

L2-norm, 168lag compensation, 84leading coefficient, 16lead compensation, 81linearity improvement, 8linear regulator problem

stochastic, 116L∞-norm, 168loop

gain, 18gain matrix, 18IO map, 6, 10shaping, 34, 81transfer matrix, 15transfer recovery, 123

LQ, 104LQG, 115LTR, 123Lyapunov equation, 118Lyapunov matrix differential equation, 142

margindelay-, 22gain-, 21, 79modulus-, 21multivariable robustness-, 234phase-, 21, 79

matrix, 311closed loop system-, 107complementary sensitivity-, 33determinant, 311error covariance-, 118Hamiltonian, 114Hermitian, 312input sensitivity-, 33intensity-, 116loop transfer-, 15nonnegative definite-, 312

325

Index

nonsingular, 311observer gain-, 117positive definite-, 312positive semi-definite-, 312proper rational-, 18rank, 311sensitivity-, 33singular, 311state feedback gain-, 105symmetric, 312system-, 14trace, 311unitary, 312

M-circle, 75measurement noise, 31MIMO, 11, 33, 153minimal realization, 163minimum phase, 38mixed sensitivity problem, 251

typek control, 252modulus margin, 21monic, 109µ, 234µ-synthesis, 289µH , 236multiloop control, 185multiplicative perturbation model, 209multivariable dynamic perturbation, 233multivariable robustness margin, 234

N-circle, 75Nehari problem, 260Nichols

chart, 77plot, 77

nominal, 32stability, 213

nonnegative definite, 312nonsingular matrix, 311norm, 167

Euclidean, 168Hankel, 173induced, 171∞-, 213Lp-, 168matrix, 171onC

n, 168p-, 168spectral-, 169

submultiplicative property, 171normal rank

polynomial matrix, 159rational matrix, 160

notch filter, 83Nyquist

plot, 18, 74stability criterion, 18

observability gramian, 174observer, 117

gain matrix, 117poles, 122

open-loopcharacteristic polynomial, 15pole, 87stability, 20zero, 87

optimal linear regulator problem, 104order

of realization, 163output

controllability, 178functional reproducible, 178principal directions, 183

overshoot, 80

para-Hermitean, 265parametric perturbation, 198parasitic effect, 32parity interlacing property, 20partial pole placement, 254peak value, 168perturbation

full, 217multivariable dynamic, 233parametric, 198repeated real scalar, 232repeated scalar dynamic, 233unstructured, 208

perturbation modeladditive, 209basic, 208multiplicative, 209proportional, 209

phasecrossover frequency, 79margin, 21, 79minimum-, 38

326

Index

shift, 69PI control, 65PID control, 66plant capacity, 31p-norm, 168

of signal, 168of vector, 168

poleassignment, 16excess, 252of MIMO system, 161placement, 16

pole-zero excess, 74polynomial

characteristic-, 14invariant-, 159matrix, Smith form, 159matrix fraction representation, 226monic, 109unimodular matrix, 158

position error constant, 62positive

definite, 312semi definite, 312

principal directioninput, 183output, 183

principal gain, 183proper, 18

rational function, 15strictly-, 16

proportionalcontrol, 65feedback, 7perturbation model, 209

pure integral control, 65

QFT, 92quantitative feedback theory, 92

rank, 311rational matrix

fractional representation, 226Smith McMillan form, 160

realizationminimal, 163order of, 163

regulatorpoles, 122

system, 2relative degree, 74, 252relative gain array, 186repeated real scalar perturbation, 232repeated scalar dynamic perturbation, 233reset time, 65resonance

frequency, 80peak, 80

return compensator, 5return difference, 108

equality, 108, 144inequality, 108

RGA, 186rise time, 80robustness, 7robust performance, 242roll-off, 32, 86

LQG, 132mixed sensitivity, 252

root loci, 88

Schur complement, 114, 313sector bounded function, 220sensitivity

complementary-, 23function, 23input-, 30matrix, 33matrix, complementary-, 33matrix, input-, 33

separation principle, 121servocompensator, 189servomechanism, 189servo system, 2settling time, 80shaping

loop-, 81Sherman-Morrison-Woodbury, 313σ, σ, 169signal

amplitude, 168energy, 168L2-norm, 168L∞-norm, 168peak value, 168

signature matrix, 265single-degree-of-freedom, 12singular matrix, 311

327

Index

singular value, 169decomposition, 169

SISO, 11small gain theorem, 211Smith-McMillan form, 160Smith form, 159spectral

cofactorizaton, 306factor, 265

spectral norm, 169square down, 161stability, 11

asymptotic-, 12BIBO-, 13internal, 13, 175matrix, 174nominal, 213Nyquist (generalized), 19Nyquist criterion, 18of polynomial, 206of transfer matrix, 175open-loop, 20radius, 219radius, complex, 219radius, real, 219region, 201

stabilizing, 14standardH∞ problem, 258standardH2 problem, 126state

estimation error, 117feedback gain matrix, 105

state-space realization, 163static output feedback, 164steady-state, 60

error, 60step function

unit, 60strictly proper, 16structured singular value, 234

D-scaling upper bound, 238lower bound, 236, 238of transfer matrix, 236scaling property, 236upper bound, 236

submultiplicative property, 171suboptimal solution (ofH∞ problem), 264SVD, 169Sylvester equation, 16

symmetric, 312system matrix, 14

trace, 119, 311tracking system, 2transmission zero

rational matrix, 160triangle inequality, 168two-degrees-of-freedom, 11typek control

mixed sensitivity, 252typek system, 61

unimodular, 158unit

feedback, 5step function, 60

unitary, 312unstructured perturbations, 208

vector spacenormed, 168

velocity constant, 62

well-posedness, 176, 263

zeroopen loop, 87polynomial matrix, 159

328

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