Advanced Control An Overview on Robust Control - sga … an overview on robust... · Advanced Control An Overview on Robust Control P ' C Scope allow the student to assess the potential

Advanced Control

An Overview on Robust Control

P

C

Scope allow the student to assess the potential of different methods in

robust control without entering deep into theory. Sensitize for

the necessity of robust feedback control.

Keywords uncertainty representations, H∞, µ synthesis, LMI

Prerequisites Nyquist criterion, gain and phase margin, LQG state space

control

Contact Raoul Herzog1, Jurg Keller2

Version 5.1

Date September 19, 2011

1raoul.herzog@heig–[email protected]

Advanced Control, An Overview on Robust Control MSE

2 Raoul Herzog, Jurg Keller September 19, 2011

Contents

1 Introduction to Robust Control . . . . . . . . . . . . . . . . . . . . . 41.1 Motivation of Robust Control . . . . . . . . . . . . . . . . . . 41.2 An Attempt to define Robust Control . . . . . . . . . . . . . . 51.3 Structure of this Document . . . . . . . . . . . . . . . . . . . 5

2 Review of Norms for Signals and Systems . . . . . . . . . . . . . . . . 63 The Nyquist Criterion and the Small Gain Theorem . . . . . . . . . . 8

3.1 Review of the Nyquist Criterion and Classical Stability Margins 83.2 The Small Gain Theorem . . . . . . . . . . . . . . . . . . . . 103.3 Applications of the Small Gain Theorem to Robust Control . 11

4 Description of Model Uncertainty . . . . . . . . . . . . . . . . . . . . 124.1 Unstructured Uncertainty . . . . . . . . . . . . . . . . . . . . 134.2 Structured Uncertainty . . . . . . . . . . . . . . . . . . . . . . 19

5 Formulation of the Standard H∞ Problem . . . . . . . . . . . . . . . 226 A Glimpse on the H∞ State Space Solution . . . . . . . . . . . . . . 257 Limitation of H∞ Methods . . . . . . . . . . . . . . . . . . . . . . . . 268 Outlook: µ Synthesis and LMI Methods . . . . . . . . . . . . . . . . 27

8.1 Structured Singular Values (SSV) and µ Synthesis . . . . . . . 278.2 Linear Matrix Inequalities (LMI) . . . . . . . . . . . . . . . . 27

9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29A.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

A.1.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30A.1.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

A.2 Exercise 1: Linear Fractional Transformation . . . . . . . . . . . . . . 47A.3 Exercise 2: Small gain theorem . . . . . . . . . . . . . . . . . . . . . 47A.4 Exercise 3: Drawback of classical stability margins . . . . . . . . . . . 48A.5 Exercise 4: Two cart problem . . . . . . . . . . . . . . . . . . . . . . 49Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

September 19, 2011 Raoul Herzog, Jurg Keller 3


1 Introduction to Robust Control

1.1 Motivation of Robust Control

System modeling is generally one of the most important tasks in engineering, andin particular in control engineering. Modeling is always a delicate task, since thephysical reality is often complicated, and all attempt to mathematically describe areal physical process involves simplifications and assumptions, e.g.:

• dynamics are often consciously neglected1 to make the model tractable, e.g.unmodeled sensor and actuator dynamics, higher order modes from large scalestructures modeled by finite elements (FEM),

• nonlinearities are often either hard to model or too complicated,

• parameters are often not exactly known, either because they are hard to mea-sure precisely, or because of varying manufacturing conditions2.

Furthermore, it is desirable to end up with simple plant models. Indeed, in moderncontrol, the controller is the output of an optimization problem, and the complex-ity of the controller is directly linked with the complexity of the plant model: acomplicated high–order plant will automatically lead to a complicated high–ordercontroller, which is undesirable.

Robust control deals with system analysis and control design for such imperfectlyknown process models. One of the main goals of feedback control is to maintainoverall stability and system performance despite uncertainties in the plant.

In general, robustness does not “come for free” from a controller designed viaoptimal control and estimation theory (observer design): a controller designed for anominal process model generally works fine for the nominal plant model, but mayfail3 for even a “nearby” plant model.

An important point of all feedback control synthesis methods is the control en-gineer’s awareness of inherent trade–offs : increasing the robustness will generallymake the controller “less aggressive”, and will thereby decrease system performance.Robust control allows to specify more or less directly the plant uncertainty, andallows to predict the possible trade–offs between robustness and closed–loop perfor-mance.

Sometimes bachelor–level courses in control may give the impression that every-thing is feasible with control, and that it’s only a matter of finding a “good” con-troller. But this impression is completely false: the plant itself implies inherent lim-itations4, especially if the plant is unstable. The achievable performance/robustness

1It’s common to say: “My model is precise up to . . . Hz.”2For serial production of devices incorporating feedback control, individual tuning is not desir-

able. The robustness of the system should be sufficient to tolerate all uncertainties and tolerancesfrom the manufacturing processes. For mechatronical systems, the manufacturing uncertaintiesare coming from the mechanical and electronical parts. Generally, there are no manufacturingtolerances inside the digital controller because software is perfectly reproducable.

3Instability or unacceptable performance degradations may occur.4e.g. by the Bode integral relation.


MSE Advanced Control, An Overview on Robust Control

trade–offs strictly depend on the plant, and cannot be overcome by any sophisticatedcontrol [1].

1.2 An Attempt to define Robust Control

A definition of robust control could be stated as:

Robust control aims at designing a fixed (non–adaptive) controller such that somedefined level of performance5 of the controlled system is guaranteed, irrespective ofchanges in plant dynamics within a predefined class.

Robust control offers a collection of powerful mathematical tools and efficientsoftware algorithms able to partly6 answer the following key questions:

Describing and characterizing plant uncertainty: What is a good way to de-scribe plant variations or plant uncertainty? Some descriptions attempt tofaithfully describe the real situation (e.g. probability distributions on physicalparameters), but unfortunately there may exist no efficient method to solve theproblem. Other uncertainty descriptions are less direct, but more convenientfor the theory (e.g. analytic solutions exist).

Robustness analysis problems: As an example consider a nominal plant and agiven stabilizing controller. The uncertainty in the plant model is defined bya class of perturbations. There is no “true” model, we have to deal with agiven set of possible plant models. Now, we would like to know whether or notthe closed–loop remains stable for the whole class of plants. This is a typicalrobustness analysis problem.

Robustness synthesis problems: Find a controller which stabilizes a given classof plants. Generally, synthesis problems are more difficult to solve than ana-lysis problems.

1.3 Structure of this Document

The document is structured as follows: In section 2, norms for signals and systemsare reviewed for the single input single output case (SISO). Section 3 recapitulatesthe Nyquist criterion and the small gain theorem, an important working horse inrobust control. Section 4 addresses the question how to describe model uncertainty.Sections 5 introduces the setup of H∞ control. It is not the goal of this document todescribe the underlying mathematical theory, which is quite demanding. Therefore,

5e.g. closed–loop stability, reference tracking performance, and disturbance rejection perfor-mance.

6It should be noted that some simple problems in robust control (e.g. exact stability determi-nation of a linear system in which several parameters vary over given ranges) have shown to beNP–hard, hence as difficult as other famous problems for which no efficient solutions are known toexist, or likely to be found.



section 6 only sketches the H∞ solution from a user point of view. Section 7 discussessome limitations and drawbacks of standard H∞ methods. Finally, section 8 givesan outlook to the actual state–of–the–art in robust control. Section 9 concludeswith some general remarks on robust control.

2 Review of Norms for Signals and Systems

Norms are used in many places of engineering to quantify the magnitude of an object(e.g. the amplitude of a signal), or to quantify the proximity of two objects (e.g.the proximity of two systems). In robust control, norms play a crucial role:

• the choice of metric used to quantify the amount of process uncertainty,

• the choice of norm used in the optimization problem associated with the con-troller synthesis. The linear quadratic gaussian LQG control is based on theoptimization of a || · ||2 norm, whereas H∞ control is based on the optimizationof a || · ||∞ norm.

The L2 (Euclidean) norm of a time–domain scalar signal u(t) is defined as:

||u||2 =

√

∫ ∞

−∞u2(t) dt. (1)

If this integral is finite, then the signal u is square integrable, denoted as u ∈ L2.For vector–valued signals u(t),

u(t) =

u1(t)u2(t)

...un(t)

. (2)

the 2–norm is defined as

||u||2 =

√

∫ ∞

−∞||u(t)||22 dt =

√

∫ ∞

−∞uT (t)u(t) dt, (3)

where the superscript T denotes transposition.A Linear Time–Invariant (LTI) system G can be described either by a state spacerealization

x = Ax + Bu

y = Cx + Du

or by its corresponding transfer function

G(s) = C(sI − A)−1B + D. (4)



Similar to signals, we would like to define different norms for systems, respectively fortransfer functions. Two systems G1 and G2 are “close” if the norm of the differenceof their transfer functions ||G1−G2|| is “small”. Two systems might be “close” withrespect to one defined norm, but “far” with respect to another norm7.

The H2 norm of a stable system G(s) is defined as the L2 norm of its impulseresponse g(t). When dealing with stochastic signals, the H2 norm of a systemcorresponds to the variance E[||y||2] of the system response y(t) when the system isexcited by a unit intensity white noise input u(t).

The H∞ norm of a stable transfer function G(s) is defined as the maximum RMSamplification over arbitrary square integrable input signals u 6= 0.

||G||∞ = supu∈L2

||y||2||u||2

(5)

Note that the H∞ norm for a system is induced by the L2 norm for signals. Thephysical interpretation of the H∞ norm corresponds simply to the maximum energyamplification over all input signals. It can be shown that for single–input single–output systems ||G||∞ equals the peak magnitude in the Bode diagram of the transferfunction G(j ω):

||G||∞ = supω

|G(j ω)| (6)

In H∞ control the performance to be optimized is defined in terms of minimizingthe H∞ norm of closed–loop transfer functions, e.g. the sensitivity function S(s)and the complementary sensitivity function T (s). This type of optimization prob-lem is also called min–max problem: H∞ control seeks to minimize the worst–casescenario, i.e. when the closed–loop function has its peak. In contrast, classical LQGcontrol minimizes the closed–loop behaviour for known input signals8, whereas H∞

control works with unknown input signals and tries to optimize the worst–case sce-nario. Therefore, the solution of H∞ control problems typically inhibits “flat9” Bodemagnitude plots (no more peak). When minimizing the absolute peak in the Bodemagnitude diagram, automatically other local peaks pop up. This effect is called“waterbed–effect”. Finally, the value of all local peaks join the value of the absolutepeak, and the response becomes “flat”. The theoretical background of this comesfrom the Bode integral theorem, see equation (12) and figure 6 page 12.

Instead of “flat” Bode magnitude plots we can also impose a given shape. Thiscorresponds to a modern version of classical loop–shaping. In this context, importantdesign parameters are frequency dependent weighting functions W (s). They allow

7With the metric induced by the H∞ norm, the two systems P1(s) = 1

s+ǫand P2(s) = 1

sare

infinitely “far”, no matter how small ǫ is. This may seem illogical because a reasonable controllerwill stabilize both plants P1 and P2 simultaneously, and the resulting closed–loop systems will benearly identical. A remedy consists in using the so–called Vinnicombe metric, which also allows totreat marginally stable and unstable systems.

8e.g. the energy of the resulting closed–loop impulse response9allpass behaviour



10−1

100

101

102

Mag

nitu

de (

abs)

10−2

10−1

100

−180

−135

−90

−45

0

Pha

se (

deg)

Bode Diagram

Frequency (Hz)

Figure 1: Example: The || · ||∞ norm of G(s) = 1s2+0.1s+1

is ||G||∞ ≈ 10.

to shape given closed–loop functions, e.g. the sensitivity S(s), by optimizing theirweighted norm ||W S||∞.

The H∞ norm of a multivariable (MIMO) transfer matrix G(s) needs the im-portant concept of singular values which is beyond the scope of this document.

3 The Nyquist Criterion and the Small Gain Theorem

The Nyquist criterion is a cornerstone of classical control, and is of fundamentalimportance in robust control. The following chapter recapitulates the Nyquist cri-terion, the small gain theorem, and shows its application to robust control.

3.1 Review of the Nyquist Criterion and Classical Stability Margins

The Nyquist criterion allows to check closed–loop stability based on the inspectionof the loop gain L(s) = C(s) · P (s), without computing the closed–loop poles, i.e.the roots of 1 + L(s) = 0. The Nyquist criterion is based on Cauchy’s argument,and says:

The closed–loop system with loop gain L(s) and a negative feedback polarity isstable if and only if the complete10 Nyquist plot of L(jω) encircles the critical pointscrit = −1 exactly NP anticlockwise times in the complex plane, where NP is thenumber of unstable (right halfplane) poles of L(s).

As a special case, if L(s) already is a stable open loop transfer function, NP iszero, and the Nyquist plot of L(jω) must not encircle the critical point −1 in order

10For the complete Nyquist plot, ω runs from −∞ to +∞.



10−1

100

10−3

10−2

10−1

100

101

frequency Hz

mag

nitu

de o

f a c

lose

d−lo

op fu

nctio

n

Waterbed effect

minimizethe peak !

magnitude will pop upelsewhere !

Figure 2: Illustrating the waterbed effect.

loop gain

L(s)

feedback with minus polarity

Figure 3: Elementary feedback system with loop gain L(s)

to ensure closed–loop stability. Figure 4 recapitulates the definition of the classicalgain margin Am and phase margin φm. For example a gain margin of Am = 2 meansthat the closed-loop stays stable even if the loop gain doubles. The phase marginφm indicates the amount of additional delay the feedback loop can tolerate beforebecoming unstable. Robust control does not work with the classical stability marginsAm and φm for two reasons: 1) there exist no analytical optimization techniques forthese margins, and 2) there are cases where the classical margins indicate a goodrobustness against individual gain and phase tolerances, whereas the feedback loopis not at all robust against simultaneous variations of gain and phase, see exercice2, page 48. For these reasons, robust control prefers as margin the critical distancedcrit between the Nyquist plot of L(jω) and the critical point scrit = −1:

dcrit = minω

(dist(L(jω), scrit)) = minω

|L(jω) − scrit| = minω

|1 + L(jω)|. (7)

It follows that the critical distance dcrit is the reciprocal of the sensitivity function



−1.5 −1 −0.5 0 0.5 1 1.5 2

−1

−0.5

0

0.5

1

real part

imag

inar

y pa

rt

Nyquist plot of L(s)

complex plane

−1

dcrit

L(j ω)

φm

L(j 0)L(j ∞)

1/Am

Figure 4: Definition of the critical distance dcrit and the classical stability margins.Here, the Nyquist plot is only drawn for positive frequencies ω > 0.

peak:

dcrit =1

||S||∞, where S(jω) =

1

1 + L(jω). (8)

Maximizing the critical distance dcrit corresponds to minimizing the || · ||∞ norm ofthe sensitivity function. Therefore, one of the natural objectives11 of robust controlconsists of minimizing the || · ||∞ norm of the sensitivity function.

3.2 The Small Gain Theorem

The small gain theorem follows directly from the Nyquist criterion: if L(s) is stableand ||L||∞ < 1, then the loop gain |L(jω)| is smaller than 1 for all frequencies ω,and hence L(jω) does not encircle the critical point −1. It follows that the smallgain condition ||L||∞ < 1 is a sufficient but not necessary condition for closed–loop

11It is important to see that sensitivity peak minimization is not the only objective in real worldproblems. In fact, minimizing only the || · ||∞ norm of the sensitivity function turns out to be anill-posed problem, because the gain of the resulting controller would be infinite.



stability. The small gain condition corresponds to an infinite phase margin φm = ∞.If ||L||∞ < 1, the closed loop stays stable even if the feedback polarity is wrong!

The small gain theorem is not limited to linear feedback control: it can evenbe generalized to nonlinear feedback control. In this case, L becomes a nonlinearoperator in time domain, and the || · ||∞ norm condition on L(s) must be replacedby an operator norm condition. This may sound abstract, but a simple applicationis the feedback of a linear dynamic system in cascade with a static nonlinearity. Inthis case, the small gain theorem yields a sector bounded12 nonlinearity as a sufficientcondition for closed–loop stability.

It makes no sense to apply the small gain theorem directly for controller syn-thesis: for good tracking performance a high loop gain is needed within the controlbandwidth13. However, the small gain theorem has a great utility for analyzingfeedback loops with unstructured uncertainty.

3.3 Applications of the Small Gain Theorem to Robust Control

Suppose that P0(s) denotes the nominal plant, and C(s) a stabilizing controller.Now consider an additive perturbation which yields a “cloud” of plants P (s) =P0(s)+∆a(s), where ∆a(s) denotes an unknown stable transfer function representingthe modeling uncertainty of P0. The question arises “how much” uncertainty theclosed loop may tolerate before becoming unstable. Figure 5 shows that the feedback

a

-C

1+P0C

a

P0

C

perturbed plant P

+

+

a

P0

-C

+

+

controller

nominal plant

additive uncertainty

feedback

seen by

Figure 5: Application of the Small Gain Theorem to additive plant uncertainty.

12also called Popov criterion.13If the controller includes an integral action (e.g. the I part of PID), the loop gain is infinite at

0 Hz, and hence ||L||∞ = ∞.



seen by ∆a is −C1+P0C

. The corresponding loop gain is ∆a ·−C

1+P0C. Since both ∆a and

the nominal closed–loop are stable we can apply the small gain theorem for thefeedback loop seen by ∆a. A sufficient condition for the stability of the perturbedsystem is therefore:

∣

∣

∣

∣

∣

∣

∣

∣

∆a ·−C

1 + P0C

∣

∣

∣

∣

∣

∣

∣

∣

∞

< 1. (9)

Using the submultiplicative property of norms ||AB|| ≤ ||A|| · ||B||, the sufficientstability condition becomes:

||∆a||∞ ·∣

∣

∣

∣

∣

∣

∣

∣

C

1 + P0C

∣

∣

∣

∣

∣

∣

∣

∣

∞

< 1 =⇒ ||∆a||∞ <1

∣

∣

∣

∣

∣

∣

C1+P0C

∣

∣

∣

∣

∣

∣

∞

. (10)

Equation (10) is a conservative bound indicating the amount of tolerated additiveuncertainty which preserves stability.

In case of multiplicative unstructured uncertainty, the set of perturbed plants isdescribed as P (s) = P0(s) · (1+∆m(s)). Using the small gain theorem, the followingstability bound can be found:

||∆m||∞ <1

∣

∣

∣

∣

∣

∣

P0 C1+P0 C

∣

∣

∣

∣

∣

∣

∞

=1

||T ||∞. (11)

A high complementary sensitivity peak value ||T ||∞ leads to a small tolerable multi-plicative perturbation ∆m. We will now discuss an extremely fundamental relation-ship, called Bode integral theorem:

∫ ∞

0ln |S(jω)| dω = π

∑

unstable poles p of L(s)

Re(p). (12)

This law can be seen as a conservation law: the integrated value of the log of themagnitude of sensitivity function S(jω) is conserved under the action of feedback.If the open–loop L(s) is stable, then the integral becomes zero. At low frequencies,in order to have good tracking performance, the sensitivity must be much smallerthan 1 (negative dB levels), i.e. ln |S(jω)| must become negative. The Bode in-tegral theorem states that the average sensitivity improvement at low frequenciesis compensated by the average sensitivity deterioration at high frequencies. This isillustrated by figure 6. If the plant is unstable, the situation is becoming worse sincethe right hand side of equation (12) is now positive. This means that the averagesensitivity deterioration is always larger than the improvement. The more unstablethe plant is, the more positive the real part of the unstable poles, the more diffi-cult the situation becomes. This applies to every controller, no matter how it wasdesigned. Unstable plants are inherently more difficult to control than stable plants[1].

4 Description of Model Uncertainty

It is worthwhile to clarify what is meant by model uncertainty. In a control systemthere are two categories of uncertainties: disturbance signals and perturbations in



10

0.1e

duti

ng

aM

go

L

1.0

2.00.0 0.5 1.0 1.5

Frequency

Figure 6: Loopshaping constraints: Sensitivity reduction at low frequencies in-evitably leads to sensitivity increase at higher frequencies. Picture taken from [1].

the plant dynamics. Disturbances are external stochastic inputs which are not undercontrol. Examples of disturbance signals are sensor and actuator noise or changesof the environment. Dynamic perturbations are model uncertainties caused by notexactly known or slowly changing plant parameters, and unmodeled or approximatedsystem dynamics. In a house temperature control, disturbances and dynamic plantperturbations would be :

• Disturbances:changing outdoor temperature, wind speed, open windows and doors, heatingdue to electrical equipment or human bodies, sun radiation

• Dynamic plant perturbations:not exactly known isolation coefficients, unknown and changing heat capaci-ties, uncertain efficiency of the heating device

The example shows that it is not always clear how to classify the disturbances.Therefore, it is not surprising that in µ–synthesis the two categories of disturbancesare handled within the same formalism. For H∞–controller design plant uncertaintyresults from approximating the real system with a mathematical model of tractablecomplexity. Additionally its physical parameters are not exactly known.

As explained in the motivation section, the aim of H∞–based controller design isto include uncertainty into controller design. To achieve this it is necessary to havea mathematical description of model uncertainty. In this section, two different typesof uncertainty descriptions will be introduced. The first is unstructured uncertainty,whereas the second is structured uncertainty.

4.1 Unstructured Uncertainty

A description of model uncertainty has to meet the following goals: it should besimpler than a physical model of the neglected system dynamics, and it should be



tractable with a formalism which can easily be used in controller design. A simplesolution is found, if all dynamics, time–invariant perturbations that may occur indifferent parts of the system, are represented with a single perturbation transferfunction ∆(s). There are many possibilities to include ∆(s) into a control system,but only the two most commonly used will be presented in the following. These areshown in figures 7 and 8. The focus is on SISO systems. Figure 7 shows ∆a(s) asan additive perturbation of the plant transfer function P (s):

P (s) = Po(s) + ∆a(s), (13)

where P (s) is the actual, perturbed plant, and P0(s) is the nominal plant.The transfer function ∆a(s) is used to describe a frequency dependent unstruc-

tured uncertainty as follows:

∆a(s) = W2(s) · ∆a(s) (14)

where the normalized perturbation ∆a(s) is any stable transfer function with:

||∆a||∞ ≤ 1 (15)

This uncertainity description is closely related to a plant transfer function repre-sentation in C, the Nyquist plot, as shown in figure 9a). ||∆a||∞ = 1 defines a circle,whose radius is scaled14 with W2(s).

a

Figure 7: Additive uncertainty

Dynamic perturbations can also be described with multiplicative uncertainty.The corresponding block diagram is shown in figure 8.

P (s) = P0(s) · (1 + ∆m(s)) (16)

In multivariable systems, transfer functions are matrices. It is known that matrixmultiplication is not commutative, so for matrix–valued P0 and ∆m, P0(s) · (I +∆m(s)) is generally different from (I + ∆m(s)) ·P0(s). As a consequence, input andoutput multiplicative perturbations have to be distinguished. Input multiplicativeperturbations are able to model actuator uncertainty, whereas output multiplicativeperturbations are used to describe sensor related uncertainties.

14Index 2 for W is commonly used for uncertainty weighting functions.



m

Figure 8: Multiplicative uncertainty at plant input

The capabilities of additive or multiplicative uncertainty representation are il-lustrated with an example. For comparison purposes the unmodeled high frequencydynamic is specified. It will be investigated how this a priori known model error canbe represented with an additive or multiplicative uncertainty.

Example:

Many plants can roughly be approximated by a first order system, e.g. PT1. Sucha model usually neglects the high frequency dynamic of the system. As it is wellknown, a control system with a PT1-plant can achieve any required closed loopperformance. Consequently, we are interested in an uncertainty description whichincorporates possible unmodeled phase loss into controller design and prevents acontroller design with unrealistic high bandwidth.The nominal plant model is

P0(s) =5

s + 1. (17)

The unmodeled dynamic is represented with a set of transfer functions with thefollowing elements:

P∆(s) =1

(τs + 1)2, τ ∈ [0.02 . . . 0.05]. (18)

The plants to be controlled are:

P (s) = Po(s) · P∆(s). (19)

For comparison purposes the unmodeled dynamic described above is representedas an additive and multiplicative perturbation. Additive uncertainties are best rep-resented in the complex plane C, i.e. in the Nyquist diagram of figure (9a), whereasmultiplicative uncertainties can easily be plotted in a Bode diagram of figure (9b).The figures show the plant’s frequency response P (jω) for different values of τ . Next,a single description of uncertainty has to be found for the specified range of valuesfor the uncertain parameter τ . To accomplish this, the additive and multiplicative∆(jω) are determined for a set of values of τ according to the following formulas:



Additive unstructured perturbation:

∆a(s) = P (s) − P0(s). (20)

Multiplicative unstructured perturbation:

∆m(s) =P (s)

P0(s)− 1. (21)

−1 0 1 2 3 4 5−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

nominal plantperturbed plant set

(a) Additive uncertainty

10−1

100

101

102

103

100

105

10−1

100

101

102

103

−300

−200

−100

0

(b) Multiplicative uncertainty

Figure 9: Plant uncertainty

The magnitude of the perturbations ∆a(jω) and ∆m(jω) are shown in figure 10.As defined in (14), frequency dependency is specified with the weighting transferfunction W2(s). The solid line in the figures is an envelope of all error frequencyresponses and ∆(s) = W2(s) ·∆(s) is therefore an upper bound for the uncertainty.

Additive weighting:

W2a(s) =7.5s

(s + 1)(s + 15). (22)

Multiplicative weighting:

W2m(s) =0.997s(s + 170)

(s + 9)(s + 135). (23)

In figure 11 the resulting uncertainty regions are plotted in the correspondingplots.

Compare the resulting regions with the set of plants in figure 9! Obviously, theuncertainty region is very large due to the fact, that the nominal plant is not in thecenter of the plant set. Since the unnecessary uncertainty regions are mainly oppositeof the critical point, it might not have a large impact on controller design. Afterω = 10 rad/s the 0–gain point lies within the uncertainty set. As a consequence, a



10−1

100

101

102

103

10−3

10−2

10−1

100

approx. error boundadditive. errors

(a) Additive uncertainty

10−1

100

101

102

103

10−3

10−2

10−1

100

101

approx. error boundmultipl. errors

(b) Multiplicative uncertainty

Figure 10: Uncertainty descriptions

−1 0 1 2 3 4 5 6−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

perturbed plant set(w)

(a) Additive bounds

10−1

100

101

102

103

10−5

100

105

10−1

100

101

102

103

−300

−200

−100

0

100

(b) Multiplicative bounds

Figure 11: Uncertainty bounds

plant phase cannot be determined anymore. In the Bode diagram of figure phasebounds are set to −270 and 90 degrees, in order to get a nice bode plot.

In a similar way, static nonlinearities (sector criterion) or uncertain dead–timecan be described with unstructured uncertainties.

There arises the question, which of the representations of uncertainty should beused. Since the optimal H∞–controller has the order of the plant plus the orders ofall the weighting functions, it is preferable to choose a representation which leadsto a minimal order weighting. In the SISO case, additive uncertainty can be recastinto multiplicative uncertainty with simple algebraic operations:

P0(s) ∆m(s) = ∆a(s) (24)

Since the additive representation includes the plant transfer function P0(s) it isexpected, that an additive representation usually is of higher degree. To illustratethis: uncertain dead–time can be fit into a multiplicative uncertainty description



which is independent of the the plant:

P (s) = Po(s) · e−sT (25)

∆m(s) = e−sT − 1 (26)

This function is shown in the amplitude plot of figure 12. It can be bounded witha DT1 transfer function.

10−1

100

101

102

103

10−3

10−2

10−1

100

101

Figure 12: Dead–time uncertainty

The chosen uncertainty representation determines which closed–loop transferfunction has to be considered in H∞–controller design in order to guarantee robust-ness with respect to stability and performance.

Additive and multiplicative uncertainty are special cases of a more general frame-work which will be explained below. First, notice that we have to distinguish twodifferent feedback loops: the feedback loop formed by the controller and the per-turbed plant, and the feedback loop in which the uncertainty ∆ resides. A generalsolution for all different uncertainty descriptions can be obtained, if the controlsystem is described in the feedback structure shown in figure 13. The plant P ispartitioned according to the dimensions of ∆.

P =

[

P11 P12

P21 P22

]

. (27)

With simple algebraic calculations it follows:

z = [P22 + P21∆(I − P11∆)−1P12] w (28)



gh

Figure 13: Standard P–∆ configuration

if (I − P11∆)−1 exists.

Define:F(P, ∆) := P22 + P21∆(I − P11∆)−1P12 (29)

F(P, ∆) is called a linear fractional transformation LFT of P and ∆. In the singleinput single output case the LFT corresponds to a bilinear transform. This represen-tation is also used for more sophisticated uncertainty representation (next section)and also to formulate the general H∞–controller design problem.

Additive and multiplicative uncertainty representations are special cases of alinear fractional transformation. For additive perturbations, the matrix P becomes:

P =

[

0 II P0

]

. (30)

For multiplicative perturbations at the plant output, the matrix P becomes:

P =

[

0 P0

I P0

]

. (31)

The example before shows that the conservative error bounds on the plant can belarge, compared to the real plant perturbation. An unstructured uncertainty modelis also not suited for perturbations which only affects a part of an interconnectedsystem. This motivates a more general representation of uncertainty which will beintroduced in the next section.

4.2 Structured Uncertainty

Structured uncertainty means uncertainty (tolerances) of concrete physical parame-ters. Some examples are listed below:

• electrical components like resistors or capacitors are always affected with tol-erances, e.g. 2% for a standard resistor.

• In a magnetic bearing system, the nominal air gap is an important parameterwhich is affected by manufacturing tolerances and by thermal growth whenthe machine is running.



• The relative permeability µr of a magnetic material is usually not very preciselyknown

The internal structure of a linear plant can be represented as a block diagram withintegrators, summators, and gains. The physical parameters are gains in the blockdiagram. It can be shown that the plant transfer function is always an LFT (linearfractional transform) with respect to every physical parameter p1, . . . , pn.As an example, consider the uncertain plant equation (17), (18) and (19) page 15.We will treat the time constant τ in P∆(s) as uncertain parameter lying in the rangeof 0.02 . . . 0.05. Suppose that the nominal value τ0 corresponds to the mid–rangevalue τ0 = 0.035, and τ = τ0 + ∆τ , where ∆τ is a real parameter varying between−0.015 . . . + 0.015. The uncertain system P (s) = P0(s) · P∆(s) = 5

s+1· 1

(τs+1)2can

be recasted with the following linear fractional transform in figure 14. Note that

Paug

diagonal uncertainty block

w z

gh

Figure 14: Structured uncertainty with an unknown time constant τ appearingtwice.

the uncertainty block ∆ in figure 14 is diagonal with the same diagonal element ∆τ

repeated twice. The reason is that the uncertain time constant τ appears twice inthe second order term P∆(s). The augmented plant Paug(s) will be of third orderwith 3 inputs and 3 outputs.Exercise : Calculate Paug(s) and show that the associated linear fractional transformyields

F(Paug, ∆) =5

s + 1·

1

((τ0 + ∆τ )s + 1)2(32)

There are available software tools capable to directly define and handle systemswith structured or unstructured uncertainty. Using the Robust Control toolbox ofMatlab the example above can be entered as follows :

>> P0 = tf(5, [1, 1])

Transfer function:



5

-----

s + 1

>> tau = ureal(’tau’, 0.035, ’range’, [0.02, 0.05])

Uncertain Real Parameter: Name tau, NominalValue 0.035, Range [0.02 0.05]

>> Pdelta = tf(1, [tau, 1])*tf(1, [tau, 1])

USS: 2 States, 1 Output, 1 Input, Continuous System

tau: real, nominal = 0.035, range = [0.02 0.05], 2 occurrences

>> P = P0 * Pdelta

USS: 3 States, 1 Output, 1 Input, Continuous System

tau: real, nominal = 0.035, range = [0.02 0.05], 2 occurrences

>> bode(P)

which gives the family of Bode plot in figure 15.

−150

−100

−50

0

50

Mag

nitu

de (

dB)

10−2

10−1

100

101

102

103

−270

−180

−90

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Figure 15: Example : structured uncertainty with an unknown time constant τ .

In contrast to unstructured uncertainty, see figure 13, structured uncertaintyleads to a feedback configuration figure 15 with a static diagonal block ∆. In addi-tion, the diagonal terms are real–valued, whereas in the unstructured case they aremostly complex–valued norm–bounded transfer functions ∆(s).

It is obvious that structured uncertainty representations are often closer to phys-ical specifications. However, the algorithms needed to tackle robust control analysisor synthesis problems with structured uncertainty are much more complicated, seeoutlook section page 27.



We disadvise from recasting highly structured parameter uncertainty as additiveor multiplicative unstructured uncertainty, because this implies a high degree of con-servativeness. Indeed, the class of perturbations may become excessively large, andthe resulting controller, if there is any complying with the robustness specificationsis likely to show poor performance.

5 Formulation of the Standard H∞ Problem

In this section the controller design problem is formulated in the H∞ formalism.The objective is to design a controller, which does not only have nominal stabilityand performance, but has this properties with all the plants represented by someuncertainty description. H∞–controller design was specially developped, to solvethis problem in a systematic manner. As system which is stable with all the plantsof the uncertainty set is called robustly stable. It has a robust performance, if theperformance specifications are met for all the admissible plant behaviors.

The control system with controller and uncertainty model is shown in figure 16.In H∞–controller design, the H∞–norm of the mapping from w to z can be mini-mized. As a consequence the control problem has to be formulated in this formalism.The inputs w are typically reference or disturbance signals, whereas the outputs zcan be the control error or the controller output. The H∞–controller design problem

Figure 16: Control system with uncertainty

cannot directly by solved in the structure of figure 16. Since performance specifi-cations and stability conditions can both be expressed as H∞–norm conditions onsome transfer functions, there are two ways to simplify the control system of figure16. Either the stability conditions are formulated in the same formalism as perfor-mance specifications, resulting in a structure as in figure 17(a), or the performancespecification is formulated similarly to the uncertainty description ∆p (see figure17(b)). For unstructured uncertainties the structure of figure 17(a) is easier to go,but for structured uncertainties, only the structure in figure 17(b) leads to an ele-gant representation. The structured uncertainty problem is solved with µ-synthesis,which is beyond the scope of this introduction. Consequently, the first approach willbe followed in the sequel.

As a result from the small gain section, it is clear that a system is robustly stable,if



(a) Robust Stability as norm condition (b) Performance as artificial plant uncer-tainty

Figure 17: Possible system representations

10−2

10−1

100

101

102

100

101

102

Figure 18: Performance Weighting Function

for additive uncertainty:

||W2 C(I + PC)−1||∞ ≤ 1 (33)

for multiplicative uncertainty:

||W2 PC(I + PC)−1||∞ ≤ 1 (34)

In a control system as shown in figure 19 the robust stability condition corre-sponds to the H∞–norm of the mapping from r to u for the additive case and fromr to y for the multiplicative case.

System performance is usually specified by conditions on disturbance attenua-tion, i.e. on the H∞–norm of the mapping from d to y, or for reference tracking onthe H∞–norm of r → e. For both mappings an acceptable performance is achieved,if the following condition on the sensitivity function S = (I + PC)−1 is met:

||γW1 (I + PC)−1||∞ ≤ 1 (35)

This condition is fulfilled, if S(jω) ≤ 1|W1(jω)|

. Consequently, a typical weighting

function W1(s) has a shape as shown in figure 18. An integrator in W1(s) forces



the sensitivity function to be zero at ω = 0. The parameter γ < 1 is used as anoptimization parameter. The larger γ the better the disturbance attenuation.

Performance and robust stability can therefore be achieved, if the two conditionsare combined to the following condition (mixed sensitivity approach):

Additive uncertainty:

∥

∥

∥

∥

∥

[

γW1 (I + PC)−1

W2,a C(I + PC)−1

]∥

∥

∥

∥

∥

∞

. (36)

Multiplicative uncertainty:

∥

∥

∥

∥

∥

[

γW1 (I + PC)−1

W2,m PC(I + PC)−1

]∥

∥

∥

∥

∥

∞

. (37)

Figure 19: Control System for mixed Sensitivity Optimization

The block diagram representation of the mixed sensitivity problem is shown in19. Here it becomes obvious, that the robust stability condition can be viewed as amapping of signals.A controller can be found by solving the following optimization problem:

Additive uncertainty:

maxγ

∥

∥

∥

∥

∥

[

γW1(s)(I + PC)−1

W2,a(s)C(I + PC)−1

]∥

∥

∥

∥

∥

∞

= 1, (38)

(Similarly for the multiplicative uncertainty)

Figure 20: LFT for H∞–optimization



With some matrix algebra, the equation (38) can be transformed into a lowerlinear fractional transformation F(P,C) as shown in figure 20. Therefore, the fol-lowing problem has to be solved:

Find a stabilizing controller C, for which:

||F(P,C)||∞ := ||P11 + P12C(I − P22C)−1P21||∞ = 1 (39)

Nowadays, various solutions to this problem are known. The two most importantare state space methods based on the solution of two Riccati equation and thesolution based on linear matrix inequalities (LMI). Special attention has to be payedon additional assumptions, that are imposed for the algorithms. Some of them willbe treated in the next chapter.

6 A Glimpse on the H∞ State Space Solution

H∞–controller design problems can be formulated in different ways, but finally, theproblem has to be brought to a representation as shown in figure 20. Its state spacerepresentation is:

x = Ax + B1w + B2u (40)

z = C1x + D11w + D12u (41)

y = C2x + D21w + D22u (42)

(43)

The system matrices are usually represented as follows:

Gs =

A... B1 B2

· · · · · · · · ·

C1... D11 D12

C2... D21 D22

. (44)

A closed optimal solution for the above general system is not yet published. Forthe H∞ problem a closed solution was first developped in [2] for the following specialcase:

Gs =

A... B1 B2

· · · · · · · · ·

C1... 0

[

0I

]

C2...

[

0 I]

0

. (45)

In a design tool the general state space description of equation (44) is transformedinto the representation of equation (45) by means of loop shifting [3]. The transfor-mation of D21 and D12 into the form with the identity matrix, requires that the two



matrices have full rank. This condition is usually violated, when weighting func-tions are strictly proper, i.e. have a zero D–matrix. To circumvent this problem,the weighting D-matrix can usually be set to a small value, without influencing thecontroller design.

The conditions for existence of a solution are:

• (A,B2) is stabilisable and (C2, A) is detectable

• D12 and D21 must be of full rank

•

[

A − jωI B2

C1 D12

]

has ful column rank for all ω

•

[

A − jωI B1

C2 D21

]

has ful column rank for all ω

Another important aspect is that the H∞ problem must have a solution atω = ∞. An unappropriate weighting at ω = ∞ can lead to unsatisfactory con-troller designs. For a closed loop system with plant as in equation (44) the directfeedthrough term (gain at ω = ∞) is:

Dcl = D11 + D12(I − DcD22)−1DcD21 (46)

where Dc is the controller D-matrix.The term D11 is modified with the controller Dc-matrix. This can only be doneto a limited extent, which is depending on the sizes of D12 and D21. Bounds aredocumented in [3].

7 Limitation of H∞ Methods

In this section some of the limitations of H∞–controller design will be summarized.

• The H∞–controller design methods offers powerful tools to solve various designproblems. Nevertheless H∞ performance measures are not always adequatefor the investigated design problem and although robustness and performancemeasures are combined in the design procedure, it does not really guarantee,that performance is robust with respect to plant uncertainty. Consequentlyit was proposed to combine H∞ robustness measures with other performancemeasures, for example measures similar to the LQR-design, the mixed H2/H∞

design. Also LMI (see chapter ’outlook’) offers many possibilities for ’mixed’design problems.

• H∞–controller design leads to controllers of high order. A high order controllerneeds more resources for its implementation and is suspect to numerical prob-lems. Therefore it is favorable to have low order controllers. The optimalsolution for a system as described by equation ((44)) has the same order as



the system itself (size of the A-matrix). The order of the system is the orderof the plant to be controlled and the sum of the orders of all the weightingfunctions, see figure 19. As already mentioned in the section on uncertaintydescription, the weighting functions have to be of minimal order. The prob-lem of high order controllers can be diminished with order reduction methods.There are several order reduction methods available. These can be appliedeither on plant before H∞–controller design or after design on the controlleritself. The designer should be cautious not to spoil the controller design with aradical order reduction. There are also suboptimal H∞–controller design algo-rithm which allow a direct design of a reduced order controller, but these arenot currently available in the commercially available controller design tools,although a direct low order controller design is doubtless the way to go.

• As it can be seen from figures 11 the uncertainty description is not tight inthe sense, that it incorporates a much larger set of plants than necessary. Thisleads to a conservative controller design, which might result in very poor closedloop performance. This problem is reduced, when a structured uncertaintydescription is used in conjuction with µ–synthesis (see Chapter ’Outlook’).

8 Outlook: µ Synthesis and LMI Methods

8.1 Structured Singular Values (SSV) and µ Synthesis

The standard H∞ minimizes an H∞ norm between input signal w and output signalz. Often these signals are vector–valued (MIMO), as it is already the case for z inthe mixed–sensitivity approach. In practice we are interested to individual transferfunctions between a scalar component wk and a scalar component zl. A large amountof conservativeness is introduced in the design when optimizing an overall H∞–norm.

The µ framework allows a more selective optimization related to structured un-certainty. A detailed description is out of the scope of this document. Basically theµ framework introduces a new norm based on structured singular values which is thenew quantity to be minimized. The solution procedure is iterative (D–K iteration)and involves a sequence of minimizations, first over the controller K (holding thescaling variable D associated with the µ), and then optimizing over the scaling D(holding the controller K fixed). The D–K procedure is not guaranteed to convergeneither to the global minimum nor to a local minimum of the µ value, but oftenworks well in practice. It has been applied to a number of real–world applicationswith success. These applications include vibration suppression for flexible structures,flight control, chemical process control, and acoustic reverberation suppression inenclosures.

8.2 Linear Matrix Inequalities (LMI)

Linear Matrix Inequalities (LMI) have emerged as powerful numerical design toolsin areas ranging from robust control (e.g. H∞) to system identification.



The canonical form of linear matrix inequality is any constraint of the form

M(v) = M0 + v1M1 + v2M2 + . . . + vnMn < 0 (47)

where v = [v1, . . . , vn] is a vector of unknowns called decision variables, and M0, . . . ,Mn

are given symmetric matrices. M(v) < 0 stands for negative definite, i.e. all eigen-values of M(v) are negative.

The scalar decision variables v1, . . . , vn are often related to unknown matrices incontrol problems, e.g. quadratic Lyapunov functions V (x) = xT P x with P > 0.Here the entries of the symmetric matrix P correspond to decision variables. It iscommon to express LMI’s in matrix form instead of the scalar formulation (47).

There are three generic LMI problems:

Feasibility problem: Find a solution v to the LMI system M(v) < 0. The setof all solutions is also called feasibility set. A simple example for an LMIfeasibility problem is the determination of stability. The system x = Ax isstable if there exist a Lyapunov function V (x) = xT P x > 0 with a negativederivative V = AT P + PA < 0. This turns out to be an LMI problem withan unknown matrix P . Another frequent example is a state space system withbounded infinity norm. This turns out to be an LMI problem.

It is easy to show that the feasibility set of equation (47) is convex. Convexityhas an important consequence: even though the optimization problem has noanalytical solution in general, it can be solved numerically with a guaranteedconvergence to the solution when one exists. This is in sharp contrast togeneral nonlinear optimization algorithms which may not converge towardsthe global optimum.

v1

v2

non-convex

constraint

convex

constraint

v1

v2

Figure 21: Convexity

Linear objective minimization problem: Minimize the linear function cT v sub-ject to the LMI constraint M(v) < 0.

Generalized eigenvalue minimization problem: Minimize λ subject to M(v) <λ B(v), B(v) > 0, and C(v) < 0.

Efficient15 interior–point algorithms [4] are available to solve these three generic

15However the complexity of LMI computations can grow quickly with the problem order. Forexample, the number of operations required to solve a standard Riccati equation is o(n3) where n

is the state dimension. Solving an equivalent LMI Riccati inequality needs o(n6) operations!



LMI problems. Many control problems and general design specifications can beformulated in terms of LMI. The main strength of LMI formulations is the ability tocombine various convex design constraints or objectives in a numerically tractablemanner [5] [6].

A non–exhaustive list of problems addressed by LMI techniques include the fol-lowing:

1. robust pole placement

2. optimal LQG control

3. robust H∞ control

4. multi–objective synthesis

5. robust stability of systems with structured parameter uncertainty (µ–analysis)

It is fair to say the advent of LMI optimization has significantly influenced thedirection of research in robust control. A widely–accepted technique for numericallysolving robust control problems is to reduce them to LMI problems.

9 Conclusion

Robust control emerged around 1980, and the progress in theory and numericalalgorithms during the last 25 years has been enormous. Efficient commercial andpublic domain software tools are available nowadays. It is possible to successfullyuse these tools without understanding the deepest details of the underlying theory.However, a minimum of theoretical knowledge is necessary, and as it is the case withany complex scientific software tools16, a lot of practical experience is needed beforebeing able to solve real–world17 problems.

In robust control, specific reasons for this are the following: Modeling is a chal-lenging engineering task, and finding an estimation the modeling uncertainty needssome experience. Additionally, either control specifications are not entirely knownin the beginning, i.e. incomplete, or real–world specifications may be very complex18

to such an extent that no direct synthesis procedure exist. No matter which con-troller synthesis method is used it remains an iterative process which also needs someexperience. In robust control, the user often needs to play around with frequencyweighting functions in order to understand the performance/robustness trade–offsof his specific problem.

One of the main achievements in robust control might be the consciousness aboutthe industrial importance of robustness. Generally speaking, system failure is notaccepted in our society, and robustness considerations will remain very important.

16e.g. tools for finite element modeling, or any other complex scientific tool17= non–academic18e.g. mixed time domain and frequency domain specifications, low controller order, etc.



A.1 Examples

A.1.1 Example 1

H∞–controller design will be demonstrated on the first order system with unstruc-tured additive uncertainty, already introduced in 4.1. The nominal plant model is afirst order system, for which any closed-loop performance is achievable, if there wereno model uncertainty. The uncertainty description is given by equation (22) andfor controller design the mixed sensitivity approach will be used (see (33))19. Theperformance requirements are: zero steady-state error and maximum closed loopbandwidth. We will investigate, how model uncertainty limits closed loop band-width.

The nominal plant is:

P0(s) =5

s + 1. (48)

In the mixed sensitivity approach, the following design problem is solved:

maxγ

∥

∥

∥

∥

∥

[

γW1(s)(I + PC)−1

W2,a(s)C(I + PC)−1

]∥

∥

∥

∥

∥

∞

≤ 1, (49)

For additive uncertainty, the weighting function is:

W2a(s) =7.5s

(s + 1)(s + 15). (50)

What is an appropriate performance weighing W1(s)? To answer that question,one has to know, how the frequency response of a desirable sensitivity functionS(s) = (I + PC)−1 looks like. If the above norm condition of is met, there is also

∥

∥

∥

[

γW1(s)(I + PC)−1]∥

∥

∥

∞≤ 1, (51)

For scalar system it is easily seen, that

∣

∣

∣

[

(I + PC)−1]∣

∣

∣ ≤∣

∣

∣

[

(γW1(jω))−1]∣

∣

∣ , (52)

The sensitivity function is bounded by the inverse of the weighting functionW1(s). Zero steady-state error is attained if W1(s) has a pole at ω = 0, i.e. if it hasan integrator.

W1(s) =1

s(53)

An unsuitable selection of W1(s) may lead to an unsolvable H∞–controller designproblem. Remember that the sensitivity function has to meet the Bode integraltheorem (12). Consequently, also W1(s)

−1 has to be in compliance with the integralequation. The weighting (53) has the drawback, that there is now limit to possibleresonance peaks of the sensitivity function. With an additional zero, which should belarger than the expected closed loop bandwidth, this drawback could be eliminated.

19The multiplicative case can similarly be solved as exercise



Furthermore, the H∞–problem has to be solvable at ω = ∞ and the technicalconditions for solving the design problem, summarized in (5) have to be fulfilled.

To be able to solve the H∞–controller design problem, the mixed sensitivityproblem has to be formulated as an LFT as shown in 22(a) and 22(b).

(a) System blockdiagram (b) LFT

Figure 22: System Blockdiagram

This can be defined in MATLAB with the following m-file:

gam=2.5;

P=nd2sys(5,[1 1],1);

W1=nd2sys(1,[1 0],gam);

W1a=nd2sys([1 0],[1 16 15],7.5);

systemnames = ’P W1 W1a’;

inputvar = ’[w; u]’;

outputvar = ’[W1; W1a; P]’;

input_to_P = ’[u]’;

input_to_W1 = ’[w-P]’;

input_to_W1a = ’[u]’;

sysoutname = ’add_sys_LF’;

cleanupsysic = ’yes’;

sysic

% now calculate the hinf controller

n_ctr=1;n_meas=1;gmin=1;gmax=10;tol=0.001;

[K_hin,clp] = hinfsyn(add_sys_LF,n_meas,n_ctr,gmin,gmax,tol);

After executing the m-file, which tries also to calculate the H∞–optimal controller,the following error message is obtained:

[a b1;c2 d21] does not have full row rank at s=0

Despite the very convenient controller design tools, it is now necessary to derive analgebraic expression for the system, shown in 22(b), to be able to understand the



reason for the error message.

z1

z(2a)· · ·y

=

W1... −PW1

0... W2a

· ·

I... P

[

wu

]

(54)

We will see, that the message shows that, when setting the weighting functions,we didn’t concern about solvability of the H∞-Problem. Therefore, some moreproblems have to be expected!

First, let us have a closer look at the system matrix add sys LF, which representsthe state space model of the LFT-plant:

Gs =

−1 0 0 0... 0

... 2.5

−2.2 0 0 0... −1.1

... 0

0 0 −15 −7.7... 0

... −3.0

0 0 0 −1... 0

... −0.8· · · · · · · · · ·

0 0.9 0 0... 0

... 0

0 0 −2.4 −0.6... 0

... 0· · · · · · · · · ·

2.0 0 0 0... 1

... 0

(55)

The pair (A,B2) must be stabilisable. The only unstable pole, due to the inte-grator in the weighting function W1 is not stabilisable, because the matrix B2 hasa zero in the corresponding row. Equivalently, it can be seen in the block diagram(22(a)), that there is no mean to stabilize a possibly unstable transfer function W1.Furthermore, D21 and D12 have to be of full rank. This is fullfilled for D21 but notfor D12. Finally, as indicated in the error message, one of the two rank conditionsis not met.

How can the design be improved, so that an H∞–solution exists? As it can beseen from (54) the above quantities are determined by the weighting functions W1

and W2a. The weighting functions W1 and W2a can be slightly modified, so that theconditions are met. For the performance weight W1, the integrator is replaced witha stable pole close to the origin, i.e.

W1(s) =1

s + 10−5(56)

This causes no perfect integral action, but this can be achieved by a subsequentcontroller approximation.With the chosen scaling W2a also the upper right block in (54), has a singular D-matrix, i.e. D12. There are two potential remedies to fix the problem: either the



plant transfer function or the model uncertainty description is modified to be strictlyproper. It’s reasonable to modify the plant uncertainty description by adding a smallzero to the transfer function. This causes only a neglectable effect to the uncertaintydescription.

W2a(s) =0.0075s(s + 1000)

(s + 1)(s + 15). (57)

The resulting m-file is:

gam=2.5;

P=nd2sys(5,[1 1],1);

% gamma is defined in workspace

W1=nd2sys(1,[1 0.00001],gam);

W1a=nd2sys(conv([1 0],[1 1000]),[1 16 15],7.5/1000);

systemnames = ’P W1 W1a’;

inputvar = ’[w; u]’;

outputvar = ’[W1; W1a; P+w]’;

input_to_P = ’[u]’;

input_to_W1 = ’[-w-P]’;

input_to_W1a = ’[u]’;

sysoutname = ’add_sys_LF’;


sysic

% now calculate the hinf controller

n_ctr=1;n_meas=1;gmin=0.5;gmax=1;tol=0.001;

[K_hin,clp] = hinfsyn(add_sys_LF,n_meas,n_ctr,gmin,gmax,tol);

With this m-file, a controller is successfully calculated. The achieved value of theH∞–norm is 0.5. The system clp is the closed loop system. Its frequency response isshown in Figure 23. The upper Bode-diagram shows the contribution of the perfor-mance measure W1(s)∗S(s) to the H∞–norm. Obviously, the performance measurecontributes the major part for ω < 5rad/s. Above that frequency the robustnesscondition, shown in the lower diagram, is dominant. When γ is maximized, thesame behavior is observed.

The parameter γ is used to tune the controller. The larger, the more bandwidthis achieved. If γ > 25 there exist no solution to the H∞-problem. In the sequel, thedesign for γ = 2.5 and γ = 5 will be compared and it will become appearant, thatit might not be optimal to completely maximize γ. We will investigate the resultingsensitivity function, gain and phase margin and the resulting controller frequencyresponse.

The comparison of the two design show the following: with γ = 5 the closed loopbandwidth is increased. The cost of this is the resonance peak in the sensitivityfunction. The is due to the bode integral relation, see also Figure 6. The largerbandwidth is achieved with a controller that show double differential action, as itcan be seen in Figure 26. Such a controller will most likely not be successful in an



−100

−50

0

To:

Out

(1)

90

180

To:

Out

(1)

−100

−50

0

To:

Out

(2)

10−1

100

101

102

103

104

0

180

360

To:

Out

(2)

Bode Diagram

Frequency (rad/sec)

Mag

nitu

de (

dB)

; Pha

se (

deg)

Figure 23: H∞–solution

industrial control application, whereas the controller for γ = 2.5 has the shape of aPI–controller with additional low pass filter. Therefore, no practical problems willbe expected. The control system with γ = 5 has another drawback: the open loophas quite a small roll-off after crossing over.How does the control system operate for the worst case disturbance? To answer thatquestion the open-loop with worst case disturbance can be analyzed, see Figure 27.For both designs, phase margin is sufficiently large and the open-loop in the high-gain freuquency range is similar to the design system.

SummaryThe simple design example shows, that

• the unstructured uncertainty description limits the closed–loop bandwidth tothe frequency range, where the plant is sufficiently well known.

• stable weighting functions are powerful instruments to achieve the requiredclosed loop properties

• special attention has to be given to the fact, that the H∞–problem is welldefined for all frequencies, especially at ω = 0 and ω = ∞. This can usuallybe achieved with minor modifications which usually have a neglectable impacton the final result.



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agni

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(dB

)

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104

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Pha

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Bode Diagram

Frequency (rad/sec)

(a) γ = 2.5

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0

10

Mag

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dB)

10−1

100

101

102

103

104

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0

Pha

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deg)

Bode Diagram

Frequency (rad/sec)

(b) γ = 5

Figure 24: Sensitivity function S(s)

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0

50

Mag

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dB)

10−1

100

101

102

103

104

105

−180

−135

−90

Pha

se (

deg)

Bode DiagramGm = Inf dB (at Inf rad/sec) , Pm = 89.8 deg (at 5.26 rad/sec)

Frequency (rad/sec)

(a) γ = 2.5

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−60

−40

−20

0

20

40

Mag

nitu

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dB)

10−1

100

101

102

103

104

105

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−135

−90

−45

0

Pha

se (

deg)

Bode DiagramGm = Inf dB (at Inf rad/sec) , Pm = 125 deg (at 11 rad/sec)

Frequency (rad/sec)

(b) γ = 5

Figure 25: Margins

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10

20

30

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10−1

100

101

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104

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Bode Diagram

Frequency (rad/sec)

(a) γ = 2.5

5

10

15

20

25

30

35

Mag

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de (

dB)

10−1

100

101

102

103

104

−90

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0

45

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

(b) γ = 5

Figure 26: Controller



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0

100

Mag

nitu

de (

dB)

10−1

100

101

102

103

104

105

−360

−270

−180

−90

Pha

se (

deg)

Bode DiagramGm = 17.3 dB (at 19.6 rad/sec) , Pm = 61.9 deg (at 4.95 rad/sec)

Frequency (rad/sec)

(a) γ = 2.5

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0

50

Mag

nitu

de (

dB)

10−1

100

101

102

103

104

105

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−270

−180

−90

Pha

se (

deg)

Bode DiagramGm = 34 dB (at 107 rad/sec) , Pm = 72.1 deg (at 8.64 rad/sec)

Frequency (rad/sec)

(b) γ = 5

Figure 27: Margins for worst case disturbance



A.1.2 Example 2

In this example, H∞–controller design with structured uncertainty will be demon-strated. The system shown in Figure 28 is a typical example for an industrialrotational control problem. A DC-motor with inertia J1 drives a load inertia J2.The positions of the two inertias are measured and the second inertia is controlledto a required position. The two inertia are coupled with an ideal spring, rotation isdamped with viscous friction in the bearings. The electric circuit of the DC-motoris modeled as an RL-Element. Model uncertainty arises from changing load inertia,varying friction and changing resistance due to temperature changes in the motor.It is not sensible to assume that all model parameters are uncertain although this isthe reality. Every uncertain parameter adds complexity to the controller design, par-ticularly if the controller is calculated with µ–synthesis. The following uncertaintieswill be modeled:

J2 = J2(1 + ∆J2δJ2), c2 = c2(1 + ∆c2δc2), R = R(1 + ∆RδR) (58)

δJ2 , δp2 and δR represent the possible, relative perturbations on the correspondingparameters. Their range is [−1, 1].

Figure 28: Model for the rotational system

Modeling the system leads to the following differential equations:

Li = u − Ri − Φω1 (59)

Φi = J1ω1 + c1ω1 + k(ϕ1 − ϕ2) (60)

k(ϕ1 − ϕ2) = J2ω2 + c2ω2 + d (61)

d is a disturbance torque acting upon the load inertia. With some algebraicmanipulations the equations are reordered as follows:

i =1

L(u − Ri − Φω) (62)



ω1 =1

J1

(−c1ω1 − k(ϕ1 − ϕ2) + Φi) (63)

ω2 =1

J2

(k(ϕ1 − ϕ2) − c2ω2 − d) (64)

For solving the structured uncertainty design problem, the system must be trans-formed into the form shown in Figure 13. This can be a tedious task if the procedureis not formalized in such a way, that it can be done with a suitable tool. A closerlook to the above equation reveals, that the uncertain parameters are either factorson variables, e.g. Ri and c2ω2, or the inverse of the parameter is a factor on analgebraic term, e.g. 1

J2(k(ϕ1 − ϕ2) − c2ω2 − d). The idea to formalize the model

construction process is, to define new variables for the algebraic terms and to repre-sent the uncertain parameter as LFT models. Afterwards a CACSD-Tool (CACSD:Computer aided control system design) can interconnect the model an build its statespace representation.

The term 1J2

is represented as an upper LFT in δJ2 as follows:

1

J2

=1

J2(1 + ∆J2δJ2)(65)

=1

J2

−∆J2

J2

δJ2(1 + ∆J2δJ2)−1

= F(MJ2 , δJ2)

MJ2 =

[

−∆J21J2

−∆J21J2

]

The term c2 = c2(1 + ∆c2δp2) is represented as an upper LFT in δp2 as follows:

c2 = F(Mc2 , δc2), Mc2 =

[

0 c2

∆c2 c2

]

(66)

and R = R(1 + ∆RδR) is represented as an upper LFT in δR as follows:

R = F(MR, δR), MR =

[

0 R∆R R

]

(67)

The LFTs block diagram are shown in Figure 29. In the figure also input variablesare defined, if necessary. All together, the system model is:

i = −1

LpR −

1

LΦω1 +

1

Lu

pR = F(MR, δR)i

ϕ1 = ω1

ϕ2 = ω2 (68)



Figure 29: LFT of uncertain parameters

ω1 = −1

J1

c1ω1 −1

J1

kϕ1 +1

J1

kϕ2 + Φi

ω2 = pJ2

pJ2 = F(MJ2)fJ2

fJ2 = k(ϕ1 − ϕ2) − pc2 − d

pc2 = F(Mc2)ω2

It’s state space representation is:

iω1

ω2

ϕ1

ϕ2

=

0 − 1LΦ 0 0 0

Φ − 1J1

c1 0 − 1J1

k 1J1

k

0 0 0 0 00 1 0 0 00 0 1 0 0

iω1

ω2

ϕ1

ϕ2

+

1L

0 − 1L

0 00 0 0 0 00 0 0 1 00 0 0 0 00 0 0 0 0

udpr

pJ2

pc2

ϕ1

ϕ2

ω2

ifJ2

=

0 0 0 1 00 0 0 0 10 0 1 0 01 0 0 0 01 0 0 k −k

iω1

ω2

ϕ1

ϕ2

+

0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 −1 0 0 −1

udpr

pJ2

pc2

(69)

Together with the system equations (69) the plant is now completely defined. Welet MATLAB calculate the interconnections of all the subsystems. In Figure 31 thebuild process is visualized. In the variable ’systemnames’ all block are specified, thesystem output names are listed in the variables ’inputvar’ and ’outputvar’ and theinterconnection are defined with ’input to . . . ’ variables. The numerical parametervalues are documented in table 1

The resulting set of plant frequency responses are drawn in Figure 30.For optimizing performance the approach shown in Figure 17(b) is chosen, i.e.

performance is formulated with a fictitious uncertainty block ∆p, also called theperformance uncertainty block. Consequently, performance has to be expressed asan input-output mapping, where performance is guaranteed when its norm is lessthan 1. Figure 32 shows the block diagram of the system to be optimized and itsrepresentation with a fictitious uncertainty block is drawn in 33 The controller inputsare the position error of the load inertia and the position of the drive inertia. Both



name nominal varianceJ1 0.1 kg m2 0J2 1.5 kg m2 50%c1 0.002 Nms/rad 0c2 0.003 Nms/rad 10%k 100 Nm/rad 0R 18Ω 10%L 2.1 10−3H 0Φ 0.1 Nm/A 0

Table 1: Parameter values

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set of plants

Frequency (rad/sec)

(a) u to ϕ1

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103

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0

Pha

se (

deg)

set of plants

Frequency (rad/sec)

(b) u to ϕ2

Figure 30: Closed-loop frequency response

measurements are corrupted with measurement noise and the plant is disturbed withtorque changes d on the second inertia. Performance outputs are the control errorand controller output u, both weighted with a suitable transfer function.Again, one has to be very careful to define a sensible H∞–optimization problem,i.e. the problem has to be well defined for all frequencies, especially at ω = ∞.Technically, this follows from the full rank condition on the matrices D12 and D21.Consequently, the mapping from any input to the two controller input must have adirect feed-through term of rank two. Both, the perturbation inputs and the torquedisturbance d have zero direct feed-through terms to the position measurement. Thereference signal r and the measurement noise n1 and n2 can contribute to a full rankmatrix D12, if the measurement noise weightings are chosen to be strictly propper.This becomes obvious, if the control system is drawn as an LFT as shown in Figure33. A full rank matrix D21, which is the direct feed-through term from the controlsignal u to the system outputs, is guaranteed, if the weighting Wu is strictly proper.Beside its technical need, weighting functions provide useful tuning knobs for thecontroller design. Often, the plant is a good low pass filter, and consequently thespectra of the measurements lack of high frequency content. This might lead to acontroller with very large gain at high frequency. Applied to the true plant with



Figure 31: System model

inevitable measurement noise, the actuator will be heavily stressed and will not havea large life-time. As a remedy, the measurement noise weighting function offer thepossibility to add high frequency content to the measurements spectra. The spectraof load disturbances are modeled with the weighting Wd. This physically motivatedweighting introduces low frequency disturbances into the system.

Disturbance attenuation and tracking error are tuned with the control errorweight We. The function will be similarly to W1 of example 1. With Wu closed-loop bandwidth can be influenced, because it can be used to limit controller gain inaround the crossing-over frequency.The weight are chosen as follows:

We =0.085 (s + 10)

s + 10−5

Wd =0.000018 (s + 1)

s + 0.001(70)



Figure 32: Control structure

Wϕ1 =0.003 (s + 1)

s + 100Wϕ2 = Wϕ2

Wu =0.1s

s + 100(71)

The frequency responses are shown in Figure 34.Now, all is defined for the controller design. In MATLAB, a system can be build

according to Figure 33 and an H∞–controller can be calculated. If the block diagonalstructure of the uncertainty matrix ∆ is not taken into account, the minimal valueof the H∞-Norm is 15.9 and the resulting control system is unstable. This shows,that for structured uncertainties, the simple H∞–approach is not a good choice. Tocircumvent the problem µ-synthesis was developed. As described in 8.1 this involvesoptimization of the H∞-Norm over all possible frequency dependent scalings of thestructured uncertainty ∆. Using D-K-Iteration, the controller, described in the se-quel is obtained.The controller frequency response is shown in Figure 35. The mapping of the controlerror r−ϕ2 to the input u is shown on the left. On the right the frequency responseof ϕ1 to u drawn. The controller is of order 22 and exhibits large gain at frequencyabove 1 rad/s. The frequency responses cannot be easily be approximated with asimple PIDT1-controller. The closed-loop frequency responses from the setpoint rto ϕ1 and ϕ2 can be analyzed in Figure 36 and the corresponding step responses arein Figure 37.

Analysis of the closed-loop performance shows, that the attained bandwidth isnot very large. This is due to the large controller gain, necessary to achieve a cross-ing over frequency above 2 rad/s. Due to the control signal weight Wu this can notbecome arbitrarily large. In order to achieved larger bandwidth, the noise weight-ings Wn or Wu have to be relaxed.M-File for plant



Figure 33: Control system as LFT

%Parameters

omega=logspace(-1,2,100);

J1=0.1;J2=1.5;DJ2=0.5;c1=0.01;c2=0.02;Dc2=0.1;k=100;R=18;DR=0.1;L=0.0021;Phi=0.1;

Ap=[0 -Phi/L 0 0 0;

Phi -c1/J1 0 -k/J1 k/J1;

0 0 0 0 0;

0 1 0 0 0;

0 0 1 0 0];

Bp=[1/L 0 -1/L 0 0;

0 0 0 0 0;

0 0 0 1 0;

0 0 0 0 0;

0 0 0 0 0];

Cp=[0 0 0 1 0;

0 0 0 0 1;

0 0 1 0 0;

1 0 0 0 0;

1 0 0 k -k];

Dp=[0 0 0 0 0;

0 0 0 0 0;

0 0 0 0 0;

0 0 0 0 0;

0 -1 0 0 -1];

P=pck(Ap,Bp,Cp,Dp);

mat_R=[0 R;DR R];

mat_J2=[-DJ2 1/J2;-DJ2 1/J2];



10−4

10−2

100

102

104

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Wd

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10−2

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102

104

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0

20

40

60

We

10−4

10−2

100

102

104

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−50

Wphi1

and Wphi2

10−4

10−2

100

102

104

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−120

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−80

−60

−40

−20

Wu

Figure 34: Weighting functions

mat_c2=[0 c2;Dc2 c2];

gam=2.5;

systemnames = ’P mat_R mat_J2 mat_c2’;

inputvar = ’[uR; uJ2; uc2; d; u]’;

input_to_P = ’[u; d; mat_R(2); mat_J2(2); mat_c2(2)]’;

input_to_mat_R = ’[uR; P(4)]’; %P(4) is current i

input_to_mat_J2 = ’[uJ2; P(5)]’; %P(5) is fJ2

input_to_mat_c2 = ’[uc2; P(3)]’; %P(3) is omega_2

outputvar = ’[mat_R(1); mat_J2(1); mat_c2(1); P(1); P(2)]’;

%Phi2 for performance, all disturbance ’outputs’, phi1 and phi2 for control

sysoutname = ’Ex2_Plant_LF’;


sysic

M-File to define control system with weghtings

% now calculate the hinf controller system

gam = 0.085;

W_e=nd2sys([1 10],[1 0.00001],gam);

W_d=nd2sys(0.0018*[0.01 0.01],[1 0.001]);

W_phi1=nd2sys(0.003*[1 1],[1 100]);

W_phi2=W_phi1;

W_u=nd2sys([0.1 0],[1 100]);

systemnames = ’Ex2_Plant_LF W_e W_d W_phi1 W_phi2 W_u’;

inputvar = ’[uPert3; dw; n1; n2; r; u]’;

input_to_Ex2_Plant_LF = ’[uPert; W_d; u]’;

input_to_W_e = ’[r-Ex2_Plant_LF(5)-W_phi2]’;

input_to_W_d = ’[dw]’;

input_to_W_phi1 = ’[n1]’;

input_to_W_phi2 = ’[n2]’;

input_to_W_u = ’[u]’;

outputvar = ’[Ex2_Plant_LF(1:3); W_u; W_e; r-Ex2_Plant_LF(5)-W_phi2; Ex2_Plant_LF(4)+W_phi1]’;

sysoutname = ’Ex2_Sys_LF’;


sysic

M-File for µ-controller design



−20

0

20

40

60

80

100From: In(1)

To:

Out

(1)

100

−360

−180

0

180

360

540

To:

Out

(1)

From: In(2)

100

Bode Diagram

Frequency (rad/sec)

Mag

nitu

de (

dB)

; Pha

se (

deg)

Figure 35: Controller frequency response

% now calculate the hinf controller system

DK_DEF_NAME = ’dk_mds’;

dkit

K_mu = k_dk3mds;

% Analysis with nominal plant

Ex2_Plant_nom=sel(Ex2_Plant_LF, 4:5, 4:5);

figure(2), clf, bode_dm(K_mu,omega)

systemnames = ’Ex2_Plant_nom K_mu’;

inputvar = ’[dw; n1; n2; r]’;

input_to_Ex2_Plant_nom = ’[dw; K_mu]’;

input_to_K_mu = ’[r-Ex2_Plant_nom(2)-n2; Ex2_Plant_nom(1)+n1]’;

outputvar = ’[Ex2_Plant_nom(1:2); K_mu]’;

sysoutname = ’Ex2_Closed_Loop’;


sysic

M-File with parameter for µ-controller design

NOMINAL_DK = Ex2_Sys_LF;

NMEAS_DK = 2;

NCONT_DK = 1;

BLK_DK = [-1 1; -1 1; -1 1; 4 2]

OMEGA_DK = logspace(-2, 3,100);

AUTOINFO_DK = [1 3 1 4*ones(1, size(BLK_DK,1))];

NAME_DK = ’mds’;

See MATLAB manual for the robust-control toolbox for a description of theparameters.



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set of closed loop systems

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(a) r to ϕ1

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20

Mag

nitu

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101

102

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0

Pha

se (

deg)

set of closed loop systems

Frequency (rad/sec)

(b) r to ϕ2

Figure 36: Closed-loop frequency response

0 2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (sec)

Am

plitu

de

(a) r to ϕ1

0 2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (sec)

Am

plitu

de

(b) r to ϕ2

Figure 37: Closed-loop step response



A.2 Exercise 1: Linear Fractional Transformation

Transforming a system into a LFT is a frequent task in controller design. Find anLFT representation (Figure 39) for the plant with a 2 degree of freedom controlleras shown in Figure 38

Figure 38: 2 Degree of Freedom control system

Figure 39: 2 Degree of Freedom control system

A.3 Exercise 2: Small gain theorem

A high order controller C(s) is approximated with a reduced order controller Cred(s).The control loop with the reduced-order controller remains unchanged, if the originalcontroller C(s) is added and subtracted in parallel to the reduced order controller.This is shown in Figure 40. With this simple block diagram manipulation, we nowhave the original control loop, which is perturbed with ∆(s) = Cred(s) − C(s).

1. Closed-loop stability should be maintained. Use the small-gain theorem toderive a condition on the norm of ∆ in order to guarantee closed loop stabilityfor the system with the reduced-order controller.

2. In which frequency range the controller approximation should be best? Usethe previously derived condition to analyse control systems for plants with orwithout integrator.



Figure 40: Controller approximation

A.4 Exercise 3: Drawback of classical stability margins

Consider a feedback system with the following loop gain

L(s) =0.38(s2 + 0.1s + 0.55)

s(s + 1)(s2 + 0.06s + 0.5).

The feedback system exhibits very good gain and phase margins, but is not at allrobust with respect to simultaneous uncertainties of gain and phase.

1. Calculate the gain margin Am and the phase margin φm.

2. Determine the peak of the sensitivity ||S||∞.

3. Determine the distance between the Nyquist plot of loop gain L(jω) and thecritical point −1.

This example shows that it is much better to assess robustness using ||S||∞ insteadof using traditional gain and phase margins. A low value of ||S||∞, e.g. implies goodgain and phase margins while the converse is not true. It follows from elementarytrigonometric relations that

Am >||S||∞

||S||∞ − 1, φm > 2 arcsin

1

||S||∞

For example ||S||∞ = 2 gives a garanteed gain margin of Am > 2, corresponding toa worst case gain margin of 6 dB, and a phase margin φm > 60.



A.5 Exercise 4: Two cart problem

Consider the following mechanical plant: two carts are moving on a horizontal planewith no friction. The control force w is acting on the left cart with mass m1, and theposition of the right cart z = x2 is measured. The carts are connected by a linearspring k. The nominal parameters are m1 = 1 kg, m2 = 1 kg, and k = 1 N/m.All parameters are affected by independent tolerances of ±10% for the masses, and±20% for the spring.

m1 m2

spring

k

control

force

w

x1z = x2

positionmeasurement

Figure 41: Two cart ACC benchmark problem

P

diagonal uncertainty

block

w z

augmented plant

gh

Figure 42: Structured uncertainty setup

1. Find a state space realization of the augmented plant P, where the structured



parameter uncertainty is represented as a diagonal feedback block

∆ =

δ1 0 00 δ2 00 0 δ3

.

The uncertainty δ1 should correspond to the uncertainty of m1, and a variationof −1 < δ1 < +1 should correspond to a variation of ±10% of m1 around itsnominal value. Similarly, δ2, δ3 correspond to the uncertainties of m2, k, andboth should also be normalized to a range of ±1.

2. Why is it inappropriate to formulate the uncertainties of this system as additiveor multiplicative uncertainties?

3. In the plant model, no damping is considered. This leads to a pole pair on theimaginary axis. Explain, why this leads to severe controller design problems,if the pole location changes due to plant uncertainty.


Bibliography

[1] G. Stein. Respect the unstable. IEEE Control Systems Magazine, August 2003.

[2] J. Doyle, K. Glover, P.P. Khargonekar, and B.A. Francis. State–space solutionsto standard H2 and H∞ control problems. IEEE Trans. Aut. Control, 34(8),August 1989.

[3] M. Safonov and D. Limebeer. Simplifying H∞ theory via loop shifting. Proc.27th Conference on Decision and Control, December 1988.

[4] Y. Nesterov and A. Nemirowski. Interior Point Polynomial Methods in Con-vex Programming: Theory and Applications. Studies in Applied Mathematics,SIAM books Philadelphia, 1994.

[5] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequal-ities in Systems and Control Theory. Studies in Applied Mathematics, SIAMbooks Philadelphia, vol. 15, 1994.

[6] P. Gahinet and P. Apkarian. A linear matrix inequality approach to H∞. Int.J. Robust and Nonlinear Control, 4, 1994.

[7] M. Green and D. Limebeer. Linear Robust Control. Pearson Education, Inc.,2002.

[8] J. Doyle, B. Francis, and A. Tannenbaum. Feedback Control Theory. McMillanPublishing Co., 1990.

[9] Robust Control Toolbox 3, User’s Guide, The Mathworks, 2008.

[10] LMITOOL: A Package for LMI Optimization in Scilab User’s Guide, 1995.

[11] D. Gu, P. Petkov, and M. Konstantinov. Robust Control Design with MATLAB.Springer, 2005.


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