IEEE TRANSACTIONS ON AUTOMATIC CONTROL. VOL. AC-32. NO. 2, FEBRUARY 1987 105

The LQG/LTR Procedure for Multivariable Feedback Control Design

Abstract-This paper provides a tutorial overview of the LQG/LTR design procedure for linear multivariable feedback systems. LQWLTR is interpreted as the solution of a specific weighted H*-tradeoff hetween transfer functions in the frequency domain. Properties of this solution are examined for both minimum-phase and nonminimum-phase systems. This leads to a formal weight augmentation procedure for the miuimum- phase case which permits essentially arbitrary specification of system sensitivity functions in terms of the weights. While such arbitrary specifications are not possible for nonminimum-phase problems, a direct relationship between weights and sensitivities is developed for nonmini- mum-phase SlSO and certain nonminimum-phase MIMO cases which guides the weight selection process.

I. INTRODUCTION

0 NE of the goals of control theory has been to capture major elements of the engineering process of feedback design under

the umbrella of a formal mathematical synthesis problem. The motivation for this goal is self evident. Once formalized under such an umbrella, the elements of engineering art become rigorous tools which can be applied more or less automatically to ever more complex design situations.

Perhaps the best known example of formal mathematical synthesis is the linear-quadratic-Gaussian optimal control problem (LQG) [ 11. This problem formalizes a specific design situation, namely the construction of feedback compensators for finite- dimensional linear plant models, with stability and least-squares performance under additive disturbances as design objectives. Needless to say, this covers only a small subset of typical overall engineering design problems. However, linear models are appli- cable often enough (particularly in early stages of a project), and least-squares objectives can be manipulated cleverly enough (via free parameters in the quadratic criterion) that LQG has proven itself useful in many diverse design applications.

Research developments over the last several years have shown that the flexibility provided by the quadratic criterion of LQG is remarkably broad. Indeed, it is possible to choose free parameters in such a way that the entire formal design process can be reinterpreted not as a least-squares error minimization problem but as a ”loop shaping” problem-that is, a problem of designing feedback compensators to achieve desirable sensitivity and com- plementary sensitivity transfer functions at critical loop-breaking points of the feedback system.

One version of this reinterpretation is the so-called LQGlLTR methodology-linear quadratic Gaussian synthesis with loop

Manuscript received October 11, 1985: revised August 26, 1986. This paper is based on a prior submission of June 1 1. 1984. Paper recommended by Associate Editor. M . G. Safonov. This work was supported in part by Honeywell internal IR&D, by NASA Amcs and Langley Research Ccntcrs under Grant NAG 2-297: and by the National Science Foundation under Grant

G. Stein is with the Honeyw,ell Systems and Research Center, Minneapolis, MN 55418 and the Massachusetts Institute of Technology. Cambridge, MA 02 139.

M. Athans is tvith the Massachusetts Institute of Technology, Cambridge, MA 02139.

IEEE Log Number 8612203.

NSF;ECS-8210960.

transfer recovery-in which loop shapes of optimal full-state regulators or filters are recovered (approximated) at plant inputs or outputs via certain specific choices of free parameters. An exposition of LQG/LTR was first given by Doyle and Stein in [ 2 ] .

Another version of the reinterpretation was developed by Safonov et al. in [ 191, where all time domain LQG problems and frequency domain Wiener-Hopf problems [20]-[24] are shown to be general weighted Hz-tradeoffs between various transfer functions of the feedback system. Weight adjustments are then suggested for shaping these transfer functions. Still another loop- shaping interpretation of LQG can be found in Gupta [25], where frequency weighting of costs is suggested to modify loop shapes.

In this paper, we provide a tutorial overview of the LQG/LTR version of the reinterpretation. However, we base the overview not on its original exposition in [2] but on the Hz-perspective suggested by [19]. We use results of [19] to find the specific H2- problem solved by LQG/LTR and discuss several useful new twists of the methodology. The twists include a formal weight selection procedure which permits essentially arbitrary specifica- tion of system sensitivity functions for minimum-phase problems, a classification of all recoverable functions in nonminimum-phase problems, and certain direct relationships between weights and sensitivities for the latter which apply to all scalar and certain multivariable cases.

Since loop-shaping interpretations of LQG look most interest- ing under the classical frequency domain design paradigm of Nyquist and Bode [3], [4], we begin in Section I1 with a very brief review of this paradigm as applied to multivariable problems. We then pose a formal H h p t i m a l synthesis problem which attempts to tradeoff the chief design functions of the paradigm. Comparing this Hz-problem with the standard LQG formulation then suggests specific choices of free parameters in the latter to solve the former. Under minimum-phase assumptions, the resulting solu- tions exhibit very nice frequency domain properties. These are described in Section IV. Properties which hold for nonminimum- phase cases are also discussed, along with current research questions. Section V discusses the nonminimum-phase case in more detail and provides a brief example. Section VI makes concluding comments.

II. FEEDBACK DESIGN IN THE FREQUENCY D O ~ ~ A I N

Ever since the basic work of Nyquist, Bode, and others, the classical approach to feedback design has followed the frequency domain perspective illustrated in Fig. 1. We are given a multivariable plant described by a rational transfer function G(s) and wish to design a compensator K(s) such that the closed-loop feedback system satisfies the following basic requirements.

I ) Stability: bounded outputs, y(s), for all bounded distur- bances, d(s), and bounded reference inputs, r(s)

2) Performance: small errors, e@), in the presence of d(s) and

3) Robustness: stability and performance maintained in the presence of model uncertainties, 6G(s), expressed in whatever form is appropriate for the real plant at hand.

As is well known, the first of these requirements imposes structural constraints on certain transfer functions of the closed-

r(s)

0018-9286/87/0200-0105$01.00 @ 1987 IEEE

106 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-32. NO. 2. FEBRUARY 1987

Fig. I .

loop system, e.g., a Nyquist encirclement count for the function det ( I + G K ) [5]. Likewise, the second requirement imposes magnitude constraints on certain transfer functions. In particular, for Fig. 1 where disturbances and commands are reflected to the output loop-breaking point, the (output) sensitivity function

S(s)=[Z+G(s)K(s)]-’ (1)

must be small for all frequencies s = j w where the disturbances and/or reference commands are large. This latter statement is a fundamental frequency domain prescription for feedback design. We will refer to it as

PI: “Make S ( j w ) small whenever d( jw) or r ( j w ) are large.” For classical single-input single-output (SISO) systems, the

meanings of “small” and “large” in P1 are, of course, to be understood in terms of the absolute values of the respective complex numbers at each frequency. For multiinput multioutput (MIMO) systems, these meanings must be treated with greater care. Complex vectors. d( jw) and r ( j w ) , will be taken as small or large according to the size of their usual Euclidean norm. Complex matrices S( jw) will be taken as small if their largest singular value 6[S( jw) ] is small, and they will be taken as large if their smallest singular value _a[S(ju)] is large. With this generalization of definitions, the design prescription PI applies to SISO and MIMO systems alike [2], [6].

The third feedback design requirement-tolerance for uncer- tainty-can also be expressed in terms of magnitude constraints. In this case, however, the transfer functions to which the constraints apply depend upon how model uncertainties are characterized. In classical SISO problems, tradition dictates that gain margins gm and phase margins pin be used to characterize tolerable uncertainty. These margins are suitable for uncertainties in the following specific form:

g(s) + M s ) =exp (L )g ( s ) = (1 + L)g( s ) (2)

where L is an arbitrary real scalar with abs(L) I gm In (10)/20 for pure gain uncertainty, or L is an arbitrary imaginary scalar with abs(L) I pm/57.3 for pure phase uncertainty. This characterization can be generalized to MIMO problems as follows:

G(s)+6G(s)=[I+L(s)]G(s) (34

where L(s) is an arbitrary stable‘ transfer function matrix with

8 [ L ( j w ) ] s r n ( w ) . (3b)

Equation (3) is sometimes called “unstructured multiplicative uncertainty.” It covers simultaneous gain, phase, and direction errors which are unknown but bounded in size. The bound m(w) indicates the maximum normalized magnitude which the model error can attain. It is typically small (m 6 1) at low frequencies but invariably rises toward unity and well above (modeling error

It can be readily verified by direct manipulation of det [ I + (G + 6G)KI (see [6]), that stability is maintained in the presence of a l l possible uncertainties characterized by (3) if and only if the “complementary (output) sensitivity function [7]”

100 percent) as frequency increases.

T ( s ) = G(s)K(s) [Z+ G ( s ) K ( s ) ] - ’ (4)

’ This assumption can be relaxed 121. [ 6 ] .

satisfies

1 a[ T( jo) ] 5- for all w . m(w)

( 5 )

This condition leads to the second fundamental frequency domain prescription for feedback design:

P2: “Keep T( jw) small wherever m(w) is large.”

Note that since T(s) is nothing more than the closed-loop command response transfer function, P2 can also be interpreted as a prescription to restrict closed-loop bandwidth to the frequency range over which the plant model is valid.

Taken together, the two design prescriptions P1 and P2 capture the basic features of feedback design as viewed from the frequency domain. We face two functions. S(s) and T(s), which both need to be small on the jw-axis. However, since

S(s) + T ( s ) = I (6)

they cannot be made small simultaneously. Rather we must trade off the size of one function against the size of the other in accordance with the relative importance of disturbance/command power and model uncertainty at each frequency.

Feedback design is thus seen as a game of essential tradeoffs between transfer functions. Of course, these tradeoffs are not always as pure as the one identified above. For example. we have ignored sensor noise, control saturation. and other bandwidth limiters in Fig. 1. These impose additional magnitude constraints on T(s). We have also restricted discussion to the output loop- breaking point. Other points give rise to tradeoffs between their respective sensitivity and complementary sensitivity functions and even to tradeoffs between functions taken from different points. Our objective here is not to cover all cases but simply to illustrate that the frequency domain viewpoint gives rise to certain basic trades which can be solved via LQG/LTR.

111. FORMAL H’-~PTIMIZATION VIA LQG/LTR

Tradeoffs between transfer functions can be formalized by posing them as function-space optimization problems. The first thing needed is a convenient criterion of smallness. Consider’

L T [ I V ] ~ = X , ~ , [ M M ~ ] ~ T ~ [MMN] . (7)

This shows that a matrix M will be small if Tr [MMH] is small. Using this latter measure for the two matrices. S(jw) and T( jw) , adding weights W ( j w ) to trade one against the other, and integrating over frequency gives the following plausible optimiza- tion problem.

Given the plant G(s), weights W(s) , and sensitivity and complementary sensitivity defined by (1 ) and (4), respectively, find a stabilizing compensator K(s) to minimize

J = \ (Tr [ S W W H S H ] + T r [ T T H ] ) do am

-0

= im Tr [“HI do (84 -0

with

M ( j w ) = [ S ( j O ) W ( j w ) 7 y j w ) I . (8b)

In mathematical terms, this represents an optimization over the Hardy space of stable transfer functions with 2-norm, i.e., an Hz- optimization problem [8], and is a specific form of more general Hz-optimization problems treated in [19], [20] and various other references.

Superscript H denotes complex conjugate transpose.

STEIN AND ATHANS: LQCiLTR PROCEDURE FOR MULTIVAMABLE FEEDBACK 107

Depending upon the properties of G(s) and the choice of W(s), optimizing compensators for (8) are not in general finite- dimensional, strictly proper, or even proper. However, under the mild restriction that G and W are themselves finite-dimensional and strictly proper. it turns out that the standard LQG problem can be used to generate a sequence of strictly proper compensators which minimize (8) in the limit and maintain closed-loop stability. To show this, we will use results of Safonov et a/. [ 191 to convert arbitrary LQG problems into equivalent H?-problems. We then select special LQG design parameters to reduce these problems to problem (8). Consider the usual LQG setup.

Given dx dt -=Ax(t)+Bu(t)+LE(t) (94

where matrices (A, B, C ) form an n-dimensional state-space representation of G(s), i.e.,

G ( s ) = C Q ( s ) B with Q(s)=(sI -A)- ' (9d)

where z ( t ) is an auxiliary response variable, and where ( ( t ) and ~ ( t ) are Gaussian white noise processes with unit intensities.

Find a controller depending only on y(7) and u ( T ) , 7 5 t to minimize

and let

Comparing (8) and (13). it is now apparent that the only remaining step needed to solve (8) is to find free parameters for (12b) such that P(s) reduces to M(s). It is easy to verify that the following choices do the trick.

Choose L and y such that

and let

H = C and p+O. ( 1 4 ~

Then

The free parameters of this problem are the noise input matrix L in (9a): the scalar y in (9b), the auxiliary response matrix H in (9c), and the scalar p in (10). In typical LQG applications, these parameters are assigned a priori physical significance (e.g., process noise, sensor noise, controlled variables, control weights), and LQG gives optimal solutions for these assignments. Such optimality is not our objective here. Rather, we want to achieve the tradeoff between transfer functions discussed in Section 11. LQG's parameters will be freely manipulated to accomplish this end.

It is well known, of course, that under mild assumptions on its parameters, the LQG problem yields a unique n-dimensional fixed-parameter stabilizing compensator K(s) as its solution [I] . Given this compensator, the LQG problem is converted into an equivalent H*-problem as follows [ 191. Note first that equations (9) can be written in the frequency domain as

Substituting u = -K(s)y yields

with

P(s) = [ H@L - HQBK(I+ GK)-'C@L - pHQBK(I+ GK)-' -pK(I+GK)-'C+L ppK(I+ GK)-' . 1

(1 2b)

Finally, substituting (12) into (IO) via Parseval's theorem yields [I91

Tr [PPH] dm. (1 3)

Note that for each nonzero value of p. the LQG solution for these choices produces a stabilizing strictly proper controller which is Hz-optimal for criterion (13). Moreover, since (13) converges to (8), a sequence of decreasing p-values produces a sequence of controllers which optimizes (8) in the limit.

It is also easy to verify that the following alternative to (14) produces another useful transfer function tradeoff:

Choose H and p such that

H+ B P

-- - W ( s ) (15a)

Then

(1%)

These choices accomplish an H k a d e o f f between the sensitivity and complementary sensitivity functions at the input loop- breaking point of Fig, 1 instead of at the output. Both choices, (14) and (15), will be referred to as LQG/LTR.

IV. PROPERTIES OF LQG/LTR SOLUTIONS

The practical value of a formal synthesis problem rests in the qualitative properties of its solutions. We will see in this section that Hz-solutions via LQG/LTR have very nice properties for the class of systems whose models are minimum phase, i.e., for G(s)'s with no transmission zeros in the closed right-half plane [ 5 ] . For nonminimum-phase models, certain integral constraints apply to the Hz-solution (and to solutions from all other design methodologies as well) whose impact is currently only partially understood.

A . Properties for Minimum-Phase Models

The compensator produced by (14) has the following well- known form:

KLQG(s) = KC(sI- A - BKc- KJC)-'KJ (16)

where Kf is the Kalman filter gain corresponding to the parameter choices (14a), and Kc is the LQ-regulator gain corresponding to (14b). Note that Kc is functionally dependent on the parameter p.

with J ( K ) as in (8) and J , ( K ) nonnegative. Optimizing this criterion for a -' For fired K(s), (13) consists of two terms: J,M) = J ( K ) + p ' J , ( K )

terms of which have finite limits. The second term of the sequence. ( p 2 J , ) e p , s sequence of decreasing p-values produces a decreasing sequence of costs, both

converges to zero.

108 IEEE TRANSACTIONS ON AUTOMATIC CONTROL. VOL. AC-32. NO. 2. FEBRUARY 1987

sensitivities never become too large. They satisfy

____ ~~

Fig. 2.

It is shown in [2] that this functional dependence produces the following limit for square minimum-phase G(s):

G(s)KLQG(s)+C+(s)Kf pointwise in s

as p-0 . (17)

Equation (17) shows that the optimal (output) loop transfer function matrix of a minimum-phase Hz-problem (8) corresponds to the loop transfer function of a Kalman filter with its loop broken at the residual point, as illustrated in Fig. 2. Moreover. (17) shows that the sequence of LQG solutions generated by (14) converges to this function ("recovers" this function) as the design parameter p becomes small.

I ) Two-step LQG/LTR Design: The above properties sug- gest a two-step approach to H2-optimal design.

Step I): Design a Kalman filter, via (14a). with desirable sensitivity, complementary sensitivity, and loop transfer func- tions.

Step 2): Design a sequence of LQ-regulators. via (14b), to approximate the functions in Step 1) to whatever accuracy is needed.

Both of these steps are easy design tasks. The first is easy because Kalman filter sensitivity, complementary sensitivity, and loop transfer functions are explicitly related to the chosen weights W(jw) . as described below. and the second is easy because it involves only repeated solutions of algebraic Riccati equations followed by inspection of S and T.

2) Relations Between Kalman Filter Functions and Weights: As a consequence of Kalman's dual equality for filters [9] and of (17). the transfer functions produced by standard Kalman filter designs can be shown to exhibit the following nice properties [ 2 ] .

For all frequencies where the weights W(jw) = C + ( j w ) L / p are much larger than unity, Kalman filter sensitivity, complemen- tary sensitivity, and loop transfer functions have the shapes

Property I): Designer-specified shapes.

U i [ ( I + C~(jw)K,)-'I =

u,[C~( jw)K, ( I+ c @ ( j w ) K f ) - ' l = 1

1

ai[ W w ) l

cri[C@(jw)K/I =ai[ W w ) l (18)

for each singular value ai.

As w + m, Kalman filter shapes satisfy Property 2): Known high-frequency attenuation.

U i [ ( I + c@( jw)K/ ) - ' ] = 1

for each singular value a;. Property 3): Well-behaved crossovers.

As the loop crosses over from high-gain regions (1 8) to low- gain regions (19), Kalman filter sensitivities and complementary

u;[(Z+ C @ ( j w ) K / ) - ' ] 5 1

U i [ C @ ( j W ) K , ( I + c+( jw)K/ ) - '1 5 2 (20)

for each singular value a; and all frequencies w. As described shortly. the first of these properties offers

essentially arbitrary freedom to shape the sensitivity function in all high-gain regions of the Kalman filter loop. The second property shows that the loop will eventually transition from high-gain to low-gain regions at an attenuation rate proportional to l/w. The frequency range at which the transition occurs (e.g., the crossover frequency) is determined entirely by the Kalman filter design. It is not altered by the later loop transfer recovery step of the design procedure. Hence. the eventual bandwidth of the feedback system is assigned and fixed in Step 1 ) of the procedure, and does not, as is often mistakenly inferred, tend to infinity with decreasing p in Step 2 ) .

The third property shows that the loop's behavior in the crossover frequency region will automatically be nice. The loop does not amplify disturbance or command errors during the transitions and retains stability in the face of modeling uncertain- ties as large as 50 percent [compare (5) and (ZO)]. The latter stability robustness margin is not adequate. of course, for the entire frequency range. At high frequencies where uncertainties can greatly exceed 100 percent, we must rely on the attenuation rate provided by Property 2) for the needed margin. As a consequence of (20). this attenuation rate actually applies not only at very high frequencies but throughout the crossover region.

3) A Formal Procedure to Shape Sensitivities: As evident from (18). in order to shape the high gain regions of a Kalman filter loop we merely need to find L , the noise input matrix in our design model. such that the ai[C+L]'s have the desired shapes. This can be done directly by experimenting with various L's, and/ or more formally by augmenting additional dynamics to the design model such that the choice becomes trivial. For example. whenever the plant has no singularities directly on the jw-axis, it suffices to add the desired shapes, wd(s), directly to the output of G(s) in the manner shown in Fig. 3. In this figure. the plant's open right-half plane poles have been collected into an all-pass factor B,(s) such that

G(s)=Bp(s)- 'G,As) (21)

where G,,,(s) is minimum phase and stable. and

B p ( j o ? ) B p ( j ~ ) H = I for all w. (22)

Signals from wd(s) are passed through the all-pass as shown in the figure. The overall state-space representation then remains stabilizable from l ( t ) and u(t) and detectable from y ( t ) and z( t ) (LQC solutions exist) and has a noise input matrix L which satisfies

c@(s)L=Bp(s)- 'wd(s) . (23)

Hence, from (22)

a~[C@.(jw)Ll=a~[wd(;w)l (24)

for all w and each singular value, as desired. Note that this formal augmentation procedure4 is particularly

simple when G(s) is stable, with Bp(s) = I . In addition. if the desired function W&) is itself a legitimate Kalman filter loop transfer function (Le., satisfies a dual Kalman inequality [lo]),

' Formal augmentation of desired loop shapes was first proposed for stable plants in [ 111. Extensions to the unstable case similar to Fig. 3 are developed

additional algebraic Riccati equation. in [12]. The needed all-pass pole factors can be constructed by solving an

STEIN AND ATHANS: LQG!LTR PROCEDURE FOR MULTIVARIABLE FEEDBACK 109

Fig. 3 .

then the filter design step may be skipped entirely be letting Kf = L .

Despite this simplicity, however. we caution that the formal procedure should not be applied indiscriminately. It produces compensators which cancel all stable dynamics of G(s) and replaces them with dynamics of W(s). Such cancellations should be minimized by constructing the desired shapes as much as possible out of dynamics of G(s), (i.e.$ by choosing noise inputs which directly drive the original state-space of G). This becomes particularly important for dynamics close to the jw-axis. Indeed, in extreme cases with poles directly on the jw-axis, it becomes mandatory. Fig. 3 would require such dynamics to be considered part of G,(s). They would then be cancelled and would produce, at best, neutrally stable closed-loop systems.

Another interpretation of the above caveat is that. unlike loop shapes for stable plants, recoverable loop shapes for plants with singularities in the closed right-half plane are not arbitrary. Rather, they must include such singularities as part of their definition. As discussed further in Section IV-B-3, the LQG/LTR procedure still applies for such permissible shapes.

All of the above properties have duals based on the free parameter choices in (15). For this case. again with square minimum-phase G(s), it has been shown that [2]

KLQ~(s)G(s)-+KC@(s)B pointwise in s

as p-+O

where Kc is the gain matrix of an LQ-regulator designed with parameters from (15a). Hence, the optimal (input) loop transfer function of an H'-problem corresponds to the loop transfer of an LQ-regulator broken at the control input point. This function can be recovered by a sequence of Kalman filter designs with parameters from (15b). A two-step design process now starts with a single LQ-regulator which achieves desirable sensitivity. com- plementary sensitivity. and loop transfer functions, and is followed by a sequence of filter designs to approximate these functions to whatever needed accuracy. Nice properties dual to (18)-(24) apply which make this process easy.

B. Properties for Nonminimuna Phase Models

When the model G(s) has right-half plane transmission zeros, the asymptotic behavior of (17) or (25) does not hold, and the two- step LQG/LTR design process fails to produce the desired functions. Nevertheless. the process still offers useful options which can be exploited in design.

1) Option 1: Avoiding the Issue: The first thing to note is that the minimum-phase requirement applies to the model of a plant, not to the plant itself. A common design trick, therefore, is to approximate nonminimum-phase plants with minimum-phase design models. This is perfectly safe provided that the resulting (deliberate) modeling errors are incorporated into the uncertainty characterization of (3). One way to do the approximation, for example, is to collect all unstable zeros into an all-pass factor analogous to (21), i.e..j

( 3 0 ) = BZ(s)G,,(~) (26)

numerically preferred, by solving generalized eigenvalue problems [13], [ 141. All-pass zero factors can be found by solving another Riccati equation. or

where G,(s) is minimum phase and

B,(jw)B,(jw)H=I for all w.

The plant G(s) is then approximated by the model G,(s). For a single zero at s = + z , this produces a normalized multiplicative error which satisfies

Note that this error is small for all w 4 z. Indeed, whenever (28) is small compared to the existing model uncertainty, nz(w) in (3). it can be added to m ( w ) with minor effects, and the LQG/LTR design process can proceed in normal minimum-phase fashion.

2) Option 2: Living with the Issue: It is evident from (28); of course, that sufficiently small values of z can produce approxima- tion errors which dominate other model uncertainties. In that case, our design trick, while still safe, becomes excessively conserva- tive. It invokes design prescriptions, P2 and (5 ) , which ignore too much of the known phase and direction structure of the modeling error.

For plants in the latter category, the W-problem offers fewer known general properties. The sequences of compensators pro- duced by (14) or (15) do not converge to nice functions such as (17) or (25 ) . They do, however, converge to Hz-optimal functions, that is. to functions which achieve the best Hz-tradeoff between S and T, subject to the inherent constraints imposed by nonminimum-phase zeros. Nonminimum-phase constraints have been interpreted only recently. and only for SISO problems, as integral relations over frequency applied to the function log [S(jw)] [15]. The relations show that sensitivity improvements (S < 1) achieved in one frequency range must be paid for with deteriorations (S > 1) over another range, with the severity of deteriorations dependent upon right-half plane zero locations. This conclusion also follows from recent results in H"-optimization theory [16].

In our H'-solutions, the frequency ranges in which improve- ments and deteriorations of sensitivity occur can be manipulated by the weights W(s). Like the nonminimum-phase constraints themselves, however. the exact relationship between W(s) and S(s) is currently available only for SISO and certain limited MIMO cases. These cases are developed and illustrated by means of example in Section V. It turns out that the shape of S ( j w ) can be directly assigned via W ( j w ) , but its magnitude level cannot (i.e., S ( j w ) is determined to within an unknown scale factor).6 Analogous results for general MIMO problems remain a goal of current research.

3) Option 3: Giving Up Hz-Optimality: A heuristic argument which explains why (17) or (25) cannot hold for nonminimum- phase problems proceeds as follows. No stabilizing compensator K(s) can use unstable poles to cancel unstable zeros of the plant G(s). Hence, the products GK or KG must retain the nonmini- mum-phase structure of G. The Kalman filter and LQ-regulator loop transfers (C@Kf or Kc+B), however, are known to be minimum phase (else they could not exhibit infinite positive gain margin [17]). Thus, neither function can be recovered.

This reasoning naturally leads to the question "What class of functions can be recovered?." More precisely, if we generalize the LQGlLTR process by replacing the Kalman filter design step (14a) with an arbitrary choice of filter gains. say Kf = F, but retain the sequence of LQ-regulators (14b). for what choices of F will a convergence result such as (1 7) apply? The answer to this question is not surprising. Any function which shares the closed right-half plane pole/zero structure of G(s), can be recovered by the generalized process. Specifically,

SISO Ha-designs 1161. Note that these limitations are precisely the same as the limitations on

110 IEEE TRANSACTIONS ON AUTOMATIC CONTROL. VOL. AC-32. NO. 2, FEBRUARY 1987

Let G(s) be square and factorable as follows:

G(s)=B,(s)B,(s)-’n(s)G,,(s) (29a)

where B,(s) and Bp(s) are all-pass factors of (open) right-half plane zeros and poles, respectively, Q(s) is a factor consisting of poles and zeros located directly on the jw-axis (not all-pass), and G,&) is minimum-phase and stable.

Choose F such that

C ~ ( ~ ) F = B , ( S ) B , ( S ) ~ ‘ ~ ( S ) WS(s) (29b)

where W,(s) is any stable, strictly proper, full rank function. Then the sequence of compensators generated by (14b)

K F ( s ) = K,(sI- A - BK,- FC)-IF (30)

satisfies

G(s)KF(s)+C@(s)F pointwise in s

as p+O. (31)

This result follows from a minor variation of the original recovery deviation for (17). The modified derivation is included in Appendix A for completeness. A formal augmentation procedure which implements the result is analogous to Fig. 3 for minimum- phase plants. Hence. the same caveats apply. More importantly, however. since F in (29) is no longer a Kalman filter gain. no LQG guarantees apply, not H2-optimality and not even stability. A separate test must be performed on the function C@Fin order to assure that it has desirable feedback properties.

V. LQGlLTR FOR NONMMINIMUM-PHASE PROBLEMS

As discussed above, the relationships between weights and sensitivities in nonminimum-phase H2-problems is currently available only for SISO and certain MIMO cases. This section provides an informal description of the SISO relationships and illustrates them with a simple example.

A . SISO Relationships

The key feature which makes minimum- and nonminimum- phase problems different is that the latter impose certain global constraints on achievable sensitivity functions. These constraints have recently been expressed in the following Poisson integral form 1151.

Consider any stabilizing compensator for the plant G(s). Let z,, i = 1, . . . , m, be the plant’s nonminimum-phase zeros, let B J S j be an all-pass factor of the plant’s unstable poles, and let Bk(s, p ) be an all-pass factor of whatever additional unstable poles are introduced by the compensator. Denote these added poles as p = ( p I p 2 - * * pk) . Assume for simplicity that these various singularities are distinct and do not occur directly on the jw-axis. Then

S,&) = S ( S ) [ B & ) B k ( S , PI1 - I (32)

is analytic and nonzero in the closed right-half plane and satisfies

jYm log [~, , ( jw)le , (w) d W = g log [ B ~ ( ~ ~ ) B ~ ( ~ ~ , p ) ~ - l

for each i (33a)

where Oi(w) are “constraint weighting functions” defined by

= Re ( z , )

[ w - Im ( z ; ) ] ’ + Re (ti) ’

with Re (z) and Im (1) denoting real and imaginary parts of z , respectively.

These equations show that the function S,,l,(jw) must satisfy the usual constraints of analyticity (Bode gain-phase relations, etc., [3]). and also an integral constraint for each zero in the right-half plane. Whenever the zero is real. it turns out that its integral constraint applies only to the magnifude of the sensitivity function. which we denote by S(w). This follows because the even property of

Re [log S,rl,(jw)] =log S,7,,(w) = log s ( w )

and the odd property of

Im [log S,,,(jw)I =arg [S,,Ajw)l

reduce (33) to the following form. Integral Constraints f o r Real Zeros:

with v i (w) = ~

2zi I j w + z , I 2 ’

For complex conjugate pairs of zeros, the integral constraints from (33) apply to both magnitude and phase of the sensitivity function and must be reduced via gain-phase relations into constraints on magnitude only. This latter step is carried out in Appendix B and produces the following results.

Integral Constraints for Complex Conjugate Zero Pairs:

with v,- = 2 Im ( z , ) [Re (ai)’ + Im (zi)’ - w’]

((jw)’+2 Re (zi)jw+Re (z,)‘+Im ( z , ) l I 2 ’ (35d)

Note that (34) and (35) together constitute a set of m implicit constraints between and the number and location of added unstable compensator poles. These constraints can be used with Lagrange multipliers to turn our original constrained Hz-optimi- zation problem in (8) (viewed as an optimization over S(jo) subject to closed-loop stability) into an unconstrained problem, ].e..

where @(w) is the magnitude of the usual weight on sensitivity, and the complementary sensitivity term in (8) has been dropped for simplicity. Necessary conditions for optimality are npw obtained by simply differentiating J , with respect to X. p , and S at each frequency. The last of these derivatives leads to the following expression for S:

[ X i v i ( w ) ] I:’

S ( w ) = 4 W ( w ) (37)

This equation demonstrates that the optimal nonminimum- phase sensitivity must be a linear combination of predetermined

STEIN AKD ATHANS: LQGiLTR PROCEDURE FOR MULTIVARIABLE FEEDBACK 111

constraint weighting functions divided by the designer's chosen sensitivity weighting function. Note, however,-that the Lagrange multiplier values Xi remain unknown. Hence, S(w) is not actually known until we solve the Hz-problem. Nevertheless, the equation lends substantial information about what sensitivity weights to choose. In particular, systems with a single right-half plane zero will have s_ensitivities proportional to Y'G/W. We can therefore compute W ( w ) explicitly to produce an entire desired shape of s(w), with only a scale factor left unknown. An example which illustrates this possibility is given shortly.

More generally. if several right half plane constraints apply. it is possible to use (37) to parameterize the weights in terms of the Lagrange multiplier values, and then to sea_rch over these Lagrange parameters until the desired shape of S(w) is achieved. This process is also illustrated below.

B. A n Exutnple

Consider the following simple SISO system with a nonmini- mum-phase zero at z = 1 + j 0 and an unstable pole at p 0.2 + j 0 :

G ( s ) =B,(s)-'G,(s) with

s-0.2 Bp(s) = - 0.2(-s+ 1)

st-0.2 c5(s)= (s+ l)(s+0.2) . (38)

According to (33), sensitivities for this plant must satisfy a single integral constraint with weighting given by

(39)

Hence, Hz-optimal sensitivities will be related to the chosen sensitivity weights by

The last formula was used to define several candidate sensitiv- ity weighting for the example. Each candidate was augmented to the plant model according to the procedure of Fig. 3, and the two- step LQG/LTR calculations were carried out. Some resulting recovered sensitivity functions and weights are shown in Fig. 4.

In Fig. 4, case 1, for example, the weight m(w) = x * m i s used to produce a flat sensitivity function. The unknown constant level of this function turns out to be s = 1.5 which is precisely the minimum H"-norm achievable via H"-optimization [ 161. Note that this example confirms a central idea in [7] that Hz-methods can be used to solve flat sensitivity (H") problems. Equation (40) provides a way to find the necessary weights for any desired sensitivity shape.

Case 2 shows an alternate choice of @'(LO) calculated according to (40) to produce a ten-fold improvement in low frequency sensitivity compared to high frequency sensitivity. This improve- ment is indeed achieved, but at the expense of deterioration from the minimum achievable flat sensitivity. Case 3 shows a choice in which the improvement is sought at high frequencies instead of low. and Case 4 shows a choice where improvements are sought at mid-frequencies.

Finally, to illustrate the general case with more than one nonminirnum-phase constraint, consider a modified version of our plant with one additional right-half plane zero, i.e.,

sensitivity in order to keep compensators realizable. Hence. all S((LI)'S in Fig. ' We have also retained a small weight ( p = 0.001) on complementary

1 return eventually to unity.

1

S ( W )

10-

f requency (rodJonsfsecond)

(b) Fig. 4.

This new plant must satisfy two nonminimum-phase constraints, one with weighting function v I from (39) and the other with weighting

0.5 w 2 + 0.25

vz(w) = ~ .

Now suppose that we again want to achieve a flat sensitivity function. According to (37), the sensitivity weighting required to do so must satisfy

This magnitude corresponds to

W( jw) = j w + z ( h , X,)

(jw + I)(jw + 0.5)

(43)

(44)

where the zero location <(X,, X?) is indexed by the multipliers and remains unknown. A brute force search to determine its value is summarized in Fig. 5. The desired flat sensitivity is attained for z = 2.5 and has an H"-level S = 3.5. As expected, this is higher than our previous level due to the second nonminimum-phase constraint. It is also interesting to note that in order to satisfy both constraints. the compensator introduces one additional unstable pole which occurs at p I = + 2.5 when the sensitivity is flat.

All of the above results confirm the basic nonminimum-phase behavior identified in [ 151 and 1161, namely that sensitivities cannot be shaped arbitrarily. They are constrained to satisfy certain integrals which translate into a minimum Hm-bound achievable by a flat sensitivity function. Improvements over this bound are possible in any finite range of frequencies but must be paid for with deteriorations in other ranges. We have seen that the ranges in which the improvements and deteriorations occur can be effectively manipulated with the weights W(s) in SISO Hz-

112 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-32, NO. 2 , FEBRUARY 1987

10 9-

7- 8-

6-

/I 5-

2=4 /---

\

/

\ \

2-

1

frequency (radions/second)

Fig. 5

problems. While no details are given here, this capability also generalizes to the class of MIMO problems with orthogonal nonminimum-phase zero directions (including, for example. systems with a single real nonminimum-phase zero or a single nonminimum-phase complex conjugate pair). This follows be- cause the all-pass zero factor BZ(s) for such problems can be diagonalized by orthogonal transformations, and the SISO results can then be applied separately in each direction. More general MIMO cases remain to be resolved.

VI. CONCLUSION

This paper has presented the formal LQGILTR design method for linear multivariable feedback systems as the solution of a specific Hz-tradeoff between sensitivity and complementary sensitivity functions in the frequency domain.

Solutions of the H2-tradeoff have very desirable properties for minimum-phase systems. For this case, a two-step design process is appropriate, beginning with a Kalman filter (or LQ-regulator) design to achieve desirable transfer functions. and followed by a sequence of LQ-regulator (or Kalman filter) designs to recover these functions to any needed accuracy. A formal weight augmentation procedure was described to define H'-weightings which achieve essentially arbitrary shapes for sensitivity functions in the first step.

While this two-step design approach does not carry over directly to nonminimum-phase problems, several options for dealing with such problems were discussed. It turns out that HZ- solutions for SISO and some MIMO cases can still be manipulated effectively through the choice of weights, and that a large class of loop transfer functions can still be recovered with the LTR approach.

These various features make LQG/LTR a very effective design tool for linear multivariable feedback systems. The major weakness of the method appears to be its restriction to design tradeoffs at only one loop-breaking point. That is, the method can trade off S ( j w ) against T(j(jw) with both defined at the output or both defined at the input. However, it cannot easily trade off these functions when they are defined at different points.8 This means that the method currently obligates designers to reflect all feedback design requirements to one of the two loop-breaking points. While such reflections cause no difficulty in SISO problems. it is easy to construct MIMO examples where they are arbitrarily conservative. This weakness is shared by all design methodologies based on the loop-shaping paradigm. Only further research and applications experience can determine whether this will remain an important shortcoming.

APPENDIX A

This Appendix provides one way to calculate all-pass pole and zero factorizations for general MIMO plants. We assume that the plant is described by the following stabilizable, detectable state- space realization with no poles on the jw-axis.

dx dt -=Ax+Bu

y = cx.

A . All-Pass Pole Factorization

Let Kf(q) be defined by the following Riccati equation:

O=AC(q) + C(q)A T + q'BBT- C(q)CTCC(q)

K d q ) = U 4 ) C T . 642)

Then one possible all-pass pole factorization is given by

B,(s)-'=lim [ I - C(s l -A)- 'K, (q) ] 643) 4-0

G,(s)=lim C(s I -A -K,(q)C)-'B. (A4)

The fact that (A3) is all-pass can be verified from the Kalman equality associated with Riccati equation (A2), namely

q-0

( I - C@K,-)(Z- C~.K,)H=Z+q'(C@B)(C@B)H, (A5)

This shows that B; I tends to identity along the jw-axis as q tends to zero. We also need to verify that (A4) is stable. which follows from well-known properties of Kalman filters, and that B; I G, = G, which follows from direct evaluation of the product, noting that G, = ( I - C@JK~) - 'G.

B. All-Pass Zero Factorization

Let K J q ) be defined by another Riccati equation:

O=A~PP(q)+'P(q)A+q'CTC--(Y. )BBTP(q)

KAq) = B T m ) . (A6)

Then one possible all-pass zero factorization is given by

Bz(s)= lim qC(sI-A-BK,(q))-'B 647) 4'"

G,,,(s)= lim K, (q ) ( s I -A) - 'B /q . 648)

In this case, the fact that (AS) is all-pass can be verified by noting that B: = qC(I - K$B) - I and by manipulating the Kalman equality associated with Riccati equation (A6) into the latter form, e.g..

4 - a

( I - K , ~ . B ) H ( I - K , ~ B ) = Z + q ' ( C 9 B ) H ( C @ B ) (A9)

yields

Z=(Z-K,~.B)-HIZ+q'C@BHC@.B](I-K,QB)- ' (A10)

which shows that B: tends to identity along the jw-axis as q tends to infinity. We also need to verify that (AS) is minimum phase. which follows from well-known properties of LQ-gains, and that B,G,,, = G, which again follows from direct evaluation of the product.

APPENDIX B

This Appendix documents a modification of a derivation in [2] to show that the class of functions in Section IV-B-3 can be

STEIN AND ATHANS: LQGXTR PROCEDURE FOR MULTIVARIABLE FEEDBACK 113

recovered in nonminimum-phase cases. First note that the factored plant in equation (29) has the following (nonminimal) state-space representation:

where the first group of states is a realization of the BplWGms factor of G(s), and the second group is a realization of the nonminimum-phase all-pass zero factor Bz. (For unstable plants, note that F must be selected to keep the system stabilizable from ((t) . Then C@B and C@F share common closed right-half plane singularities, as required in equation (29).)

It follows from this structure and from krown asymptotic properties of LQ-regulator gains [ 181 that

PK, (P) -+WCI 01 E UCm

as p-+O (B2)

where U is an orthonormal matrix and C, is a "minimum-phase version" of the plant's output matrix, such that

C@(s)B= B,(s)C,,@(s)B and C@(S)F=B,(S)C,@(S)F. (B3)

One application of the matrix inversion lemma to (30) plus the limiting behavior (B2) now shows that the compensator KF(s) has the limit

K F ( s ) 3 ( C m ~ ( s ) B ) - ' C m X ~ s ) F @4)

where

One more application of the inversion lemma applied to CmE together with (B3) yields

C,E(s) = [BZ(s)- '(I- C ~ ( s ) F ) - ' B z ( s ) ] C , ~ ( f ) . (B5)

Finally, substituting this into (B3) and (EM) gives the desired result:

APPENDIX C

This Appendix converts the nonminimum-phase contraints for complex conjugate zero pairs from equations (33) into magnitude- only constraints. For notational convenience. let the zero pair under consideration be described by

z l = R + j I z 2 = R - j l w i t h M 2 = R 2 + I 2 , (C1)

and let

log [ L - ( j w ) l = A ( w ) +jB(w) (C2)

log tB,(z,)B&, p1l-l =a+jb . (C3)

Then, using the fact that A ( w ) is an even function, the two constraints from (33) for zI and zz, respectively, lead to the same single constraint on A (w), namely:

[ A ( W ) V I ( W ) dw = TU CT

(CW - 0

with v I ( w ) = Ol(w) + e l ( - w )

Also, using the fact that B(w) is odd, the two constraints from (33) lead to the same single constraint on B(w):

(C5b)

However, because the function S,(jw) is analytic in the closed right-half plane, A ( w ) and B ( w ) in these integrals are not independent. Rather, B(w) is determined by A ( w ) in accordance with the well-known Bode gain-phase relation [3, ch. 141:

Hence, (C5a) can be rewritten as

and reordering the integrations yields

im A ( w ) ; [; S o v{(v) dw= - rb. (C7) 1 2 -

- 0

According to (C6), the bracketed term in this last equation must itself correspond to the imaginary part of an analytic function whose real part along the jw-axis is wflo). It can be readily verified that

2 Is f ( s ) = s 2 + 2 R s + M 2

ib such a function with the following real and imaginary parts:

Substituting the imaginary part of this expression into (C7) gives the desired second constraint on sensitivity magnitude.

with v2(w) = 21(" - w2)

I ( jw)z+2Rjw+M2)2 '

ACKNOWLEDGMENT

The nonminimum-phase ideas in his paper are the result of various discussions with J . Freudenberg, N. Lehtomaki, J . Doyle, and S. Mitter.

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Gunter Stein (S’66-M’69-F’85) received the B.S.E.E. degree from GMI. Flint, MI, and the M.S.E.E. and Ph.D. degrees from Purdue University. West Lafayette. IN. in 1966 and 1969, respectively.

Since 1969. he has been with Honeywell. Inc.. Minneapolis, MN. where he is currently Chief Scientist of the Systems and Research Center. He is also Adjunt Professor in Electrical Engineering and Computer Sciences at the Massachusetts Institute of Technology. Cambridge.

. ..