An overview of extremal properties for robust control of interval plants

Automat/ca, Vol. 29, No. 3, pp. 707-721, 1993

0005-I098/93 $6.00 + 0.00

Printedin GreatBritain.

© 1993PergamonPress Ltd

An Overview of Extremal Properties for Robust Control of Interval Plants* M. D A H L E H , t A. TESI:~ and A. VICINO§

Interval plants enjoy remarkable extremality properties for robust stability and performance analysis, as well as control design of systems with parametric, Hoo and sector bounded uncertainties. Key Words--Interval plant; parametric uncertainty; H= uncertainty; robust stability; robust performance.

engineering insight from the designer. The difference between the exact model and a simplified one is represented as a perturbation to the simplified model. For example, an infinite dimensional system is usually modeled by a lumped system approximation, and the neglected part of the system acts as a perturbation. In order for a control design to be successful, it is necessary that it performs well on the actual physical process, which differs from the nominal model by the presence of perturbations. This leads to the requirement of robustness, which demands that the control design attains reasonable performance specification for the nominal model and the class of possible perturbations. Uncertainty in the underlying model is usually composed of parametric and non-parametric uncertainty. The parametric uncertainty is represented by a vector lying in some prespecified uncertainty set. For example, the uncertain parameters can be the coefficients of a transfer function description, or the elements of system matrices in a state-space realization. Non-parametric uncertainties are used to represent unmodeled or difficult to model dynamics; they are usually given as norm bounded perturbations. The past few years have witnessed a revival of the classical problem of analysis and synthesis of control systems in the presence of parametric uncertainty leading to many breakthroughs such as Kharitonov's Theorem (Kharitonov, 1978) and the Edge Theorem (Bartlett et al., 1988). Stability is one of the key issues in feedback control, and thus it is not surprising that robust stability against parametric perturbations represents the main topic of a large number of papers that appeared recently in the literature. New philosophies and approaches were advanced that

Almlraet--This paper provides an overview of recent results o n the problem of robust stability and performance of feedback control systems in the presence of plant perturbations. The widely studied class of interval plants is considered, and an effort is made to cover different types of plant perturbations by incorporating, in addition to the interval plant description, unstructured norm bounded perturbations. Extremality results for these classes of uncertain systems are provided in a unifying framework. Robust performance of control systems is addressed in the paper and a number of extremal results are given for the computation, of the structured singular value function for feedback systems with interval plants. Also, classical frequency response analysis for interval plant-controller families of open loop transfer functions is surveyed, with the aim of providing extremal results for the computation of specification parameters, such as phase or gain margins, and sensitivity and complementary sensitivity peaks. 1. INTRODUCTION

IN MODELINGPHYSICAL systems for control purposes, it is necessary to generate mathematical descriptions that are tractable. A very accurate model that describes the physical process exactly may turn out to be useless from the analysis and design standpoints. By constrast, sacrificing the precision of the model for the sake of simplicity may lead to unacceptable design. The careful balance between capturing the true behavior of the physical system, and generating workable models demands a great deal of skill and *Received 10 September 1991; revised 12 June 1992; received in final form 19 November 1992. The original version of this paper, not presented at any IFAC meeting, was accepted for the Automatica Special Issue on Robust Control (Vol. 29, No. 1), but could not be included due to lack of space. It was recommended for publication in revised form by Guest Editor J. Ackermann. $Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, CA 93106, U.S.A. Dipartimento di Sistemi e Informatica, Universit~t di Firenze, Via di Santa Marta, 3-50139 Firenze, Italy. §Dipartimento di Ingegneria Elettrica, Universit~t di L'Aquila, 67040 Poggio di Roio, L'Aquila, Italy. 707

708

M. DAHLEH et al.

supplied effective solutions to many seemingly difficult stability robustness tests (Barmish (1988); Dorato (1987); Dorato and Yedavalli (1990); Polis et al. (1989); ~iljak (1989)) for surveys of some of the results in this area and the proceedings of recent workshops specifically oriented to these topics (Bhattacharyya and Keel (1991); Hinrichsen and Martensson (1990); Milanese et al. (1989)). The common goal of most contributions is to derive non-conservative stability tests and assess how far is a nominally stable control system from instability. This last issue, known in the literature as the stability margin problem, has been investigated for cases where system transfer function coefficients are perturbed independently, when a linear dependence is assumed (Bhattacharyya (1987); Hinrichsen and Pritchard (1988); Tesi and Vicino (1988); Barmish (1989); Qiu and Davison (1989); Cavallo et al. (1991)) or when a nonlinear structured dependence is assumed (de Gaston and Safonov (1988); Sideris and Pefia (1988); Vicino et al. (1990)). Regarding stability tests, a large body of work has been carried out for the class of interval plants, i.e. systems represented by transfer functions given as ratios of interval polynomials. Initial work on this problem was conducted in Chapellat and Bhattacharyya (1989) where the problem of robust stabilization of an interval family was studied. It was shown that for any fixed controller a fixed number of one dimensional sets of plants must be stabilized in order to ensure robust stability of the interval feedback system. Also, several stronger results have been obtained when special structures are assumed for the controller (Barmish et al. (1990); Barmish and Kang (1991); Ghosh (1985); Hollot and Yang (1989); Rantzer (1992)). In Chapellat et al. (1990) the more realistic problem of an uncertainty model incorporating both parametric and nonparametric perturbations was studied and the results demonstrated that the small gain robust stability condition can be checked for a fixed number of one dimensional sets of plants. Modeling unstructured uncertainties as sector bounded perturbations leads to the study of robust strict positive realness. This important issue addressed in many papers (see e.g. Bose and Delansky (1989); Chapellat et al. (1991a); Dasgupta (1987); Dasgupta et al. (1991); Vicino and Tesi (1991)) is relevant to robust control in view of its implications for robust absolute stability (Chapellat et al. (1991a); Dasgupta et al. (1991); Tesi and Vicino (1991b)) and convergence analysis of recursive identification algorithms for adaptive control (Anderson et al. (1990); Tesi et al. (1991)).

In addition to robust stability, problems of robust performance in the presence of parametric uncertainty are becoming the focal point of much research in the robust control area. Along these lines (Bartlett, 1990b) analyses the extremality properties associated with steady state errors, while in Bartlett et al. (1992) examples are reported on the absence of Kharitonov type results for several step response performance specifications. In Fu (1990) and Tesi and Vicino (1991c) extremality results are derived for the frequency response of polytopic or interval families of rational functions, providing a basic tool for extending classical design techniques to interval feedback systems. Much of the present success of the parametric approach is restricted to analysis issues. Synthesis of controllers for systems affected by parametric uncertainty remains to a large extent an open and difficult problem. Preliminary results on the regularity of the robust design problem with respect to the controller coefficients were obtained in Tesi and Vicino (1991a) and Vicino and Tesi (1990). On the other hand, in the case of systems with unstructured norm bounded perturbations, the development of synthesis techniques is more successful. Recent progress in the areas of H~ (Ba~ar and Bernhard (1991); Doyle et al. (1989); Francis (1987); Kwakernaak (1991); McFarlane and Glover (1990)), and Ii (Dahleh and Pearson, 1987) control design resulted in systematic and powerful methods for the synthesis of controllers with a prescribed level of unstructured perturbations. For structured norm bounded perturbations the techniques of ~ analysis are largely used (see Doyle (1982); Packard (1988)). The present paper is aimed at reviewing recent contributions in the parametric robustness field. Three basic objectives inspire its development, and will be subsequently pursued. The first objective is to bring the techniques of robust stability closer to addressing real world problems by incorporating different types of uncertainties. The second objective is to emphasize that the "worst case" plants of an interval family from a design standpoint, are certain fixed one dimensional set of plants (Kharitonov segments) and that for some specific perturbation and controller structures these critical segments can be replaced by a finite number of plants (Kharitonov vertices). Both of these sets are independent of the order of the plant. The third objective is to establish connections between the explicit parametric representation paradigm, and the general structured uncertainty formulation used in the /z theory setting, and to show that whenever interval parametric uncertainty is

Extremal properties for robust control present, the first approach provides in a natural way a remarkable complexity reduction of the general problem. The paper is organized as follows. Section 2 introduces notation and preliminaries. Section 3 presents basic results on robust stability in the presence of structured and unstructured uncertainty. Section 4 addresses the robust performance problem. Section 5 deals with the robust strict positive realness problem and the related robust Popov criterion for absolute stability. Section 6 reports recent results on continuity of the stability margin problem for interval feedback systems. Concluding remarks are drawn in Section 7.

A set of system configurations we will be concerned with is represented in Figs. l(a)-(d). We denote by ~ a family of proper interval plants, i.e. N(s)

= D(s-'--)' N(s)

-a-

~ d~fl,

D(s) ~ ~I

where )¢'t and 50~ are families of interval polynomials, i.e.

d~flI = {N(s) :N(s) = Z bi Si, i=O

b~ ~ [bi-, b+], i = 0 . . . . .

}

,

= s" + ~'~ ais i, i=O

ai ~ [aT, a~-], i = O, . . . , n - 1}. The transfer function C ( s ) = N c ( S ) / D c ( s ) is a controller and A/(s), i - - 1 , 2 account for norm bounded plant perturbations. They are stable operators such that ¥o~ -> O.

Notice that the norm bounded uncertainties Ai(s) may be located at different points of the loop to represent different uncertainty structures. For example, the configuration of Fig. l(b) reflects output sensor errors and/or neglected high-frequency dynamics. For a given interval polynomial, we define two

-b-

GA

iil , i -e-

m},

n--I

= D(s):D(s)

I A , ( j o ) I -< 1,

2. NOTATION AND PRELIMINARIES

~= lG(s):G(s)

709

I I

.1

-d-

FIG. 1. (a) Interval feedback system; (b) interval system with multiplicative norm bounded perturbation; (c) interval system with additive norm bounded perturbation; (d) interval system with coprime factor norm bounded perturbation.

710

M, DAHLEH et al.

subsets playing a key role in robust stability and performance analysis. We will make specific reference to the numerator family, but similar definitions hold for the denominator family. The first set NK includes the four Kharitonov polynomials, representing four corners of the interval family in coefficient space WK= {N,(s), N2(s), N3(s), N4(s)}, where Nl(s) = bo + b-~s + b~s 2 + b f s 3 + b4s 4 + bss 5 + • . . ,

the above observation that an interval plant is completely characterized in the frequency domain by a four-parameter family ~R TJR= { G ( s ) : G ( s ) = N ( s ) / D ( s ) , N(s) • WR, D(s) • ~R}. Based on the sets A/K, Wxs, ~K, ~KS of the numerator and denominator families we can construct the following subsets of transfer function of CgK= { G(s): G(s) = N(s) O(s----)' N(S) eWK, D ( s ) e ~ K } '

N2(s ) = bff + blS + b 2 s 2 + b ~ s 3 + b'~s 4 + b~s 5 + • . . ,

N(s) ~:s = G(s): G(s) = O ( s ) ' N(s) •

N3(s) = b~ + b~(s + b£s 2 + b 3 s3 + b-~s 4 + b~s 5 + . • . ,

D(s) • ~

or N(s) • NKS, D(s) • ~X}-

N4(S) = bo + b~s + b f s 2 + b 3 s3

+ b ; s 4+ b~s 5 + • . . . The second set, called .AcKS, is made of four segments of polynomials, i.e. convex combinations of Kharitonov polynomials with coinciding odd or even parts WKS= {SIN, i = 1. . . . .

4},

where the four segments are SiN = ( N ( s ) : N ( s ) = ;~N,(s) + (1 - ~.)Nj(s), ~. • [0, 1]}, i=1,...,4,

j=i+l-4[(i+l)/5J.

We will often be concerned with the frequency behavior (s = j o 0 of an interval polynomial. In this case, ~ can be replaced by the following reduced two-parameter family (Dasgupta, 1988) 2¢"R = {N(s):N(s) = N°(s) + ]t,,eNc(s) +Z,.oNo(s), Z.... )...o e [ - 1 , 1]}, where Ne(s) = ~[N3(s) - N4(s)] = ½[N2(s) - N,(s) l, go($ ) = ½[N3(s ) - g 2 ( s ) ] = -12[N4(s ) - g l ( s ) ] ,

represent the so-called "even" and "odd" part of the interval polynomial, whereas

The set ~ is called the Kharitonov (vertex) set of ~ and
At) : Ai • a , } ,

4

N°(s) = 14•

Ni(s),

i=1

represents the "nominal" polynomial of the family. The subset JV'R is the intersection of the two-dimensional plane containing the four Kharitonov segments of the family with the hyperrectangle representing N~ in the coefficient space. Kharitonov vertices and segments represent the vertices and the edges of this reduced family in the plane (X.... ),.,o). It is clear from

.........

................

i

C ,a-............. l

FIG. 2. Feedback loop with block diagonal uncertainties.

Extremal properties for robust control where At is a ball in the space of H® real rational functions, denoted by RH®, with radius 1 A t = {A:A ~ RH®, IIAII®---1).

A =

A n -+-diag [~'d,e,

~'d,o, ~'n,e, ~'n,o]"

The relative interconnection matrix M can be computed via straightforward block scheme manipulations of a state space realization of the nominal plant G°(s)=N°(s)/D°(s), and turns out to be given by

M(s) = (M11(S) Mt2(s)~ \M21(s)

M22(s)]'

(1)

where Mlt(S ) = D°(s) -~

-De(s) -Do(s)

amounts to verifying that

+ Dc(/O,)D(jo,) * 0, Vto---0

Observe that an interval plant description can be cast in this general setup. In this case A(s) becomes a real diagonal matrix A s whose entries represent coefficient perturbations

-De(s) Ne(s)Oe(s) No(s)De(s)\ -Do(s) N~(s)Do(s) No(s)Do(s)) 0 0 0 0

M12(s) =

D°(s)-l[N°(s)D,(s) N°(s)Do(s) Ne(s) No(s)]t, M2~(s)=D°(s)-t[-1 - 1 1 11, M22(s) = D°(s)-lN°(s) = G°(s). If a feedback controller C(s) is added to the interval plant, the closed loop may be represented as a feedback interconnection of the diagonal uncertainty matrix A R with a transfer matrix Met(S) which is a special rank 1 matrix accounting for the nominal closed loop (Chen et aL, 1992) Mcl(S) = e°l(s)-lOc(s)V(s)Wt(s), where

V(s) = [De(s ) Do(s) -C(s) -C(s)]', W(s) = [ - 1 - 1 Ne(s) No(s)]', and P°t(s) = O°(s)Dc(s) + N°(s)Nc(s) is the nominal closed loop characteristic polynomial. We know that the closed loop is robustly stable if and only if P°t(s) is Hurwitz and det (I - M¢t(jt0)A R) :/: 0, VA :JAil -< 1, Vto _>0.

(2) Notice that this condition is exactly equivalent to the well-known "zero exclusion" condition on the closed loop characteristic polynomial Pd(s)= Nc(s)N(s) + D¢(s)D(s). This condition, which is widely used in the literature on parametric robust stability as well as in the present paper,

711

and

VNe)f~,

De~.

ca)

In fact, by rewriting Pct(s) as Pc,(S) = P~,(s) + Xn,eNc($)Ne($) + Xn.oNc(s)No($)

+ ~.d..Dc(s)De(s) + ~d.oD~(s)Do(s), the equivalence of conditions (2) and (3) is established by using the fact that det (I - P°t(s)-~Dc(s)V(s){W'(s)AS} )

= 1 - P°t(s)-tDc(s){W'(s)AS}V(s). 3. ROBUST STABILITY OF LINEAR FEEDBACK SYSTEMS WITH INTERVAL PLANTS AND UNSTRUCTURED NORM BOUNDED UNCERTAINTY

3.1. Extremal results for general classes of controllers In this section we review extremal results on robust stability of the feedback systems represented in Figs l(a)-(d). To do this, we need a preliminary lemma on the sum of two rectangular sets in the plane. Let R1 and R 2 be two rectangles in the plane. Let Ri2 be the sum of the two rectangles and 0R12 its boundary. Now, for any vertex of R~, consider the set given by the sum of the vertex with the four edges of R2. Let R~,s denote the sum of the four sets generated in this way. By changing the role of R~ and R2, we can generate a different set R~s. It is easy to show that the set Rks ----R~ t3 R~ has the following property.

Lemma 1. aR12 ~_Rks. Consider first the case when no norm bounded perturbation affects the interval plant. This is a case which has been widely studied in the recent literature. We report below the extremal robust stability result first given in Chapellat and Bhattacharyya (1989) for the system of Fig. l(a). An easy proof is included which will be instructive of the type of arguments that are used to generate many of the extremal results that are presented in this paper. From now on, we will drop the dependence of rational functions or polynomials on the variable s whenever no ambiguity arises.

Theorem 1 (interval perturbation). A controller C stabilizes the feedback system in Fig. l(a) for G ~ ~i if and only if it stabilizes the system for G e (~s.

M. DAHLEH et al.

712

Proof. Necessity is obvious, so we only prove sufficiency. Consider the closed loop characteristic polynomial Pc,(j~o) = Dc(jo~)D(jog) + N~(joON(]~o),

Ne,N'~, D e ~ . As a first observation, notice that closed loop stability for any G • ~:s guarantees that the leading coefficient of Pd(jw) does not vanish for all N • .Acl, D • ~)1. Consider now at a given frequency w the value set Ip(w) of the numerator and denominator coefficient space uncertainty sets under the map Pd(jw): Rm+n+l--)C. This set is a polygon given by the sum of two rectangles R~ and R2 Rl = {¢: ~ = D¢(jo))D(jog), for some D • ~ } ,

R2 = {¢: ¢ = N~(jo~)N(jo), for some N e N,}. Denote by I~'(a)) the image set of the Kharitonov segment set ~ s in coefficient space under the map Pd(ja)). From Lemma 1 we have

alp(w) ~_lk'(o).

wo.

(5)

By (4) and (5), we conclude that 0 ~ ala(og), Vto. Since 0glp(0) and the map Pcl(jto) is a continuous function of 09, it follows that 0 ~ Ip(w), '¢w. This relationship implies stability of P¢l(jto), for all N • 2¢i, D • ~i.

Remark. As it can be quickly verified by the proof above, Theorem 1 holds even if Nc and Dc are complex coefficient polynomials. Consider now the case when both an interval plant and norm bounded perturbations are present. We will specifically treat the cases of multiplicative, additive, and coprime factor unstructured perturbations affecting an interval plant. The following theorem states a parametric version of the well-known small gain theorem for the case of multiplicative perturbations (see Fig. l(b)). Theorem 2 (interval and multiplicative perturbations). A controller C stabilizes the feedback system in Fig. l(b) for G e ~ and A such that IIt~ll-< E if and only if it stabilizes ~f~s and e < 1 / m a x IIGC(1 + /

Ge~s

GC)-IlI~.

Lemma 2. For a given stable proper transfer function F(s)= N(s)/D(s) and a given positive number r, statements (i) and (ii) are equivalent

(i) IlF(s)ll~ < r, (ii) IF(0)[ < r and rO(jto) - eJrN(jto) 4=O,

(6) Vw, VV • [0, 2~r]. By an exact analogy to the previous multiplicative perturbation formulation, we consider the case of additive perturbations (see Fig. l(c)). The following theorem, first given in Chapellat et al. (1990) can be proven using the same idea as in Theorem 2.

(4)

Consider the case to = 0. It is easy to check that if ~,:s is closed loop stable, the image set Ik~(0)----Ip(0) is a segment of the real axis not including the origin. Moreover, stability of P~l(jw) for G ~ ~ s implies by the Mikhailov criterion of stability (My~kis, 1975) that

0

A quick proof of Theorem 2 can be constructed by using the extension of Theorem 1 in the preceding remark and the following lemma, which allows one to convert an H~ norm inequality into a one parameter family of stability problems.

Theorem 3 (interval and additive perturbations). A controller C stabilizes the feedback system in Fig. l(c) for G • ~ and A such that IIAII-
< 1// m a x IlC(1 + GC)-~II~,. Ge~ks

Finally, we consider another type of unstructured perturbations, along with the usual interval plant description that we are using. Let G1 = N/F, G2 = D/F, where N • NI, D e ~ , and F is a fixed polynomial. Defining a plant G = G J G 2 , we have that G e ~ . Let us now make the following assumption.

Assumption. G1, G2 ~ RH~, and coprime for all coefficient values in the prescribed intervals. We want to analyse robust stability of the closed-loop configuration composed of a controller connected to an element of the family

~=

{Ga=(G2 + A2)-~(G1 +A~):~--~2e c, ~ , and A = [A1, A2] eRH~, with

IIAII~-< E}.

The configuration is depicted in Fig. l(d). The following theorem (McFarlane and Glover, 1990) gives necessary and sufficient conditions for stability of the system in Fig. l(d).

Theorem 4. C stabilizes the feedback system in Fig. l(d) for all G a e ~ if and only if C

Extremal properties for robust control

is referred to Rantzer (1992) for further results and details on this topic. Consider a Kharitonov segment of plants, for example

stabilizes ~ , and sup [C(1 + G C ) - ' G f ' ] a ~ t [ (I + G C ) - ~ G f ~ J

E_I.

~<

Using the ideas of Theorem 2 md an extension of Lemma 2, one can prove the next theorem.

Theorem 5 (interval and coprime factor perturbations). C stabilizes the feedback system in Fig. l(d) for all G a e ~ a if and only if C stabilizes ~ s , and

N,(s)

G~3(s; 3.) - 3.O2(s) + (1 - 3.)O3(s) '

3.2. Extremal stability results for special classes of controllers In this subsection we examine extremal robust stability results for some special classes of controllers. By imposing further hypotheses on the structure of the controller, stronger results than those of the previous subsection are possible to obtain. It turns out that in several cases stability of the endpoints of a Kharitonov segment of a feedback system may be sufficient for stability of the entire segment. In this case, robust stability of the closed cloop system may be inferred by that of a finite number of feedback systems, including plants of the Kharitonov vertex set , ~ of the interval family. Consider the feedback system of Fig. l(a) with the compensator C given as (7)

The following result given in Barmish et al. (1990) states conditions for robust stability of this class of interval feedback systems.

Theorem 6. The closed loop system in Fig. l(a) is stable for G e ~ if and only if it is stable for G•~. We point out that in Hollot and Yang (1989) it is shown that all the vertices of the interval family suffice to check robust stability of the interval feedback system. Moreover, for the case where C(s)=-ko, it has been shown in Ghosh (1989) that a subset of ~ : made of eight plants is sufficient to guarantee robust stability of the interval feedback system. The basic argument on which Theorem 6 is based is that stability of "certain" segments of polynomials is implied by stability of their endpoints. A brief discussion of this property will be given next since it will be used in the development of the paper. The interested reader

3. ~ [0, 11,

and the corresponding "segment" of closed loop characteristic polynomials when the structure (7) is assumed for the controller Pet(s; 3.) = ko(s + fl)N,(s) + s V(s + 0:) x [3.Dz(s) + (1 - 3.)D3(s)1 ,

I[C(1 + a c ) - l a ~ a ] l l SUPs [ (1 + G C ) - I G ~ 1 JL <

l s+fl C(s) = kos v s + o:"

713

3. e [0, 11.

(8)

Consider the map Pd(Jto;'):R-->C defined by (8). The image of this map in the complex plane for a fixed to is a segment, whose endpoints Pd(Jto; 0) = P13(Jto), and Pd(jto; 1) = P12(jto) correspond to two Kharitonov plants of ~:. Let us denote by arg[P(jto)] the phase of the complex number P(jto) at a given frequency to. Assuming that /13 and PiE are stable, we know that arg [P12, (Jto)] and arg [P13Jto)] are monotonically increasing functions of to. It can be shown that (Barmish et al., 1990): d

arg P12(jto) > 0

and

d ~to arg P13(jto) > 0

do) arg Pd(Jto; 3.) > 0,

V3. •

[0, 11.

The following Lemmas 3 and 4 give sufficient conditions such that stability of a segment of polynomials is ensured by stability of its endpoints. To state these lemmas, some additional notation is needed. Consider a stable polynomial Po(s) of degree l. Consider the segment of polynomials defined by

~x = { P(s) : P(s) = Po(s) + 3.Dp(s), 3. • [0, 11}, where Dp(s), the perturbing polynomial of the "nominal" polynomial Po(s), is of degree less than I. Denote by P'(s) a stable first order polynomial and by Pe(s 2) an even polynomial. We can now state the next Lemma 3 (Barmish et al., 1990) a simple proof of which is reported in the Appendix.

Lemma 3. Let ~x be a segment of polynomials defined by a nominal polynomial Po(s) and a perturbing polynomial Dp(s) given by Op(S) = ashe'(s)Pe(s2), where o is a real number and h a nonnegative integer. ~x is stable if and only if its endpoints Po(s) and Po(s) + Dp(s) are stable. Notice that (8) is a segment such that

Po(s)

= el3(S) = ko(s + ~ ) N I ( s ) + s V(s + o0D3(s ),

Op(s) = ZsV(Oz(s) - O3(s))(s + re),

M. DAHLEH et al.

714

where Dp(s) has the same structure as that assumed in Lemma 3. This justifies Theorem 6, which follows immediately from Lemma 3. The following lemma provides a different sufficient condition for stability of a segment of polynomials and represents a special case of a more general result given in Rantzer (1992).

Lemma 4. Let ~x be a segment of polynomials defined by a nominal (complex coefficient) polynomial Po(s) and a perturbing polynomial Dp(s) such that d d--warg [Dp(jO~)] --
3.3. Extremal stability results for more general classes of plants In this subsection we show how the previous results can be generalized in a straightforward manner to analyse a special class of systems with multilinear interval perturbation structure. We start by giving appropriate definitions. Let ~3~ represents a family of proper plants, defined as follows ~=

NIN2 G:G-D1D2,

Ni

i Di ~} eP¢'~, e ~ ,

where 2¢'~ and ~ ( i e {1, 2}) are families of interval polynomials, i.e. 2¢'~= { N i : N ~= £ bqs ~, j=0

Then ~x is stable if and only if its endpoints Po(s) and Po(s) + Dp(s) are stable. Several vertex type results may be inferred by application of Lemma 4, in the same way as Lemma 3 leads readily to Theorem 6. For example, it is immediate to verify that ~K suffices for checking robust stability of the interval feedback system if the controller C(s) is such that Nc(s) = koNa(s)N~(s2), Dc(s) = S VDa(s)De(s2 ),

b~ e [b~, bff],j = 0 . . . . . n--I

~)~ = Di : D i = s n+ ~ aiy, j=0

aij e [a/)-, a~], ] = 0 . . . . .

Theorem 7. A proper controller C with structure (9) stabilizes the strictly proper interval plant ~i in Fig. l(b) for all A such that -< e if and only if it stabilizes ~gK and

IIAII

n - 1, i = 1, 2}.

In a fashion similar to Section 2 we can define the sets W~¢, @~¢, WKS, i ~KS, i and

(9)

where Na(s) and D~(s) are anti-stable polynomials, i.e. polynomials with all roots located in the right half plane and N~(s 2) and D~(s 2) are even polynomials. Some further advances in the direction of proving results similar to Lemmas 3 and 4 for different or wider classes of perturbations are in Barmish and Kang (1991) and Rantzer (1992). We now pass to consider the case when the structure of the controller is given by (9) and unstructured uncertainty is present in the feedback loop. By using Lemma 4, we can obtain the following theorem proven in the Appendix for the configuration of Fig. l(b).

m, i = 1, 2},

C~s=

NIN2 G : G - D I D 2 , Ni e WiK, D i E ~KS or N i E ,N'iKS, D i E ~iK}.

We will also consider the coprime formulation. For that purpose we define

G~-

N1N 2 F

,

factor

DID 2 G~=--, F

where N i e A;] and DiE ~ . Defining a plant G=G1/G2, we have that G e ~P. We will assume that G¢, G~ e RH~, and coprime for all parameter values in the prescribed intervals. As in the previous section, we want to study robust stability of the closed-loop configuration composed of a controller connected to an element of the family . . . G ,~

u3~P= G p = (G~ + A2)-I(G~ + tx,).G---~2e ~ f , and A = [Al, A2] ~RH~, with

IIAII~---e}.

• < 1// mGa~ x IIGC(1 + GC)-~II~. In a straightforward manner we can arrive at the following.

Remark. By considering Theorems 3 and 5, similar vertex results to those stated in Theorem 7 can also be obtained.

Theorem 8. (I) (Interval perturbations.) A controller C stabilizes the feedback system for

Extremal properties for robust control G e f#r if and only if it stabilizes the system for G ~ ~#~s. (II) (Interval and multiplicative perturbations.) A controller C stabilizes the feedback system for G e ~f and A such that 11:'11-E if and only if it stabilizes~J~s and • < 1/max

Remark. The result (I) of Theorem 8 appeared in Chapellat et al. (1991b). Similar results can be obtained for a more general class of system comprised of sums and products of interval polynomials. 4. EXTREMAL PROPERTIES FOR THE ROBUST PERFORMANCE PROBLEM

IIGC(1 + GC)-IlI~.

4.1. Robust stability/performance with structured H~ perturbations In this subsection, we consider the problem of robust stability and performance for system configurations where uncertainty is represented by interval plants and structured H® perturbations. To this end, we make reference to the scheme shown in Fig. 3(a), where the interconnection matrix M represents the closed loop system for a fixed configuration of plants. We assume that the uncertain plants are represented by q interval plants ~ , i = 1. . . . . q. If nominal plants are considered, then M

(III) (Interval and additive perturbations.) A controller C stabilizes the feedback system for G e ~r and A such that IIAII-< • if and only if it stabilizes ~ s and • < 1/max

715

IIC(1 + GC)-IlI~.

(IV) (Interval and coprime factor perturbations.) C stabilizes ~ P if and only if C stabilizes ~t~s, and

o~ ][ C(1 + GC)-'G Ill I su ( I + G C ) _ , G ~ jL< •-1.

•AP

.........................

I "0,1- i °"..

l!

1

M

I1... ..Arl j

I/J

M

Z

• . .At V ......................... ' A ° R-

-b-

MP

I I' ......................... -c-

,A,~p

-d-

FrG. 3. (a) Feedback loop with block diagonal uncertainties; (b) feedback loop with interval plants and block diagonal uncertainties; (c) feedback loop with complex block diagonal uncertainties; (d) feedback loop with mixed complex-real block diagonal uncertainties.

716

M. DAHLEH et al.

represents the nominal closed loop system. Denote by T~w(G~. . . . . Gq, A) the transfer function from the exogenous input w to the output variable z, in terms of which the performance requirement is usually expressed: Tzw(Gl . . . . .

Gq, A)

Before proceeding to the main results of this section, recall the definition of structured singular value /z (Doyle, 1982) of an interconnection matrix M relative to a given block structured perturbation A I~(M(jw))

----M22 + M 2 1 A ( / -

MllA)-lM12 •

We say that the system of Fig. 3(a) achieves robust performance if the following conditions are satisfied (1) The system is robustly stable for all G i • ~ , i = 1. . . . . q, and all A • D~(r). (2) supa,•~,.;=, ..... q,a•of(~) IITwz(G1 . . . . ,

Gq, M I I ~ -<

1.

It is well known (Doyle et al., 1982) that in the structured uncertainty context, the robust performance problem as defined above is equivalent to an "augmented" robust stability problem, obtained by closing the loop from z to w through an additional fictitious uncertainty block belonging to the class A~. Due to this equivalence between robust stability and performance, we will generally restrict attention to robust stability in this section. Let us consider the scheme of Fig. 3(b). Every configuration including interval plants can be represented in this setup, where A is defined as before and Ap accounts for parametric perturbations and belongs to the set of perturbations DG(q) = {diag (G, . . . . .

Gq): G; • q3~'}. (10)

We can also define a reduced set of parametric perturbations as DKGs(q) = {diag(G, . . . . .

:

SU

AE]D~(r ) ( O ( A ) :

det [1 - M ( j ~ o ) A ( j w ) ] = 0}-'. The following result is an immediate consequence of the above observations and relates the structured singular values of interconnection matrices of Figs 3(c) and (d). Theorem 9. For a given configuration including q interval plants

sup I t ( M P ( j w ) ) = # ( M ' ( j t o ) ) ,

Vto,

(12)

MP • ~ I

where the structured singular values /u(.) are taken with respect to the perturbation structures DlV(r) and DYe(r; 4q), respectively. As a first observation, note that the expression on the right hand side of equation (12) involves a "mixed" complex-real /z computation. In fact, Theorem 9 says that each interval family brings to the /z problem four real uncertainties. Actually, this result can be strengthened (Dahleh et al., 1992b). To this end, consider the configuration of Fig. 3(d). By observing that the numerator of det [ I - M ' A up] is, for any fixed complex matrix A, a multilinear function of the real parameters ~.i, i = 1 , . . . , 4q, the following theorem, which is the main result of this section, can be proven.

Gq):G; • Cg~s}. (11)

Figure 3(c) shows the same scheme as that of Fig. 3(b), after closing the parametric uncertainty. Two membership classes for the interconnection matrix M e, which now depends on Ap, can be defined according to (10) and (11)

Theorem 10. The following statements are equivalent: (a) system in Fig. 3(b) is stable for all A p • Da(q) and all A e Dr(r); (b) system in Fig. 3(b) is stable for all Ap • DKas(q) and all A • D~(r).

E 1= { M p : A p • D G ( q ) } , Vt~KS = { M p :A p • D~s(q)}.

For frequency domain analysis, by using the four-parameter representation of interval plants illustrated in Section 2, the general system in Fig. 3(b) can be transformed into that of Fig. 3(d), where A "p belongs to the mixed complexreal perturbation class D vv defined as DYe(r; 4q) = {diag (A,/~1 . . . . .

/~4q) :

A • D~(r), X,• [-1, 1]}. Again, the interconnection matrix M' in Fig. 3(d) accounts for the nominal closed loop system.

Of course, Theorem 10 can be restated in terms of/a as sup ~(MP(jo)) = sup I~(MP(j~o)), M e • -4~1

Vto.

M p • ~KS

The above result states that when dealing with robust stability and performance of systems subject to structured H~ and parametric interval perturbations, interval plants can be replaced by the corresponding Kharitonov segments. Not all the possible combinations of Kharitonov segments (32q) are usually necessary to be accounted for in computing the worst case ~. For many problems "smaller" subsets are generally

Extremal properties for robust control sufficient. This reduction depends on the topology of the configuration considered. Before ending this section, we state a robust performance vertex result given in Dahleh et al. (1991a), which refers to the case when the model uncertainty can be lumped into one single interval plant, subject to one unstructured uncertainty block (see Fig. l(b)), and the controller C has the structure (9). For this special configuration, assuming that w is identified as the system output disturbance and z as the system output, we have the following theorem.

Theorem 11. A compensator C solves the robust performance problem in the feedback system of Fig. l(b) for all G • ~ if and only if it solves the problem for G • ~:. As a final comment on the technical machinery needed to obtain the results presented in this section, we observe that using the structured singular value # allows one to convert the parametric performance problems into robust stability conditions, so that the basic tools introduced in Section 3 can be usefully employed for proving Theorems 9-11.

4.2. Frequency response of interval plantcontroller families of transfer functions In this subsection some results on the locus of the Nyquist plots of an interval plant-controller family of transfer functions are illustrated. These results are presented since they provide extremal results for some of the most used performance parameters in classical control design, such as phase and gain margins, and sensitivity and complementary sensitivity function peaks. Moreover, they allow us to deduce several robust strict positive realness results and provide alternative views of many of the results stated in the previous sections. Before stating the main result of this section we will introduce new sets and maps. Consider the family of loop transfer functions

= { L : L = CG, G • c~}.

(13)

For a given w, define the map L(jto) which associates with each G • ~ the complex value C(joo)G(jw). Of course, this map is defined only if D~(jw)D(jw) 4: O. Denote by IL(W) and l~(w) the value sets in the complex plane of the families ~ and ,~s, respectively. 0IL(m) denotes the boundary of the set IL(t0). Let us assume for the moment that D~(jto)D(jto) 4=O, VD • ~ . Under these conditions, the set IL(tO) is bounded and the following theorem can be proven.

717

Theorem 12. Let to be such that D(jto)D¢(jto) 4=

O, then alL(o) _ I~'(to).

Remark 1. A self-contained proof of Theorem 12 is given in Tesi and Vicino (1991c). In Bartlett (1990a) and Fu (1990) a result similar to that of Theorem 12 can be found. More precisely, in the latter reference it is shown that the boundary of the value set, of a rational function whose coefficients are affine in a parameter vector, is determined by the edges of the uncertainty box in parameter space. Once this "edge" result is established, the fourparameter representation of an interval plant introduced in Section 2 allows one to prove and understand Theorem 12 in an alternative way.

Remark

2. Theorem 12 holds true even if the hypothesis D(jto)Dc(jto)~0 is replaced by the weaker assumption D(jo))Dc(jto)+ N(jto). N~(jto) 4=0, Vto (no pole-zero cancellation along the imaginary axis). Important conclusions can be drawn from Theorem 12. Observe that at each frequency to such that no transfer function of the open loop family Z¢ has a pole at s =jto, the image set boundary 0IL(W) is determined by the Kharitonov segment set ~ s . Hence, the locus of Nyquist plots of transfer functions of the family is bounded by the set of plots corresponding to the 32 Kharitonov segments of plants ~ s - An immediate consequence of this fact is that extremal phase and gain margins, or maximum loop sensitivity and complementary sensitivity peaks for an interval feedback system are achieved at a plant belonging to one of the 32 Kharitonov segments of the interval family. 5. ROBUST STRICT POSITIVE REALNESS AND ABSOLUTE STABILITY FOR INTERVAL PLANTCONTROLLER FAMILIES In this section we give some extremal results for robust strict positive realness (SPR) of families of interval plant-controller transfer functions (13). In particular, we consider the problem of real shifted positive realness of rational functions. By real shifted strict positive realness of a rational function L we mean SPR of the function k - l + L, with k a given real number (we will denote this property by kSPR). Robust SPR and kSPR are mainly motivated by problems of convergence/performance analysis of adaptive schemes and robust absolute stability of nonlinear systems. We briefly introduce the latter issue which is the natural extension of the classical Lur'e control problem

718

M. DAHLEH et al.

to the case where the linear plant is replaced by an interval plant. Let us consider an interval Lur'e feedback system having a stable interval plant-controller family LP in the feedforward path and a memoryless continuous nonlinear (possibly) time varying function ~p(t, y) in the feedback path, subject to the so called "sector condition", i.e. such that tp e Ok where Ok = {~: tp(t, 0) = 0 and O<--ycp(t, y) <_ky 2,

case when C is a fixed order polynomial, i.e. C(s) =-Nc(s). In this case the following theorem providing a vertex result for the kSPR problem can be proven (Dahleh et al., 1992a). Theorem 13. Given a strictly proper interval plant ~d and a fixed polynomial C(s)=-No(s), a necessary and sufficient condition for kSPR of the interval plant-controller family LP is that ~ic is kSPR.

0 < k < o % Vt eR}. The robust Lur'e control problem consists of studying stability in the large of the closed loop system for any tp e Ok. If the system is stable in the large for all tp e Ok and all L ~ ~, then the interval Lur'e system is said to be absolutely stable (AS) in the sector [0, k]. A sufficient condition for absolute stability of the interval Lur'e system is provided by the robust version of the well-known "circle" criterion (~iljak, 1969), which states that the system is AS if the family L¢ is kSPR, i.e. if for all L e L P the following inequality holds k -1 + Re [L(jto)] > 0,

Vto.

By using Theorem 12 we easily get that if the Lur'e systems corresponding to the Kharitonov segments of the interval plant satisfy the circle criterion, then the whole interval Lur'e system is AS. Several types of vertex results can be obtained for the kSPR property whenever k = ~ and/or special structures of the controller C are assumed (see Bose and Delansky (1989); Dasgupta (1987); Chapellat et al. (1991a); Vicino and Tesi (1991)). Here, we briefly summarize some of the most important of them. First, we consider the SPR property, i.e. the case k -~ = 0. Regarding SPR of an interval plant, Dasgupta (1987) and Bose and Delansky (1989) solved the problem, reaching the conclusion that ~ is sufficient for SPR of ~ . Later references (Chapellat et al. (1991a); Vicino and Tesi (1991)) have shown by different approaches that only eight out of the 16 Kharitonov plants guarantee SPR of ~ . Moreover, denoting by ~ : the set of loop transfer functions corresponding to ~ , it turns out that -~i~ is sufficient for SPR of the whole family ~ (Vicino and Tesi, 1991). For the case when k - ~d: 0, it can be shown (Chapellat et al., 1991a) that if C is a proportional controller, then kSPR of ~ : is necessary and sufficient for kSPR of 2g. If the sign of the real shift k -~ is known, then only a subset of ~ including 12 Kharitonov plants suffices for ensuring kSPR of the family (Vicino and Tesi, 1991). In what follows, we discuss in more detail the

The basic property underlying this result is that the SPR and kSPR properties are equivalent to a one parameter family of stability problems. Again, Lemma 4 and convexity of the positivity property of polynomials allows one to prove the theorem. The compensator assumed in the above theorem is important because it allows us to give a robust version of the well-known Popov criterion for absolute stability of Lur'e systems. We recall that the Popov criterion for a Lur'e system with a stable linear feedforward path G states that the system is absolutely stable in the sector [0, k], if there exists a real q verifying the following positiveness condition k -~ + Re [(1 +jtoq)G(jto)] > O,

Vto.

Theorem 13, which obviously holds for C ( s ) = 1 + sq, allows us to state a robust version of the Popov criterion for interval Lur'e systems as the following. Corollary 1. A Lur'e control system is absolutely stable in the sector [0, k] for all G e ~ if there exists a real q verifying the Popov condition for G e ¢~:. 6. REGULARITY PROPERTIES OF THE STABILITY MARGIN FOR INTERVAL FEEDBACK SYSTEMS The real stability margin (or radius) of a control system is a key concept in the parametric control field. It represents a measure of the minimum perturbation which destabilizes a system designed to be stable in nominal operating conditions. It is easily recognized that it is the inverse of supo, #(to) for a system where the perturbation structure includes real parameters only. Methods for determining this quantity for the case where plant transfer function coefficients are linear functions of physical real parameters may be found in Bhattacharyya (1987), Hinrichsen and Pritchard (1988), Tesi and Vicino (1988) and Qiu and Davison (1989), whereas de Gaston and Safonov (1988), Sideris and Pefia (1988) and Vicino et al. (1990) provide algorithms for the case of nonlinear (polynomial or rational) dependence.

Extremal properties for robust control In this concise section, we report results on regularity of the stability margin problem with respect to controller coefficient perturbations, an issue which is motivated by what follows. Suppose that the designer chooses a specific controller, which guarantees a prescribed stability margin and ensures robust stability of the closed loop system for a given uncertainty set in plant parameter space. Suppose that the controller coefficients c are slightly perturbed with respect to their nominal design value c = c °. Can the designer give any guarantees that the controller continues to achieve its objectives, i.e. robustly stabilizing the system. Engineering intuition suggests that a negative answer to this question would be considered as an indicator of a poor control design. In this perspective, the regularity assessment of a robust control problem with respect to input data, i.e. controller parameters, becomes a mandatory requirement which must be preliminarily verified before a robust control design can be performed. Recently, the possibility of existence of discontinuities of the stability margin on input data has been pointed out in Barmish et al. (1990), where some examples are reported, but no analysis is attempted. In Vicino and Tesi (1990) a technique for assessing regularity is provided for the case when linear dependent perturbations enter the plant coefficients. The results in Vicino and Tesi (1990) can be specialized for interval feedback systems, showing regularity of the stability margin under fairly weak assumptions on the compensator structure. From a design standpoint, the optimally robust controller problem can be formulated as follows. Find a controller coefficient vector c* such that

c* = arg lmax c~H p(c)}

(14)

where p(c) represent the stability margin of the system relative to a controller c and H is the domain of stability of the nominal system in the controller coefficient space. Any parameter vector c* solving the optimization problem (14) defines an optimally robust compensator C(s; c*). We notice that in general the function p(c) may have local maxima, and, even worse, it may be discontinuous (Barmish et al. (1990); Vicino and Tesi (1990)). The following theorem (Tesi and Vicino, 1991a) which is the main result of this section, shows that for the class of interval perturbations, under weak conditions, the stability margin is continuous in the controller coefficients.

Theorem 14. (i) If m = 0, i.e. the plant family is an all pole

719

family, p(c) is continuous Vc e H such that C(s;c) has no multiple purely imaginary poles. (ii) If m -> 1, p(c) is continuous Vc ~ H. A simple procedure for designing an optimally robust controller for an interval system is given in Tesi and Vicino (1991a). The procedure turns out to be effective in applications where the objective of the design is to adjust optimally only few parameters of the controller with fixed structure. 7. CONCLUSIONS

In this paper, a view of the main results on robust control of SISO systems where uncertainty models include interval plants and unstructured as well as structured H~ perturbations has been delineated. An effort has been made to show that various aspects of robustness theory can be dealt with by using common analytical tools. Recently developed connections of purely parametric robustness results with the # analysis approach to robust stability and performance of uncertain systems have been described in some detail. The general picture arising from all these specific issues shows that as far as frequency domain design is concerned, any interval plant can be replaced by a fixed number of one dimensional families of plants, irrespective of the order of the plants and of other kind of uncertainties present. Also, it turns out that for certain specific structures of perturbations and controllers the robust stability and performance problems admit vertex type results. It is clear that while in principle these results are very appealing, the restrictive conditions under which they are derived represent a severe limitation from an applicability viewpoint. REFERENCES Anderson, B. D. O., S. Dasgupta, P. Khargonekar, F. J. Kraus and M. Mansour (1990). Robust strict positive realness: characterization and construction. IEEE Trans. Circ. and Systems, 37, 869-876. Barmish, B. R. (1988). New tools for robustness analysis. Proc. 27th IEEE CDC, Austin, TX, U.S.A. pp. 1-6. Barmish, B. R. (1989). A generalization of Kharitonov's four polynomial concept for robust stability problems with linearly dependent coefficient perturbations. IEEE Trans. Aut. Control, 34, 157-165. Barmish, B. R., C. V. Holiot, F. J. Kraus and R. Tempo (1990). Extreme point results for robust stabilization of interval plants with first order compensators. Proc. ACC, San Diego, U.S.A. Barmish, B. R. and H. I. Kang (1991). Extreme point results for robust stability of interval plants: beyond first order compensators. Proc. IFAC Symp. on Design Methods for Contr. Syst., Zurich, Switzerland. Barmish, B. R., P. P. Khargonekar, Z. C. Shi and R. Tempo (1990). Robustness margin need not be a continuous function of the problem data. Systems & Control Letters, 15, 91-98.

720

M . DAHLEH et al.

Bartlett, A. C. (1990a). Nyquist, Bode, and Nichols plots of uncertain systems. Proc. ACC, San Diego, U.S.A. Bartlett, A. C. (1990b). Vertex results for the steady state analysis of uncertain systems. Proc. 29th IEEE CDC, HI U.S.A., pp. 436-441. Bartlett, A. C., C. V. Hollot and L. Huang (1988). Root locations of an entire polytope of polynomials: it suffices to check the edges. Mathematics o f Contr., Sign. and Syst., 1, 61-71. Bartlett, A. C., A. Tesi and A. Vicino (1992). Vertices and segments of interval plants are not sufficient for step response analysis. Systems & Control Letters, 19, 365-370. Ba§ar, T. and P. Bernhard (1991). H~-Optimal Control and Related Minimax Design Problems. Birkhauser, Boston. Bhattacharyya, S. P. (1987). Robust stabilization against structured perturbations. Lecture Notes in Control and Information Sciences, 99. Springer Verlag, Berlin. Bhattacharyya, S. P. and L. Keel (Eds) (1991). Robust Control. CRC Press, Boca Raton. Bose, N. K. and J. F. Delansky (1989). Boundary implications for interval positive rational functions. IEEE Trans. on Circ. and Systems, 36, 454-458. Cavallo, A., G. Celentano and G. De Maria (1991). Robust stability analysis of polynomials with linearly dependent coefficient perturbations. I E E E Trans. Aut. Control, 36, 201-205. ChapeUat, H. and S. P. Bhattacharyya (1989). A generalization of Kharitonov's Theorem: Robust stability of interval plants. IEEE Trans. Aut. Control, 34, 306-311. Chapellat, H., M. Dahleh and S. P. Bhattacharyya (1990). Robust stability under structured and unstructured perturbations. IEEE Trans. Aut. Control, 35, 1100-1108. ChapeUat, H., M. Dahleh and S. P. Bhattacharyya (1991a). On robust nonlinear stability of interval control systems. IEEE Trans. Aut. Control, 36, 59-67. Chapellat, H., M. Dahleh and S. P. Bhattacharyya (1991b). Extremal manifolds in robust stability. Report UCSB-ME91-3. Chen, J., M. K. H. Fan and C. N. Nett (1992). The structured singular value and stability of uncertain polynomials: a missing link. A S M E Winter Annual Meeting, Atlanta U.S.A. Dahleh, M. A. and J. B. Pearson (1987). Ll-optimai controllers for MIMO discrete-time systems. IEEE Trans. Aut. Control, 32, 314-323. Dahleh, M., A. Tesi and A. Vicino (1991a). Robust stability and performance of interval plants. Systems & Control Letters 19, 353-363, Proc. 30th IEEE CDC, Brighton, U.K., pp. 435-436. Dahleh, M., A. Tesi and A. Vicino (1992a). On the robust Popov criterion for interval Lur'e systems. Proc. 31st IEEE CDC, Tucson, U.S.A. pp. 2808-2809. Dahleh, M., A. Tesi and A. Vicino (1992b). Robust stability/performance of interconnected interval plants with structured norm bounded perturbations. Proc. 31st IEEE CDC, Tucson, U.S.A. pp. 3169-3174. Dasgupta, S. (1987). A. Kharitonov like theorem for systems under nonlinear passive feedback. Proc. 26th IEEE CDC, Los angeles, U.S.A., pp. 2062-2063. Dasgupta, S. (1988). Kharitonov's theorem, revisted. Systems & Control Letters, 11, 381-384. Dasgupta, S., P. J. Parker, B. D. O. Anderson, F. J. Kraus and M. Mansour (1991). Frequency domain conditions for the robust stability of linear and nonlinear dynamical systems. IEEE Trans. on Circ. and Systems, 38, 389-397. de (3aston, R. R. E. and M. G. Safonov (1988). Exact calculation of the multiloop stability margin. IEEE Trans. Aut. Control, 33, 156-171. Dorato, P. (Ed.) (1987). Robust Control. IEEE Press, NY. Dorato, P. and R. K. Yedavaili (Eds) (1990). Recent Advances in Robust Control. IEEE Press, NY. Doyle, J. C. (1982). Analysis of feedback systems with structured uncertainties, l E E Proc., Part D, 129, 242-250. Doyle, J. C., K. GIover, P. P. Khargonekar and B. A. Francis (1989). State-space solutions to standard He and H~ problems. IEEE Trans. Aut. Control, 34, 831-847. Doyle, J. C. and G. Stein (1981). Multivariable feedback design: concepts for a classical/modern synthesis. IEEE

Trans. Aut. Control, 2,6, 4-16. Doyle, J, C., J. E. Wall and G. Stein (1982). Performance and robustness analysis for structured uncertainty. Proc. 21st IEEE CDC, Orlando, U. S. A., pp. 629-636. Fan, M. K. H. and A. L. Tits (1986). Characterization and efficient computation of the structured singular value. IEEE Trans. Aut. Control, 31, 734-743. Francis, B. (1987). A course in H~ control theory. Lecture Notes in Control and Information Sciences. SpringerVerlag, Berlin. Fu, M. (1990). Computing the frequency response of linear systems with parametric perturbation. Systems & Control Letters, 15, 45-52. Ghosh, B. K. (1985). Some new results on the simultaneous stabilization of a family of single input, single output systems. System & Control Letters, 6, 39-45. Hinrichsen, D. and A. J. Pritchard (1988). New robustness results for linear systems under real perturbations. Proc. 27th 1EEE CDC, Austin, TX, U.S.A. pp. 1375-1379. Hinrichsen, D. and B. Martensson (Eds) (1990). Control o f Uncertain Systems. Birkhauser, Boston. Hollot, C. V. and F. Yang (1989). Robust stabilization of interval plants using lead or lag compensators. Systems & Control Letters, 14, 9-12. Kharitonov, V. L. (1978). Asymptotic stability of an equilibrium position of a family of systems of linear differential equations. Differential'nye Uraveniya, 14, 1483-1485. Kwakernaak, H. (1991). The polynomial approach to H~-optimal regulation. Lecture Notes 1990 CIME Course on Recent Developments in H~ Control Theory. Springer Verlag, Berlin. McFarlane, D. C. and K. Glover (1990). Robust controller design using normalized coprime factor plant descriptions. Lecture Notes in Control and Information Sciences. Springer Verlag, Berlin. Milanese, M., R. Tempo and A. Vicino (Eds) (1989). Robustness in Identification and Control. Plenum Press, NY. Mygkis (1975). Advanced Mathematics for Engineers. Mir, Moscow. Packard, A. (1988). What's new with #: Structured uncertainty in muitivariable control. Ph.D. Dissertation, UC Berkeley, U.S.A. Polis, M. P., W. OIbrot and M. Fu (1989). An overview of recent results on the parametric approach to robust stability. Proc. 28th IEEE CDC, Tampa, U.S.A., pp. 23-29. Qiu, L. and E. J. Davison (1989). A simple procedure for the exact stability robustness computation of polynomials with affine coeffcient perturbations. Systems & Control Letters, 13, 413-420. Rantzer, A. (1992). Stability conditions for polytopes of polynomials. IEEE Trans. Aut. Control, 37, 79-89. Sideris, A. and R. S. S. Pefia (1988). Fast computation of the multivariable stability margin for real interrelated uncertain parameters. Proc. ACC, Atlanta, U.S.A. Siljak, D. D. (1969). Nonlinear Systems: The Parameter Analysis and Design, Wiley, New York. Siljak, D. D. (1989). Parameter space methods for robust control design: A guided tour. IEEE Trans. Aut. Control, 34, 674-688. Tesi, A. and A. Vicino (1988). Robustness analysis for uncertain dynamical systems with structured perturbations. Proc. 27th IEEE CDC, Austin, TX, U.S.A. pp. 519-525; IEEE Trans. Aut. Control, 35, 186-191. Tesi, A. and A. Vicino (1991a). Design of optimally robust controllers with few degrees of freedom. Proc. IFAC Syrup. on Design Methods o f Control Systems, Zurich, Switzerland. Tesi, A. and A. Vicino (1991b). Robust absolute stability of Lur'e control systems in parameter space. Automatica, 27, 147-151. Tesi, A. and A. Vicino (1991c). Kharitonov segments suffice for frequency response analysis of interval plant-controller families. Proc. Int. Workshop on Robust Control, San Antonio, U.S.A. pp. 403-415. CRC Press, Boca Raton. Tesi, A., G. Zappa and A. Vicino (1991). Analysis and filter

Extremal properties for robust control design for sirict positive realness of families of polynomials. Proc. of the 30th IEEE CDC, Brighton, U.K., pp. 31-36. Vicino, A. and A. Tesi (1990). Regularity conditions for robust stability problems with linearly structured perturbations. Proc. 29th IEEE CDC, Honolulu, U.S.A. pp. 46-51. Vicino, A. and A. Tesi (1991). Robust strict positive realness: new results for interval plant plus controller families. Proc. of the 30th IEEE CDC, Brighton, U.K. pp. 421-426. Vicino, A., A. Tesi and M. Milanese (1990). Computation of nonconservative stability perturbation bounds for systems with nonlinearly correlated uncertainties. IEEE Tram. Aut. Control, 35, 835-841.

APPENDIX

Proof of Lemma 3. Let us assume, by contradiction, that there exists an unstable polynomial belonging to the segment flax. Hence, there must exist at least two pairs (3.~', w~'), (3.3, to~) such that P(joJ*, 3.*) = P~o(w*) + jP~o(W*) + 3.*e(jw*) h xP'(jo*)P~(-w?2)=O,

to?@o,

i=1,2.

(15)

721

can be easily verified that Condition (2) implies

,r~'(to*) P:;'(to*)l

l+t0 L ~

Po°(o*)J <0"

(16)

Finally, from (16) and Conditions (2) and (3), we get f'(to*) >0, and, hence, since f(to) is a smooth function, it cannot have more than one zero V3. e [0, 1]. This completes the proof.

Proof of Theorem 7. Since by Theorem 2 we know that Kharitonov segments suffice for determining robust stability of the entire family, we need only show that the maximal H® norm of the closed loop transfer function W = GC(1 + GC) -l along any Kharitonov segment of %:s is attained at one of the two Kharitonov vertex plants corresponding to its endpoints. To this purpose, consider one segment defined by N,(s) G(s; 3.) = Ds(s ) + 3.[D2(s) _ D3(s)l,

3. ¢ [0, 1].

(Similar arguments hold for all the Kharitonov segments of plants.) We have to show the implication max (llW0(s)ll®, IlWl(s)ll®) < 1/E ~llW(s;3.)ll®
V3.¢[0, 1],

(17)

where

Let P'(s) be given by

W(s; 3.) = G(s; 3.)C(s)(1 + G(s; 3.)(C(s))-',

e'(s)=(/J+ffs),

o6~ER,

0¢~>0,

and let us assume that h is even (a similar proof is obtained when h is odd). Consider the following function f(w) - U(to)[P~,(to)a'to - Po°(to)#l, where U(to)-o(jw)hP~(--W2). Notice that, since Po(s) is Hunvitz, except for trivial degeneracies, U ( w ) ~ 0 , Vw. It can be easily checked that there exists a pair (3.*, w*) satisfying (15) only if w* satisfies the following conditions (1) f(to*) = O,

(2) to*Wo(to')P°(w*) >0,

~ . , N,(s) / .

Wo(,)= ,.ts~~

N,(s)

U +~

_., ~N,(s)(1+ w,(~) = ctsj +

-'

c(s)) ,

b_~ N,(~) c(s))

-'

"

By the hypotheses on the left hand side of (17) and Lemma 2, the following two polynomials are stable for any fixed real

y ~ [0, 2~r] Po(S) = N,(s)Nc(s)[1 - Ee/V] + Oc(s)D3(s ), P,(s) = N,(s)N¢(s)[1 - Ee3v] + D¢(s)D2(s ). Consider now the segment of polynomials

(3) otto*u(to*)Po(to*) < 0.

Now, consider the derivative of the function f(to) with respect to to, i.e.

i f ( w ) - U(to)[Peo(to)ot + P~,'(to)otl~o(to)ot- Po°'(~)#] + u ' ( w ) [ ~ ( w ) o ~ t o - po°(W)p].

From Condition (1), f'(to) evaluated at to = to* is given by

{

f'(to*) = otU(to*)P~(to*) 1 +

°* ) e',',,,'(to*)]/ to,FPo(to t ~ p o(to,), j j .

By the interlacing property of the stable polynomial Po(s), it

AUTO 29:3-J

Px(s) = Po(s) + 3.[P,(s) - Po(s)],

3. e [0, 11.

By Lemma 4, since Nc(s ) and Dc(S ) have the structure (9), Px(s) is stable V3. ¢ [0, 1]. This implies that Px(jw) = Nt(jw)N~(jw)(I - Ee/y) + D~(jto)D3(jto)

+3.Dc(jto)[D2(jto)-D3(jw)]*O, Wo, We[0,1). (18) Since (18) holds for any y e [0, 2~], the second statement of (6) in Lemma 2 is satisfied for the transfer function W(s; 3.), V3. ¢ [0, 1]. Since the first requirement of (6) can be trivally verified for W(s; 3.), the theorem follows from I.emma 2.