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OPTIMIZATION OF HIGHLY UNCERTAIN FEEDBACK SYSTEMS IN Hm
Mohamed Seddik Djouadi
Department of Electrical & Cornputer Engineering McG ill University, Mont réal
July 1998
A Thesis submitted to the Faculty of Graduate Studies and Research in partial fulfilment of tbc requirements for the degree of Doctor of Philosophy National Library BiMiothéque nationale 1*1 of Canada du Canada Acquisitions and Acquisitions et Bibliographie Services senrices bibliographiques 395 Wellington Street 3%. me Wellington OttawaON K1AW CMawaON KlA ONI Canada Canada
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The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom ît Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. To my beloved parents, my brothers Sofiane, Fares, WaiI and Mehdi. my dear wife, and to the memol of Professor George Zames. Abstract
This dissertation presents a new mathematical frarnew~rkto optimize performance of mrilti- input mult i-output feedback systems subject to large dynarnical uncertainty. Optimal per- formance is represented by two-disc type optimization problems defined in appropriate function spaces. These optimizations are interpreted as shortest distance minimizations in special vcctor valued MI spaces. Charaterization of various predual and dual maximiza- tions shows existence of optimal solutions. Alignment conditions are exploited to show that the optimal solution is flat or "allpass". therefore generalizing a result obtained previously for single-input single-output systems. Detailed analysis gave further quantitative results, in particular absolute continuity of extrema1 measures is proven. This lead to extrema1 identities which provide a test of optimality. A novel operator theoretic framework is next developed. Key multiplication operators acting or1 particular vector-valued Hardy spaces are introduced. Subsequently, the optimizations arc shown to be equal to the induced norms of specific operators. The latter are Banach spacc projections of multiplication operators, therefore analogous to the Sarason operator wcll known in the standard Hoc theory Further computations show that these operators are in fact combinations of multipIication and Toepltiz operators. Explicit formulas for the optimal controllers are provided through existence of maximal vectors. Then "infinite ma- tri? reprcsentation with respect to a canonical basis is given, and the norms of the relevant opcrators are approximated by special matrix norms. These results are further generalized to unstable systems using coprime factorization tcch- niques with similar conclusions. Relation to the standard two-block Hoc problem is inves- tigated in the context of duality and operator theory. The optimal solution is then shown to bc flat. implying that a well known Hankel-Toeplitz operator achieves its norm on its discrcte spcctrum for (possibly) infinite dimensional systems. Finally. the optimal robust disturbance attenuation problem for continuous time-varying plants subject to continuous time-varying uncertainty. is shown to reduce to finding the smallest &cd point of a two-disc type optimization problem under continuous time-varying control laws. Duality is then applied in the context of nest algebra of causal stable systems, and shows existence of optimal continuous time-varying controilers. It is also shown that for tirneinvariant nominal plants under time-varying uncertainty, continuous time-varying control laws offer no advantage over time-invariant ones. Résumé
Cette dissertation présente une nouvelle approche mathématique dans le but d'optimiser les performances des systèmes mutli-entrées multi-sorties sujets à de large incertitudes dy- namiques. Ces performances optimales sont représentées par des problèmes d'optimisation du type "deux-disques" posés dans des espaces fonctionnelies appropriés. Ces derniers sont interpretés comme problèmes de minimization des distances les plus courtes dans des espaces HF particuliers de fonctions vectorieiles. La charactérisation de maximums dans les espaces dual et pré-duai montre l'existence de solutions optimales. Des conditions d'alignement sont exploitées pour montrer que ces solutions ont la propiété d'être "passe- tout" et présque "passe-tout", et par conséquent généralisant le même résultat obtenu lors de travaux antérieurs pour les systèmes une seule-entrée une seule-sortie. Une analyse détaillée nous permet d'obtenir plus de résultats qualitatives. en particulier, la continuité absolue de mesures extrémales, qui impliquent l'existence d'itentités extrémales pouvant servir de tests d'optimalité. Urie nouvelle approche théorique opératorielle est ensuite developpée. Des opérateurs de rniiltiplications définis sur des espaces vectoriels de Hardy sont introduits. Par la suite, on montre que les critéres d'optimization considérés sont en fait égaux aux normes induites d'opérateurs spécifiques. Ces derniers sont des compositions de projection d'espaces de Ba- nach et d'opérateurs de multiplication. D'autre calculs montrent que ces opérateurs sont des combinaisons d'opérateurs de Toeplitz et de multiplication. Des formules explicites pour les contrôleurs optimales sont alors déduites. En plus, une représentation par rapport A une base canonique ainsi que des approximations par des normes de matrices sont données. Ces résultats sont ultérieurement généralisés aux systèmes instables en utilisant des téchniques de factorisation coprimes, avec des conclusions similaires. La relation avec le problème à deux-blocs HS est étudiée dans le contexte de la dualité et la t.liéorie des opérateurs. On montre que la solution optimale est aussi "passe-tout" impli- quant qu'un opératuer de Hankel-Toeplitz pour les systémes (possiblement) de dimension infinie attaint sa norme sur son spectre discret. Finalement. on démontre que le problème de rejection optimale et robuste des perturbations pour les sytèmes continus variant dans le temps sujets a des incertitudes dynamiques con- tinues variant aussi dans le temps. se réduit à trouver le plus petit point fixe d'un problème d'optimisation "deux-disques". La dualité est appliquée dans le contexte d'algébre triangu- laire des systèmes causales et stables. On démontre ainsi I'existence de contrôleurs optimales variant dans Ie temps. On montre aussi que pour les systèmes nominaies invariants dans le ternps avec des incertitudes dynamiques continues variant dans le temps ne sont pas meilleurs que ceux invariant dans le temps. Acknowledgements
1 would like to express my deepest gratitude and sincere appreciations to my supervisor Profcssor George Zames. This thesis would have never been possible without his generous financial support, deep insights, continuous guidance and encouragement, rigour and high standards. It was indeed an honour to be his student and Iearn from him in many aspects. 1 ani indebted to Professor Charalambos Charalambous for his enthusiastic guidance, depth of tliought and feedback in completing the thesis after Professor Zames' passing. 1 would also like to thank Professor Peter Caines for communicating to us his enthusiasm in doing research, and who with Professor Charalambous rnanaged Prof. Zames' affairs after liis passing. 1 would also like to express my deep gratitude to Professor Robert Vermes fkom the Math- emat ics and S tat istics Department of McGill University, for many academic and social discussions during rny studies. 1 am also indebted to Professor Le Yi Wang for arranging a one year visit to the Depart- ment of Electrical and Cornputer Engineering of Wayne State University, for his generous financial support and helpful and lively discussions. 1 wouId Iike to extend my appreciation to Professor Bertram Schreiber of the Department of Mathematics of Wayne State University, for introducing me to tensor products and for his generosity in spending his time on detailed technical discussions whenever 1 needed.
1 would like to thank past and present graduate students at the Center for Intelligent Machines (CIM) for many enlightening and lively discussions, in particular. Mojtaba Ah- rriadi, Farhad Aghili, Kaouthar Benameur, Shen Gang, Sylvain Gendron, Paul Hubbard, Lin Lin, James Owen. Fazal Ur Rahman? Yuanjun Wei! and my officemates Shuping Qing and Hamid Taghirad. .4CK,WOWLEDGEhf ESTS
Finally. 1 would like to express my sincere gratitude to my dear wife, for her understanding, patience. and her continuous and loving support through the years of my studies. Claims of Originality
Solut ions to the single-input single-output (SISO), and the multi-input multi-output (MIMO) versions of the two-disc problem. as well as the MIMO extension of the optimal robust disturbance attenuation problem (ORDAP). are characterized using Banach space duality theory. Dual and predual descriptions of these problems are derived. Duality theory is used to prove a "flatness" and approximate "flatness" result for the .MM0 extension of ORDAP. Extrema1 measures in the dual for the SISO and MIMO versions of the twu-disc problem, as well as the MIMO extension of ORDAP, are shown to be absohtely continuous w.r.t the Lebesgue measure. A consequence of this result: is that extremal identities which provide a test of optimality are derived. It is also shown that optimal solutions are dual extremal functions for apprcximation problems in predual spaces.
a New operator theoretic Erameworks for the SISO, and MIMO versions of the two-disc problem. as weU as the MIMO extension of ORDAP. are deveIoped. Key multipli- cation operators are introduced, together with special vector vaIued Hardy spaces. It is shown that the induced norms of these multiplication operators coincide with the norm involved in the optimizations. a Optimal performances for these problems are shown to be equal to operator induced norms. The operators are computed explicitly, and are shown to be a combination of multiplication and Toeplitz operators. a Under continuity assumption, existence of maximal vectors is proven, leading to an explicit formula for the optimal controller. a An infinite matrix representation with respect to a canonical basis is given: together with general formulas and a recursive algorithm to compute the columns. Approxi- mation of the operator norm by particular norms of matrices is also provided. a ORDAP for unstable systerns subject to unstructured uncertainty is shown to reduce to a &cd point problem. Banach space duaiity charactrizing the solutions. and their attendant properties is given. An operator theoretic solution is also described. a Relation to the standard two-block Hm problem is explored. Duality is applied to characterize the solutions, and to show a "flatness" and an approximate .'flatness" result. -4 test of optimdity is given. Finaiiy a Harikel-Toeplitz operator (quantifying optimal performance) for (possibly) infinite dimensional systems is shown to achieve its norm on the discrete spectrum. The ORDAP for continuous time-varying plants and feedback is reduced to a two- disc type problem- Duality is applied to show existence of an optimal continuous time-varying feedback control Iaw. It is also shown that for nominal time-invariant plants subject to continuous-time varying dynarnical uncertainty, continuous time- varying control laws offer no advantage over time-invariant control ones. TABLE OF CONTEXTS
TABLE OF CONTENTS
Résumé ......
vii
Claims of Originality ......
LIST OF FIGURES ...... xiv
Xotation and iMat hematical Preliminaries ......
CHAPTER 1. Introduction ...... - ......
1.1. Overview ...... 1 Problems Description ...... 1.2.1. The Two-Disc problem ...... 1.2.2. The Optimal Robust Disturbance Attenuation Problem (ORDAP) . . . 1-2.3. An Optimal Robust Model-Matching Problem ......
1.3. Thesis Organization and Outline of Results ...... - 1.4. Related Literature Review ......
CH-WTER 2. Duality Structure of ORDAP ...... 2.1. An Equivalent Distance Problem in MIOF (c,,,)...... 2.2. Banach Space Duality Structure and Existence of Optimal Solutions . . . . 2.3. Absolute Continuity and Existence of an Extrema1 Identity for the Optimum 2.3.1. Alignment in the Dual ......
2.3.2. Dual Extremal Property and Extremal Identity for the Optimum , . . . TABLE OF CONTEXTS
2.4. Extension to Strictly Proper Plants ...... 27 2.5. MIMO Extension of ORDAP ...... 29 2.5.1. Duality Structure of MIMO Extension of ORDAP ...... 29 2.5.2. Allpass Property of the Optimum: Alignment in the Dual and Extremal Identity for the Optimum ...... 31 2.6. Properties of the Optimal Behavior of the ORDAP ...... 36
CHAPTER 3. Operator Theoretic Sotution to ORDAP ...... 40 3.1. -4 Key Multiplication Operator ...... 40 3.2. Exact Operator Theoretic Solution ...... 47 3.2.1. Optimal Performance ...... 47 3.2.2. Optimal Controller ...... 56 3.3. Computation of the Operator Norm ...... 57 3.4. Approximation by Norms of Finite Rank Operators ...... 65 3.5. Operator Theoretic Solution to the MIMO Extension of ORDAP ...... 69 3.5.1. Another Key Multiplication Operator ...... 69 3.5.2. Exact Operator Theoretic Solution for the ORDAP of MIMO Systems . 73
CHAPTER 4 . ORDAP for Unstable Systems ...... 77 4.1. Introduction ...... 77 4.2. Problem Formulation ...... 78 4.3. BanachSpaceDualityStructure ...... 83 4.3.1. E.xistence of Optimal Solutions ...... 83 4.3.2. -4llpass Property, Absolute Continuity and Existence of an ExtremaI Identity ...... 85 4.4. Operator Theoretic Solution ...... 89
CHAPTER 5 . Relation to the Standard Two-Block HbDProblem ...... 91 5.1. Duality Structure of the Problem ...... 92 5.2. Allpass Property and Alignment in the Dual ...... 93 5.3. Optimal Solution via Operator Theory ...... 98
CHAPTER 6 . ORDAP for Continuous Timevarying Systems ...... 102
xii TABLE OF CONTENTS
Definitions and Notation ...... 102 6.1. Introduction ...... 103 6.2. Problem Formulation ...... 103 6.3. Duality Structure and Existence of an Optimal Solution ...... 108 6.1. Time-Varying versus Time-Invariant Control laws ...... 112
CHAPTER 7 . Conclusions ...... 115 7.1. Synopsis ...... 115 7.2 Directions for Future Research ...... 116
... Xlll LIST OF FIGC'RES
LIST OF FIGURES
Feedback Control in Presence of two Sources of Disturbances . . . . . 6
Feedback Control in Presence of Plant and Dist urbance Uncertaiuty 7
Two-Degree of Freedom Feedback Control Scheme ...... 8 Notation and Mat hemat ical Preliminaries
9- @ denotc the fields of real and complex numbers respectively. If x E @ theu rE Cenotes the cornplex conjugate of x. MIMO and SIS0 are acronyms for multi-input multi-output and single-input single output respect ively. w.r.t. is the abbereviation used for "with respect to". The symboi means "by definition". I stands for the identity map. " denotes the complex conjugate transpose of a vector or the adjoint of an operator. de- pending on the context. while denotes the transpose. 1.i.m stands for limit in the quadratic mean. < - . - > denotes either the inner or duality product depending on the context. If B is a Banach space then B' denotes its dual space. For an n-vector < E &. where @, denotes the n- dimensional complex space, I(-f is the Euclidean norm. c,,is the space of n x n matrices A, where IA[ is the largest singular value of A. Gy?. G, and C2, denote the complex Banach space of 2n-vectors C7 C = (~1): Cl,
(? E wit h respect ively, t hc norms Clearly. q, is the dual space of qnaand vise-versa. C2nxn- GnxnqnFn. CZnxn and q;En denote the complex Banach space of 2n xn matrices
4. -4 = ( ) , Ai Ao E Ci n7 rith rnpectirely the foilowing norms
where STr(Ai) = T~~C~(AIA~)$= x:=l aj(Ai)?is known as the trace-class norm. and a,(A,) are the singular values of Ai- II II,,, is the nuclear norm. - &, ,, is the dual space of ",UX, [64, 521, and vise-versa since GnX, and -,UX, are finite dimensional and t herefore reflexive.
3 denotes the unit disc of the complex plane, D = {z E @ : lzl < 1). Ll9 denotes the boundary of 9,X) = {z E C : [zl = 1). If E is a subset of mythen ECdenotes the complement of E in m. m denotes the normalized Lebesgue measure on the unit circle LPDy .m(ûQ)= 1.
rn iL.C. is the label used for "Lebesguc almost everywhere". For a rnatrix or vector-valued function F on the unit circle? IF1 is the real-valued function defincd on the unit circle by ~~l(e'')= ~~(e")l,0 E [O: 2a). If X denotes a finite dimensional complex Banach space, U(X)?1 5 p 5 oc? stands for the Lebesgue-Bochner space of weakly measurable, (see [41]for the definition), and p-th power absolutely integrable X-valued functions on under the norm
Il- ess sup 11 f (eiu)ll,~, for p = m (O. 10) e~p.2~) wiiere j E -P (X):and 11 - Jix- denotes the norm on X. The standard facts about the theory of Fourier series [42] have been carried over to the vector valued case [48, 601. If / E L'(x)? (!LL(X))contains al1 U(X)spaces for p > 1). the k-th Fourier coefficient is defined by fk jm f (~)z-~dm. (X).1 5 p < x. is the Hardy space of X-valued anal-vtic functions on the unit disc Dt viewed as a closed subspace of V (X). In fact these spaces can be realized as
@(X)={f €ILP:fk=o if k<0} (O.11)
The scalar version? Le., X = @ of the flP and W, 1 5 p 5 oo, will be denoted by LP and HP respectively. For X = C,, or G,,, the spaces ILP (X) and W (X), will be denotes by P(X)and HP(X) respect ively. In particular H~(X)(resp. ?&(X)) denote the subspace of Hp(X) (resp. W (X)) given by {F E HP(X) (resp. W(X)) : J:" ~(e")dm= O). While H(x)P (resp. P(X))are the spaces obtained by taking compiex conjugates of al1 functions in W (X) (resp. HP(X)). C(X)denotes the space of continuous functions on the unit circle with values in X. under the sup norm (0.10). Xn inner function is an analytic function f in the unit disc. such that 1 f (z)l < 1 and 1 f (ci')[ = 1 almost everywhere. An outer function is an analytic function F in the unit disc of the form
wherc IL is a rcal valued integrable function on the unit circle and X is a complex number of niodulus 1 [42]. In HO"? an outer function is stable and minimum phase (has no zeros in 14 < 1)- A niatrix G is inner if it is analytic and ~*(e'"~(e") = I almost everywhere, where 1 is NOTATION rLXD MATHEM-\TICAL PRELIMKNARIES the identity matrix. A matrix F is outer if F(z) is fidl rank for al1 lzl < 1 [70]. A matrix G is co-inner if its tranpose is inner. A matrïx F is ceouter if its transpose is outer. CHAPTER 1
Introduction
1.1. Overview
The design of control systems is usually based on mathematical models which are in general rough approximation of real physical plants. this in twn gives rise to uncertainty in the mathematical description. In this thesis we are concerned with the problem of controlling uncertain dynamical systems via feedback. There are two, and only two reasons for using feedback. The 6rst is to reduce the effect of any unmeasured disturbances acting on thsystem. The second is to reduce the effect of any uncertainty about the system dynamics. The approach we take here is the Hm control theory initiated by Zames in his scminal paper [77]. The reason is that HOC theory facilitates to pose the problem of feedback design in the frequency domain in a rigorous mathematical framework, i-e., a well posed optimization problem in appropriate function spaces. In its origin it was coricerned with two fundamental problems namely, Uncertain Disturbance Attenuation and siniultaricous Plant and Disturbance Uncertainty Attenuation. The first problem received a vast attention in the control community for more than a decade. It is now well understood and known as the standard Hm problem (see for e.g.. [31, 321 and references therein). The second problem lies at the heart of robust control theory. it deals with unstructured plant uncertainty casily described in terms of frequency responses but virtually intractable in a state-space framework. Specificall_v, it includes the Two-Disc Problem [20, 521, in which there are uncertain disturbances at the plant output and the controlier input: the Optimal Robust Disturbance Attenvation Problem [??, 4, 30, 561, where an uncertain plant 1.1.2 PROBLESIS DESCRIPTIOX is subject to disturbances at the output; and an Optimal Robust Model-Matcking Problem [77, 521, where an uncertain plant is required to foiiow a nominal model. in the next section we describe these problems in a somewhat greater details.
In view of the isometric isomorphism between HOC on the unit disc and Hoc on the right-haLf plane we will confine ourselves to the unit disc. The resuits obtained apply to both discrete and continuous time systems.
1.2. Problems Description
1.2.1. The Two-Disc problem. The two-disc problem is concerned with distur- bance rejection in a feedback system subject to a pair of uncertain disturbances representing, for example. output and sensor noises, without the simplifying assumption that these are miitually orthogonal [20, 551 (see Figure 1.1).
output disturbacc / filter known plant \
FIGCRE1.1. Fcedback Control in Presence of two Sources of Disturbances
The plant Po is assumed to be known, and the external disturbances dl and d2 are uncertain. t hat is. assumed to lie in weighted balls in H2(@, ): more precisely
wlierc LV and V are outer functions in HoC(GXn). The objective is to find a feedback control law C which stabilizes Po and optimally suppresses the effect of di and d2 on the system output y. This problem is equivalent to [20, 521 1.1.2 PROBLEMS DESCRIPTION
1.2.2. The Optimal Robust Disturbance Attenuation Problem (ORDAP). In ORDAP a stable uncertain plant P. lying for example in the uncertainty set %(Po,V) below, is subject to disturbances at the output (see Figure 1.2)
FIGURE1.2. Feedback Control in Presence of Plant and Disturbance Encertainty
The objective is to find a feedback control law which provides the best uniform attenuation of uncertain output disturbances in spite of uncertainty in the plant mode1 [77,521. A more detailed description will be given in Chapter 2.
1.2.3. An Optimal Robust ModeI-Matching Problem. The t hird problem we are intcrested in is to what extent feedback reduces uncertainty? It has been shown in [77] that the optimal weighted sensitivity is a monotone increasing fuction of the open-loop uncertainty Le.. increasing uncertainty reduces the ability of feedback to further attenuate disturbances. A dual question is. given a certain level of open-loop uncertainty. what is the best achievable uncertainty reduction if a feedback control is to be used? More prccisely. if a plant P lies in a bal1 of uncertainty about some nominal value Po, that is
what is the smallest radius of any closed-loop uncertainty that can be achieved by a single feedback control law? To avoid the trivia1 answer to this question that zero closed-loop uncertainty can be achieved by disconnecting the system from the input (sec Figure 1.3): it is suggested in [77],(see also [52]).ta adopt the invariant plant scheme. Le., the nominal ciosed-loop system is normalized 1.1.2 PROBLEMS DESCRLPTIOX
FIGURE1.3. Two-Degree of Freedom Feedback Control Scheme
to be equal to the open-loop system Po. Rom Figure 1.3
The closed-loop map K belongs to HLIO(Gxn)and can be expressed as 1771
K - Po = (1+ PoQ)(I+ APQ)-'AP : where Q = C(I+ P,c)-'
= AmPo (1-5) where AP = P - Po.and A, E HW(Gxn) represents the closed-loop multiplicative plant uncertainty. Expression (1.5) shows that our problem is a particular robust model-matching problem. whcre the mode1 to match is the nominal plant Po. Following f77,521. the potential of feedback to reduce plant uncertainty can now be for- nially stated as the minimizat ion of the "worst-case" WI-weighted closed-loop mult iplica- tive uncertainty radius suppag(po,r-)IIWIA,llm over ail robustly stabilizing nominal plant invariant control laws. Mathematically, this problem is trarislated into the following mini- niization
It was shown in (Lemma 2.2 [52]) that expression (1.6) is equal to We will see in Chapter 2 that expression (1.7) is a particular version of ORDAP. as is expression (1.2) [52] in the SISO case. Therefore, the acronym ORDAP will be used to denote al1 the three problems described above. It should also be noted that ORDAP includes problems involving questions of weil posedness of feedback system 1681-
1.3. Thesis Organization and Outline of Results
It is weU known that ORDAP can be reduced to a parametric version of the two- disc problem [4, 301. where a standard two-block Hm approximate solution was proposed. Zames and Owen showed that these approximations can be infinitely poor. and therefore they stressed the need for an exact theory for such problems. In this thesis we give exact explicit solutions based on duality and Banach space operator theory. The objective is to provide an exact mathematical kamework to optimize performance of dynamical systems subject to uncertainty that may be large. In particular. by analyzing wel posed optimiza- tions such as (1.2) and (1.7). we are able to quantiS. exactly in terms of operator theov the ability of feedback to reduce uncertainty. in Chapter 2 we consider two optirnization problems for MIMO systems- The first is the hlIMO version of the two-disc problem (1.2). the second is the MIMO extension of ORDAP whicri includes the optimal robust mode1 matching problem discussed in the previous sec- tion. 111 the SISO case these two problems are mathematically equivalent [52]. Then we give a slightly different duality structure from the one developed in [79, 521. which leads to an exact operator theoretic solution. The main difference is that we work in P(G,,,) iristcad of LOC(~,,,), and therefore we are able to exploit useful properties which corn- pIcx analytic functions enjoy. First. it is recognized that the optimal performance index ( 1.2) can be expressed as the shortest distance between a vector function ( a ) and a rubspace D = ( ) in rame special norm ( V and W are weiglitings and U is the inner part of the nominal plant). In particular. we characterize the predual and tlie dual space of respectively Iti- (c,,,) and its restriction to the continuous functionso as well as the preannibilator and annihilator of S. The shortest distance is then related to the supremum of an extrema1 functiooai in both cases. the first in a specific IL1-space. 1.1.3 THESIS ORG.L\4nZ.4TION .Ili OUTLLVE OF RESLXTS
and the second is the space of complex measures where the supremum is actually achieved. It is then showen that under quite general conditions the extremal measure is absolutely coutinuous with respect to the Lebesgue measure. A consequence of this result is that the optimal performance/controller can be interpreted as a dual extremal function for an optimization problem in an !Li-space, and satisfies an extremal identity which provides a test of optimality. The search in the convex programming algorithm proposed in [52] can therefore be Iimited to functions which satisfy certain norm constraints, resulting in faster corivergence and less memory usage. Similar results are derived for the MIMO extension of ORDXP. It is shown that a feedback which optirnally reduces the uncertainty arising Gom out put dist urbances and plant uncertainty exists. and satisfies an extremal identity. The weighted sensitivity is then shown to be flat or "alipass", and if the feedback is "almost" optimal an approximate "allpass" condition holds. In Ctiapter 3 we develop a new operator theoretic framework to characterize explicitly the solutions to ORDAP. First? key muitiplication operators acting on particular vector-valued Hardy spaces are introduced. We show that their induced norms have the merit to coincide with the norms of the re1evant weighted sensitivity functions for ORDAP and its MIMO extension. Next. we prove that optimal performance for ORDAP is equal to the induced norm of an operator which is analogous to the Sarason operator [63]?which solves the stan- dard one block HW problem 1331. Under continuity assumption of the problem data. this operator not only quantifies optimal performance for feedback systerns, but also provides an explicit formula for the optima1 controller t hrough existence of maximal vectors. The solution we propose depends on an interplay between function theory' Banach space duality and operator theory. The operator in question is a Banach space projection of a multipli- cation operator. The projection is then explicitly computed showing that our operator is in fact a combination of multiplication and Toeplitz operators. This leads naturally to an "infinite matrix" representation of the operator with respect to a canonical basis. General forniulaç and a recursive algorithm are then proposed to compute the columns of the in- finitc matrix. Moreover, it is shown that the norm of the operator can bc approximatcd bÿ the norms of finite rank operators (i-e.. matrices) obtained essentially by truncating the original infinite matrix representation- The latter provides "ncarly3 optimal solutions, and accuracy increases with dimension. Our approach holds in quite general conditions, unlike 1.1.3 THESIS ORGI-Lh'IZATION .LXD OCTTLINE OF RESL'LTS
the convex programming approach of f521,which relies upon discretization of the unit circle into a finite number of points, as well as uniform Lipschitz continuity of the weightings and the i~inerpart of the nominal plant. These results are further generalized to the MIMO extension of the ORDAP. Another key multiplication operator is introduced, and it is shown that the matrix norm involved in the performance index for the MIMO extension of ORDAP is in fact an operator induced norm. We give the duality structure of the problem. and an exact operator theoretic solution is then proposed. Existence of maximal vectors is also shown. It turns out that given a certain level of open loop uncertainty. then the best achievable uncertainty reduction by a single feedback control law is equal to the norm of a weil defined operator. In Chapter 4 we extend the results of Chapters 2 and 3 to unstable systems using coprime factorization techniques. First we show that the ORDAP for unstable systems subject to unstructured uncertainty is equivalent to finding the smallest fixed point of a certain func- t ion. Therefore generalizing the same result which holds for stable systems. Duality is then applied to show existence of solutions. and the optimum is shown to be flat and satisfying an extrema1 identity. Similady, a solution based on operator theory is given. In Chapter 5 we show that our theory has further implication for the standard two-block HOC problem. The latter was shown to provide a good approximation to ORDAP for the limit- ing case of .'almost" complementary disturbance and plant uncertainty weightings [54, 521. Duality theory is then applied showing that the convex programming algorit hm proposed in 1521 provides another numerical solution different from the usual weil known E-iterations (sec for e-g.. [73,43, 311). Moreover, in [43] the authors showed that optimizing the HOC rnixed sensitivity is equivalent to characterizing the spectrum of a "Hankel-Toeplitz" oper- ator. In particular tbey proved that for rational plants under certain conditions. the norm of the Hankel-Tocplitz operator is achieved on the discrete spectrum, i.e. at an isolated cigenvaluc wit h finite multiplicity. Taking a complet ely different approach we generalize their rcsult to infinite-dimensional systems. We prove first that under specific conditions tlic optimum is flat. and then as a consequence we show that the norm of Hankel-Toeplitz is also achieved at some isolated eigenvalue.
1x1 Chapter 6 we formulate the ORDAP for continuous-time varying plants subject to time- vary ing unstructured uncertainty. We show t hat for causal time-varying systems, ORDAP 1.1.4 RELATED LITERaTbTRE REVIEW
is equivalent to finding the smallest îixed point of a "two-disc" type optimization problem under time-varying feedback control laws. The space of causal and stable continuous plants is viewed as a continuous nest (or triangular) algebra of operators acting on the Hilbert space of finite energy inputs to finite energy outputs. Duality is then applied to show exis- tence of optimal continuous time-varying control laws. It is also shown that for continuous time-invariant nominal plants time-varying control laws offer no improvement over time- invariant feedback control laws and hence settling an open question in 1521. In Chapter 7 we conclude with few remarks and directions for future research.
1.4. Related Literature Review
In the late twenties and early thirties Black and Nyquist realized that feedback could reduce the effect of distortion and improve the accuracy of amplifier circuits. Since then lieuristic (classical) lead-lag cornpensators relying on frequency response techniques were developed to stabiiize and reduce sensitivity of industrial servodesign. In early eighties Zarnes. [7?] proposed a mathematical fiamework which captures the essence of the classical ideas. and where lead-lag compensators can be derived from a well posed optimization prob- lem instead of the classical rules of thumb. This marked the emergence of the HbC feedback control theory. which has been a very popular area of research for more than a decade. The pri~naryattention as noted earlier was the standard Hm problem, which basically deals with uncertain disturbance at tenuat ion. Solutions based on input-output representation (see for c-g.. [31, 32, 291 and references therein) as well as state-space techniques (see for e.g.. 138, 21, 691 and references therein), which are presumably computationaily more efficient. were proposed. The standard two-block HMproblem, which includes the mixed-sensit ivity problcm. was studied by Kwakernaak [47]. Jonckeere and Verma [73, 431 solved the prob- lem in ter~nsof the norm of an operator with a Hankel-Toeplitz structure. The spectrum of such an operator consists of a continuous and a discrete part. therefore making the corn- piitation of its norm difficult. Zames and Mitter [78] presented a method of computing the spectra and eigenvectors for infinite-dimensionai systems subject to continuous weightings. More recently Fagnani (241 analyzed the problem using the geornetric approach based on Krein space theory as developed in the Ball-Helton theory [3].Georgiou [35] showed that the cornputation of the gap metric between two systems is equivalent to a special version of the two-block Hm problem. Shortly thereafter, Georgiou and Smith [36,371 were able to provide expiicit formulas for the optimal radius of stability and the corresponding optimal coritroller in terms of a Hankel operator and its maximal vectors when the problem data are cont inuous. ORDAP was iormulated by Zames (771 and considered by Francis and Bird [4, 301. They showed t hat under certain conditions the optimal robust disturbance attenuation was bounded below and above by the positive fixed-points of two functions. These two functions corre- spond to parametric versions of the standard two-block Hoc problem. Owen and Zames 152, 79) showed that this approximations can be infinitely poor. and therefore there is a need for an exact theory for such problems. In particular using Banach space duality theory they showed that under specific conditions there exists a solution to ORDAP, which satis- fies a flatness condition and is unique in the SIS0 case. A numerical convex prograrnming algorithm based on duality was proposed, together with a two-block HOC approximation in
the case of "almost complementary" weightings, i.e, IIW*Vllm = E << 1. However duality was riot fully exploited, and the operator theoretic structure of the solution has still not been described. Moreover, there has been no closed form solution reported in the literature. Applications of ORDAP has been reported, for example, in electro-mechanical systems. and more specifically in positive torque feedback in motion control: when fine motion tracking in presence of large disturbances or complex loads as found in robotic manipulators, and/or plant uncertainty is required 11). Also in control of harmonic drive systems under con- strained and free motion applications [71],and aircrait flight control (e-g., control of the longitudinal motion of an aircraft) [23],to name a few. -4nalysis of time-varying control strategies for optima1 disturbance rejection for known time- invariant plants has been studied by Shamma and M.A. Dahleh [66], Chapellat and M. Dahleh [7].A robust version of these problems was considered in [44, 45, 651. Feintuch and Francis studied bot h the weighted sensitivity and weighted mixed-sensit ivity problcms in the time-varying case [25, 261. Al1 these authors showed that for time-invariant nominal plants. time-varying control laws offer no advantage over time-invariant ones. The ORDAP for discrete time-varying systems and feedback laws has been considered by Owen [52]. He characterized the problem in the predual space, and proved tbat in some induced norm topology the same conclusion holds. CXAPTER 2. DUXLITY STRUCTURE OF ORDAP
CHAPTER 2
Duality Structure of ORDAP
In this Chapter we give a similar duality structure to [52, 791, which will prove to be more useful for our purpose. ORDAP and its MIMO extension are first interpreted as shortest distance minimizations in special vector valued Hm spaces. Then, using duality theory uTeprove existence of optimal solutions. Under the continuity assumptions of the problem data. we establish extrema1 identities for both ORDAP and its MIMO extension. Align- ment conditions are then exploited to show that the optimum for the MIMO extension of ORDAP is flat or "ailpass". Its optimal behavior is finally investigated.
Following Owen and Zames [52, 791, in the ORDAP described in section 1.2.2 , the plant frequency response p(eiO)is assumed to be uncertain and lie in a weighted bal1 in the frequency domain described by the set
where Po is the nominal model, V is a fixed stable and invertible transfer function, and X is a variable stable transfer function. A physical interpretation is given in the book of Doyle et al. [20]. The output disturbances d lie in the set
where W is outer in Ha(G ,, ). The worst case weighted sensitivity function S relating the disturbance signals u to the CHAPTER 2. DUALITY STRUCTZi'RE OF ORDAP
Wl-weighted output y can be expressed as [77, 4, 30, 521
where Q = C(I + p0Q)-l7 and WI E Hm(&x,) is outer. The optimal worst case transmitted disturbance energy denoted by fi, for the set of plants %(Po.V) can be expressed as [30,521
For rion-invertible Po, this non-linear non-convex optimization problem can be converted to finding the smallest positive fked point of the function [4, 52, 791
In the SIS0 case (2.5) takes the following form
~(r)= inf ess sup (IW~(I-P.Q)W(~'~)~+~~P.QV(~'~)I) QEHm(Gxn) BE[o.~~)
wliich is identical to the two-disc problem performance index (2.10) below. As pointed out by Owen (p. 33, [52]):clearly in the MIMO case (2.10) is an upper bound for (2.5). If W. V are cornmensurate. i.e.,
where w, and II,arc scalar valued HOo functions: A E HOC(&xn): the optirnization problem (2.5) can be simplified iurtùer. In iact (2.5) takes the following form (p. 28' [52]) 2.2.1 .hi EQüTV.4LE-c'T DIST-LXCE PROBLEM IN W' (cm,,)
inf ess sup max (1 W~(I- P,Q) w(~'*)cI+ ~~P,Qv(~")cI) (2.8) QfHa (Gx n ) 6~[0,2?r) ICI 5 1
for some W2E Hw(Gxn)-
inner/outer factorizations, and absorption of V into the fiee parameter Q yield (p. 28: WI)
inf ess sup max (I(W- u WQ)(~'*)CI+ rlp~(e")~~) QEHm(cxn) e~[0,27r) ICI 5 1
where W. W. V E Hm(&,") are outer functions. and U E Hm(Gxn)is inner. Likewise
(1.2) cari be written in the form (p.33, (521)
-4s said earlier (2.10) is an upper bound for (2.9). and the two expressions are identical in the SIS0 case. Hence we will consider first the optimization problem (2.10) which is a little casier.
2.1. An Equivaient Distance Problem in Nim (Gnxn)
bP(qnxn) was defined earlier as the Banach space of bounded analytic functions in the unit disc 9 with values in C&,,. the norm is given by
where K =
Since K is analytic on 9.then (IKIIp(ynxn)is a subharmonic function [41],hence satisfies 2.2.2 BAXXCH SPACE DUALITY STRUCTURE A.XD E-YISTEKCE OF OPTIMAL SOLCTIOXS
the maximum principle, and therefore
Then (2.10) is equivalent to
PO= inf QEffffi(Gxn) (r)-("F)~=YGnxr, 1
(2.13) is the shortest distance from to the subspace S = ( ur ) Hm(hxn) of FF(CIC:,,, ). As in [52, 791 if we assume that (Al)( W* w + v*v)(~") > 0. VO /B [O. 27r). Then S is a closed subspace of P(G,,,): and there exists an outer spectral factor A of l@'~i'+ P-3. Moreover B has the following equivalent representation
where R = ( :i ) RI = L'WA-~R2 = YI-' so that RR = I, Lrbesgue almost everywhere (A ax.). that is, R is inner. The above assumption initially excludes strictly proper plants. however we will invoke a procedure which aIlows our theory to be extended to them. Existence of solutions is next obtained. using a duality structure and some of the ideas in (52, 771.
2.2. Banach Space Duality Structure and Existence of Optimal Solutions
Recall that L' (qnyn)denotes the space of weakly Lebesgue measurable (see [41]).and absolutely integrable functions defined on the unit circle i39 with values in qny-=,under the norm
where G = ( z: ) E lL1 (Gnyn).and STrG. = T~(G:G,)$ = E: q(G*),Tr denotes the trace. and u~(G~)ihe singular values of Gi:i = 1.2. 2.2.2 BXXACH SPACE DUALITY STRUCTURE .LI EXISTENCE OF OPTIMAL SOLUTIONS
Let Ihz (qnyn) be the quotient space L' (qnX, )/$(qnyn) under the quotient norm
riiere C = ( z: ) . [CI denates the coset ruacistcd to C. It has been shown in [Il, 64, 52, 791 that ILm (G,,, ) is isometrically isomorphic to the dual space of IL1 (Gnyn). Since HP(qnxn) may be viewed as a subspace of L~(cn ,, )' and $,(qnyn)is the preannihilator of HP(en,,) in IL' (qnFn ): then MF' (G,,,) is iso- nietrically isomorphic to the dual space of e(qnyn)and we write
given by
Let '9 be the subspace of @(qnyn)defined by
where @ denotes the direct sum of two subspaces. 1 is the identity map, and
The following Lemma characterizes t hc preannihilator (or preorthogonal) of S.
Lemma 2.1. l9 is the preannihilator of b in (qnYn).
PROOF.Since W: ) is the preannihilator of EP(c,,,), the Lemrna follows fkorn Lemrna 1 in [79, 56, 551. O 2.2.3 .ABSOLUTE CONTINUITY .LhW EXISTENCE OF .W EXTREMAL IDENTITY FOR THE OPTIh~IU4!
From a standard result in Banach space duality theory relating the distance fiom a vector to a subspace and an extremai functional in the predual (Tbeorem 2, p. 121 [49]),we deduce the following Theorem
Theorem 2.1. Unde+ assumption (A1): that is, ( W*w +v*v)(eie) > O. VB E [O. 2n). there exist at least one optimal Q, such that
min (7)- RQ Q€=- Gnxn1 7- -1 - (-nxn)
where [I - Ills denotes the coset norm in 'S.
In general the supremum on the left-hand side of (2.21) is not achieved by any [G]E~ S. The best we can Say at this point is that, it follows by the BishopPhelps' Theorem [SI. that the set of functions F E !HP (q,,,)for which the supremum in (2.21) is attained is norm dense in !IP(G, ,, ). However if we assume (A2)W. Ri, R2 are continuous on the unit circle. as is the outer spectral factor A of **Li/ + p*v.and po > pOotwhere
i-e.. when the open unit disc analyticity constraint is removed. (in the scalar case (n = 1): p, = IJ min(/w(ei0)l. I~(e")l)ll,), then the optimal solution is Bat [52, 79, 561, and the suprcmum in question is always achieved. This motivates the following section.
2.3. Absolute Continuity and Existence of an Extrema1 Ldentity for the Optimum 2-23 ABSOLL'TE CONTWUITk' .LYD EXISTENCE OF AN EXTREMAL IDENTITY FOR THE OPTIMUM
2.3.1. Alignment in the Duai. Recall that C!(C,,,) denote the space of contin- uous iunctions on aU with values in Gnxn. If f = ( :: ) eIGnxn) we set
The dual space of of C!(G,,,) is given by the space M(@",) of qnyn-valued functions of bounded variations on ll9 under the norm (11, 52, 79, 551
wher and w, denotes the surn of the total variations on [O. 2n) of al1 entries vl)y2 of vi and u3- Let A(C:,,, ) be the space of U&,,,-vahed functions which are continuous on the closure of the unit disc % and analytic in 9.A(C:, ,, ) is known as the disc algebra. The annihilator of A(@:,,, ) is completely described by the vector-valued version of the F. and M. Riesz Theorem [61]which yields
The dual space of A(C;, ,, ) is then given by
under the quotient norm
Sext define the subspace Sc = B n e(C&,,). It follows fiom Lemma 3 [79, 551 that the ariiiihilator of Sc is given by 2.2.3 .L\BSOLUTE COSTIi\(iTTY ,hiEXISTENCE OF AN EXTREMAL IDENTITY FOR THE OPTIMUSI w here
Ir1 the foIlowing Lemma we establish that the distance Gom the vector function i:) to the subspacc Sc is the same as to S.
Lemma 2.2. Under assumptions (A1) (FV*w + V'V)(eiB) > O. VB E [O, 2a). (A2) W.Ri, R2 are continuous on the unit circle? as is the outer spectral factor A of W*W + v*V, and p0 > p.,. The folloming holds
PROOF.The first equality follows Gom the same argument used to prove Lemma 2 [79,56: 551. The second equality follows from Theorem 1 (page 121, [49]). O
The key observation now is that the maximum in (2.29) is achieved by a measure u, abso- lutely continuous with respect to the Lebesgue measure. It will then follow by the Radon- Nikodÿm Theorem that the supremum in (2.21) is in fact achieved in %.
Lemrna 2.3. Under assumptions (Al) (W*W+ V*V)(ei8) > O, VO E [O;2a). (A2) W, Ri,R2 are continuous on the unit circle, as is the outer spectral factor A of W* ICfe have where II - llLs denotes the coset nom in '-S. PROOF.By Lemma 2.2 there eUsts a measure [fio] E 8:: Il[Co]ll t herefore I/[A.] /l(,4clnxn = 1. The extrema1 measure vo can be written as for some v' E M(CKn),and G3 E ~g(@n.n)- Ncxt suppose that E is a Borel subset of XI,such that X,(E) = 0, and m(E) > O. Define ~*(e") = G;R(~")for 9 E E 2.2.3 ABSOLC'TE CONTI-CUITY AND E-XISTENCE OF AN EXTREMXL IDENTITY FOR THE OPTIMU41 then k E L=(w.. q,,,) and which implies TT(G~G~)(~")= O. m a-e. on E. and from a standard result hom matrix theory G3(eLo)= O, m a-e. on E, but since G3 E z:(&,n):then Gf must be identically the zero matrix. Therefore the maximum in (2.29) remains po: if SC is restricted to Sk. w here but 9; = (RC!(G ,, ))'. thereiore by duality contradicting our hypothesis (A2) (i.e., p > p,). Hence m is absolutely continuous with respect to 5,. and by the Radon-Nikodym Theorem there exists F E L' However by the Lebesgue decomposition Theorem for vector measures (p. 31. there is a measure Wm absolutely continuous w.r.t. m, and a singular measure 5, such that Expressions (2.39) and (2.40) imply &, = 0, and therefore fi, is absolutely continuous w.r.t. m. and hence (2.30) hoids. O 2.3.2. Dual Extremal Property and Extremal Identity for the Optimum. -A well known fact in the theory of HP-spaces of scalar valued functions is that every coset in L'/HA contains a representative of the least possible coset nom 122, 39, 341. More precisely, given a kernel f E L1.consider the linear iunctional qjf on Hm defined by 4, : h - J,?' fh(eie)dm. Knowiiig that Hm 2 (L~/H;)*. find a kernel function fo E L1,such that fo - f E HA,and Ilfoll~~= min Ilf +911~1 g€ Hc! 4 dual extremal function is a function h E HOC, Il hlla < 1 such that [59, 22, 341 We will show t hat t hese remarkable properties hold for L1 (Gnyn) /a:(qnyn ). and are in- heritcd by "S. The optimal solution can then be interpreted as the dual extremal function of an optimization problem in L' (en?,). Lemma 2.4. [13] (a) Giuen [FI E I@ (qnyn)?there ezists ho E m;(qnyn) such that (b) Under assumption (Ai?),given [FI S' them exists xo E X such that F + x, zs then an eztrernal kernel or function for [FI. PROOF. (a) There exists a sequence {h,}?==, c C;(~~F~)such that n-oclim IIF + hzllv ) = Il [FI 11% (qnyn) (2.43) irnplies that the measures (hndm) have uniformly bounded total variations: by Xlaoglu's Theorem [49] the sequence {h,dm) has a weak* cluster point say: dp. Hence Vg E e(qn,,) But {h,} c H:(G~,), therefore J:" ~r(~*h~)(e")dm= 0, Vg E ~(q,,,)and Vn irnplying #' ~r(~*(e")d~)= O. Thus dp annihilates A(C;, ,, ), by the F. and M. 2.2.3 XBSOLUTE COXTLWTTY .Lhm E-SISTEXCE OF .hi EXTREMAL IDENTITY FOR THE OPTIMUSI Riesz Theorem we deduce dp = hodm, for some ho E a:(qn:,) This gives and hence (a) holds. (b) There erdsts a sequence {CI - RR*)xn + Ry,) E X where xn f LL(caY,nxn ) 7 'Yn E -1 HO(Gxn)such that and then by the same argument as (a) { (1- RW)z, + Ry,} converges in the weak* topology. we have for Vg E A(@:, ,, ) since R and 1 - RR* are continuous by assumption (A2). Again by the Radon- Nikodym Theorem there exist hlv2€ L1(qp:,) such that However (1 - RI??)* = 1 - RE. and FR= I together with (2.48) imply that By (a)tliere exists ho E E~(G~?,) such that (2.41) holds. On the other liaiid. by uniqueness of the weak' limit we have 2.2.3 -4BSOLUTE CONTLXUTTY hXD EXISTEXCE OF .;LI EXTREMGL IDENTIR FOR THE OPTIMUM thus R'h2 E Hg(&,, ), and (2.47) irnpiies Putting xo = (I - RR')hi + R(Fh2)E X, we get Combining Lemma(@*w 2.3 and 2.4, we obtain the foilowing Theorem which looks familiar in the theory of extrema1 problems. Theorem 2.2. Under assumptions (AI) + v*v)(eie)> 0. Ve 18 [072n). a (A2) W.Ri, R2 are continuous on the unit circle. as is the outer spectral factor A of Mr'W + v'V. and p, > pO0. F, E [FI E' S. ll[F]lllc= 1: is an extrema1 kemel for [FI:and Qo is optimal if and only if u/ierc Fo = ( :: ) . The optimal perfomance ( i2::0 ) is then a duai eztrernal funcfion for the appmzimalion problem minlex I[F + f llii cc;?, ). PROOF.-0nly if' by assumption 3Fo E~ 8: IIFollLicqSni= 1: and QoE Hm(&,,) such that 2.2.4 EXTENSION TO STRICTLY PROPER PLANTS but the integrand Integrat ing impiies equality mist hold throughout. This combined wit h flatness imply ~nax(STr(F~~):ST~(F.~))(~'~) = 1, m a.e. ..If' suppose that (2.50) holds; integrating it yields Which is equal to 2n Tr{ ((W*.O) + Q'R*) ~)(e")dm,VQ E Hm(&,, ) tience equality must hold throughout and Q, is then optimal. Remark 2.1. Theorem 2.2 shows that the search in the convez programming algorithm proposed in Chapter 4 [52] can be restricted to functzons which satisfy (or rnax(Fi(eiO),~*(e~*)) = 1? m a.e., in the scalar case): and hence results in faster algo- dhm convergence with fess memory usage. Moreouer, expression (2.50) may be used as a test for optirnality. 2.4. Extension to Strictly Proper Plants In gencral there does not exist an optimal control law for both ORDAP and the two- disc problem as shown in Theorem 3.5 in [52]. Strictly proper plants may be interpreted 2.2.4 EXTEXSION TO STRICTLY PROPER PLANTS as tiaving jw-axis zeros at w = m for continuous-time systems (or at z = 1 for discrete- time systenis). This is an intrinsic problem with Hm,it arises because the optimal control system has a transfer function with point discontinuities on the jw- axis (or unit circle) at the offending ju-axis zeros (respectively unit circle zeros) [62].This has noting to do with the specific algorithm being employed. The failure arises from intrinsic ill-conditioning of the HS-problems when there are jw-axis zeros. Nevertheless Owen in [52]showed that if we consider the following modification of (Al)?we can allow strict propriety (Al') The outer factor of Po takes the form HY, where H is an invertible function in HX and Y is a strict proper rational outer function whose zeros on the unit circle are in finite nurnber . In addition, there exists an c0 > O such that the weights W and V satis@ w*w(c'')+ v'v(eiB)> Ve E [O, 24. However as mentioned earlier, for a large class of systenis strict proprieiy rules out the existence of an optimal control law and its attendant properties. In these situations it is riieaningful to analyse the properties of '.nearly2 optimal solutions, when the performance is very close to the optimum. This can be achieved by by using the foiIowing simple conformai niapping of the s-plane (or unit disc by using s = s) - s+a s-a ,y=- . then s = - ltbs. 1 - bS which perturbs the jw-axis (unit circle) including w = x. moving the offending points sliglitly into the interior of the right-hall plane (or unit disc). and the following holds 1621 s = lim S a40 b-O Then for al1 G E Hm(&,, ) 2.2.5 MIMO EXTEXSION OF ORDAP It follows that if Q, is the optimal solution of the perturbed problem then Another method is to select suitable weighting functions which ca~celthe offending zeros. Any jw-axis or unit circie zero and infinite zero of R may be canceiied by a suitably selected weighting Say W', having poles at the jw-axis and infinite zeros of Po. If Po has zeros at infinity of multiplicity m. an appropriate multiplier would be W' = (1 + a)".for some sniall E > 0. and has the properties that (see f62]for details) i) W'P, E Hm. ii) li~,~W'P, = Po 2.5. MIMO Extension of ORDAP In this section we study the duality structure of ORDAP for MIMO systems. The optimal performance index is then given by (2.9), i.e., (I(w - UWQ)(~")C]+ T~YQ(~O)CI)(2.54) Urider assurnption (Al) for the weights w and Y.Le.. (Al) ( W* w+V*Y)(eie) > O, VB E [O?2a). as shown in Section 2.1. (2.54) can be transformed into t lie following optimizat ion problern ' - PO- inf ess sup max (I(W - RIQ)(~")cI+ ri R~Q(~") follows is similar to- that of the previous sections with a modified matrix norm. 2.5.1. Duality Structure of MIMO Extension of ORDAP. Define P(G,,,) (rcsp. lEF(enxn)) to be the Banach space consisting of 272 x n essentially bounded (resp. 2-23 MIMO ESTEXSION OF ORDAP analytic) functions defined in D with values in the space Gnxn endowed with the nom CEG Expression (2.55) implies that ORDAP for MIMO systems reduces to an optimization prob- lem of the form PO = inf Q€Hw(Gxxn) Recall that G:, is the space of 2n x n matrices wit h entries in C endowed wit h the nuclear norm (1 - Ij ,,, defined as follows. if Al. A:! E @nx then the nuclear norm ( :: ) is defined to be the infimum of al1 sums xk(sk 111 yk[lT, xt E G.yk E qn such tliat [52] Yow define t Le Banach space LI (q,U:, ) of weakly Lebesgue integable q;Xn-valued func- tions endowed with the norm It lias been shown in [52] that Lm (&,. ) is isometrically isomorphic to the dual space of L1(q:: ). Since IF (CZnxn) may be viewed as a subspace of ILm (C2nxn ). and Wi(CZ&) is thpreannihilator of WjO (Fnx) in L1(q:zn ). then HP(CZnxn ) is isometrically isomor- ptiic to the dual space of @((IZ$,): where @(q;~,)4 lLl(Cj~:,)/@(~,~~,). The prcannihilator of S is given by 2.2.5 MIMO EXTENSIOK OF ORDAP The quotient norm in 5 is denoted by II - IlS and is given by = V [FI E II [FI IIS inf ]IF + hllL~ Theorem 2.3. Under assumption (Al) ( W*w + Y*v)(eio) > 0: Q0 E [O. 27). There ezlsts at least one Q, E HOC(&,, ) such that r po = inf ess sup max (I(W - R~Q)(~")c/+ IR~Q(~'~)cI) (2.63) QEH3C(Zx=1 e.qo.2z) ICI 5 1 2.5.2. Allpsss Property of the Optimum: Alignment in the Dud and Ex- tremal Identity for the Optimum. In this section we assume that assumption (A2) Iiolds for W. RI and R?: that is. R~~ R~ are continuous on the unit circle. as is the outer spectral factor A of W*W + v'C. Let e(C2, ., ) denote the Banach space of continuous Gnxn-duedfunctions defined on aL) under the norm It will be showo that the dual space e(G,,,)' is isometrically isomorphic to a space M\lLpnXn)/-nue consisting of qX,-valued compIex bounded measures defined on the unit circle. .4ccordingly. let v = ( I: ) t M(C;E:, 1- and introdure the following bilinear forn on Let wy be the sum of the total variations on [O, 8) of al1 entries of UI,~.By the Radon- Nikodym Theorem. there exists a vector function G,,€ L'(&~~:W,).r = 1. 2 such that (2.67) is reduced to The norm on M(Q',n',,) is now defined to be PROOF.Since @2,,, is the dual space of q;:, by Lernma 3.3 [52] and Theorem 2.10 171. Cznxn is finite dimensional_hence reflexive and it follows that q::, is the dual space of G,,, . Then by Singer's Theorem (p.398 [67]) Lct A(C2nx ) be the space of &,,-valued functions which are continuous on the closure of the unit disc 3.and analytic in 9.As previously the F. and M. Riesz Theorem characterizes the annihilator of A.(Cînxn ) as The dual space of A(C~~~~) is the quotient space 2.2.5 MIMO EXTEXSION OF ORDAP undcr the quotient nom for v E M(@?,UX, ) ?? = ((1- RR*)M(@?~',,) t RH~(@,~,))~R~(,:,} (2.73) Then under assumptions (Al) and (A2) Lemma 2.2 holds ako for pi. i.e., there exists at Ieast one optimal Q, E Hm(&,, ) such that Sext define ' - POO- QEe(Ginf X, -fiQ (O) 'C i-e.. when the open unit analyticity constraint is removed. We show that the optimal solution in this case is also flat or allpass. and an approximate flatness condition holds. Theorem 2.4. linder assumptions r (-4 I) (@*w+ i"V)(e1*) > 0' VO E [O. 27r). r (-42) W.R,. R, and A are continuous on the unit circle. If IL; > then i. Any optimal Q, E HX(GXn)in (2.57) satisfies the patness or "allpass" condition ma (I(W - R~Q,)(~")CI+ 1 R~Q,(~")cI)= . Ve l~ [O! 2~) (2.78) ICI L 1 2.2.5 LMI~IOEXTLYSIOX OF ORDAP ii- If (Q,}rzs any sequence Hm(&,,) such that The condition > &, zs sharp. PROOF. i. Let 4, be the extrema1 functional corresponding to the extrema1 rnea- sure vo which achieves the maximum in (2.76): Le., 4,(-)=< vol - >. where dpo = Gdu(0).G = ( u: ) s !L1(GEfl). Let & br the rootinuous Iinear extension of t~ L~(C~~~~) which vanishes on 8,. Theo max (I(W - &Q~)(~~~)cI+ I%Q~(~~~cI) ICI 5 1 nuc 2.2.5 SUMO ESTENSIOS OF ORD.4P where cf+(.) = &(O) satisfies dw)= !l&[b.qc;;;, ) = 1- ( ~~:~:~) nuc Moreover 4, is such that Suppose now that E is a Borel subset of ?l9such that @(E)= O and m(E)> 0. Define the foilowing essentially bounded matrix valued functioa and IC = 0: for 8 E EC which implies that G3(ei0)= O m a.e. in Et but since m(E) > O. G3 must be identically zero. Therefore the maximum in (2.76) is achieved on the set {u E M(C;::n) : v(û) = J,;,,(I - RR*)dvJ(8). vf E M(c;:,uc,,)}/%~ which is the an- nihilator of Re(&,,) implying by duality = and then contradicting our assumption. and thus i. must hold. ii. Follows from the same argument used to prove ii. Theorem 2 [79]. cl Remark 2.2. The argument used in the proof of i. aboue (c-j. section 2.3.2) shows thut the eztremal meusure v, is absolutely continuous with respect tu the Lebesgue measure. Hence the supremum in (2.65) às achieved at some function F E 5' IIIF]lls = 1. Moreouer Lemma 2.4 hoids for cosets in @ (q;:,) and S. Therefom the optimum ( ) - RQO 2.2.6 PROPERTIES OF THE OPTIMAL BEHAViOR OF THE ORDAP satisfies the following Theorem analogous to Theorem 2.2, and which prouides a test for the optimum. Theorem 2.5. Under assumptions (Al) (W'W t v'v)(eie) > O? V6 E [O7 27r). (AS) W. Ri, Rz are continuous on the unit circle, as is the outer spectral factor A of w*w+ v*v. und pk > poo, Fo = ( z:) € [FI € 5. [1(4111= 1 is an utremai kernel for [FI; and Q, is optimal if and only if max (l(w - &Q,)(~")CI+ I&QO(~'*)~I) m a.e. 7 1 -uc ICI L 1 - (-nxn) The optimal performance ( wi2t00) is then a duai utremal junction for the appmz- PROOF.The proof is similar to the proof of Theorem 2.2 except instead of ~nax(STr(F,~).STr(FO2))! use 2.6. Properties of the Optimal Behavior of the ORDAP In this section we generalize properties of the optimal behavior of the ORDAP obtained in [52]for SIS0 systems to MIMO systems. Note that for the MIMO extension of ORDAP. 1; is scaled by a parameter r. then assumption (A2) is strengthened to (AZ') The outer spectral factor of w'* + r2Y*Y is continuous for any positive T. and the outer part of Po is invertible in HW(&xn). 2.2.6 PROPERTIES OF THE OPTIhlAL BEHAWOR OF THE ORDAP Rccall that in the MIMO case ORDAP assumes the following form The following result generalizes Theorem 3.6 [52]. Theorem 2.6. 1. Under ossumptions (AI): (w*w+Y*v)(~") > O. V9 f [O. 2r)). and (AZ') aboue. Ifp; > O. then there ezists an optimal feedback control lmstabi- lizing euery system in 23(Po,V) such that 2. If in addition pi > PL,, then there ezists at least one optimal controller which stabilizes euery system in %(Po,V) and achieues (2.88). In this case, the magnitude of the weighted sensitivity function for the nominal plant Po satisfies PROOF. 1. It follows from Theorem 2.3 that for each positive r. there exists Q E Hx(G ) for the optimal performance index represented by x(-)in expression (2.5). By Theorem 2.1 [52], is the smallest hed-point of x(-). If Q, is the optimum for x(P',>. then which implies that 2.2.6 PROPERTIES OF THE OPTIM-IL BEH-AVIOR OF THE ORDAP and also which implies since IIXVPoQo((, 1. thus (1- XVP,Q~)(~")is invertible for each B E [O. 2s). Letting we get 2.2.6 PROPERTIES OF THE OPTIMAL BEH.4ViOR OF THE ORDAP 2. By 1. there exists at least one optimal controller wbich stabilizes every system in %(Po,V). Therefore Y(/LD) = PO = inf ess sup max (1 W(I - P,Q)(~'*)cI+ ~~~VPOQ(~") implies Assumption (42') implies that (42) holds for w and rV for each positive r. Hence al1 conditions for the application of Theorem 2.4 are met. Thus, Qo satisfies proving that (2.89) holds, and we are done. 3.3.1 A KEY 1IULTIPLICATIOS OPER4TOR CHAPTER 3 Operator Theoretic Solution to ORDAP In this Chapter we give explicit solutions based on operator theory to ORDAP and its MIMO extension. The solutions depend on operators defined on special vector-valued Hardy spaces. These operators analogous to the Sarason operator for the standard Hm problern. Thcy allows us to quantify optimal performance for feedback systems and provides explicit formulas for the optimal controllers. We start by introducing a particular multiplication opcrator which plays a key role. Next. we characterize dual and predual spaces of certain vector-valued H' spaces. We show then that optimal performance is equal to the operator norm of a specific operator. which is a projection of a multiplication operator. We show in part icular t hat flatness implies existence of maximal vectors. The projection is explicitly computed. showing that the operator is in fact a combination of multiplication and Toeplitz operators. We derive an infinite matrix representation for the operator. General formulas ürid a recursive algorithm is proposed to cornpute the columnsl and an approximation by norrns of finitc rank operators or matrices is obtained. Finally. similar operator theoretic frarnework is carried to the MIMO extension of ORDAP in different Hardy function spaces. 3.1. A Key Multiplication Operator Lct iL2 (q,) denote the space of weakly Lebesgue integrable c,-valued functions de- fined on the unit circle m,with the norm 3.3.1 A KEY MULTIPLICATION OPERATOR RecalI that Gn is the complex Banach space C2, with the norm .Vote that (1 - !IL2(Yn) is not given by any Hilbert space norm' so that IL2 (c,) is not a Hilbert space. but is indeed isomorphic to the Hilbert space 0? (&, ): where G, is the 2n Euclidean space which norm is given by (0.3). Let = ( zl ) E (qnxn). the multiplication operator associated with a. and map- ping the standard Lebesgue space ~?(@n)into k2 (en)is denoted by M4. More precisely. In the next Proposition we show that the operator induced norm II M4 [( is equal to l[O1(z, (qnxn 1. and in fàct it suffices to work in the standard Hardy space H~(&). Proposition 3.1. Let 8 E iHP"(~,,,)~ then 1. PROOF. 1. Equality (3.4) follows from a measure theoretic argument: first note that II M4II 5 II @ = (:: ) then we have For the reverse inequality, let @,, T = 1, 2? have singular values arranged in de- creasing order 1O,1 = 2 or2 2 - - - > orn > O: T = l, 2: and singular value decomposition where Urand V, are unitary n x n-matrix valued functions, and Moreover for each B E [O. 27r) the columns of vr(ei")'r = 1.2, form a complete orthonormal basis of eigenvectors for each of a:@,. r = 1,2. The first column of V, (elo). -VI,. r = 1: 2, corresponds to the largest singular value of O, (ei6).JO, (ei6) 1, s E [O,2~). Yow let E > O; then by definition of the supremurn there exists a set E c XJof positive Lebesguc measure such that 3.3.1 A KEY MULTIPLICATION OPERATOR Let XE be the characteristic function of E. and define Therefore we have 27 ll*ll(n= (I*1gI + 1~29/)~(e'~)d~ Replacing and a2 by their singular value decompositions. yields Since Lii and fi are unitary, we obtain then 3.3.1 A KEY Ml..7LTLPLICATIOK OPERATOR Thus Since E is arbitrary. then IIM+II > 11*115- (-,,, 1, and we conclude that JIM+II = ll@ll~-(q~~~)- 2. Clearly by 1. 'V Let f (2) = En=-,,a,zn be a trigonometric polynomial in L'(Q)' where an E & and Il f llLqG) 5 1. Then Multiplying by the N-th power of eie, yields But since the trigonometric polynomials are dense in L2(@,) then f trigo. poly. Thereforc 3.3.1 A KEY MULTIPLICATIOX OPERATOR so (3.5) must hold and we are done. O Since q, is finite dimensional, then the dual space of lL2 (G,) is clearly given by the Banach space iL2 (qna) and vise-versa (111. It follows that iL2(Cn) is reflexive and Recall that (G,) (Hl?-(ql)) is the closed subspace of e? (U& ) (IL2 (C$cK))coosisting of I(L2(Gn) IlL2(c-))-bounded pairs of analytic functions in the unit disc. II - (11 - ,n Proposition 3.2. Let I@(G,) and @(@2"nby) be defined as above. then the following hold 1. @(c&) = (@(qn,ax))* 2- LP(qna) = (W(Gn))* Hence Kf (G,)and !Hl? (qna)are reflezive Banoch spaces. Before procceding to give the proof. we need some additional notation and definitions. The tensor product L? @-, is isomorphic to the linear space of finite surns x:=, fj(eie)xj~ \vit h f, E L? and xj E û&. L~ @ 4,is dense in iL2 (Gn) and the norm of the latter is a crossnorm on the subspace L2 O G,:i.e.. [48] for f E L' and x E qn.Since Gn is finite dimensional. then L2 @ cncan be identified witli L2 (q,). Likewise. H' @ cncan be identified with (qn). PROOF. 1. The dual space of 9dL (q,) can be identified with its associated space H' 82*qnax (Theorem 4 [48]):where H202. qnais equal to Hz 8 qna.but under the associated norm 3.3.1 -4 KEY MZILTIPLIC.4TIOX OPERTOR Note that @(qna)can be regarded as a subspace of (w(qn))*[48]. Define the identity mapping then i is continuous. By Theorem 5 [48].@ (qnm) = (-Il (Gn ))'. Le.. the associated norm II - 112- and the norm of & (QnX)coincide, if and only if the positive Riesï projection P+ in L'(-!=) is bounded, i.e., if f E L2(qna) has Fourier coefficient ( fn},"=-,, fn E Gnz?then P+ f E L2(qna) and has Fourier coefficients {gn}F==_, such that gn = O: Vn < O and gn = jn, VTZ3 0. We set g = P+ f. Then P+ is a linear operator on IL2 (qF) with values in Hf (qna).The negative Riesz projection P- is defined analogously, it associates to f its negative Fourier coefficients, i.e., P- f = h, where h has Fourier coefficients (hn)r=-, such that hn = O, Vn 2 0. and h, = fn. Vn < O. P- will be used later to prove a forthcoming Theorem. Since the trigonometric polynomials with coefficients in enmare dense in L2(qnaY)? it suffices to prove that IIP+ f IIL2(qn=) 5 KlJf 11L2(qn~), 3K > 0: Vf, f trigonometric 3.3.2 EXACT OPER4TOR THEORETIC SOLUTION Hence and so P+ is bounded. 2. Follows in a similar manner. In the next section we characterize the optimal solution in terms of an operator which is arialogous to the Sarason operator for the standard one block HbO problem. 3.2. Exact Operator Theoretic Solution 3.2.1. Optimal Performance. Let ll be the orthogonal projection onto the closed subspace @(û&) a RH^(&) of @(Gn).where @ (û&) 8 RH~(Q) is understood to be the orthogonal complement of RH?(& )' RH^(@, )'. Then Il is a bounded linear operator on i3F (qn). Next. define the following key operator II associates to each f y / E H2(Q)! its best approximation in the subspacc RH2(& ) in the II - IILzcqn -norm. Indeed by duality we have - max hERH-(Lm) (y),-h WGn) llm2(q"-) 5 1 3.3.2 EXACT OPERATOR THEORETIC SOLUT103 Therefore - max LqCn) II~IIL~(~~,) 5 1 The Iast equality follows since Ii f E (RH*(@,))' defines by itself a linear bounded functional on (RH'(@, ))l by Proposition 3.2, and the fact that closed subspaces of reflexive Barlach spaces are also reflexive. Z plays a central role in finding an explicit solution to our problcm through the foilowing Theorern which quantifies optimal performance. Theorem 3.1. Let p, be the the optimal performance indez defined by expression (2.10). Then under assumptions (-41) and (AZ), that is, (-41) (W-w + V*V)(eïe)> 0: Vt? E [O. 27r). (A21 W7Ri. R2 are continuous on the unit cimle, as is outer spectral facto+ A of w*M' + - - V'V. and po > po,, where so that the optimum is allpass. Then the following hold: i) p, is equul to the operator induced norm of E: namely ii) There ezists a mazimal vector for E,2.e.. f E H~(@,)of L~(@,)-nom 1 such that Bcfore proceeding to prove Theorem 3.1, we need the following Lemma. 3.3.2 EXACT OPERTOR THEORETIC SOLUTION G being in the dual of @? (Gn). Hence For the reverse inequality, V6 > O. there exists f E H2(G): Il f llr?(G ) = 1 such that T herefore 3.3.2 EXACT OPER4TOR THEORETIC SOLC'TIOS Since 6 > O is arbitrary. we get (3.17) and (3.18) imply that (3.16) hotd, and the Lemma is proved. Xest. we turn to the proof of Theorem 3.1. PROOF.Note that VQ E Hm(GXn)'Vg € II2(&), we have Qg E lY2(@n).i.e.. QH'(G)C H2(&). hence HRQg = O. Next. letting f E HZ(&) with norm IIf IIL?(c,) 5 1. WC have then 3.3.2 E-UCT OPERATOR THEORETIC SOLCTIOK Siiice (3.19) holds Vf E H2(&).Il f llL. By Lemmas 2.3. 2.4 and Theorem 2.2 there exists [FI = [ ( ) ] eL S. witi. F iiaving L' (qnyn)-norm 1. and such that By the proof of Theorem 2.2. rnax(S~r~~(ex@).STrF?(eie)) = 1. rn a-e.: and there exists h E H2 such that lh(ei0)I2= max(STrFi(eZB).STTF~(~")). m a-e.. (p. 53. 1421). But then Ph E L2(qnyn).IIFhllL:cqy, 1 = 1- Moreover Fh E (RH~(Ç,,))~.since 2 z Tr(G*R*(I- RR*) ( i:) h + C*RRgoh+ ~*~~h)(e")drn Soting tliat R'(I - RF)= O. R'R = I. G'g, E H:(&,,). (since go = RFE L~(&,,); go E H: n L2(C,,) = H?(G,,)). Therefore 3.3.2 EXACT OPEFUTOR THEORETIC SOLETIOX' Likewise R'g E H:(&,,) Therefore Fh E (RH*(&,, )) l. Xow Iet In denote the n x n identity matrix. then Exploiting an idea of Young (761, the subspace (RH2(&,, ))' consisting of functions each of whose colurnns belongs to (RH'(&)) can be identified with the tensor (RN'(& )) '@&. and will henceforth be denoted so. Likewise ~'(q,,)2 H~(&) €3 G. We have and lias norm Ilh @ I,ll = IlhlJr~llInll= 1. similarly Fh E RH?(&))^@& and has nom 1. The member on the left-hand side of (3.22) is equal to lierice we obtain 3.3.2 EXACT OPERATOR THEORETIC SOLUTIOS wliere ‘Dl is the multiplication operator from the left by , acting from H2(@,)@G and < -.. > denotes the inner product in IL2 (G,,, ). Hence Expressions (3.20): (3.21) and (3.26) imply that p, = 11211: and equality in (3.25) and (3.26) must hold inferring existence of a maximal vector. O Remark 3.1. Assumption (A2) (continuity and flatness or allpass property of the op- timum) is in fact not necessary for (3.14) to hold [12, 17, 16, 181. Howeuer we needed (-4.2) to proue existence of a man'rnal vector, through absolute continuity of the eztremal nieusure v, and a factorization argument. For SIS0 systems, (n = l)?if ORDAP originates from the optimal robust mode1 match- ing of section 1.2.3, then the nom of the operator Z7 namely 11EIl, provides the optimal uncertainty reduction achieved by a single feedback control law. 3.3.2 EdL4CT OPERATOR THEORETIC SOLUTIOS The orthogonal projection ll plays a crucial role in solving our problem as well as the standard two-block problem, e.g., mixed sensitivity problem. It is thus important to characterize it explicitly, by the following Theorem. Theorem 3.2. Let ri be the orthogonal projection fiom (c,) tu Hl? (q,)eRH2 (6 ) : tlren PROOF.For f E (G,). let us compute (I - RP-~')'f = (I -RP+R*)(I - RP+R*)f = (1- RP+R*- RP+R*+RP+R*RP+R")f = (I -2RP+R*+RP+R*)f since R'R = I and P: = P+ = (1 - RP+R*)f (3.32) hencc (1- RP-R') is indeed a projection. Clearly the adjoint (I - RP+R*)* of (I- RP+R') is equal to (1 - RP+R*) itself. so that (1- RP- R') is an orthogonal projection. Xcxt. we show that the nul1 space of (1- RP+F),Ker(I - RP-R*) = RH2(@,). Let f E Ker(I - RP-R*): then Since R*f E ~'(@n)then P+Rf E H'(&) and therefore (3.33) implies f E RH'(^). Herice Ker(I - RP+iP) c RH2(@,). Convcrsely, let f E H2(@,),then thus Rf f Ker(1 - RP+F):so Ker(1 - RP+F) > RH*(&), and therefore Ker(I - RP+R*)= RH2(&). 3.3.2 ESXCT OPER4TOR THEORETIC SOLCTIOS Next . we characterize the image space of I - RP+F. Assume that f E El!? (q,) e (& ). th~f = fi - Rf2 for some fi E m(q,),f2 E H2(Q;i), and f is orthogonal to Rf2. We have and Xote that P+R*fi E H2(G).and (1- RP-R') fi IRP+P fl since since the orthogonal decomposition is unique, (3.34) and (3.37) imply then (3.38) implies that f E Image(1- RP+IP). so w(G,) 8 RH2(@,) C Image(1- RP'R'). Con\vrsely. suppose that f E Image(1 - RP+R'), that is. tliere exists g E IM?(Gn)such that tlicri cleariy f L RH^(@^). so f E m(Gn) RH*(&)' thus Image(1 - RP+R*) c - 4 rr (G,) e RH2(& ). and equality follows. O The following CorolIary follows then from Theorem 3.1 and 3.2. Corollary 3.1. The optimal perfomance indez p, is given by the followzng expression 3.3.2 E,YACT OPERATOR THEORETIC SOLCTIOI; and them f E H2(G),11 f 11L2(G) = 1 such that 3.2.2. Optimal Controller. Theorem 2.1 implies that there exists a vector func- tion Moreover by Proposition 3.1. may be viewed as a multipiication operator ,Mq acting from Hz(@,) into I@ (en) such that The first inequality in (3.45) holds since the orthogonal projection II acting from H'(@,) into El2 (G,) 8 RHZ(& ) has induced norm less that or equal to 1: to see this consider for It follows that Remark 3.2. For SIS0 systems, (n = 1): f is a scalar function and we ca72. diuide through to obtain a formula for q: which determines iE uniquely. 3.3.3 COMPUT-4TIONOF THE OPER-4TOR XORM Therefore we get hence The optimal controller Co = Q,V-'(1 - QoV-lPo)-' achieues then optimal robust perfor- mance. For the general case (MIMO systems), the optimal controller is in fact hzghly non-unique, and a more involved procedure ts required in order to compute it. Roughly speaking, the relation ik f = Ef determines only a rank 1 part of the function iEt and hence the optimal parameter QO. We need suficiently many independent directions to determine the whole symbol. A good starting point me the papers by Young, Peller and %il [76, 58, 571. 3.3. Computation of the Operator Norm In t his section we show how it is possible to compute the norm of the operator Z and lience the optimal performance index p,. First? note that every f E H'(@,). has a power series expansion f (z) = CEC=,zkfk7 for sorne fk E C,., . Therefore. (z%,}. j = 1.2.. . . , n: k = 0.1.2.. . . ; forin a basis for H?(@,). where e,? j = 1' 2,. . . , n is the usual canonical basis of c.To determine a representation of Z with respect to the basis {zkej}. j = lo2:. . . n: k = 0.1.2.. . . : it suffices to study the action of E on each element of the bais Say, for example. zGel for a fixed k. We have then for j = 1'2:. . . 'n ahre w, is tlie j-th column of W. In the represntation of a, the j-th column will be the representation of IIyjzk with respect to the basis {zLei},i = 1'2,. . . 2n: 1 = 0: 1: 2.. . - : of (c,). Hence. we can interprate Ilwzk,for each k. as the matrix function whose columns are projections of the columns of WZ* onto the subspace RH^(&)^. With this in mind' 3.3.3 COMPUT-4TION OF THE OPERATOR XORM the action of E on f is then determined by Therefore with respect to the canonical basis 1 z1: . . . z" : . . . ; f has the following repre- sentation Each column in (3.51) is the image of a basis vector after iipplying z. These columns arc dctcrmined by studying the action of Z on the basis vectors (2):j = 0.1.2:. . . as cxplained above: the k + 1-th column corresponding to the representation of zzk with respect to the basis 1. zl,z2. . . . : is then formed by the coefficients of :zk in GnXnwith W - RIP+FFV respect to 1. z1: z2:. . . . To compute these coefficients consider. Z = -R2P-qW since W. Riand R? belong to HS(Gxn).then they aii admit powe; series expansioossin the unit disc Thcn we have Ri = CL=-, rlkzk.and We get also 3.3.3 COMPUTATION OF THE OPERATOR SOR\I WZJ - R~P+R;WZ~ Now each column Et3 = : j =071,2,.-.. -R2 P+R;lVzJ Hence straightforward computations yield .i\t the n-th step, we have Remark 3.3. The previovs formulas show that when the first column is computed, al1 the remaining columns follow easily. However f~oma computational point of ,uiew, it is more eficient tu compvte these columns recursively. Hence the following recursiue algorithm. 3.3.3 COMPGT.ATION OF THE OPERATOR 'rOR41 Recursive Algorithm: for j = 0.1.2,.. . .n,. -. where The operator t bas then the following infinite matrix representation with respect to the basis 11. r. z2.. . . }: 3.3.3 COAIPUT.4TIOX OF THE OPERTOR XORM If f E ~~(@i):f(z) = CgozJfj.f, E Gothen the action of B on f is determined by To computc the norm of 3 it is a bit easier to work with the adjoint operator Z' of E. First note that the correspondence E = (a,k)l,c=o00 is clearly linear and weIl behaved with respect to adjoints. and Z = (bjk)z=o,where bjk = utj: for al1 j'k 2 O. Let f E H2(Q). be a maximal vector of L2(G)-norm 1 for t. f exists by Theorem 3.1. As previously. it follows by duality that Z solves the following approximation problem - max IlZ*l*(w*_ 0) hdm/ g€ RHZ(Cn) L IlhIl~(~~)a 1 Thcrefore. equality must hold throughout. and h = ( ) (q~) is a maximal vector for the adjoint 2'. Next, we show that h is such that 3.3.3 COMPUTATION OF THE OPER4TOR SORM That is 1hl(ei")l = Ih2(eio)l,rn a.e. If hk = CEohklzl$ for hkl E L = 1, 2; we will get Equality (3.59) is important since it allows us to bring in some Hilbert space structure as shown by (3.60). Indeed alignrnent in the dual (3.57) implies that equality (3-61) below holds. and for some go E H2(@,) we have Applying Cauchy-Schwarz iaequality, we get t hus tience equality must hold throughout, and from (3.63) and (3.64) we deduce 3.3.3 COPUlPù'TATION OF THE OPERATOR NORXI hmce lhi l(eie)= Ih2((eie),m a-e.. and (3.59) and (3.60) must hold. 3C since b' = (blk)l,k=,, where bjc = aij; j. k = 0,1.2.. .. : then bjk = ('bIk : 'b,k) = ('all . 'a;,). Let us recall that so that Thereiore E'h = Cr==, € H~(@,)'where Hk = xzobjkhj. k = 0.1,2:. . . . Thus Hcnce computation of the operator norm reduces to finding coefficients hk,, n = 1.2. . . . : X: = 1.2: satisfying (3.60) such that the sum on the left-hand side of (3.71) is maximized. Example: Consider the nominal plant and weighting functions The outer spectral factor is then given by 3.3.3 COMPUT.4TION OF THE OPERATOR NORV The iriner and outer parts of the plant are respectively given by Thcrefore Ri = 0.5U(z),R2 = 0.866. and W(z)= 0.5. The first column is given by the second column by finallÿ the third colurnn 3.3.1 APPR0,YIhIATION BY NOR\IS OF FIh1TE RA3X OPERATORS Then the operator Z in t his case takes the following form 3.4. Approximation by Norms of Finite Rank Operators In what foilows we will show that it is possible to approximate the norm of Z by the norms of finite dimensional operators or matrices. Indeed. if we restrict 5 to finite dimerisional subspaces of its domain, then we will obtain a sequence of finite rank operators whose norms are %early" optimal. in the sense that they approach the norm of Z as the dimension of the initial spaces increases. More explicitly, let Pm denote the orthogonal projection £rom H*(& ) to the subspace spanned by {1. r. z2. . . . .zm) Span{l. r. . . . . rm}: and Pm the orthogonal projection from (q,) onto the span of { 1. r. r2... . . rm} over G,. Clearly. 11 Pmll < 1: to prove that IIP, II 5 1, note that Pm solves the following approximation problem. for F E l@ (G,)!IIFIIL:(q, 1 5 1 IIFIIL~~~>2 min IIF - ~IIL~?,) gf Span(1.z. 2a max 1 F'*h(ei")dm! (by duality) (3.73) 2 max Ilhllz~(~~=)5 1 h E Span{l, z, . . . , zm} 2 IIPnFIIç.(qn) where Span{l' z, . . . .zm}l is the orthogonal complement of Span{l, z, . . . ,zm }. Tlius I!PmFIIV(Z,,l< IIFIIï(qn).for ail F in the unit bail of @(a&), implying llPrnll 5 1. The rank of Pm, rank(P,), defined as the dimension of the image space of Pm: is such that rank(Pm) 5 m. The operators Zm = PmZPm?m = 0: 1.2, -.. : have rank iess than or equal m. Consequently. each operator Smcan be represented by a finite matrix formed by the n(m + 1) first columns and 2n(m + 1) rows of the infinite matrix representing E in expression (3.55). Example: For m =1, the previous example gives Sext. we show that in fact, as m tends to infinity, tmconverges to & in the strong operator topology, i.e.. for f E H*(&), Il f llL?tGl 5 1, we have that lim lit f - Pm"Pmf llL2 Irideed. by adding and subtracting Pm=f , the triangle inequality implies Since @t, is finite dimensional, by the Riesz-Fischer Theorem (cf. Lecture VI. [40]) the two terms on the right-hand side of (3.77) tend to zero: and thus (3.76) must hold. In particular under assumptions (Al) and (-42): there exists a maximal vector f € H~(@,). Ilf IlL?(; = 1. and Iim IIE f - IFmEPm f llLz(qn 1 = O rn-oc However llL2(q,) L~(G> (3.79) JIPmEPnf ,n 5 IIPmII(IEPmf 5 II'P'IIIIf II 5 II'II 3.3.4 APPROXIMATION BY NORSIS OF FINITE RANK OPERTORS T herefore (3.78) implies IlZll - Lm llPmEPmfllL.(q.,=O m-oc Since P,EP, is finite rank. then it has a maximal vector fm E H2(G).11 fml[r?(G) = 1 for each m. It follows then Thus IlPmtPmil -+ IJtllas rn -+ ~o'or equivalently. IIPmzPmfmlls(q,, - 112 f llLz(q, >- Moreover. as noted earlier with respect to the canonical basis {l,z,. . . zm}, IP,EPm is rcpresented by a matrix obtained by truncating the infinite matrix representation (3.55) of -. L. i-e.. Qom alm ... Umm The columns of Z, can be computed recursively by using the previous recursive algorithm. Tlie samc procedure used to compute the norm and maximal vector for the adjoint E* provides the norm and maximal vector for the adjoint of Em: Z;? obtained by taking siniply the adjoint matrix of (3.82). Now, we claim that the maximal vector, of the ad- joint Z;, for each m. enjoys property (3.60). To prove the claim, let hm = (h:. ) (PmRSpan{l. z. . . . . zm})'? where (PmRSpan{l. t. . . . . rm})lis the orthogonal complement of P,RSpan{l'z.. . . .zm}' and i(hmllL2(2n) = 1. Then the same argument used in the pre- vious section. (for E). yields min - gm~Sp~{l.~,...,="') L2(G,) 3.3.4 APPROXIMATION BY XORVS OF FLXiTE R4XK OPER4TOS.S which is clearly by Cauchy - Schwarz inequality Hence equality holds throughout. and (3.84) and 3.85) imply that hm satisfies (3.60). Le., where hk, = CEohk,,lz'. for htmVi E @1 k = 1,2 and 1 .I is the Euclidean norm. In the scalar case, (n=l). Zz is a (m+ 1) x 2(m + 1) matrix. Its maximal vector hm = ... is a 2(m + 1) vector, with the property that Partition Eh as (Z:, ? q,)? where zm?k = 1,2: are (m + 1) x (m+ 1) matrices. 3.3.0 OPERATOR THEORETIC SOLUTION TO THE MIMO EXTEKSIOX OF ORDAP Then the norm of 2, reduces to computing or equivalently. by passing to Z, ClearIy. such norms are bounded above and below, by respectively IEIm]+ IZ2,j and - max(lZlml 1~~~1).Le. Remark 3.4. It is therefore possible to approzimate as closely as desired the norm of - i. and hence p,. by computing the special matriz norrns dejîned by (3.88) or (3.89). From expression (3.81). we see that ezplicit computation of the maximal vectors of Zm, (or 3;): vil1 shed light on the form of the mm-mal vector of Z: and therefore the optimal controller Co. 3.5. Operator Theoretic Solution to the MIMO Extension of ORDAP 111 this section we give a solution based on operator theory for the ORDAP for MIMO systerns along the Iines of section 3.2. The key observation here is to interpret the norni (2.56) as an operator induced norm in special vector valued Hardy spaces. 3.5.1. AnotherKey MultiplicationOperator. Let L'(&,,) de- note tlic Banach space of Lebesgue square integrable (and analytic) &,,-valued functions on the unit disc under the norm wliere 1 - 1 denotes the largest singular value. Likewise define lL2 (en,,)(Hf(c,..)) to be the Banach space of &,,-valued and Lebesye square integrable (and analytic) functions on the unit disc endowed with the norm, for 3.3.5 OPERATOR THEORETIC SOLUTION TO THE MIhIO ESTESSION OF ORDAP acting Erom L?(&, ,) (or H'(&, ,) ) into IL2 (C2,,, ) (or !FI" (C2n ) ). more precisely Clcarly. LW,+ is a bounded linear operator. We show in the next Proposition that the operator iriduced norm is equal to II@II-. Proposition 3.3. Let PROOF. 1. First we show that IIM+II 5 II t herefore 5 ess sup max (1+r(eZe) Conversely, letting E > O, then there exists E c X17 m(E)> O such that (3.97) Choose g = XE - I E L2(Gx,), where XE and 1 aret respectively. the characteristic function of E and the Identity map. Then 1191122(n,, , = m(E)5 1 and However the right hand side is such that 3.3.5 OPERATOR THEORETIC SOLUTION TO THE MIMO EXTENSION OF ORDXP hence which together with (3.96) implies that (3.93) holds. 2. Equality (3.94) foilows from the same argument used to prove Proposition 3.1 since the trigonometric polynomiak with coefficients in GXnare dense in L'(L,, ): and then multiplying by a suitable power of z gives the result. O The dual space of IL2 (c2,,, ) is given by IL2 (q;:, ) and vise-versa' hence IL2 (&, ,) is reflexive [Il].In the next Proposition we characterize the dual space of @ (G*,,,). Proposition 3.4. Let il@ (c,,,) and El? (q::,) defined as above. Then 1. PROOF.Follows fiom the same arguments as in the proof of Proposition 3.2 since c2,,, and q,":,,are finite dimensional and then the positive Riesz projection P+ is a bounded linear opcrator. O 3.3.5 OPERATOR THEORETIC SOLUTIOX TO THE SIIMO EXTENSIO': OF ORDAP 3.5.2. Exact Operator Theoretic Solution for the ORDAP of MIMO Sys- tems. Let II' be the orthogonal projection on the closed subspace I@ (Cln n )BRN~(G,, ) of 3i (@?, ,, )' where (&, ,) 8 RH^(&. ) is the orthogonal complement of &H2(&, ). (RH'(&,, ))'-. Here orthogonality is understood to be with respect to the inner product of matrices, Le.. A, B E Ilif (enxn) are orthogonal if and only if Thcn II' is a bounded Iinear operator on l@ (&xn)- Next. define the following operator Clcarly. II' associates to each f, f E Hz (& ,, ) its best approximation in the min - max h€ Ril2(P,, ,) (y),-h ~~(Gnxn ) llFlIi?<~&>5 1 - max 12TJ'*IIf( 7 ) f (eie)dm = IIFIIL~Ç~~~)5 1 The following Theorem quantifies optimal performance in terms € Er dong the lines of Tlicorcm 3.1. Theorem 3.3. Under assumptions (Al), (A2) and pi > &, the following hold: 73 3.3.5 OPERATOR THEORETïC SOLUTION TO THE MI-MO EXTENSION OF ORDAP i) zs equal to the operator induced nom of S'? namely ii) Tf~ereezists a maximal vector for El. i-e.. f E H2(GXn),Il f Il L2(G,n = 1 such that PROOF. 1. The proof is similar to the proof to Theorem 3.1. First note that VQ E HCS(Gxn).and VG E Hz(&,,), we have QG E HZ(&,,), Le.. QH2(@ixii)C H2(G,,). hence ~'RQG= O. Let F E H2(@lxn),with norm IIFilr?cn,, 1 5 1. we have 5 min G€H2(GX,) G€H2(Gxn) ~*(C2nxn) ~2 (cari x n ) Consequently S UP (3.106) GE!~'(Gxmin n 1 (0)F-fiG IIFIIL~c,,~~)5 1 72- (~2, r x n ) For t hc reverse inequality, note t hat and flatncss impties = 1, m a.e. 3.3.5 OPERTOR THEORETIC SOLUTION TO THE MIMO EXTENSION OF ORDAP Then there exists a function h E H2 such that lnequalities (3.107) and (3.111) imply that 112'11 = 2. FoIlows from (3.107) and (3.110). The orthogonal projection II' is also given by Theorcm 3.2, Le., wlicrc this time I is the identity map oii Hf(@?,,,). P+ is the positive Riesz projection froni k' ) into @ (enxn)-The developments of Sections 3.3 and 3.4 carry over, and the norm of E' can be approximated as close as needed by the norms of a sequence of rint rices. However we need n independent maximal vectors to determine completely the opt irrial performance (and thus the optimal controller) - RQ., which in this case 3.3.5 OPERATOR THEORETIC SOLUTION TO THE MIMO EXTENSIOX OF ORDAP is also in general non-unique. It is worth noting that, if ORDAP corresponds to the optimal robust mode1 matching problern described in section 1.2.3. then for MIMO systems, 11 2'11 is the best uncertainty reduction achieved by a single feedback control law. CHAPTER 4 ORDAP for Unstable Systems 4.1. Introduction In this Chapter we generalize Our previous results obtained in the context of stable systems to (possibly) unstable systems. We consider a general family of linear time-invariant plants described by a bal1 of uncertainty centred about a nominal model, and subject to uncertain external disturbances. For ORDAP. the objective is to find a feedback control law which provides the optimal uniform attenuation of uncertain output disturbances for a11 plant in the ensemble considered. This problem is basically different kom the problem of optimizing the radius of stability of unstable systems solved for instance in [50, 361 with a soniewhat larger class of uncertainty. The latter problem is in turn equivalent to optimizing nominal performance, while our problem deals with optimizing performance for a whole ciass of plants including the nominal model. Giveii an uristable nominal plant Po, it is suggested in [??] to use a two-stage design proccdurc for sclecting an appropriate controller. First select a stabilizing controller C for Po and thcn select a stabilizing controller for P where P = Po(l+ CP0)-' which optimizes performance. However it is shown in [74] that the resulting configuration is not optimal in gcncral for the original plant Po. i.e.? the two stages are not independent. In fact they are independent if and only if Po is strongly stabilizable, Le., stablizable by a stable controller. This is thc case provided the number of poles of Po between any pair of real zeros inside the unit disc is even [?4]. In this Chapter we take a direct approach leading to an exact 4.42 PROBLEM FOLMULATIOX characterization of optimal performance/controller along the Lines of Chapters 2 and 3. Some of the results in this Chapter appeared in 112, 151. 4.2. Problem Formulation Let Po denote the nominal plant and let its fiequency response domain neighbourhood consists of al1 plants P which belong to the uncertainty set , ) = {(I+ VA)Po: V" E Hm(@Rxn),A is Linear time - invariant (LTI) (possibly unstable) transfer function in Hi t herto, suppose (A3) Po does not have poles on the unit circle m. Then it is shown by Verrna et al. (721 that there exists z stabilizing controller for the uncertainty set C(Po,V) if and only if (A4) al1 plants in C(Po, V) have the same number of unstable poles. WC henccforth assume so. It should be noted that in C(Po, V)we allow the locatious of the unstable poles of the plant to Vary. The model (4.1) bas been reported, for example, as a model of an unstable aircraft in longitudinal motion [23].The nominal plant was an infiriitc dimensional transfer function with an unstable pole and a delay. In fact, many simplifications were made to obtain such an "abstract model" for an aircraft, and the uriccrtainty set (4.1) was found appropriate to take into account the unmodeled dynamics. It was also shown there that stabilization, robustness to unmodeled dynamics, and optimal sensitivity reduction for this problem cmbe put into the optimality criteria (see expression (4.12) below) considered in this Chapter. The weighted sensitivity function is as previously givcn by The nominal transier function Po admits right and left coprime factorizations as follows 1741 where Mot No' M, and & are in Hm(Gxn)and there exist Xo? Y.' X, and Io alI in HX.(& ,, ) satisfying the generalized Bezout identity (741 It is well known that the controllers which stabilize Po can be parametrized in the form 1741 wliere (3 E Hm(Gxn) and determinant of (yo- NoQ). det(Io - NoQ) # O. C must be constrained so as to stabilize al1 plants P E C(Po,V). In mathematical language this is summarized in the following Lemma. Lemma 4.1. C robustly stabilizes the farnily of plants C(Po7V) if and only if IIVNo(Xo+ PROOF.It follows from Proposition 2.1 [46] that C robustly stabilizcs C(P,: V) if and only if IIVPoC(I + P0C)-' 11, < 1. However Iieiice C stabilizes C(Po?V) if and only if IIVNo(Xo+ QM,) 11, 6 1. O Expressions (4.2) and (4.6) imply then S = w((1tP.C)+AVP,C)-' = W(It P.C)-' (1 + AVP,C(I + P~C)-')-' = W(Y, -N~Q)&(I+AVN~(X~+Q~~)-' (4.9) follows since straightforward computations show that 4.4.2 PROBLEM FOFWULATIO-U' The worst case weighted sensitivity function denoted by Si is thus The optimal worst case transmitted disturbance energy, p, is then given by the foiiowing expression -4s in [4, 30, 521 this non-linear non-convex constrained optimization problem can be coriverted to a hed-point probiem, this is stated in the following Theorem. Theorem 4.1. Suppose No zs not invertible in HOQ(&,,). Under assumptions (A.3) a71d (-44) the following hold 1. i) and ii) are equiualent i) C is a robustly stabilizing controller for the set of unstable plants described by C(Po,V), and ess sup sup (1 W(Y, - N~Q)M,(~")cI + rlVNo(Xo+ QM,)(~'*) 2. Suppose there etists a Ql E Hm(&,,) svch that IIVNo(Xo + QM,)II, < 1- If on optimal Q E Hm(&,,) exzkts in (4.15) for r equal to the smallest fied point of 7 and ~(0)> O, then the optimal worst case weighted sensitivity /I is equal to the 1.4.2 PROBLEM FORMCiLATION srnailest fized-point of the function inf ess sup (~w(~-N~Q)~?~(~'~)<~+~~VN~(-~~+QM~)(~'~)<[) '(') = QEHw(& rn ) 0~[0,2r) Remark 4.1. The ezi.~tenceof a fized-point for the function q(r)follows from the fact - that q : [O. SC) - [O. m) is continuous non-decmasing in r. g(0) = infqEHm(G,,) IIFV(Y, - No~)-&~olil,iis> O. and for we have g(r)-r < O, By a Theorem fiom elementanj analyszs there ezists at least one fied- point in [O?oc). The continuity of ~(r)- r also implies that the set {r E [O!m) : g(r)-r = 0) is closed and there must ezist a minimum element. PROOF.The proof of Theorem 4.1 is identical to the proof of Theorem 2.1 in [52]?and therefore is omitted. O The optimal robust performance index can be simplificd further. For this manoeuver we need the concept of CO-innerCO-outer niatrices. A matrix A E HM(%,,) is termed CO-inneror co-outer if its transpose is inner or outer respectively. Moreover for cach matrix -4 E H°C(Gxn). there exist CO-innerand CO-outcrmatrices A, and A,, such that Tlicn &l0 E Hw(Qxn) has a CO-inner co-outer factorization = M~M~O;M,' MCO E H Since by assumption (A3) Po has no poles on the unit circle m' then M=. lias an inverse in Ho0(Gxn)[75], and can be 'absorbed' in the free parameter Q. Likewise No lias an inner-outer factorization. No = N,N,, N,, No, E HbC(GXn).WNOl and VNo 4.4.2 PROBLEM FORMUL-\TIOS have inner-outer factorizations given respectively by uiW and V,V. Expression (4.17) can be written as inf ess sup sup (((wI,- u~wQ&)(~")c~+ J(VX, - ~/QM~)(~")C() QG(&[0,27;) ICI L 1 where W.Y E Hm(@nxn) are general outer functions. Wi E Hw(Cxn) is inner. and ifaE Hr (GXn)is a co-inner function. Letting w$; = Wl: and Yx, = VI.we get inf ess sup SU^ (~(W~-U~WQM~)(~'~)CI+I(V~-~Q~~)(~~~)CI) QE~TGx~8~[0,2x) ICI 5 1 The two-disc problem takes the form inf Il 1 wsl + PlIlcc C proper and stabilizing wlicre W. V E Hm(@,,,) are outer and T is the complementary sensitivity function relating the controller input disturbance to the system output. Mathematically S traightforward computations yield inf ess sup (1 W(Y~- N~Q)~,(e'")l + IVNo(X0 + Q&)(eie) 1) (4.23) QEIIYP,x 0~[0,2~) wliich is equivalcnt to inf ess sup (I(u;w~M: - WQ)(~~')[+ I(v~~ + VQ){~")I) (4.24) Q n 0~[0,2r;) 4.4.3 BAVACH SPACE DUALITY STRUCTC'RE As seen before the optimization problem (4.24) is a little simpler than (4.20), in the SIS0 case they are identical. In this Chapter we will illustrate how our theory generalizes to un- stable systems by considering merely problem (4.24) for simplicity. Of course, generalization to (4.20) follows in a similar fashion. 4.3. Banach Space Duality Structure 4.3.1. Existence of Optimal Solutions. As in Chapter 2 we assume that (A5)there exists 6, > O such that (W*W t v'v)(eiB) 2 6,, V6 ,B [O, 2x1. This excludes strictly proper plants Po but as in Chapter 2, ou;- theory can be extended to them. Expression (4.23) is the equivalent to p = inf QEHa(Gxn) E" "1 (Y,,,) Expression (4.25) is the shortest distance from the vector ( ) to the subspace Sl = w here so that TT= 1. rn a-e., Le.. T is inner; and Al can be 'absorbed' into the free parameter Q. In the following Lemma we characterize the preannihilator lSi of S Lemma 4.2. The preannihilator of Siis giuen by 4-43 BXVACH SPACE DUALITk' STRUCTURE P ROOF. Consider and lz7Tr(KT. G)(I - TT)( z: ) @(eie)dm = O QG E !L1(Gyn) Since H=(&, ) ~~isthe annihilator of H: (& ,, )& in L' (1c,rnyn) and h>z(~;.K,')(I - TT)(='')= O m a.e. Once again by duality we deduce the following existence Theorem. Theorem 4.2. Under assumptions a (A3) Po does not have poles on the unit ci~clem. a (A4) al1 plants in C(Po,V) have the same number of unstable poles. a (AS) there ezisfs 60 > O such that ( W'W + ~*p)(e'"2 60, 'IO E [O, 2x). Then there exists at least one optimal Q, E HS(Gxn) which satzsfies which by duality 2s equal to rup IlzTTr(W;. V;) (tzio)dm1 II [fl ILS,5 1 where II - Illsi denotes the coset nom in LS i. 4.3.2. Allpass Property, Absolute Continuity and Existence of an Extrema1 Identity. In this section we assume that (A6) .4?~. Ac&0.Al. Tl' T2' VI and Wl are continuous on the unit circle. Dcfinc the subspaces Sic and respectively by 111 the fo1Iowing Lemma we characterize the armihilator of S 1,. Lemma 4.3. The annihilator of Si, is given by PROOF.The annihilator of A(C,.,,,)~I&:where A(& ., ) is the disc algebra on the unit disc of functions with values in C,, , is given by HA(&., )M, since if G E H:(G.,) then 4.4.3 BANACH SPACE DIiXLITY STRUCTURE Hence ~QI~(&xn)M& - z-du(e) =GU,, GER^(&,,) o du(8) = (I - ~z-)dd(e)+ TGM,~~,3 U' E IW(C~!~~)(4.35) (since T(I- TT)= O, TT= 1) The following Lemma establishs a similar result to Lemma 2.2. Lemma 4.4. Under assumptions (A3) Po does not have poles on the unit circle m. (A4) al1 plants in C(Po,V) have the same number of unstable po,les. 1-45) there ezists ho > O such Mat (W'W + v*v)(eie)2 6,. W8 E [O? 27r). (A6) M,~Mcor Al- Tl:Tz7 VI and Wl are continuous on the unit circle. There emkts at feast one optima/ Q, E HOa(&,,) such that wliere II - Ilsl is the coset nom in S:,. 1 c PROOF.The proof is similar to the proof of Lemma 2.2, and therefore ia ornitted. O If we remove the open unit disc analyticity constraint on Q, we get = inf IIIWl - T~QM,~+ IV1 - ~2~M,-jlllm Q€e(Gx n ) 4.4.3 BANACH SPACE DC'ALITY STRUCTURE Xext Theorem establishes that under specific assumptions the optimum is also flat (or --all pass" ) . Theorem 4.3. Under assumptions (.43), (A4)t (AS) and (A6), ifp > p'. then i. Any optimal Q, in (4.36) satasfies the flatness (or "allpass") condition ii. If {Qn)n=l is any sequence in Hffi(C,,,,) such that Ti~econdition p > is sharp for both conclusions in the sense that if /I = then there ezists kt/, V and Po for which (4.39) and (4.41) are false. PROOF.Theorem 4.3 is a gencralization of Theorem 2 [79] (or Theorem 3.4 in 1521). Thc proof is completely analogous to it, and therefore omitted. O Remark 4.2. CVhen Theorem 4.3 holds, duality theory developped in Chapter 2 cany over to unstable systems. Moreover Lemma 2.4 holds for cosets in 'Si, i-e., for al1 [FI Si there ezists x, E Xi such that IIIF]llls, = 11 F + xoll;~(qnyn p The maximum in (4.37) is then achieved by a measure absolutely continuous with respect to the Lebesgue measure. Lemma 4.5. Under assumptions (-43) Po does not have poles on the unit circle m. (-44) al1 plants in C(Po:V) have the same nvmber of unstable poles. (AS) there ezlsts 6, > O such that (W'W + q'v)(ei0) 3 ho, VO E [O, 2a). (AG) M~, Al Tl:T2, Vl and Wl are continuous on the unit circle. 4.4.3 BAN-ACH SPACE DUALITY STRUCTURE (4.42) PROOF.The Lemma follows from the same arguments used to prove Lemma 2.3. O Remark 4.3. The results of Lemma 4.5 including mazimization in the predual, in cun- trast with minimum with respect to the parameter Q E Hm(Gxn)in the dual lead to a dual pair of numerical solutions in the conuez progmmming method developed in Chapter 4 [52]. These numerical solutions ayproach the optimum from opposite directions and produce an estimate of p. w1 - Ti QOM, Similarly to Theurem 2.2? the dual eztremal vector function and the eztremal kernel f, achieving the mm-mum in (4.42) satisfy identity (4.43) below. Conuersely, any vector function and kernel satisfying (4.43) are eztremal. Hence the fol- lowing Theorem providing a test of optirnality. Theorem 4.4. Under assurnptions (A?) Po does not have poles on the unit circle m. (A4) all plants in C(Po.V) have the same number of unstalle poles. (-45) there edts 6, > O such that (W'W + ~'~)(e")2 VO E [O: 2x). (AG) &, McO,Ai, Tl T2:VI and Wl are continuous on the unit circle. If p > pl then, Fi = ( :: ) E FI E~ ilfi kq,n) - /I[F]l/ls,= 1: is an eztnmal kcrnel for [FI; and Q, E HOC(Qxn) is optimal if and only if 4.4.1 OPERATOR THEORETIC SOLUTION holds Lebesgue almost euerywhere. wi - ~1~2% The optimum is a dual extrema1 function for the approximation prob- 4.4. Operator Theoretic Solution In this section we generalize the operator theory fiamework developped in Chapter 3 to possibly unstable systems. Define iil to be the orthogonal projection onto the closed subspace L2(c,) 9 TH*(&) of L2(G,). where L2(q,) 8 TH2(G)is undentood to be the orthogonal complement of TH'(@, ). (TH' (@, ))L- Next define Xote that ril associates to each f. f E M;H~(& ) its best approximation in the siibspace TH?(@,) in the norm of k2(q,,, ). Again by duality min (;)f-h - ma h€TlP(C" ) L2) -1 IlFIl~q~~-)sl max (4.46) IIFIIL~(,~--~) 5 1 12- (%,, .Pl 1 The following Theorem quantifies optimal performance aIong the lines of Theorem 3.1. Theorem 4.5. Assume that assumptions 0 (A3) Po does not have poles on the unit circle m. (.44) al1 plants in C(P,, V)have the same number of unstable poles. a (A5) these ezists 6, > O such that ( W'W + v'V)(eiO) 2 6,: Vtî E [O, 2n). 4.4.4 OPERATOR THEORETIC SOLUTION (A6) M,: Mc0, Ai' Ti T2,VI and Wl are continuous on the unit circfe. hold, if p > p', then 1. ii. There exists at least one rnuxîrnd vector f E M;H~(c) of L*(L)-nom 1 such that PROOF.Follows from the same arguments used in the proof of Theorern 3.1 modulo minor modifications. D We saw in Chapter 3 that IIi = I - TP+T* is the orthogonal projection onto the subspace I;' (c,) 6 TH^ (@, ). The norm of the operator El can therefore be computed by the same numerical procedure developed t here. In particular Iri the scalar case (n= 1) The optimal parameter Q, is given by Q, = Q,M,'. The optimal controller Co can be cornputed from Thoptimal feedback control law Cominimizes the largest W-weighted output signal energy which results from any external disturbance d E D to the output of any plant P in the uncertainty set C(P,, V)!Le., achieves optimal robust performance. CHAPTER 5. RELATION TO THE STMiiARD TWO-BLOCK H" PROBLE4I CHAPTER 5 Relation to the Standard Two-Block Hm Problem -4s noted in Chapter 1 Jonckheere and Verma showed that the problem of optimizing the H rnixed-sensit ivi ty is equivalent to characterizing the spectrum of a "Hankel-Toepli tz" operator [431. In particular they proved that for rational plants under certain conditions. the HX performance is reached at an isolated eigenvalue with finite multiplicity. More recently Georgiou and Smith using normalized coprime factorizations showed that the problern of optinlizing the radius of stability in the gap metric is equivalent to a special version of the standard two-block HOC problem [37J.In particular they were able to provide expiicit formulas for the optimal radius of stability and the optimal controller in terms of a HankeI operator and its maximal vectors and corresponding eigenvalues when the problem data are continuous. In this Chapter, under the continuity assumption, we generalize some of their results to the standard two-block HOo problem by developing a duality approach parailel to the one in Chapter 2. In particular we show that for MIMO (possibly infinite-dimensional) systems the optimum is Bat and the Hankel-Toeplitz operator discussed earlier achieves its riorm on the discrete spectrum. Zames and Mitter presented a method of computing spectrums. eigenvalues and eigenvectors for generaI systems subject to continuous weight- irigs [?BI. It should be noted that the standard two-block Hm problem provides a "good" approximate solution for OFLDAP in the case of *'almost7'cornplementary weightings W and V. Le.. IIW'VllcK: = E << 1 [52, 541. For simplicity we consider Iinear time-invariant stable plants. The unstable case can be settled similarly. 5.5.1 DU-ALITY STRUCTbRE OF THE PROBLEM The performance index under consideration is therefore where Po E Hm(&,,) is the nominal plant. Wl and Vl are outer weighting functions in H(IC(G,,). -4s shown in Chapter 2. under assumption (Al) (@w + v*V)(ezB)> 0. VO E [O. 27r). Where Y and w are outer functions in Hx(rt,,,) ( see expression (2.9)). This problem is equivalent to 0 = inf QEHOO(Gxn) -RQ (:) CXI where R E (CZnxn) is inner: and W outer in Hoc (& ,, ). Recall that WOO (C2,,, ) is the Banach space consisting of pair of bounded analytic 2n x n matrix-valued functions on the unit disc 9. under the norm Expression (5.2) is the shortest distance from to the subspace S = RHoC(& ,,) iri the (CZn )-norm. 5.1. Duality Structure of the Problem Define ILOO (Ga,, ,) to be the Banach space of bounded matrix-valued functions on the unit circlc XJ under the norm (5.3). Whereas E' (q,,,)is the Banach space of @2,,,- valued integrable functions defined on with the norm It is well known in the theory of vector-valued LP-spaces t hat the dual of ILL(Q,, ,) is isornctrically isomorphic to ILm (G,,,) [11j7since G,,, is the dual space of q,,, [64j, and vise-versa since these spaces are finite dimensional. To every functional $ on IL1 (Q,,,) 5.5.2 ALLPASS PROPERTY AND =\LIGXMENT O; THE DC'AL there corresponds a vector function Kd related to q5 through the following bilinear form The sanie argument used in Chapter 2 yields the preannihilator of S where = ((1- RR*)L~(G,,,) R~:(G,.))~~:(G,,,) Hence the following existence Theorem. Theorem 5.1. Under assumption (Al) (W'W + v*v)(eie)> 0, Vt? E [O. 2n). There exists at least one optimal QoE Hm(&,,) such that 5.2. Allpass Property and Alignment in the Duai Let e(G,,,) denote the space of C2,,,-valued functions which are continuous on XI uiider the sup-norm (5.3). The dual space of e(G,,,) is isometrically isomorphic to the space MI(@;, ,, ) of bounded qnxn-valued measures under the norm 5.5.2 ALLPASS PROPERTY .AND ALIGXMEXT THE DUAL where w, is the total variation on [O. 27r) of ail entries of v. and Gu,,E LL(@nxn,w"). r = 1.2. If 4 E C(&, ,, )'. then the isometric isomorphic is given by the bilinear mapping Define the subspace Sc = S n CO,,,): then the annihilator of Scis given by where Under assumption (A2)W. Ri.R2 are continuous on the unit circle, as is the outer spectral factor Al of MSr*Cv + PY. The following Lemma establishes that the distance from € C(@2nxn)to Sc is the Lemma 5.1. Under assumptions (Al) and (A2) the following hold PROOF.The first equality follows from a straightforward modification of the proof of Lcrnma 2 [79], the second fiom Theorem 1 (page 119: [49]). U 5.5.2 ALLP.4SS PROPERTY AND ALIGX-VIEXT I'i THE DCAL Wheu the open unit disc analyticity is removed Then a similar resuit to Theorem 2.4 follows Theorem 5.2. Under assumptions (-41) and (AZ), if 0 > ,& then i. Any optimal Qu E HbC(Gxn)satisfies the allpass condition ii. If {Q,,)F=,is any sequence in HS(&,,) such that The condition P > Poo is sharp for 60th conclusions in the sense that if P = Poo, then there exist W:V and Po for which (5.1 7) and (5.19) are false. PROOF.Follows by the same argument used in the proof of Theorem 2.4. The conter example given after Theorem 2 [79]can be used to show that condition fl > fl, is sharp. 0 Remark 5.1. The same argument used in Chapter 2 shows that the extrema1 measure u, in (5.15) is absolutely continuous with respect to the Lebesgue measure. More precisely, llrere ezists a uector function Fm E EL(Gnxn) such that duo = F,dm. and hence dv, = F,Gdm. Therefore the supremum in (5.9) is achieved by the coset [f,]= [F,G]. -4gain a similar argument to Lemma 2.4 shows that there ezists a vector function F, E [fol:such thal Il[fo]llrt.= IIFollLicGnXn1 = 1. A similar identity to (2.50) (Theorem 2.2' Chapter 2) is deduccd. It should be noted that the duality theory deueloped here for the standard two- block Hm probfem fits into the convez progrumrning algorithm given in [52], and therefore provzdes another numerical solution diflerent from the usual well known c itenztions (sec for: cg.. [43]). Theorem 5.3. Under assumptions 5.5.2 ALLPASS PROPERW AND ALIGNBIEXT IX THE DUAL 0 (A2) W.RI, R2 are continuous on the unit citcle, as is the outer spectral factor Al of LV*W + Y*Y. If fi > ooOso that patness holds, then Fo = ( a; ) E [FI E' 9' II[F]IIL~= 1. 2s an eztremal kernel for [FI, and Q, is optimal if and only if W - &&O The optimal is then a dual eztremal function for minlEx II F+ f Ilsi cc; = R2Qo ,n x, 1. PROOF."Only if' by assumption 3Fo E' S. IIFOllLicqnxn1 = 1. and Qo E HDE(Gxn) stich t hat the following alignment condition holds but the integrand Tr {((w'.O) + QER') F,} (eie)5 (Iw - RIQ.[' t IR~Q,~*)~((ST~F~~)~+ (ST~F~~)')(eio) (5.22) rn a-e., by Cauchy - Schwarz inequality iritegratirig implies equality must hold throughout. This cornbined with fiatness imply i0 ((s~r(F~1)'+ (STr(F,*) 2 )~(e' ) = 1. rn a-e.. 5.5.2 ALLPASS PROPERTk' .;L'm ALIGNMEiST LX THE DCAL .'If' suppose that (5.20) holds, integrating it yields hence equality must hold throughout and Q, is optimal. Remark 5.2. In the SIS0 case, (5.20) reduces to wh ich implies Sirrce (1~~~(eiB)['+ ~~,~(e~~)l~)(e'~) = 1. m a.e., the set E = {O : ~,~(e'~)= ~,~(e") = O} has Lebesgue measure O. However integrating (5.22) and (5.23): and since equality must hold (in the Cauchy-Schwarz inequalzty): we get for some positive constant y. This shows that F,I and Foicannot vanish on a set of positive rneasure unless Q, G O. But this would give a non-jlat solution for IWI non-constant. Ezpression (5.24) determines Q, uniquely. 5-53 OPTIMAL SOLUTIOS \;I-4 OPERATOR THEORY 5.3. Optimal Solution via Operator Theory Let again II be the orthogonal projection ont0 the closed subspace I@ (Gn) E RH2(& ) of 8(C2n): where @(@2,) is understood to be the Hardy space of C2,-valued functions defined on 9. under the Hilbert space norm Define the operator E2 by We obtain a Theorem similar to Theorem 3.1. Theorem 5.4. Under assumptions - (AI) (W'W + PV)(eZB)> 0' VO E [O. 27r). r (-4.2) W.Ri, R2 are continuous on the unit circle, as is outer spectral factor Al of wtw -+- Pi? If 0 > O,, (2. e., Patness holds), then ii. There exists a mazirnal uector f f H2(@,) of L2(& )-nom 1 such that PROOF. i. FolIows either from the cornmutant lifting Theorem [28], or from slight changes to the proof of Theorem 3.1. Note that flatness is not necessary for (5.27) to hold. ii. Follows from a similar argument used in the proof of ii. in Theorem 3.1. 5.5.3 OPTIMAL SOLUTION MX OPERATOR THEORY Theorem 5.1 implies existence of a vector function ik E HP' (C2,,,) such that siricc llnll 5 1. It foliows as previotisly that Under assumption (Al) there exists a square inner matrix (R, Rl)?where RI = )? (\ :;: I such that [8] R;W and RTL W belong to Lm(&,,). Using spectral factorizations of the entries of the riiatrix on the Ieft-hand side of (5.32) one can show that there exist inner matrices Ur€ Hx(GLxn) (Theorem Z [43, 731: Chapter 8 [29])such that Then we rnultiply the rnatrix on the right-hand side of (5.32) by the inner matrix Mi and obtain 5-53 OPTIMAL SOLUTION Vi.4 OPEUTOR THEORY where G. f2 E HOC(G,,) and U E Hm(&,,) is inner. Our optind performance index /3 has therefore the following "Hankel-Toeplitz" formuiation 173, 431 where rGr.Gand TQ**are respectively the Hankel and Toeplitz operators with symbols U'G and R'R (respectively). More explicitly if we let 8 = U"G to simplify the notation, then where H2(@,)' is the orthogonal complement of H~(&) in L'(& ), and P-!P+ respectively tlit iiegative and positive Riesz projections [73,43]? Le.. It is well known that re-re + T'-R has a spectrum with continuous and discrete parts (51,43, 781. The continuous part corresponds to the essential spectrum. Under assumption (A2). R; W is continuous and the operator Te is compact [51].The spectrurn of Te*re + Tn-n is formed by the essential spectrum plus isolated eigenvalues with finite multiplicity. WC show that under assumpt ions of Theorem 5.4, the operator re=ïe + Tn* achieves its norm on the discrcte spectrum, that is and thercfore gcneralizing the same result obtained for rations1 plants and weightings in [43] to infinite-dimensional plants. 5.5.3 OPTIMAL SOLUTION VIA OPERATOR THEORY Prernultiplying (5.30) by Mi(R, RI)' we get and applying their respective adjoints yields Since multiplication by inner matrices preserves the L~ and ?c-norms? by passing to them from (5.33). (5.34) and (5.39) we obtain hence f is a maximal vector for re. re +Twn, in fact f is the eigenvector which corresponds to its maximal eigenvalue Am,. Hence we proved the following Corollary to Theorem 5.4. which generalizes Theorem 6 in [43] obtained for finitedimensional systems. Corollary 5.1. Under assurnptions of Theorem 5.4: the opemtor re. re achieves its norm on its dzscrete spectrurn that is A method of computing discrete spectrum and eigenvectors of moted Hankel-Toeplitz opcrators for SIS0 infinite-dimensional systems subject to continuous weightings is pre- scnted in [78]. CHAPTER 6 ORDAP for Continuous Time-Varying Systems Definitions and Notation 0 B(E,F) denotes the space of bounded linear operators from a Banach space E to a Banach space F. endowed with the operator norm. L?[o.x) the standard Lebesgue space of essentially square integrable functions de- fined on the interval [O. cm). DI, the family of delay operators defined for h 3 O by: and for h < O by: whcrc q,,) is the characteristic function of the interval (0:cm). An operator A E B(E.F) is said to be time-invariant if. for al1 h 2 0, A satisfies the operator equation (71 ADh = DhA P, the usual truncation operator, which sets al1 outputs after time T to zero. An operator A E 'B(E,F) is said to be causal if it satisfies the operator equation: The subscripts '.,,", .'=" and the superscript "'I"denote the restriction of a subspace of operators to its intersection with causal. strictly causal (see [27] for the definition), and tirne-invariant operators respectively. "'"stands for the adjoint of an operator or the dual space of a Banach space depending on the context. "-"? denotes convergence in the weak' topology. 6.1. Introduction In this Chapter we consider the Optimal Robust Disturbance Attenuation Problern for continuous time-varying systems subject to time-varying unstructured plant uncertainty. and therefore generalizing previous results obtained for discrete time-varying systems by Owen 1521. Hcre the plant uncertainty set is described by a weighted sphere in the algebra of bounded linear operators fkom L' into L~ instead of HOo, and the feedback control laws are allowed to be tirne-varying. In particular we show that for causal continuous (possibly time-varying) systems. ORDAP is equivalent to finding the smallest fixed point of a "two-disc" type optimizaton problem under time-varying feedback control laws. This in turn is expressed as a shortest distance in a special Banach space of bounded linear operators reiated to a functional maximization in its predual space. Therefore proving existence of optimal time- varying control laws. It is also proved that for continuous time-invariant nominal plants tinie-varing control laws offer no improvement over time-invariant feedback control laws and liencc settling an open question in (Chapter 8 [52]). 6.2. Problem Formulation Let Po E 'B,,(L~[o,oc)~L2[0, cc)) be the nominal (possibly time-varying) plant, and deriotc the set of plant uncertainty by (?(Po:V) = (P E %,(L2(-m. m), L'(-OC+ cm)): P = XVP, + Po, (6-5) X E 'B,(L~(-w. oc), oc: Co)), llxll c 1) 6.6.2 PROBLEM FORMUL.4TIOX where V is a causal stable. linear time-invariant weighting function. The ORDAP can be shown to be equivalent to finding the optimal worst case sensitivity function with respect to disturbances and plants in C!(Po,V), achievable by a feedback control law. Mathematically this is equivalent to wlicre MT is a causal stable. Linear, time-invariant weighting function. Expression (6.6) can bc u-ritten as inf Q E IB,(L2, L') If a particular controller Q achieves a "worst-case" weighted sensitivity function less than some r > O then Expression (6.8) is equivdent to which yields u-hich is in turn implied by (see Chapter 5 of [27]) 6.6-2 PROBLESI FORMljL.4TIOX Hence Let < be the function defined for r E [O1 11 as follows: then < is a continuous. positive. non-decreasing function of r. We get a similar Theorem to Tfieorem 6.1 for discrete time-varyïng systems in [52] Theorem 6.1. [14] 1) Let Po be time-invariant and p0 as above: if the= exists an optimal Q E %,(L2,L2) for each r f [O. 11 in the ezpression (6.121, then p, is equai to the srnailest fized point of 2) If Po is time-uarying then p, zs bounded above by the srnailest fized point of c(r). PROOF.The proof follows by slightly modi-ing the proof of Theorem 6.1 in [52].Let Q E 'S,(L~.L2) be such that f E L2 Suppose by way of contradiction that there exists E > O such that Fix 6 > O and consider the following construction of a function F E L?[o:cc) Step 1: let ro = 0, and select f 1 f L2[0,oo), Il f 11 Lz 5 1, such that f 1 has compact support (i.e. closed and bounded in the topology of the reals) supp(fi), where the support of fi is defined to be the smdlest closed subset of [O, m) outside of which fi vanishes, and satisfies (6.14) for f = fI and r = O. Since VP,Q~~E LZIO,cm) there exists a positive real number TI > Isupp( f 1)l. where Jsupp(f 1)1 is the length of supp(f i). and such that Step k: select fk E ~~10,m) with compact support such that 11 fkll L? 9 1' Pr&-,fk = O. fk satisfies (6.14) for f = fk and T = O. Define rk > Isupp( fk)( and such that Next define If 6 is chosen sufficiently srnall with respect to IIVP~QIIand c. we get Dcfine X E B,(L', L~)to be the foIlowing contractive, causal. finite rank linear operator from L' - L' t I Xk = 9- and 11 VP,Q/~II Lz + l' 6.6.2 PROBLEM FORMWLATIOX If 6 was chosen sufficiently small for large enough N (6.21) contradicts (6.14) and so claim 1 is proven. Claim 2: The proof Claim 2 follows exactly as in the proof of CIairn 2 (in the proof of Theorem 6.1 1521) by making the same modifications to prove CIaim 1 above, and therefore will not be repeated here. Claim 2 implies that Since Po. W and V are causal tirne-invariant. and D-,QD, causal. we get Tiic rest of the proof is similar to the proof of Theorem 2.1 in [52J,and therefore omitted. Cl Hericc the optimization (6.7) for time-invariant nominal plants reduces to Xext we proceed to give the duality structure of the problem, wilich shows existence of an optimal Q for p,. 6.6.3 DUXLITY STRUCTURE XhrD EXISTENCE OF .LX OPTIMAL SOLUTION 6.3. Duality Structure and Existence of an Optimal Solution Let f2 be the Banach space ~~[O'rn)x L*[O? m) under the norm: The vector iunction ( "':,:"') can be viewed as a muitïpiication operator from L'[O. x)into L2 with the operator induced norm: Therefore the optirnization problem (6.12) can be expressed as a distance problem fkom the vector iunrtion ( ) belonging to T3(L2.L~) r~ the rulapace S = of S(L'. L'). -4s previously: we assume that Then therc exists an outer spectral factor Ai E Hm. invertible in HE such that AT(iw)Al(iw) = M;'(iw) W(iw)+ Vn(iu)V(iw). Then Ai Po as a bounded linear operator in B,(L~.L2) has a polar decomposition UiG.where Ui is a partial isometry and G a positive operator both defined on L? (Theorem 4.39 (191).Next we assume (A7) U1 is unitary and G" E 23,(L2.L2). Let R = ( ) A;'OI. assumption (AT) implies that the operator FRf T3(L2:L*) has a bounded inverse, this ensures closedness of S. According to Arveson (Corollary 2, [2],see also [27]).the self-adjoint operator FRhas a spectral factorization of the form: R'R=A*A, where A, A-'€ 23,(L2,L2) 6.6.3 DUALITY STRUCTURE .LYD E.XISTENCE OF AN OPTIMAL SOLUTION Define R2 = RA-', then R;R2 = 1. and S has the equivalent representation. S = R?'s,(L'. L*). After absorbing A into the fiee parameter Q, the optimization problem (6.12) is then equivalent to: Let L? be the Banach space L' x L' under the norm: The following Lemma characterizing the dual space of C2 foilows kom [ll]: Lemma 6.1. Let IL2 and L* defined as above. then the follouring hold .(p)* ~2 .(L2)* 2 iL2 Hence al1 these Banach spaces are reflezive. Define N(Bi: B2) to be the Banach space of nuclear operators mapping the Banach space Bito the Banach space B2 under the nuclear norm. Recall that an operator A : BI - B2 is said to be nuclear if it has the representation [IO] -4f = En < F,'. f > en. where en E B?, E B; (6.29) and C, IICII - IIen II < m (6.30) where < . - > is the duality product and the nuclear norm is defined to be wherc the infinmm is taken over al1 possible representations of A. The trace of the nuclear operator A is denoted by &(A) and is defined by This suni is well defined and can be shown to be independent of the representation [IO]. The foIlowing Lemma applies to the Banach spaces L* and L*. 6.6.3 DUALITY STRUCTURE .&\Tl EXISTENCE OF OPTIMAL SOLUTION Lemma 6.2. [7, 101 if 4 E N(L?E~ If, the isornetric isornorphism = is given by: O(A)=< B1A >= trA'B = trB7A, where A E N(L',L'). B E 'B(L',c') Next we need the following Definition. Definition 6.1. [9] A nest zs a chain IV of closed subspaces of a Hilbert spuce 31 containing {O} and 31 which is closed under intersection and closed span. The triangular altjebra or nest algebra 7(h/) is the set of al1 operators T such that TN N for euery element N in M. Given a collection {N,) of subspaces of a Hilber space, V, N, denotes the closed iinear span and A, Na denotes intersection. For N belonging to a nest n/, define where N' < iV means N' c N, and N' > N means Nt 2 N. The subspaces N 8 N- are called the atoms of N. If there are no atoms, JV is called continuous. Now let us define for each t E (-oc9 OC). Mt (1 - Pt)(L2),i.e.. the subspace of L' consisting of al1 functions f E L' such tbat f = 0' a.e. on (-m1t). Then M -A (Mt. -X < t < OC) is clearly a continuous nest since The space B,(L~:L?) can be viewed as a nest algebra since causal operators leave M invariant. i.e.. for al1 operators A E Z3,(L2, L2),AMt C ML,for al1 Mt E M. The Banach space N(L', L~)is nothing but the well-known space of trace-class operators from L' to L~.and that the nuclear norm in this case reduces to the trace-norm (641. Applying Theorem 16.6 [9], we get the following Lemma 6.6.3 DUALITY STRUCTLX AND EXISTENCE OF A3 OPTIMAL SOLUTIOK Lemma 6.3. ,Vc(L2: L2) is the preannihilator of 3,(L2, L'). PROOF.The preamihilator of 'B,(L'?L2), %BC(~',L*) is defined by Let n/ft be the closed Linear span of {MTE M : T < t ): but since M is a continuous nest: then Mc = Mt. It follows Lom Theorem 16.6 [9] that L'Bc(~2.L') = N=(L*.L~). O Now let S, be the subspace of N(C2,L'(O. m)) defined as follows wtiere p is the canonical projection of L2(-a.3~) into L2[0.m). and LZ is defined to be the Banach space L2(-oo?00) x L2(-cm, m) endowed with the norm of L'. Define the following subspace of N(L2?IL2) Lemma 6.4. iS is the preannihilator of S in N(L~.L'). PROOF.Let T E B(L2[0.m), Ç2).then < T'A >= 0' VA E' S a t~(~((U(1-RzR;) IL? T + pNR; T)= O V@ E N(L~,L2[0, m)) . N E N,(L',L2) o(I - R2R;) Ir? Tp= O and RG ILz Tp E 23,(L2,L2) by Lemma 6.3 o(I - R2R;) IL? Tp = O and R2R; Içz Tp E R~%,(L'.L~) oT E R~'B,(L*.L~) Lsirig a standard result from Banach space duality theory relating the distance from a vector to a siibspace and an extrema1 functional in the predual (Theorem 2. Chapter 5.8 [49]). we deduce the following Theorem Theorem 6.2. Under assumption (-47) Ui is unitary and Gkl E 'B,(L2,L'). 6.6.4 TLMEV.UtYiXG VERSUS TIME-IXV'VUULYTCOXTROL LAWS There ezists at least one optimal Q, E 23,-(L2,L2), i-e., a linenr time-varying contml law such that: min QE'Bc(L2.L2) (a)-... 6.4. Time-Varying versus TirneInvariant Control laws In this section. we assume that the nominal plant Po is time-invariant. It follows also that R2 is also time-invariant. Define the following performance index when the controllers are constrained to be linear time-invariant: In the next Theorem. we show that no advantage is gained in performance if we allow our controllers to be time-varying, and thus generalizing the same result obtained for discrete time system in [52, 531. Theorem 6.3. If Po is linear, causal. time invariant and assvmption (A7) holds. then PROOF.Since 23:(L2: L2) c 'B,(L~.L*)? it is obvious that po 5 ~2. To prove the opposite inequality let Q f 23,(L2, L~) such a Q exist by Theorem 6.2. Using an idea of Chapellat and M. Dahleh [7]:let IL > 0. Sincc llDull 5 1. Va- we have that Vn > O 6.6.4 TI-ME-VARYING VERSUS TIhfEmV.4RI-iLYT COXTROL L-ALVS Using time-invariance of W and R2, we get Define for al1 n 2 O t hen (6.45) yields IIQn(h)fl 5 11Q11, t/7L > O and then II R2Qn(h)llis uniformly bounded by IIR21111QII. Moreover since the predual space of 'B(L2,C2) is N(L2. IL2 ) : by Alaogluk Theorem there exists a subsequence of {R2Qnk(h))which converges in the weak* topology to some vector function R E 3(L2,L'),and we write By (6.47) and a property of weak* limits, we have By definition of the weak* limit: VA S. O =< R2Qn,(h),A >-< R1A > as k -+ r' hence R = R?Q(~).for some ~(h)E B,(L2:L2). The adjoint of Dh:DL is defined by Vf E L' Xow VA E N(L'. IL2 ) we have For al1 h > O. we showed that Finally for al1 n > 1 define the sequence R~Q, = R~Q(~).By (6.52) it is bounded. t herefore tliere exists Q such that R2Qnk -* R~Qand R2Q satifies (6.53) as before. Again using properties of nuclear operators it is easy to see that R~Q~~D~-D~R~Q~~-* R?QD~- R~Q. Csing exactly the same procedure as in [7]to prove that R~Q,,Dh - Dh-Qnr -* O shows tliat R?Q is time-invariant. but since Rz is time-invariant. Q must also be time-invariant and this completes the proof. 17 CHAPTER 7 Conclusions 7.1. Synopsis This thesis is concerned with the effcct of feedback on disturbance and plant uncer- tainty that may be large. Basically three problems were considered, the two-disc problern. the optimal robust model matching, and an optimal robust model matching al1 under the acronym ORDAP. These ~roblernswere intractable by established methods in the standard H" theory, and the duality theory developed in the work of Owen and Zames does note characterize explicitly exact solutions. in particular when certain Lipschitz conditions do not hold. This provided the motivation of the study undertaken in this thesis. We started by a re-examination of the duality theory of OFLDAP working in the smaller HP space (rather than the larger LX). ORDAP was then interpreted as a distance minimization, allowing predual and dual rcprcsentations under a continuity assumption for the latter. Useful geometric properties wcre deduced, for example, an extrema1 identity for optimal perforrnance/controller which provides a test of optimality: the extrema1 function in the predual space satisfies a norm constraint. The same results were carried out to the MIMO extension of ORDAP. Align- nient conditions relating the closest vector function in the distance minimization t.o the rriaxirrial elcment in the dual optimization were obtained. Various qualitative properties wcre deduced, in particular flatness of the optimum, and if a feedback is "nearly" optimal thian approxirnate flatness condition holds. Then a novel operator theoretic framework -"r .r -2 DIRECTIONS FOR FUTURE RESEARCH was developed using vector function and Banach space operator theory. The optimal so- lution was shown to be equal to the nom of a certain operator. analogous to the Sarason operator and which is a combination of multiplication and Toeplitz operators. The norm of this operator gives the best uncertainty reduction by a single feedback control law. An explicit formula was given for the optimal controller in the SIS0 case. Infinite matrix rep- resentation of the operator with respect to a canonical basis was provided, together with approximation by norms of finite rank operators. lntroducing new vector-valued Hardy spaces enabled us to interpret the norm involved in the MIMO extension of ORDAP as an induced norm. Then a solution based on operator theorg was also given and similar results hold. SimiIar ideas were generalized to MIMO unstable systems using coprime factorization tech- niques. after showing that ORDAP for this kind of systems could bc also reduced to an irnplicit two-disc type optimization. Duality structure shows t hat the probiem fits into the convcx prograrnming method developed in [52]. Similar properties of the optimum were &O derived. Relation to the standard two-block HW problem is examined in the context of duality. allowing to deduce qualitative properties for the optimum, e.g., flatness, extrema1 identity, cas well as quantitative results concerning the norm of a Hankel-Toeplitz operator. Finally analysis of ORDAP for continuous time-varying plants is exarnined in the context of nest algebras of causal stable systerns. Geometric characterization by Banach space duality showed existence of optimal solutions, and a comparison of time-varying versus time-invariant control laws is provided. 7.2. Directions for Future Research ,An immediate direction for future work is to compute the following special matrix norm " r .-r -2 DIRECTIONS FOR FUTb23E IIESE=\RCH where A = (Ai . Az), are n x n matrices with complex vahed entries. Or equivalently iilirre B = ( B: ) . Bi,?are n r n matrices This will eiuble exact cornputation of the norms of 3, (or Z:)? and hence approximate the nom of 3 as closely as desired. Explicit cornputation of the maximal vector of Z, will shed light on the form of the maximal vector - of z. and therefore the optimal controller. -4nother research project is to produce an "adaptive" version of the results described above, i-e.. a feedback law which adapts in response to evolving identification data. In particular, our results provide the optimal non-adaptive controller in the sense of sensitivity reduction. If this controller does not meet specifications, then an adaptive controller is needed. Roughly speaking, during the process of controiling any uncertain time-invariant or slowly time- varying plant, for which the set of uncertainty is not static, as time progresses it becomes possible to observe more identification data pertinent to the uncertain plant. At each time a finite number of observations have been made. and the remaining set pertains to systems whicli sat isfy the initial conditions and conform to these observations. 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