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Information to Users INFORMATION TO USERS This manuscript has been reproduced from the mMlmrnaster. UMI films the text directly from the original or subrnitted. Thus, sorne îhesis arid dissertation copies are in typewriter face, whiîe othen may be from any type of cornputer printer. The quality of this reproduction is dependent upon the quality of the copy submittad. Broken or indistinct print, oolored or poor quality iHusttatioris and photographs, print Meedthrough, substandard margins, and impro~er alignment can adversely affect reproduction. In the unlikely event that the author did not serrd UMI a complete manuscript and there are rnissing pages, these will be Md. Also, if unauthorked copyright material had to be removed, a note will indicate the deletion. Oversize materials (6.9-, maps, drawings, cham) are repmduced by sectiming the original, begiming at the upper ieft-Md anet and ccmtinuing from left to right in equal sedons with small werlaps. 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Be11 & Howell Information and Leaming 300 North Zeeô Road, Am Arbor, MI 48106-1346 USA ôû&521-0600 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 The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant a la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, dismbute or seii reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic foxmats. la forme de microfiche/film, de reproduction sur papier ou sur format électronique. 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
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