Contributions to Time-Varying Linear Control Systems
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Ilchmann, Joachim : Contributions To Time-Varying Linear Control Systems Druckausgabe: llchmann, Achim: Contributions to time-varying linear control systems / Joachim llchmann. - Ammersbek b. Hamburg : Verl. an d. Lottbek, 1989 (Wissenschatfliche Beitrage aus europäischen Hochschulen : Reihe 11, Mathemalik ; Bd. 2) Zugl.: Bremen, Univ., Diss., 1987 ISBN 3-926987-09-X Wissenschaftliche Beitrdge aus EUROPAISCHENHOCHSCHULEN REIHE11 Mathematik BAND:2 Joachimllchmann ContributionsTo Time-VaryingLinear GontrolSystems CIP-Titelaufnahmeder DeutschenBibliothek llchmann,Achlm: Contributionsto time-varyinglinear control systems / Joachim llchmann.-Ammersbek b. Hamburg: Verl.an d. Lottb€k,1989 (WissenschattlicheBeitrage aus europaischon Hochschulon : Rsihe 1 t, Mathemalik; Bd. 2) Zugl.:Bremon, Univ., Oiss., 1987 lsBN3-926!187-09-X NE:Wissenschaftlicho Beilrage aus europaischen Hochschulervl I @Verlagan der Lottb€k- PeterJensen - Februar1 989 Ammersbekbei Hamburg tsBN 3-926987-09-X tssN 0934-0505 Druck:Fotodruck an der Uni;Hamburg Umschlagentwurf: Pet€r Jensen Mathematics Subject Classification (1980) Prlrnrry:93C05, 93C50, 34A30, 34D1 0 Secondery:93805, 93807,9381 0, 93817, 93D05,93D1 5 Kcyworde: time-varinglinearfinitgdimensionalcontrolsystems, algebraic description, disturbance decoupling,stabilizability, robustness, differential matrix Riccati equation, controllatXlity,lime-varying subspaces, system equivalence, (A,8)-invariance, nonin- tore{ing,exponsntial stability, oxpon€ntial dichotomy, stability radius Contents 0 Introduction I 1 Controllabilit5r and Observability for State Space Systems 1.0 Introduction............ 1.1 Basic notations and definitions 5 1,2 Controllabilityinto subspaces.......,..... 7 1.3 Dual and adjoint relationshipsbetwen controllability, reonstructibility, reacha- bility and observability ta 1.4 Controllability indicesand a canonicalform . 19 1.5 Timevarying subspaces,the controllableand the unreconstructiblefamily ..... 24 Difrerential Polynomial Matrix Syetema - An Algebraic Approch 33 2.0 Introduction....... 3g 2.1 Ditrerential polynomial matrices 35 2.2 Polynomialmatrix system6,solution vector spaces and systemequivalence ..... 45 2.3 Controllabilityand observability ............ 5b 2.4 Input-outputmap and formal transfermatrix .,... 57 2.5 ,M[D]-rightmodules and the input module . ........ .......... 60 2.6 Invariants of systemequivalence resp. similarity 65 2.7 Characterizationsof systemequivalence 69 S Disturbance Decoupling Problems - A Geometric Approach 72 3.0 Introduction.... .. ...... 72 3.1 (,4, B)-invariant time-varying subspaces T4 3.2 Algebraiccharacterization of (4,8)-invariance ...... 79 3.3 Dualitybetween (,4,8)- and (C,.4)-inva.ria.nce........ 8l 3.4 Disturbancedecouplingproblem........ ........... 89 3.5 Controllability subspacefamilies .. 92 3.6 Noninteractingcontrol 94 4 Stability, Stabiliability, Rnbustneas, and Differential Riccati Equations S7 4,0 Introduction.,..... gT 4.1 Suficientconditions for exponential stability ...... .......,.,........ gg 4,2 Stabilizability of systemswith exponentialdichotomy flO 4.3 Bohl exponent and Bohl transformations 116 4.4 The structured stability radius f20 4.5 The perturbation operator f2g 4.6 The associatedparametrized ditrerential Riccati equation 130 4.7 Dependenceof the solutionPe+(r) on the parmeter p ,,... ,......... 143 4.8 Nonlinear perturbations and robust Lyapunov functions . , 14S References 160 Subject Index 156 Symbol Index 16t Introduction In this theoielinear finite dimensional timevarying control eystems in statespace form i(t) = A(t)"(t)+r(r)r(r) (0.1) v(t) = c(t)c(t) reep.in diferentialoperator description rQ)Q) = Q(l))(u) (0.2) s = v(D)(z)+w(D\(u) ae, studied. Roughly speaking, three differert mathenatical techniquc (algebraic, geometric and analytic) are used to analyse variour control thmretic probleme of tirne-va.rying slstems, Algebraic approach Rosenbrock(1970) introduced the well-known setting of timeiava^riant syetemsin diferentia.l operator description(0.2). He etudiedthe queetionunder which onditions two consta.ntcontrol systemsrepresented by (0.2) have the sa.medyna.mics and the sa,meinput-output behaviour, Thn_is the problem of (strict) system equivalene. Fuhrmann(1926) and (1927) analyzedthie problem via model theoretic tools. Thia enabledhim to rssociatea caonical state apacemod- ule with any frtorization v(z)P(z)-1QQ) t w(z) oI a proper ratioaal transfer matiix. so far the analysisof the problem of systen equiralencefor tineinva^riant system! wa^sdone ia the Ireqtercv domain. Pemebo (1927) wu the first who etudied syetem equivalencein the tdme domain,his basicidea waeto coneidersolution eetaofthe systemequatioas, Thio approachwas systematically exploited by /linncAsen and Prdtzel-Woltcrs (1980) to obtain a eeli-contained thmry of system equiwalencein the time domain. They derived an algebmic crircrion of systen equivalence, defined and chumterized contrcllability and, obsenability, and precented z canoni- cal state spacemodel eimilar to Fuhrmann,omodel. concerning timeaarying systems,for a long time there has beena wideepreadecepticism whether an algebraictreatment in the style of xalman would at all be poosible.In partitular it war not clearhow to extend transform techniques.There weresome attempts to rtudy time-varyin8eque. tions of the form (0.2), cf. Ylinen (1975) and (19s0), Kamen (1926). Ilchmann, NirrLqer-and Schmale (1984) were guided by the time-invariant approach of Hinrichsen arul Prdtzel-Wolters (1980) when they generalized the concept of system equivalence for time-mying eystems. They considered 6y8tem matrices defined over a certain skew polynomial nng and introduced the no- lion, oI 'full'difcrential operators. This set them in a position to generalize, for a fairly rich clus of timewying systeme of the form (0.2), the tineinvarimt resulte of Hin"ichsen and Prdtzel-Wolters(f984), Theee results ae preented in the first halfofChapter 2 ofthis thesis. A module thmretic approach to diferent definition ol stflctuml indicee of time-invariant state spee systems wae given by Minmer arul Prdtzel-Wolters (1979). Via polynomial nodules ild their minimal bases they proved the equality oI contmllability indices, minimal irulices, geomet- ic indies nd. dynanical irulices. Prdtzel-Wollers (1981) continued this work to chuacterize Brunovskj-equivalence for time-invariatrt systems of the form (0.2). Guided by this approach md using the skew polynomial ring introduced in llchmann, Ninberger and Schmale (1984) I generalized the resulte of Mdn zner and Prdtnl-Wolters (1979) nd Prdtrel' I,t/ollers (1981) for timeva,rying eystems (se Ilchmann (1985a)). The characterization of mdni mal bases ol right ebew polynomial modulee stends a result of lomey (1975). It is possible to define a tronsler matritit the time domain and to u6e thi8 to churcterize eystem equivalence. Different invarianle with respect to system equivalence reep. similarity were defined and their equality wu shown. Thie is presented in the second half of Chapter 2 of this thesis. Geometric apprmch In the late sixties Basile and Mano (f 969) md Wonham and Morse (1970) developed the concept of /,4,8)-invaiant subspacee to solve decoupling md pole assignment problens for multinriable 6ystems. Latet Wonham (1974) established the eo called gometric approach. This approach was generalized for nonlinear systeme by flirschhorn (1981), Isidord, Krcner, Gori-Giori anrl Monaco (1981) and for infinite dimensional eyatens by Czrlcin (1985), (1986), to name a few. In Chapter I of this thesis time-uorying eubepnces zte studied. This turns out to be the appro- priate framework to extend the linear timeinvaimt geometric approach lo piecewise analytic state spee systems. If only analytic systems are considered this approach is a specialization of nnatural" the nonlineu setting. However the concept propoeed here is more for time-varying systems (diferential geometry is not ueed) and the class of piecewise analytic systeme is richer than the class of malytic 6ystems. In Chapter I of thi6 thesis I present the results which were essentially given in /lcfimcnn (1985b) and (1986). The concepte of (A,B)- and (C,A)-inuaiance are introduced, characterized and their dual relationships are shown. By using these results the solnbility of the dieturto,ne decoupling prcblem nd, the nonintemcling prcblemis characterized. Analytic rpproech Concerning exponential stability of rystems of the form n$) = A(t)z(t), ,>0 (0.3) thre mpects ue etudied in this thesie: for short, sufrcient conditions for exponential stabilitg; sufacient and necessuy conditions fior the stabilizability of systens (0.1) by state fedback; mbuslneceof stability. It is well-known that if, for all , > 0, the spectrum of A(t) is lying in the oper left half plane "slow and the parametervariation of A(t) is enough", then (0.3) is exponentiallystable. see e.g. Rosendmlc(1963), coppel (1978). However,these results are qualitative. h llchmann, owens and Prdtzel-wolters (1987b) we derived quantitatire rcsulte, Thie means,upper bounds for the eigenvaluesmd for the rate of changeof.4(r) which engureexpotrential stability of (0.3) ae determined.This is presentedin Section4.1 of the presentthesis. Ikeda, Maeclcand Kodama (1972) md (1975) rtudied the problem to etabilizea timevarying system (0.1) by state feedback. F\rthermore they gave a sufrcieat condition which guaantees that (0.1) is stabilizable by deterministic state estimation fedback. lt Ilchmann anil Kern (1987) these problems were analysed in case that the eystem (0.3)