Robust Control: Proceedings of a workshop held in Tokyo, Japan, June 23 – 24, 1991 PDF
Preview Robust Control: Proceedings of a workshop held in Tokyo, Japan, June 23 – 24, 1991
Lecture Notes in Control and Information Sciences 183 Editors: M. Thoma and W. Wyner S. Hosoe (Ed.) Robust Control Proceedings of a Workshop held in Tokyo, Japan, June 23 - 24, 1991 Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo HongKong Barcelona Budapest Advisory Board L.D. Davisson • A.GJ. MacFarlane" H. K w a k e r n ~ J.L. ~ s e y " Ya Z. Tsypkin • AJ . Vit~rbi Editor Prof. Shigeyuki Hosoe Nagoya University Dept. of Information EngIneering Furo-cho, Chikusa-ku Nagoya 464-01 IAPAN ISBN 3-540-55961-2 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-55961-2 Springer-Verlag NewYork Berlin Heidelberg This Work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the fights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992 Printed in Germany The use o fregistered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-rea/Iy by a,uflaors Offsetprinting: Mercedes-Druck, Berlin, B~'okbindidg: B. Helm, Berlin 6113020 5 4 3 2 1 0 Printed on acid-free paper PREFACE The workshop on Robust Control was held in Tokyo, Japan on July 24- 25, 1991. From 10 countries, more than 70 reseachers and engineers gathered together, and 33 general talks and 3 tutorial ones, all invited, were presented. The success of the workshop depended on their high level of scientific and engineering expertise. This book collects almost all the papers devoted to the meeting. The to- pics covered include: Hoo control, parametric#tructured approach to robust control, robust adaptive control, sampled-data control systems, and their ap- plications. All the arrangement for the workshop was executed by a organizing com- mittee formed in the technical committee for control theory of the Society of Instrument and Control Engineers in Japan. For financial support and continuing cooperation, we are grateful to Casio Science Promotion Foundation, and many Japanese companies. Shigeyuki Hosoe C O N T E N T S Out l ine of the W o r k s h o p P a p e r s • H.KIMURA (J, J~)-Lossless Factorization Using Conjugations of Zero and Pole Ex- tractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 • D.J.N.LIMEBEER, B.D.O.ANDERSON, B.HENDEL Mixed H2/H~ Filtering by the Theory of Nash Games . . . . . . . . . . . . . . . . . . . . . 9 • M.MANSOUR The Principle of the Argument and its Application to the Stability and Robust Stability Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 • L.H.KEEL, J.SHAW, S.P.BHATTACHARYYA Robust Control of Interval Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 • B.BAMIEH, M.A.DAHLEH, J.B.PEARSON Rejection of Persistent, Bounded Disturbances for Sampled-Data Sys- tems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 • Y.OHTA, H.MAEDA, S.KODAMA Rational Approximation of Li-Optimal Controller . . . . . . . . . . . . . . . . . . . . . . . . . 40 • M.ARAKI, T.HAGIWARA Robust Stability of Sampled Data Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 • S.HARA, P.T.KABAMBA Robust Control System Design for Sampled-Data Feedback Systems . . . . . . . 56 • Y.YAMAMOTO A Function State Space Approach to Robust Tracking for Sampled-Data Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 • F.-B.YEH, L.-F.WEI Super-Optimal Hankel-Norm Approxmations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 • J.R.PARTINGTON Robust Control and Approximation in the Chordal MeLric . . . . . . . . . . . . . . . . . 82 • R.F.CURTAIN Finite-Dimensional Robust Controller Designs for Distributed Parameter Systems: A Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 • 3.-H.XU, R.E.SKELTON Robust Covariance Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 VIII • T.FUJII, T.TSUJINO An Inverse LQ Based Approach to the Design of Robust Tracking System with Quadratic Stability ................................................. 106 • T.T.GEORGIOU, M.C.SMITH Linear Systems and Robustness: A Graph Point of View .................. 114 • M.FUJITA, F.MATSUMURA, K.UCHIDA Experimental Evaluation of H~ Control for a Flexible Beam Magnetic Suspension System ...................................................... 122 • K.OSUKA Robust Control of Nonlinear Mechanical Systems ......................... 130 • A.S.MORSE High-Order Parameter Tuners for the Adaptive Control of Nonlinear Systems ................................................................. 138 • H.OHMORI, A.SANO Design of Robust Adaptive Control System with a Fixed Compensator .... 146 • S.HOSOE, JF.ZHANG, M.KONO Synthesis of Linear Multivariable Servomechanisms by Hc¢ Control ....... 154 • T.SUGIE, M.FUJITA, S.HARA ]-/oo-Suboptimal Controller Design of Robust Tracking Systems ........... 162 • M.KHAMMASH Stability and Performance Robustness of ~I Systems with Structured Norm-Bounded Uncertainty .............................................. 170 • S.SHIN, T.KITAMORI A Study of a Variable Adaptive Law ..................................... 179 • M.FU Model Reference Robust Control ......................................... 186 * T.MITA, K.-Z.LIU Parametrization of H~ FI Controller and Hoo/H2 State Feedback Control ... 194 i A.A.STOORVOGEL, H.L.TRENTELMAN The Mixed H2 and Hoo Control Problem ................................. 202 • D.J.N.LIMEBEER, B.D.O.ANDERSON, B.HENDEL: Nash Games and Mixed H2/H~ Control ................................. 210 • J.B.PEARSON Robust g1-Optimal Control .............................................. 218 ( J , J ' ) - L O S S L E S S F A C T O R I Z A T I O N U S I N G C O N J U G A T I O N S O F Z E R O A N D P O L E E X T R A C T I O N S Hidenori Kimura Department of Mechanical Engineering for Computer-Controlled Machinery, Osaka University 1. Introduction A matrix G(s) in L** is said to have a (J, J')-losslessfactorization, if it is represented as a product G(s)=O(s)rl(s), where O(s) is (J, J')-Iossless ]11] and rl(s) is unimodular in rl**. The notion of O, J')- lossless factorization was first introduced by Ball and Helton [2] in geometrical context. It is a generalization of the well-known inner-outer factorization of matrices in H"* to those in L**. It also includes the spectral factorization for positive matrices as a special case. Furthermore, it turned out that the (J, J')-lossless factorization plays a central role in H"* control theory [1][4][7]. Actually, it gives a simple and unified framework of t l " control theory from the viewpoint of classical network theory. Thus, the (J, J')-lossless factorization is of great importance in system theory. The (J, J')-lossless factorization is usually treated as a (J, J')-spectralfactorization problem which is to find a unimodular n(s) such that G-(s)JG(s) = rl~(s)jrl(s). Ball and Ran [4] first derived a state-space representation of the (J, J')-spectral factorization to solve the Nehari extension problem based on the result of Bart et.al. [5]. It was generalized in [ 1] to more general model-matching problems. Recently, Green et.al. [8] gave a simpler state-space characterization of the (J, J')-spectral factorization for a stable G(s). Ball et al. 13] discussed some connections between the chain-scattering formulation of H** and (J, J')-lossless factorization. In this paper, we give a simple derivation of the (J, J')-lossless factorization for general unstable G(s) based on the theory o f conjugation developed in [9]. The conjugation is a simple operation of replacing some poles of a given rational matrix by their mirror images with respect to the origin by multiplication of a certain matrix. It is a state-space representation of the Nevanlinna-Pick interpolation theory [ 11] and the Darlington synthesis method of classical circuit theory. The (J, J')-lossless factorization for unstable matrix, which is first established in this paper, enables us to solve the general I1° ° control problem directly without any recourse to Youla parameterization [12]. In Section 2, the theory of conjugation is briefly reviewed. The (J, J')-lossless conjugation, a special class of conjugations by (J, J')-lossless matrix, is developed in Section 3. Section 4 gives a frequency domain characterization of the (J, J')-lossless factorization. Section 5 is devoted to the state space representation of the (J, J')-lossless factorization. The existence condition for the (J, J ')-lossless factorization is represented ill terms of two Riccati equations, one of which is degenerated in the sense that it contains no constant tema. Notations : C(sI-A)" B+D := [A-~--1 R-(s) =R T(.s), R'(s) = RT(~), tC IDJ, ~A) ; The set of eigenvalues of a matrix A. RU:x r ; The set of proper rational nmtrices of size mxr without poles on the imaginary axis, Rllmxr; The set of proper stable rational matrices of size m×r. 2. Con juga t ion The notion of conjugation was first introduced in [9]. It represents a simple operation which replaces the poles of a transfer function by their conjugates. The conjugation gives a unified framework for treating the problems associated with interpolation theory in state space [10]. In this section, the notion of conjugation is generalized to represent the relevant operations more naturally. To define the conjugation precisely, let G(s) =[ A~DB I, A~ R.-. De R--r (2 . t ) be a minimal realization of a transfer function O(s). Definition 2.1 Let a(A) = A1UA2 be a partition of o'(A) into two disjoint sets of k complex numbers At and n-k complex numbers A2. A matrix e(s) is said to be a AFconjugaror of G(s), if the poles of the product G(s)O(s) are equal to {-A l} UA2 and for any constant vector ~, ~ 0, ~(sl - A)' 3 Premultiplication of (2.11) by (sl - A)" t and postmultiplication by (sI - A)aa l yield Idt (sI - Al)" 1Bt = (st - A)" l BO(s). (2.12) Here, we used the relation N~ (sl - At)" ~ B~ = (sl + A'r)" t B,, which is due to (2.9) and the second relation of (2. I0). The condition (2.5) implies that Mt is non-singular and (At, 130 is controllable. Taking X=NIMI "l in (2.10) yields (2.4). Since MI-IAcMI=AI, Ac should be stable. It is clear that (^¢, BDc)=(MflAtMt, MIBI) is controllable because (Ab BI) is controllable. Due to (2.9), we have Be=NIBI=NIMI'IBDc=XBDc, from which (2.3) follows. The derivation of (2.5) is straightforward. Remark. Theorem 2.2 implies that the conjugator of G(s) in (2.4) depends only on (A, B). Therefore, we sometimes call it a conjugator of (A. B). Remark It is clear, from (2.7), that the order of O(s) given in (2.6) is equal to the rank of X which is equal to the number of elements in Lt. 3. (J, J ' ) -Loss l e s s C o n j u g a t o r s A matrix O(s)~ RL" is said to be (J, J')-unitary, if tm*r)x(p*q) O-(s) J O (s) = J' (3.1) for each s. A (J, J')-unitary matrix e (s ) is said to be (J, J')-tossless, if O'(s) J O (s) g J', (3.2) for each Re[s]->0, where j = [ I,, 0 ] j ,=[ I~ 0 ] m>_p, r_>q (3.3) 0 -I, , 0 -I~ , A (J, J')-lossless matrix is called simply a J-lossless matrix. A conjugator of G(s) which is (L J')-lossless is called a (J, J')-lossless conjugator of G(s). The existence condition of a J-lossless conjugator, which is of primary importance in what follows, is now stated. T H E O R E M 3.1. Assume that (A, B) is controllable and A has no eigenvalue on the jc0-axis. There exists a (J, J')-lossless stabilizing (anti-stabilizing) conjugator of (A, B), if and only if there exists a non-negative solution P of the Riccati equation PA + ATP - PBJB'rp = 0 (3.4) which stabilizes (antistabilizes) A, :-- A- BJB'rp. In that case, the desired (J, I')-lossless conjugator of G(s) and the conjugated system are given respectively by L -JBT [ I I , C - DJBTp D ( 3 . 5 ) where Do is any matrix satisfying D~J D, = J'. (ProoJ) Since (2.4) implies X(sl + AT ) = (sl - Ac)X, O(s) in (2.3) is represented as O (s) = Oo (s) De, O 0 ( s ) = I + C c X ( s I - A c ) "j B, A c = A + B C c X . (3.6) Assume that P is a solution of the Lyapunov equation P a , + AcT P+ X" r c,'rJ Co X =0 . (3.7) Then, it follow that O~ (s) J O0(s) = J + BT (-sI-AcV) a (XrC~ J + P B) + (J C¢ X + B'r P) ( s t - A¢) "~ B. Since ~ is stable and (A¢, B) is controllable from the assumption, O(s) is J-unitary if and only if J C, X + Br I, = 0. (3.8) In that case, we have O~(s)JOu(s)= J- (s + ~BT(sl - A~)"P(sl -A c)4B Hence, in order that Oo(s) is J-Iossless, P should be non-negative. From (3.8), (3.7) is written as (3.4) and O0(s) in (3.6) can be written as O0(s) = I- Jffr(sl+ Ar)'IPB, from which the representation (3.5)f ollows. Now, we shall show a cascade decomposition of the (J, J')-Iosslessc onjugation of (A, B) according to the modal decomposition of the matrix A. Assumethat the pair (A, B) is of the form As! A,., The solution P of (3.4) is represented in conformable with the partition( 3.9) as p__. [ P,, PIa] PITS P22 . (3.10) L E M M A 3.2. If the pair (A, B) in (3.9) has a (J, J')-lossless conjugator O(s) of the form (3.5), then O(s) is represented as the product O(s) = O,(s) OKs) (3.11) of a J-lossless conjugator Ol(s) of (At t, Bt) and a (J, J')-lossless conjugator 02(s) of (An, 82) with Bz=Ba+Ptff'hB, , where P~a is a peudo inverse of P,, in (3.10). Actually, e~(s) and O-Xs) are given respectively as [-J B; ' I 4 B; ' ( 3 . 1 2 ) where ~ = PH - P ~ { ~ a and D is any constant (J, J')-unitary matrix. (ProoJ) Since P ~ 0, we have Pt2(I - P~2Pza)= 0. Therefore, UPU'r =[ A0 Paa0 ], (U.I)'rB =[_~] . U=[ 10 "PtalP~] From this relation and (3.4), it follows that 0 Pal Aal A~ 0 AT 0 P~ 0 P22 0 P22 (3.13) where Azz = Aat + P~aPTaAn - A~zl'~ff'rz. Hence, t~AH + A~IA - AB~]BT A = 0. Since ~ ~ 0, Or(s) in (3.12) is a J- lossle_ss conjugator of (A,. B~). Also, it is clear that Oz(s) given in (3.12) is a (J, J')-lossless conjugator of (Aa:. B9. Since (3.11) implies that Az~ = BzlBT A, we have e~(s)O-~s) = 0 -A~ Pza-lh 19 = = e(s). -sa~ -J~I I t - jdu- ' The proof is now complete. 4. (J, J')-Lossless Fa c t o r i z a t i o n s A matrix G(s) e RL~',,.o x c*-*.) is said to have a (J, J')-losslessfactorization, if it can be represented as G(s) = O(s) l'I(s), (4.1) where O(s)¢ RL~'m,,r)x(v+*) is (J, J')-lossless and rI(s)is unimodular in H ' , i.e., both rI(s)and lI(s)" * are in RH~,-,0,, c,-,o • It is a generalization of the inner-outer factorization of H" matrices to L" matrices.
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