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Local-Global Double Algebras for Slow H- Adaptation: Part 11- Optimization of Stable Plants Le Y

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Local-Global Double Algebras for Slow H- Adaptation: Part 11- Optimization of Stable Plants Le Y IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 2, FEBRUARY 1991 143 Local-Global Double Algebras for Slow H- Adaptation: Part 11- Optimization of Stable Plants Le Y. Wang, Member, IEEE, and George Zames, Fellow, IEEE Aktruct-In this two-part paper, a common algebraic framework is In this section, the double algebra will be E,, with local norm introduced for the frozen-time analysis of stability and H" optimiza- PJ.) and global norm It II(l(,). tion in slowly timevarying systems, based on the notion of a normed Suppose that W,,W, E IS, represent two weightings, and algebra on which local and global products are defined. Relations E represents a strictly causal plant. It is standard that between local stability, local (near) optimality, local coprime factoriza- G E, tion, and global versions of these properties are sought. feedbacks which are globally stabilizing in IS,, i.e., maintain all The framework is valid for time-domain disturbances in I". H"-be- closed-loop operators in E ,, can be parametrized by a compen- havior is related to I" input-output behavior via the device of an sator QEE, which gives a sensitivity (I- GQ),and a approximate isometry between frequency and time-domain norms. weighted sensitivity S E , In Part 1, some of the main features of normed double algebras were introduced. S = - Part 11 establishes an explicit formula linking global and local sensitiv- W,(Z GQ)W,. (2.1) ity for systems with stable plants, where local sensitivity is a Lipschitz- continuous function of data. Frequencydomain estimates of time-do- Denote W,G by G, and suppose that it has a local factoriza- main sensitivity norms, which become accurate as rates of time variation tion approach zero, are obtained. Notions of adaptive versus nonadaptive G,= U@ (robust) control am introduced. It is shown that adaptive control can GO"' achieve better sensitivity than optimal nonadaptive control. It is demonstrated by an example that, in general, H"-optimal where U and Goutare locally inner an! locally outer in E,, interpolants do not depend Lipschitz continuously on data. However, for some U, > U, i.e., for each t E Z, U,(u,(.))EH" is inner &suboptimal interpolants of the central (maximal entropy) type and ( GOu),( U,( a)) is outer. AAK ' EH" are shown to satisfy a tractable Lipschitz condition. We are given a sensitivity S' E E, which locally interpolates W, W, at U, i.e., for which there exists Q,E E, such that I. INTRODUCTION SI= WZW, - U@Q, (2.2) NE of the aims of this work is to obtain a paradigm of where is smaller *an -W,W, in CL,(*). Q is now chosen to 0adaptive feedback in the H" context and a comparison of ,?' adaptive versus nonadaptive feedback. A prerequisite for such a locally realize S'. The simplest case' will be considered here, in comparison would appear to be some means of computing which Q is chosen to satisfy optimal or nearly optimal performance under time-varying n weightings, in order to determine whether the best that can be S'= W,W, - U@Gout@(QWl). (2.3) achieved with updated information is better than the best that can This choice of Q can be described as a local product of be achieved without. Frozen-time analysis provides a way of noncausal operators, provided the domains of definition of the obtaining approximate optimization, which can be used to con- various local functions are first extended to noncausal operators. struct an elementary paradigm. However, as freezing is involved Previously, these domains included the space E, of operators in several operations, including inner-outer factorization and with kernels k E 16( - 00,00) satisfying the causality constraint optimal H" interpolation, it can involve some messy bookkeep- k(t, 7) = 0 for t < 7. ing. Our formalism seeks to tidy up this process. Definition 2.1: Henceforth, the definitions o,f local product The same notation will be used here as in Part I. K @ F,transform norm p,(K), and rate d:p)(K)mare extended to operators with (possibly noncausal) kernels k(t, t - PI. ADAPTIVEDESIGN BY bCALINTERPOLATION (*))el;(- QO, 00). (The definitions as originally stated can be The main objective here is to synthesize a global sensitivity extended intact.) from a prescribed local one, which may be locally optimal or Q satisfying (2.3) is now explicitly given by nearly optimal, and to determine how well the global (optimum) solution approximates the local (optimum) one. The double-alge- Q := [ GQ8 ( W2W1- S')] W;' (2.4) bra symbolism allows an explicit description of this problem. provided the inverses in (2.4) exist, where Gp may be non- Manuscript received December 4, 1989; revised June 26, 1990. Paper causal with Fourier transform in L:. The problem is to deter- recommended by Associate Editor, J. Hammer. G. Zames is with the Systems and Control Group, Department of Electri- mine whether (2.4) is stabilizing and makes the (true global) cal Engineering, McGill University, 3480 University Street, Montreal, P.Q., Canada H3A 2A7. 'This case occurs when the global products in (2.2) are accessible to L. Y. Wang is with the Department of Electrical and Computer Engineer- computation. Alternatively, the global products can be replaced by local ing, Wayne State University, Detroit, MI 48202. ones. The ensuing inequalities then involve more terms, generated by the IEEE Log Number 9041414. additional local factors, but the essentials remain similar. 0018-9286/91/0200-0143$01.00 0 1991 IEEE 144 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 2, FEBRUARY 1991 sensitivity S a good approximant to S' for slowly-varying G, mentioned at the beginning of this remark approach each other 4,and S'. by (2.9). Assumption 2.1 for Theorem 2.1: (2 Ip 5 00, U > 1) Proof of Theorem 2.1: As Q satisfies (2.4), it has the and W;' are in E, form ;tydL)@ -U. Sufficient conditions for the local interpolation (2.2) and as- Q= {(Gout)~@(U@@(W2Wl-S'))}W;' sumption b) to' hold are that there exists U, > U for which Since (Gout)@, U 63 ( W, W, - S'),and W; ' are in Bu by (2.29 The zero vector! of in z 1 IU, con- 0 (F-m), I assumptions a) and b) and the local interpolation condition (2.2), tain the zero veciors of z)_for each t E Z (where {€ is a U,( Gn and EOis closed under the @ and products, Q is in Eu,i.e., zero vecior of K(z) ai z if z)l = 0) - K( Q stabilizes G in gu. are bounded in an annulus b') I U,(z)-' I, I Cyt(z)-' I The bounds (2.7), (2.8) are obtained from the inequalities U II z I IU,, uniformly in t and z. 0 (3.9), (3.14) of Part I, respectively, applied to the expressions It will be recalled that the following constants were defined in S' = W2W, Gw63 (GQ @ F) and S = W2W1 Part I - - G ,(GP @ F) where the identifications G + - G w, K -+ G$) @Fand F+ W2W1 aremade. 0 A. Local H" Adaptive Optimization A natural idea for adaptive compensation is to make the weighted sensitivity S, at time t E Z depend on the local behav- ior of the weighted plant inner and outer factors U,,G;', and' weighting W, := ( W2 Wl),. That triple of local operators consti- tutes the data available at t about the plant in the form of a Theorem 2.1: Q defined by (2.4) stabilizes G in B. If G,, nominal model and a band of uncertainty. The data may be G 9, and S' are slowly varying, then the weighted sensitivity available a priori or acquired through identification or a combi- S E is explicitly given by nation of both. In frozen-time adaptive design the controller generates a local approximation S: to S, based on that data. The adaptation law can be represented by a map Y ' where 9':2 x (IT;), s; = Y'(fi,,@,), U, > U. F := W2Wl-S'andG,=U@Gout. If the data varies slowly, Thecrem 2.1 provides a basis for frozen-time adaptation provided S' also varies slowly. A suffi- Moreover, the difference )I S(I ,) - pU($:) =: A(t)has the upper bound cient condition for this is that Y' be Lipschitz in its variables, i.e., that there be constants l$?, g@ for which the inequality A(t) 5 K(,~){K$(~,U) + (2.7) ll$; - i:-lllH~oI&J)llfi, - fi,,-lI1 + @, - R-111 and )I SI1 ,) also has the lower bound holds with )I * )I representing 11 (Ip or, in more compact matrix notation U0 where (2.10) B(p, (2.8a) a):= ~~(6~)8s) where a,([8,]) means [du(8i,)] and A(:) is the row matrix [A$), X'L)]of Lipschitz constants. In the case of variable rates, it diP)(F) 5 a,( k2@')+ au($'). (2.8b) will be assumed the Lipschitz constants hold independently of Remark 2.1: It follows from (2.7) that in a variable rate rate. situation (see Section 111-D in _Part I), if the rates approach 0, the One can try to design S' by local Er; optimization, which frequency-domain norm pu(S;) of the local sensitivity S: ap- gives an optimal weighted sensitivity Stpt satisfying proaches an upper bound on the time-domain norm 11 SI( ,) of the global sensitivity S. If S' actually satisfies the uniform radial growth condition defined in Section 111-C of Part I, then it follows from (2.7), which would imply (2.8) and the inequality that A(t) is bounded by In the case in which is a Lipschitz function of data, Theorem 2.1 gives a global sensitivity S which approximates the local optimum.
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