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Widths and Uncertainty Principles for LTI and Slowly Varying Systems

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Widths and Uncertainty Principles for LTI and Slowly Varying Systems IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 9, SEPTEMBER 1994 1827 Fast Identification n-Widths and Uncertainty Principles for LTI and Slowly Varying Systems George Zames, Fellow, IEEE, Lin Lin, and Le Yi Wang, Member, IEEE Abstract- The optimal worst-case uncertainty that can be that identification speed depends on the size of the region achieved by identificationdepends on the observation time. In the of uncertainty, which can be quantified using a measure of first part of the paper, this dependence is evaluated for selected metric complexity such as an n-width or a metric dimension. In linear time invariant systems in the 2’ and H” norms and shown to be derivable from a monotonicity principle. The minimal fact, the dependence of accuracy on time will be characterized time required is shown to depend on the metric complexity of by two notions of n-width which are related to the standard the a priori information set. Two notions of n-width (or metric notions of Kolmogorov and Gel’fand. dimension) are introduced to characterize this complexity. The paper is divided into two parts. The first part, consisting In the second part of the paper, the results are applied to of Sections 11-IV, is concerned with linear time invariant systems in which the law governing the evolution of the uncertain elements is not time invariant. Such systems cannot be identified systems. In the second part, consisting of Section V, the accurately. The inherent uncertainty is bounded in the case of results will be extended to obtain uncertainty principles for the slow time variation. identification of slowly varying linear systems. Slowly varying The n-widths and related optimal inputs provide benchmarks linear systems are of interest in adaptive control because from for the evaluation of actual inputs occurring in adaptive feedback a certain point of view they are the most general ones for which systems. an input-output theory is useful. In particular, identification of uncertain elements has predictive value only if their future I. INTRODUCTION behavior is like their past or, at worst, approximately like DENTIFICATION speed is one of the main elements their past. If a “black-box’’ system changes substantially in I affecting the performance of adaptive’ control systems. relation to the length of time needed to identify it, however, Speed can be defined in terms of the minimal observation then accurate identification is inherently impossible. This fact time needed to achieve a specified accuracy in input-output is expressed through uncertainty principles, which relate the behavior or, altematively, the maximum accuracy achievable inherent uncertainty to the n-widths mentioned above. in given time. Here, optimal speed will be computed for certain representative problems. It will be shown that the optimal Previous Results speed that can be achieved in identification depends on the The information-based approach to worst-case identifica- a priori information, independently of the particular algorithm tion was introduced by Zames [4], [3] using concepts of E- that may be used. (The optimal speed can only be achieved dimension (inverse n-width) and €-entropy. Recently there has for certain optimized inputs. Its main relevance in adaptive been a revived interest in this subject. We note especially the control will be in providing a benchmark against which the related works of Tse, Dahleh, and Tsitsiklis [5] and Helmicki effectiveness of input ensembles that are actually present in et al. [6] as well as various results on worst-case identification, feedback systems can be compared.) e.g., Makila [7], Gu et al. [8]. A mathematical framework for It has been emphasized elsewhere [3], [2] that the concept slowly linear time-varying (LTV) system control was given in of metric complexity can provide a unifying framework for the [9], [lo], and there are results on LTV identification in [ll]. subjects of identification, adaptive feedback, and organization. Portions of the present paper originally appeared in [l]. For example, feedback can be viewed [2] as an agent for the reduction of complexity. One of the objectives here will be to establish a direct link between identification speed and Notation complexity. With that in mind, identification problems will be C, R, Z and Z+ denote the complexes, reals, integers and considered in which the a priori information locates the plant nonnegative integers. in a region of uncertainty in a metric space. It will be shown P[a, b], 1 5 p 5 00, -00 < a 5 b < 00 denotes the space Manuscript received February 26, 1993; revised October 12, 1993. Recom- of sequences of real numbers f(t),t being an integer in the mended by Associate Editor, A. Vicino. interval a 5 t 5 b, satisfying IlfllP: = [E,”=,If(t)l”]’/P < 00 G.Zames and L. Lin are with Department of Electrical Engineering, McGill University, Montreal, Quebec, Canada H3A 2A7. for 1 5 p < CO, and Ilflloo:= SUPtE[,,blIf(t)l < 0O. This L. Y. Wang is with Department of Electrical and Computer Engineering, notation is extended in the usual way to the cases where the Wayne State University, Detroit, Michigan 48202 USA. interval has one (or both) end points missing, such as [a, b), IEEE Log Number 9402994. or is (semi) infinite, such as [a, 00). ‘See [2] for a formal definition of input-output adaptation, based on the facts that performance is a function of accuracy (or uncertainty) and accuracy P[,,,] is the truncation operator on P, defined by increases with time. (P[,,,~f)(t):= f(t) for t E [n, m],and 0 otherwise. 0018-9286/94$04.00 0 1994 IEEE 1828 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 9, SEFTEMBER 1994 IlSll~is the norm of the largest function in the subset S of Say the output is observed on a time interval [to, tl) which a normed space L, i.e., IJSIIL:= sup { lllcll~: k E S}. may be infinite. For our reference problem it is important not to SI[tl,t2)is the subset of functions of S mapping Z to R assume at the outset that the input is zero before to, i.e., not to with support in the interval [tl,t2) of Z, i.e., Sl[tl,t2):= rule out the possibility that prior excitation might speed up the s n P[tl,t2)S. identification. It is that possibility which makes the problem Null (@):= {k E X:@(k) = O}. nontrivial, and there are several reasons for not excluding it: Hp(Dr)denotes the Hardy space on D,: = {z E (I: : IzI < If observations start while the plant is in operation and the T}, T > 0. IIHllm,r: = supZED,IH(z)l is the norm defined on initial conditions at time t, are not known, the response H"(Dr). to them acts as a noise component in the output. Until that HW(D):= H"(Dr)l,=l. noise decays to some small level, say at to, (to > t,), U*is the dual space of U. observations of the output often convey no information T:U -+ U,is the shift operator, i.e., (Tu)@)= u(t - 1). about the plant. The useful output observation interval d" denotes the Gel'fand n-width. starts at to, whereas the input is free to be shaped earlier, 0" denotes the identification n-width. after t,. p(-) denotes the variation rate of a time-varying system. If the input is designed to contain a deterministic dithering component, say an almost-periodic function, then knowl- 11. FORMULATION:TIME-INVARIANT CASE edge of the component completely determines its past, which can be viewed as a known prior excitation for the We will consider discrete-time systems represented by con- purposes of experiment. One would like to know whether volution operators of the form K:U + Y such a prior excitation could be beneficial. 00 Inputs which start prior to the observations will play a y(t) = Ck(r)u(t- 7-), t, 7- E Z (1) role in "moving window" adaptive problems (see Section r=O 111-B). where IC(.) E L, under the following assumptions: If the input u E U is known and fixed, then on the basis of i) U,Y are normed linear spaces of functions Z -+ W the observations of the output Ku(t)in [to, tl),the location representing inputs and outputs respectively. Initially, we of the true kernel, ktrue, is narrowed down from the a priori assume that the sets U and Y are equal to P(--co,CO). data set Sprier to a smaller set, S(ktrue) (Later this will be relaxed so as to include the case U = S(ktrue)={k S,ri,,(Ku-KtrUeu)(t)=O WE(~O,~I)}. Y = P(-co, CO). Inputs start at -CO to allow situations E in which the system is running before observations (2) begin.) It will be assumed henceforth that the a priori data set satisfies the following. ii) L is a normed linear algebra consisting of causal weight- Assumption I: Sprier is a closed convex symmetric (i.e., ing functions Z+ + R acting on input pasts. The set L is k E Sprier -k E Sprier) subset of L. contained in Z1[O, m), ensuring that (1) is well defined. +- If the estimated kernel, kest E K is optimally chosen for Moreover the norm on L satisfies 11 . IIL 5 Const.11 . 11p. (The H" norm is an example of such a norm.) S(ktrue), then it can be shown3 that the norm of the worst-case uncertainty, i.e., Identification is concerned with estimating the kernel k E L from observations of the output y and input U, given the a e(u):= sup inf sup llk-k,,tllL (3) priori information that the true kernel lies in a subset Sprier of ktrueESprlor kES(ktrue) L.
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