Copyright © IFAC Adaptive Systems in Control and Signal Processing. Grenoble. France. 1992 LOWER INFORMATION BOUNDS FOR AN ADAPTIVE CONTROL PROBLEM A. Judltsky* and A. Nazin** *IRISAIINRIA Campus de Beau/ieu 35042. Rennes. France **lnstituJe!or Control Science. Pro!soyuznaya 65.117342 Moscow. Russia Abstract. An adaptive tracking problem is considered for a linear stochastic SISO control plant. Several information bounds are obtained under a variety of conditions imposed on the disturbances and control strategies. eywords: adaptive control, optimal algorithms, lower bounds, efficient control strate- gies. 1 Introduction Suppose that a control plant is defined by the following difference equation N Yn = - 2...: aiYn-i + Un _l + €n for n > 1 _=1 (1) where (Un). (Yn), (€n) are the scalar sequences of con- trols. outputs and unobserved disturbances respectively. Let a = (a I, ... , aN) T be the vector of unknown param- eters (T denotes a transpose symbol). We assume that the order N of the plant is known a priori as well as the open set A E JRN which contains the vector a of true pa- rameters. The control objective is to follow the reference trajectory (y?) as precisely as possible. The values of Yn and Un are observed. This problem was studied in many papers (see. for in- stance. the survey of Astrom et a1. (1973) as well as mono- graphs by Caines (1988) and Goodwin and Sin (1984)). Several adaptive control algorithms have been developed and the conditions of their convergence have been deter- mined. On the other hand. less attention has been paid to the study of the rate of convergence for these algorithms and to the design of optimal methods (see. for instance, Goodwin and Sin (1984)). In the present paper we study the minimax informa- tion lower bounds for this problem. This issue is relevent to the adaptive contro problem in the following respects: lower bounds give limits imposed by nature on the pos- sible performance of algorithms. and. thus, provide some relevant characterizati on of the problem; they also provide us with some absolute scale of optimality (in the informa- tion sense (Tsypkin. 1983. 1984) for control algorithms proposed to solve the problem. For the adaptive stabilization problem, i.e. when == 0, this problem has first been studied by Nemirovskij and Tsypkin (1985) for a multi-dimensional plant. In the pre- vious work (Nazin, Juditsky, 1991) we obtained a some- what analogous bound, using a different approach. The present results constitute a generalisation of bounds, ob- tained in (Nazin, Juditsky, 1991), for the case of the track- ing algorithm. Obviously, they correspond to the bound presented by Nemirovskij and Tsypkin (1985) in the case where == o. Heuristically speaking, the main point of the paper is that "tracking toward any given trajectory cannot be 191 ea&ier than tracking toward the zero path". Though beinl!, quite intuitive, this result is still suprising. Indeed, we can reformulate it in the following less evident way: one cannot design any bounded trajectory to get the tracking error smaller than in the case of the tracking of the zero trajectory". Of course, the tracking problem, put this way looks rather unusual, but it seems to be of some interest. It should be note that the bounds are obtained on a very wide class of control strategies, and not only on the classical class of linear control algorithms. We cover also randomized control strategies and substantially weaken the conditions on disturbances. 2 Problem Statement Suppose that the initial conditions YI-N, ... ,Yo are ran- dom and Po is their distribution in JRN. The control Un is chosen at time n as a value of the Borel function Un=Un(YI_N.···'Yn;17) , (i.e. JRN+n X 3 --+ JRl) (2) of the available observations and, possibly, an additional random variable 17 that takes values in the measurable space (3,U) and randomizes the controls. 17 is assumed to be independent of the disturbance sequence (€n). We denote by P'1 the distribution of 17· Let us call the totality U = {P'1,(un(-))ln O} the control strategy, where the functions Un (-) are defmed in (2). Every pair (a,U) generates a random process (Yn) with the probability measure in the corresponding measur- able space. The mathematical expectation with respect to this measure is denoted by Ea,u, We shall use a symbol E to denote the expectation when its value does not depend on both a and U. We can state the adaptive control problem for the plant (1) in the following way: find a control strategy U = U· such that for any parameter vector a E A n -12...: lim - Ea,u' (y, _ n __ oo n '=1