A discrete-time nonstationary linear control system is considered to be given by the algebraic ... The control system is subject to a bounded additive noise.
SYST|MS CONTROL UTTERS
ELSEVIER
Systems & Control Letters 31 (1997) 33-40
Linear adaptive control for nonstationary uncertain systems under bounded noise A l e x e i V. K u n t s e v i c h a'*, V s e v o l o d M. K u n t s e v i c h b a Institute for Mathematics, Karl-Franzens University of Graz, 36 Heinrichstrasse, A-8010 Graz, Austria b V.M. Glushkov Institute for Cybernetics, National Ukrainian Academy of Sciences, 40 Prospekt Akademika Glushkova, 252650 M S P Kiev-207 (Ukraine) Received 22 December 1995; received in revised form 23 July 1996; accepted 10 March 1997
Abstract
A discrete-time nonstationary linear control system is considered to be given by the algebraic difference equation in the state space. The control system is subject to a bounded additive noise. Uncertain parameters of the system take their values on the given polytopes which evolve in time. The objective is to generate a linear feedback, which provides the minimization of a given performance criterion in adaptive way. In general, the control problem is reduced to the convex programming one of an insignificant computational complexity. Therewith, the control problem can be solved analytically in the case of interval set-valued parameter estimates. © 1997 Elsevier Science B.V.
Keywords: Uncertainty; Adaptive control; Robustness; Set-valued estimate; Minimax problems
1. Introduction
The theory of the worst-case deterministic (setmembership) parameter identification, in the presence of a bounded noise, iaas seen a recent burst of activity (see [23, 19, 14, 22, [7, 14, 8, 5, 9]). In modern robust control, the starting point for control system analysis and design is a nominal plant model and (norm) bounds on model uncertainty. Practical algebraic robust identification algorithms that have been designed up to now could be grouped as follows: (1) those ones which have been carried out (particularly in [14]) with the object of obtaining exact set-valued estimates of the form of convex polytopes;
* Corresponding author. Tel.: (316) 380-51-73, e-mail: alex@ bedvgm.kfunigraz.ac.at. Research of this author was partly supported by Fonds zur F6rderung der wissenschaftliehen Forschung (FWF) under Lise-Meitner Stipendium (Grant M00331-MAT). 0167-6911/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved PH S0167-6911(97",00026-1
(2) those ones that make use of approximate set-valued estimates, particularly of ellipsoids [3, 2, 15, 16,4, 5, 18]. These methods are perfectly developed and proved their efficiency• However, there is very low activity in set-membership parameter identification for nonstationary plants. To solve a general control problem for a system with uncertain parameters having got a priori setvalued estimates one necessarily faces a minimax problem arising with the primary objective (performance criterion) and the necessity to take into account all the possible values of system parameters. Making use of any control objective, a control designer has to solve a minimax problem that often appears to be a convex one (depending on a prior objective function). There are various ways how to reduce such a problem to one of insignificant computational complexity. It is worth to mention here the book [1]. Making use of Lyapunov method, the authors of the
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A.V. Kuntsevich, V.M. Kuntsevich/Systems & Control Letters 31 (1997) 33 40
book obtain finally a problem of convex function minimization subject to a constraint in the form of semipositiveness of a symmetric matrix. There is no room to compare the approaches used, thus, the authors have to restrict themselves with pointing to the difference in the approach used in [1] and that one suggested by the authors in [ 12, 11 ]. In general, there is a strong connection between convex programming and robust control synthesis. One can find the proof of this connection in [6, 21, 12, 20, 1, 7]. Robust adaptive control might present the direction for further development. The main aim of this line of research is to combine the identification and the control design in a mutually supportive way, from the point of view of robust performance. In common practice, a control system is subject to a noise which is assumed to be bounded. We also assume that the system parameters are time-varying, therewith, the rates of change are bounded and the set-valued estimates are given for the starting discretetime moment. The main purposes of the present paper are: (i) to present the algorithm for adaptive control design in general case, and (ii) to prove the implicit formula for the optimal control in the case of interval parameter estimates. Therewith, the authors essentially use polyhedral set-valued parameter estimates and the convexity of the performance criterion. Starting with the simplest case we extend the obtained results to the general one.
2. Single input design A single input control system is described by the difference vector-matrix equation in the state space
n =
.
(2)
It is assumed that fn is bounded: fn~0, on its rate of change, F, E ~ C ~m is a noise bounded by the given box ~. System (24) is defined by a set of equations Ai, n - l X n - 1 ÷ Bi, n - l U n - 1 ÷ fi, n--1 = X i , . ,
i=l,m,
n = 0 , 1,...,
(25)
where A i,n-~ and Bz,,-1 are the ith rows of the respective matrices, fi,,-1 is the ith coefficient of the vector Fn-I. Thus, the set of Eqs. (25) corresponds to Eq. (12) to within the notations.
un
In other words, the problem of optimal control design takes the form of minimization of the distance ~r from the state X,+I of the given standard system at the discrete time n + 1 to the set of possible states of system (24) (compare with the way problem (9) was stated). Let us start with the following introductions. Denote Li,, = (A~n,B~n)T,
i = 1,m,
where AT l,n and BY i,n are the ith row of the matrix A,, resp. B,. The estimate for the vector Li.n is a convex polytope ~i,n which could be presented by its vertices L~, as follows: ~'i,n = conv{L~n: k = 1,Ki, n},
where Ki,, is the number of the vertices of the polytope ~--i,n. Next, let L n be the matrix consisting of m rows L'f i---- 1,m. A set of coefficients of the matrix L, in the form of an m × m-dimensional vector is estimated by ~ n = ~--1,n X ~-~2,n X ' ' '
X ~--rn,n.
Let Lnk denote the kth vertex of ~, and K~-lli=~,mKi, n denote the total number of the vertices of
A.V. Kuntsevich, V.M. Kuntsevich / Systems & Control Letters 31 (1997) 33-40
the polytope ~n. Substituting (An, Bn) by Ln one can rewrite problem (26) in the form: min
max
U. LnE9~.;F,E~
~o(X,, Un, Ln, F,).
max
C, is an m-dimensional column vector of the controller parameters. In this case, problem (11 ) takes the following form:
find
As the function ¢p(-) is convex in Ln and in Fn (see (27)), the maximum is necessarily reached at some vertex L~ of the polytope ~n and at some vertex F~ of the polytope ~?. Therefore, L.E~.; F.E~:
~o(Xn,Un, Ln,Fn) = q~k,s(Xn,Un),
¢r
min C
where (Pk,s(Xn, Un) = (p(Yn,
U,,Lkn,Fn~).
AnEg~n;
* T * where (An) is the ruth row of the matrix An, H"II denotes any vector norm. Sets 9.In and bn are interval ones (~n is abox in ~m). Let An and bn be the centers of these sets, respectively, and let us represent 9.In and bn in the centralized form: o
o
o
bn = bn + 8bn.
(29)
which is equivalent to (26) (see [11, 12]). This problem can be efficiently solved with the recently designed nonsmooth optimization routine [10]. From the viewpoint of computational complexity the use of any particular vector norm, i.e., f~, #e or {~, in (28) has no significance. It is worth to cite the book [1] where nearly the same reduction is con:fidered. Let us also cite the books [ 13, 14] where another method for solving the control problem with single input and with A n = 0 has been suggested. Let us consider now two particular cases, namely, (i) control systems with single input and interval setvalued estimates of uncertain parameters, and (ii) interval control systems with square matrix Bn of the diagonal form. In both cases, it is assumed that there is no noise.
bn E ~1 are interval ones presented in central&ed form (32) and the control system defined by their o o centers An and bn is controllable, i.e., 0 ~ bn, Vn >>-0, then the minimizer for problem (31 ) is o
•
o
Cn = -(bn)-l(An -An). See appendix for the proof.
5.3. Unperturbed interval MIMO systems with diagonal matrix Bn Consider the class of multi-input control systems given by (24) with nonsingular diagonal matrix Bn:
Bn=diag{bi, n}im=l, bi,n~LO, Vn>~0, i = l , m ,
1...:]
and with the matrix M n of canonical form (2). Here AnT will denote the mth row of the matrix An and An will denote the given parame,ter vector of standard system (8), when the matrix A takes the form (30).
o
o
9~n = An + ~9~n,
f~n = Bn + 8 ~ n ,
the minimizer for the synthesis problem
(30) • • •
4~
min
max
Cn AnE~[,; BnE~n
a2,n
...
(34)
and sets 9.In, f~n are interval ones presented in their centralized form
Assume An takes the canonical form
0
(33)
Theorem 2. I f the matrix B, satisfies conditions (34)
5.2. Interval single-input systems without noise
An ~
(32)
Theorem 1. I f set-valued estimates 9.Inc ~m and
find u,,
(31)
bnEbn
~In = An + 8~n,
Finally, the control problem takes the form:
min{y: ¢&,~(Xn,Un) 0 (recall the condition 0 ~ b ) . Inequality (A.5) is the correct one, if one of the following inequalities are fulfilled: (A)TCI < 0,
-
-
C1 < 0.
(A.6) (A.7)
Assume inequality (1.6) takes place. Therefore,
IIA-+/~C~ II < II - A-4-/~C~ 11 and ,45~ argmax~A q~(-), because ~o(A-)< ~p(-A-) (recall here that 91 is a box), i.e., the value ~p(-A-) at the vertex, which is opposite to A-, is greater than at 8A =.~.
-b°A+Gb < ~A+C~ and /~ ¢ arg max~b q~(-), because ~#(/9)< q ( - b) and the maximal value of ~0 is reached at 8b = -/~. Thus, both considered cases lead to the contradiction. []
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