Improving transient response of adaptive control systems using ...

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 9, SEPTEMBER 1994

Improving Transient Response of Adaptive Control Systems using Multiple Models and Switching

between multiple models can be used to improve performance in adaptive systems, even when a single model alone is sufficient to assure stability.

Kumpati S. Narendra and Jeyendran Balakrishnan 11. STATEMENT OF THE PROBLEM

Abstract- A well-known problem in adaptive control is the poor transient response which is observed when adaptation is initiated. In this paper we develop a stable strategy for improving the transient response by using multiple models of the plant to be controlled and switching between them. The models are identical except for initial estimates of the unknown plant parameters. The control to be applied is determined at every instant by the model which best approximatesthe plant. Simulation results are presented to indicate the improvement in performance that can be achieved.

I. INTRODUCTION Direct and indirect methods for the adaptive control of a linear time-invariant plant are currently well known [l]. Numerous globally stable algorithms have been developed over the last decade for the so-called “ideal case” that result in zero steady-state tracking error. Extensive computer simulations of these algorithms have revealed that the tracking error is quite often oscillatory, however, with large amplitudes during the transient phase. In this paper, we suggest one method of improving the transient response in the ideal case by using multiple identification models. The unknown plant parameters are assumed to belong to a compact set S . N adaptive identification models with identical structures, but with initial estimates of plant parameters uniformly distributed in S, are used. Each identification model 1, is paired with a parameterized controller C,, to form an indirect controller IC,. The control strategy is to determine at each instant the model whose identification error is the minimum according to some criterion function and use the corresponding controller to control the plant. Consequently, the problem is to determine an appropriate criterion for the choice of the model at any instant and demonstrate that the resultant switching scheme yields a stable system with improved performance. An important consideration in such schemes is the stability of the overall system. It is shown that any arbitrary switching scheme yields a globally stable system, provided that the interval between successive switches has an arbitrarily small but nonzero lower bound. This result provides the freedom to choose the switching scheme entirely on the basis of performance. In this sense, the nature of the problem is distinctly different from that normally encountered in the adaptive control literature. The idea of switching between multiple adaptive models has been used in the past by Middleton et al. [2] to alleviate the problem of stabilizability of the estimated model in indirect control and later adapted by Morse et al. [3] for model reference adaptive control when the relative degree of the plant is unknown. While these results used switching techniques to solve global stability problems in adaptive control, the thrust of this paper is to demonstrate that switching

The basic problem we wish to address is closely related to the model reference adaptive control (MRAC) problem treated in standard textbooks on the subject (e.g., [ 11). The plant to be controlled is linear and time-invariant, with input U : R+ + R and output yp: R+ -+ R related by

where W p ( s ) = k,(Z,(s)/Rp(s)) is the transfer function of the plant. Here k, E R is nonzero, and R p ( s )and Z,(s) are monic coprime polynomials of degree n and m, respectively, with m < n, and unknown coefficients. We make the standard assumptions [ l ] about W,(s) that i) the order n, ii) the relative degree n* n - m, iii) the sign of k, are known, and iv) Z,(s) is Hunvitz. The reference model to be followed is linear and time-invariant, with input r: R+ + R (which is piecewise continuous and uniformly bounded) and output ym: R+ + R related by

where W , (s) = k , (2, ( s ) / R , (s)) is the transfer function of the reference model. Here k, E R is nonzero and R, (s) and Z, (s) are monk coprime Hunvitz polynomials of degree n and m, respectively. The standard MRAC problem is to determine a differentiator-free control input U(.) such that all signals in the overall system are y,(t) - y,(t) converges bounded and the control error e,(t) to zero asymptotically. This problem was first solved in 1980 in [4]-[7] and has been widely used in the design of adaptive systems. Our additional (qualitative) requirement is that e, ( t ) stays within reasonable limits and settles down to an acceptably small value within a reasonable time.

e

111. DESCRIPTIONOF THE ADAPTIVE CONTROL SYSTEM The proposed control system (see Fig. 1) consists of N identical indirect controllers In particular, N adaptive identification models {I,},”=1, with identical structures but different initial estimates of the plant parameters, are used. Corresponding to each 1, is a controller C, and the identification and control parameters of each pair IC, are tuned simultaneously as described in this section. While all the identification models operate in parallel, only one of the controllers is connected to the plant at any instant. The decision as to which controller to connect to the plant at every instant is determined by the switching scheme, which is described in Section V. Parameterization of the Plant: The plant is parameterized as follows. Define W I , w2: R+ + R”-’ by

{lC‘,}El.

hi = A w l

Manuscript received December, 1992; revised June 17, 1993 and January 5, 1994. This work was supported in part by National Science Foundation Grant ECS-8921845. The authors are with the Center for Systems Science, Department of Electrical Engineering, Yale University, New Haven, CT 06520-1968 USA. IEEE Log Number 9402729.

+ ZU

where (A, I ) is an asymptotically stable system in controllable

0018-9286/94$04.00 0 1994 IEEE

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 9, SEPTEMBER 1994

MODEL

IDENTIFIERS e

1

-

1

1

(b)

CONTROLLERS

Fig. 2. Internal structure of the adaptive control system: (a) identifiers (b) controllers. Fig. 1. Structure of the adaptive control system with N identifiers {Ij},"=, and controllers {Cj},"=, .

8* are computed from p" as

canonical form, with A

X(s) = det(s1-

A) = Xl(s)Z,(s)

(4)

The output of each Cj is given by

for some monk Hunvitz polynomial Xl(s) of degree n - m - 1. Since 2, (s) and Rp(s) are coprime, there exist unique polynomials a(.) and p ( s ) each of degree n - 1 satisfying the Bezout identity where O3

( k 3 , 0;

, BO,,

0;

)T

is the j t h estimate of 6" with error

vector 8, -0'. While all the identifiers share the same regression vector J,all the controllers share the vector g (see Fig. 2). A The control error e , ( t ) = y p ( t ) - ym ( t )can then be expressed as There exist unique constants @; k p / k m E R, p; E Rn-', a: E R, a; E satisfying Po' P I T ( s l - A)-'Z = p(s)/X(s) and a: + a l T ( s l - R ) - l Z = k p a ( s ) / X ( s )such that yp can be expressed as

Fnp1

+

e , = Wm(s){P;BTW}

(12)

where e"(t)5 0 ( t )- 0*,and 8 ( t ) E { 81( t ) ,. . . ,ON ( t ) }is the control parameter vector used at time t. With the above definitions, the *T A manner in which the parameters I ; j ( t ) and 8 j ( t ) are tuned can be yp = Wm(s){P0*u P;'Wl a;yp cy1 W Z } = W , ( s ) { p * T W } (6) described. A where w = ( U , wT, yp, w : ) ~ and p* (Po', p r T , a:, a ; T ) T . Dynamical Adjustment of Identification and Control Parameters: Motivated by (IO), we define the following closed-loop estimation Since p* is a constant, yp can be rewritten as errors in terms of I;, and 8,

+

+

+

e

where G W,(~)l~~{w}. Structure of the Identifiers: The N identifiers puts

have out-

A

are the j t h estimates of yp and p * , respectively. where GP, and Defining the j t h identification error as e13 = gp, - yp and the parameter error vector

fi,

A

= $3 - p * , we obtain

Structure of the Controllers: If p* is known and the control input A is chosen as u*(t) = O*Tg(t)where g = ( T , wT, yp, w : ) ~ and 8' A ( k * , SI', e,*, O ; T ) T E R2", it follows from (6) that yp W , ( S ) T= ym (assuming zero initial conditions). The elements of

and €6, = (Q,,F,&, , EOo, , ) T . In addition to the identification errors eI, , the parametric errors €0, are also used in the adjustment of and 8, as shown in (14). When i 3 ( t ) p* and O,(t) 8*, these errors are identically zero. Hence, they determine the extent to which $,(t) and 0 3 ( t ) satisfy the relation (10). In Section IV,it is shown that the following adaptive laws, together with any allowable switching scheme, force the identification errors eI, ( t ) ,the closedloop estimation errors ce,(t) and the control error e , ( t ) to tend to zero asymptotically

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 9, SEPTEMBER 1994

Iv. GLOBALSTABILITY WITH ARBITRARY SWITCHING SCHEMES The adaptive laws (14) indicate how the identification and control parameters are to be adjusted at every instant, but do not determine the sequence of controllers to connect to the plant. The principal result of the paper guarantees that, as far as stability is concerned, it does not matter what this sequence is: The overall system will be globally stable for any arbitrary switching sequence, provided that the intervals between successive switches have a nonzero lower bound T,,, > 0, which can be chosen to be arbitrarily small. This result is intuitively reasonable since it is known that each one of the N controllers {IC,} results in global stability if used alone. The proof does not follow directly, however, since switching between stabilizing controllers need not result in a stable system. The proof is based on the methods in [ l ] and [4] for the case of a single model (i.e., N = 1).Since all the identification models and the adaptive control laws are identical, the overall system is globally stable if the switching settles down after a finite time at any one of the indirect controllers. We prove that the same is the case with an arbitrary switching sequence between the N controllers, even if switching never stops. Let {%}go denote the instants at which switching takes place, so that T, + CO, and Tz+l- T, > T,,, > 0. The input to the plant is u ( t ) = OT(t)g(t) where 6 ( t ) is chosen by the switching scheme from the set (6, The proof proceeds A by contradiction. We assume that the signal w p = ( U ? , yp, w ? ) ~is unbounded and show that this leads to a contradiction. The proof is given in Steps 1 4 . The results of Steps 1 and 2 are standard and follow the same arguments used in [ l ] and [4]. The results of Steps 3-5 are based on two propositions which are extensions of well-known results in adaptive control theory (involving bounded signals) to the case of unbounded signals. The propositions are stated without proof since they follow along the same lines as in the case of bounded signals. If a signal f E C2 and f E C", it is well known that limt-" f ( t )= 0. Proposition 1 is an extension of this fact. Proposition 1: Let f : R+ + R and w : R+ -+ R" be continuous signals. If f(t) = P(t)O(sup,,, 1 1 ~ ( ~ ) 1 1 ) where E C2 is continu-

(t)}El.

ll~F)ll)>

ous, and f(t) = O(SUP,