stability of discrete nonlinear systems under nonvanishing perturbations: application to a nonlinear...

IMA Journal of Mathematical Control & Information (1999) 16,2341

Stability of discrete nonlinear systems under nonvanishing perturbations: application to a nonlinear model-matching problem

CBSAR CRUZ-HERNANDEZ? Electronics & Telecommunications Depamnent, CICESE,

PO. Box 434944, Sun Diego, CA 92143-4944, USA

JOAQU~N ALVAREZ-GALLEGOS AND RAFAEL CASTRO-LINARES$ Department of Electrical Engineering, CINVESTAY

RO. Box 14-740, 07000, Mexico, D.E, Mexico

A stability analysis of discrete-time nonlinear systems subjected to a class of nonvanishing perturbations is presented. It is shown that, if the equilibrium point of the unperturbed system is uniformly asymptotically or exponentially stable, then the state trajectories are ultimately bounded if some conditions on the perturbations are satisfied. A region with guaranteed stability for the perturbed system is also given. The results obtained are applied to the model-matching problem associated with a perturbed discrete-time nonlinear system.

Keywords: Discrete nonlinear systems; perturbed systems; model-matching problem; stability robustness.

1. Introduction

Stability of equilibrium points is usually analysed in the sense of Lyapunov. The founda- tions of the theory are well established, and it is an indispensable tool in the analysis and synthesis of nonlinear systems. An introduction to the classical Lyapunov stability theory for difference equations with applications to control systems is given by Kalman & Bertram (1960). In particular, a practical technique, which generates an open set of initial conditions and specifies bounds on the input amplitude, that guarantees bounded states for all time is described by Degroat et al. (1992). On the other hand, Byrnes et al. (1993) have shown that any nonlinear system with Lyapunov-stable unforced dynamics can always be globally stabilised by smooth state feedback if suitable controllability-like rank conditions are satisfied.

An important problem in the analysis of stability is the robustness with respect to mod- elling errors, aging of physical components, uncertainties, and disturbances that exist in any realistic problem. The perturbation effect at the equilibrium point can be null or not. The first case is called vanishing perturbation; this means that the equilibrium point of the perturbed system remains the same as that of the equilibrium of the unperturbed one. The second case is called nonvanishing, for which equilibrium points of the perturbed and unperturbed systems are not the same. However, the perturbed system may not have an

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@ Oxford University Press 1999

equilibrium at all, in which case we cannot study the problem as the stability of equilibria any longer. So, the best we can expect are ultimately bounded state trajectories if the perturbation is small in some sense.

In this paper we discuss the stability of discrete-time nonlinear systems subjected to a class of nonvanishing perturbations, when the unperturbed system has a uniformly asymptotically or exponentially stable equilibrium point. We show that the solution trajectories stay bounded if the perturbation satisfies some conditions. We parallel, for discrete- time nonlinear systems, some results given by Khalil (1992) for continuous-time nonau- tonornous systems. However, in the discrete-time case there are some important technical differences.

The results obtained, which include a conservative estimate of a parameter region that guarantees stability, are applied to the nonlinear model-matching problem for discrete systems.

Given a plant and a model, the model-matching problem consists in finding a compensator such that the input-output behaviour of the compensated system is similar to that of the model. For nonlinear continuous-time systems, a number of results in relation to this problem have been obtained by means of differential-geometric concepts (Benedetto & Isidori 1986) or via differential-algebraic methods (Moog et al. 1991). The so-called structure algorithm is used by Isidori (1985), and the zero-dynamic algorithm is utilized by Benedetto (1990). The same algorithm is used by Castro & Benedetto (1990) and Benedetto & Grizzle (1994) to study the asymptotic nonlinear matching problem. Also, the nonlinear model-matching problem with internal stability of the closed-loop system has been considered by Byrnes et al. (1988) for single-input single-output systems, and by Huijberts (1990). (1992) for multi-input multi-output systems.

For nonlinear discrete-time systems, very few publications have been devoted to the model-matching problem. For example, Kotta (1989) designed a dynamic state compensator so that the input-output behaviour of a nonlinear discrete-time system matches that of a linear model. In Kotta's (1995) paper, a compensator is found which, beginning from a fixed time instant, makes the outputs of the compensated system equal to the outputs of a nonlinear model, provided that their inputs coincide, though the asymptotic convergence of the output tracking error is not considered in that work. The approach used in our paper allows us to obtain the asymptotic convergence, and also to guarantee a stability region, for the state trajectories of the perturbed closed-loop system.

On the other hand, the stability and stabilization of discrete-time nonlinear systems have also received some attention. The stabilization of a nonlinear discrete-time control system to a specified equilibrium point by means of an adequate state feedback is considered by Tsinias (1989), whereas an output-feedback stabilization problem is studied by Tsinias & Kalouptsidis (1990). where a deterministic disturbance signal affects the dynamics of the system. More recently, the global stabilization of discrete-time nonlinear systems has been addressed by Kazakos & Tsinias (1994). Likewise, Cruz & Alvarez (1994a) carry out a stability analysis for a discrete-time nonlinear system fed back with a linearizing controller, subject to vanishing perturbations. CNZ & Alvarez (1994b) make a first attempt at stability analysis for discrete nonlinear systems under nonvanishing perturbations.

The present paper is organized as follows. In Section 2 we present a stability analysis for the perturbed nonlinear system in the case when the unperturbed system has a uniformly asymptotically or exponentially stable equilibrium point. In Section 3 we apply these re-

STABILITY OF PERTURBED DISCRETE NONLINEAR SYSTEMS 25

sults to a model-matching problem associated with a perturbed plant. Finally, in Section 4, we draw some conclusions.

2. Stability of perturbed systems with nonvanishing perturbations

We first consider a nonautonomous discrete-time nonlinear system, called the unperturbed system, described by

x(k + 1) = f ( k , x (k ) ) ,

where x (k ) E L3' = { x E W n : Ilx 11 < r } , with r > 0, is the state and f : Z+ x Br -t Dr. Z+ denotes the set of nonnegative integers, and llxll is the norm of x. Let p(k, ko, xo) denote the solution of (2.1) corresponding to the initial condition x(k0) = xo and evaluated at the kth instant of time (k 2 ko). Let Brqb denote the set Br.k, = { ( k , x ) E Z+ x W n : llxll < r, k 2 ko}. A point xo E W n is called an equilibrium of (2.1) if f (k , xo) = xo, for all k 2 0, i.e. if xo is a f i e d point of the map f for all k 2 0. Thus, if this condition is satisfied, then p(k, ko, xo) = xo for all k 2 ko 2 0. We make the following assumptions on system (2.1):

(i) The origin x ( k ) = 0 is an equilibrium point, i.e. f ( k , 0 ) = 0 for all k 2 ko. (ii) There exist r > 0 and t > 0 such that, for each initial condition (ko, xo) E Brqr,

there exists exactly one solution p(k, ko, xo) for all k 2 ko. (iii) f ( k , .) is continuous for all k. Then p(k, ko, .) is also continuous for each pair

( k , ko), with k 2 ko. (iv) The origin x (k ) = 0 is an isolated singular point.

We shall that f belongs to the class £ ( f E E ) to indicate that f satisfies the four conditions above.

DEFINITION 2.1 (Locally positive definite and decrescent functions, Vidyasagar 1993) A function V : Z+ x Wn -t W is called a locally positive definite function (lpdf) if V ( k , 0) = 0 for all k 2 ko and there exist a constant r > 0 and a function a! of class K such that

V ( k , x ) 2 ~ ( l l x l l ) ~ Vk 2 0. Vx E Br.

V is decrescent if there exist a function B of class K and a constant r > 0 such that

We give the following results concerning different types of stability of the equilibrium of (2.1).

THEOREM 2.2 (Stability, Vidyasagar 1993) The equilibrium x = 0 of system (2.1) is stable if there exists a definite function V : Z+ x W n -+ W, and a constant r > 0 such that

(i) V ( k , x ( k ) ) is an lpdf, and (ii) A V ( k , x ( k ) ) A Vk+l - Vk = V ( k + 1 , p(k + 1, k , x ( k ) ) ) - V ( k , x (k ) ) < 0,

Vk 2 0 , Vx (k ) E Br.

If, in addition, V is decrescent, then x (k ) = 0 is uniformly stable.

A definite function V(k, x) that satisfies properties (i) and (ii) of Theorem 2.2 is known as a Lyapunovfunction for system (2.1)

THEOREM 2.3 (Asymptotic stability, Vidyasagar 1993) The equilibrium x = 0 of system (2.1) is uniformly asymptotically stable if there exists a decrescent lpdf V : iZ+ x Rn + W such that

AV(k,x) GO, Vk 3 0 , Vx EB,\{O}.

THEOREM 2.4 (Exponential stability, Vidyasagar 1993) Suppose that there exists a definite function V : Z+ x Rn + R and constants cl, c2, cs, r > 0 and p > 1 such that

Then x = 0 is an exponentially stable equilibrium of system (2.1).

The following notion on the ultimate bound for the solutions of system (2.1) will be used to study the stability properties of a class of perturbed discrete-time nonlinear systems.

DEFINITION 2.5 (Ultimate bound) The solutions of system (2.1) are said to be uniformly ultimately bounded if there exist positive constants b and c, and for every cr E (0, c) there is a constant T = T(a), such that

The constant b in (2.4) is known as the ultimate bound.

Let us now consider the so-called perturbed system

where f (., a ) is the map that defines the dynamics of the unperturbed system (2.1). and suppose that f E E. Then

Notice that if g(k, 0) = 0, the perturbed system (2.5) has an equilibrium point at the origin. In this case we can analyse the stability of the origin as an equilibrium point of the perturbed system. But, when g(k, 0) # 0, the origin will not be an equilibrium point of the perturbed system anymore. Therefore we cannot study the problem as the stability of equilibria any longer. In the first case we say that (2.6) is a vanishing perturbation term and, in the second case, it is nonvanishing.

Our goal in this paper is to study the following problem: Consider the perturbed system (2.5). Assume that f E E and that the origin is a uniformly asymptotically or exponentially stable equilibrium point for the unperturbed system (2.1). Under which conditions on the perturbed function (2.6) does there exist an ultimate bound for the solutions of the perturbed system?

Before going to solve the previous problem, we need to give a result of existence of the ultimate bound for the solutions of system (2.1). We make the following assumptions.


( A l ) There exists p > 0 such that the equilibrium point x = 0 of system (2.1) is uniformly stable on B,.

(A2) There exists a continuous function V : Z+ x Br -+ R, V(k , a) such that

for 0 < p < a;l(al ( r ) ) , for all k 2 0 and for all x E Br, where ai(.) (i = 1,2,3) are class K functions defined on [0, r ) .

We then give the following theorem that can be seen as the discrete-time version of the result stated by Khalil(1992) for the continuous-time case.

THEOREM 2.6 Consider the unperturbed system (2.1). Assume that A1 and A2 hold. Then there exist a class K function q ~ , a class L function a, and a finite time k l , depending on x(k0) and p, such that the solutions of (2.1) satisfy

Proof: By assumption Al , we can restrict our analysis to the solutions starting outside Bp. Let p < r be a positive number, and define a time-dependent set

contains the ball (x E B, : llx 11 < a; '(al (p)) ) because the condition llx 11 < a;' (a1 (p)) implies that az(llx 11) < a, (p) and, since V(k , x ) < a2(llx ll), then V(k, x ) < a1 (p). Furthermore, Qk.p C Bp = (x E Rn : llx 11 < p} . For this, note that, if x E QkSp, then a ~ ( l l x l l ) < V(k. x ) < adl lx l l ) < ~ I ( P ) , and therefore a~( l l x l l ) < ~ I ( P ) , so llxll < P.

Since p < a l l ( a l ( r ) ) and p < r , we can choose p such that p < a;'(al(p)). This implies that q ( p ) < a l (p ) , and then we have that a&) < a1 (p) < a1 (r) . Furthermore, for any x(k0) such that Ilx(k0) 11 < a;' (a l ( r ) ) , we can choose p close enough to r such that Ilx(ko)ll < a; ' (a l (p)) . Let q = a;'(az(p)) < r , and define a time-dependent set

The set Qk,,, contains the ball { x E Br : Ilx l l < a;' (a1 (q))) and, because a2(p) < a1 (p), then Qk,,, C S2kVp. On the other hand, B, = (x E Br : IIxII < p } c Qk., because IlxII < p implies that az(llx 11) < az(p) and, from V(k, x ) < az(llx ll), we have that V(k, x ) < az(p). Furthermore, a, (llxll) < V(k , x ) < az(p) implies that llxll < ar1 (a2 (p ) ) = q, and therefore Q k , , c B,,.

We thus have that B, C Qk,,, C B,, C BP C Br and Qk,,, C Q k q p C Bp C Br. Consider now the sets

It is clear that SZk,,-, 1 .(2R.q-Lc. These sets have the property that any solution of (2.1) starting inside either set cannot leave it, because of assumption A2. Therefore, if Ilx(k0) 1 1 < a r ; ' ( ~ ~ ~ ( ~ ) ) and x(ko) $ B,, the solution x ( k ) will belong to Qk,,- , for all k > ko. For a solution starting inside of Qk,,-,, but outside of Qk,,-,, let kl be the first time it enters Qk,,-,; k l could be ko (if the solution starts inside Qk.,-,) or ~o (if it never enters Rk,,-,). We have, for all k E [ko, k l ) , that

where a is a class K function defined on [O, a1 ( p ) ) . If we consider the above auxiliary equation, A V ( k , x ( k ) ) < - a ( V ( k , x ( k ) ) ) < 0 implies that V ( k , x (k ) ) decreases mono- tonically and, because V ( k , x ( k ) ) > 0 , then limk,, V ( k , x ( k ) ) = c > 0 exists. From the same auxiliary equation we have that a ( c ) = 0 and, since a is of class K, then c = 0 . The argument also applies in the case when zero is reached in a finite number of steps. Since the estimate is uniform with respect to ko, there exist a function p of class K , a number I] > 0 , and, for each x(ko) satisfying Ilx(ko)ll < V I = a;'(al ( r ) ) , a function o of class L such that

Since p(llxoll)o(k - ko) -, 0 as k + m, there must be some finite time after which p(llxoll)a(k - ko) < for all k . Therefore kl is finite; that is, the solution must enter the set Qk,,-,, in finite time. Once inside the set, the solution remains inside for all k 2 k l . Therefore, V ( k , x (k ) ) < a1 (v ) , Vk 2 kl , and Ilx(k)ll < I], Vk 2 kl . 0

REMARK 2.1 In the particular case when x(k ) = 0 is an exponentially stable equilibrium point of system (2. l ) , we have that

with constants c > 0 and p < 1, and p(((xo(l)a(k - ko) = J(c2/cl)llx(ko)llPk.

Consider again the unperturbed system (2.1) and suppose that assumption (A2) holds with p = 0 . This assumption then becomes:

(B 1 ) There exists a C I Lyapunov function V : Z+ x L3, + W for the unperturbed system (2.1) that satisfies

The assumption B 1 implies that the origin of the unperturbed system is a uniformly asymptotically stable equilibrium point. The existence of a Lyapunov function with the properties (2.7) and (2.8) is guaranteed by a converse theorem of Lyapunov stability theory (Vidyasagar 1993). In addition, we introduce the following assumption on the bound of the perturbation term (2.6).


(B2) The perturbed function (2.6) satisfies the uniform bound

with 4 ( x ) = ,/a, for some positive constants I and 6.

The right-hand side of inequality (2.9) allows us to distinguish vanishing and nonvanishing perturbation terms. If g(k , x ) is a continuous function in x , uniformly in k , then it is bounded in some compact subset of Br; therefore it may be bounded by some constant I. The components of g(k , x ) that vanish at the origin can be limited by the term s ~ ~ ( x ) .

THEOREM 2.7 Suppose that the assumption B 1 holds. Then, for Ilx(ko) ll < cr,'(crl ( r ) ) , there exist constants f, 8 > 0 such that the solution x(k ) of the perturbed system (2.5) is uniformly bounded for k 2 ko and uniformly ultimately bounded with an ultimate bound which is a function of i and 8 for any perturbed function that satisfies B2 with 1 < i and 6 < 8.

Proof: We use V ( k , x (k ) ) from B 1 as a Lyapunov function candidate for the perturbed system (2.5). The forward difference function A V along the trajectories of this system is

where the subscript of AV indicates that the difference is related to the corresponding equation, and where

Since V ( k , x ( k ) ) is a C' function in x(k ) for all k , 6 V ( k , ~ ( k ) ) ~ satisfies

where I g is a positive constant. Now using condition (2.8), we have that

and by assumption (B2),

If 6 satisfies the bound

with a1 = - $Ig > 0 for 8 < l / l g . Hence

30 CESAR CRUZ-HERNANDEZ ETAL.

is n dass K function of l and 6, with i satisfying the bound

From T h e m r n 2.6 it follow that. if I/x&)JJ e a;' IT)), then the snlution of tht per- ttrrbcd systcm (2.5) is uniformly bounded for k 3 kg nnd u n i f m l y ultimately b u n d e d w i ~ h an ultimate bound b which is a fuuncli~n of I and 6. ihat is. b = n;'{a2{fl)] . So. the ultimata bound of thc solution of the perlurbd syarem (2.5) is

We now study the stability of the perturbed system (2.5) when the origin is an exponentially stable equilibrium point of the unperturbed system (2.1). For this end, consider the assumptions:

( C l ) There exists a C' Lyapunov function V : Z+ x Br + R for the unperturbed system (2.1) that satisfies

el llx(k)l12 < V(k . x W ) < c211x(k)l12, (2.10)

AV(k, x(k)) < -c311x(k)l12, c l , c2, c3 > 0, (2.1 1)

for all (k , x (k ) ) E Z+ x Br. (C2) The perturbation term (2.6) satisfies the bound

Ilg(k xW)ll < 1 + 811x(k)l12, (2.12)

for some positive constants 1 and 6, for all k >, 0 and for all (k , x(k)) E Z+ x Dr.

Then we have the following theorem.

THEOREM 2.8 Suppose that assumption C1 holds. Then, for all JJx(ko) l l < J O r , there exist constants i, 6 > 0 such that the solution x(k) of the perturbed system (2.5) has an uniform ultimate bound, for all k 2 ko, which is a function of land 8, for all perturbation terms that satisfy C2 with 1 < i and 6 < 6. Proof: Along the lines of the proof of Theorem 2.7 one can again show that the solution of the perturbed system (2.5) is uniformly bounded for k > ko with ultimate bound

c3 - c ' 0 1 r 2 , 0 < B < l . b(,u(l,8)) = ~ f i , where 8 < 8 < -, 1 < 1 < -- I , cz 1,

STABILITY OF PERTURBED DISCRETE NONLINEAR SYSTEMS 3 1

3. Application to a nonlinear model-matching problem

We now consider a discrete-time nonlinear system depending on a real-valued parameter vector E E Wq, called the perturbed plant described by

where the state x E X , an open set in Wn, the input u is inside an open set 2.4 in Pm, and the output y belongs to an open set y in Pm, The mappings f : X x U x Wq + X and h : X 4 Y are supposed to be analytic. The corresponding unperturbed discrete-time nonlinear system, called the unperturbed plant, is defined as

i.e. { (x , U ) = f (x , U ; 0) for all ( x , u ) E X x U . In addition, we consider a model described by

X M ( ~ + 1) = f ~ ( x ~ ( k ) , u ~ ( k ) ) , M :

Y M ( ~ ) = ~ M ( x M ( ~ ) ) ,

where the model state X M E XM (an open set in BnM), the model input U M E UM (an open set in Rm), and the model output y~ belongs to an open set YM in Wm. Also, the mappings f~ : XM x U M + XM and h~ : XM + yM are supposed to be analytic.

In this section we present an analysis of a model-matching problem associated with the perturbed plant P ( E ) . Under some additional conditions on the unperturbed and the perturbed plants, and assuming that the equilibrium point of the unperturbed closed-loop system is asymptotically stable, some stability properties of the perturbed closed-loop system are determined.

3.1 A nonlinear model-matching problem

Let us consider the unperturbed plant P and the model M . The compensator utilized to control the unperturbed plant is a discrete-time nonlinear system described by

with state xc E XC (an open set in WV), input x and U M , and output u . Here fc and hc are analytic mappings defined on a suitable open and dense subset of Xc x X x UM. The compensated unperturbed plant (3.2)-(3.4) defines a new control system, with input U M ,

output y p o c , and state .? = ( x , x c ) , described by

32 C ~ S A R CRUZ-HERNANDEZ ETAL.

where

We assume that system (3.2) evolves in a neighbourhood of an equilibrium point xO; that is, around (xO, uO) E X x U such that f(xO, uO) = xO, with {u(k) = u0 : k 2 0) being a (constant) input sequence. For this sequence there exists another (constant) output sequence {y(k) = h(xO) = : k 2 0). In the same way, let the equilibrium point of (3.3) be denoted by (xk , u;) E XM x UM. We are interested in finding a compensator C for the plant P which, irrespectively of the initial states of P and M, makes the output y(k) of P asymptotically converge to the output y ~ ( k ) produced by M under an arbitrary input u (k). The discrete nonlinear model-matching problem addressed here is defined as follows.

DEFINITION 3.1 (Discrete-time nonlinear asymptotic model-matching problem, DAMMP) Given the unperturbed plant P and the model M around their respective equilibrium points (xO, uO) and (xk , u;), and a point (x(O), x ~ ( 0 ) ) E XO x X; c X x XM, find an integer v and a compensator C with initial condition xc(0), such that the output of the compensated unperturbed plant yp,c converges asymptotically to the output y ~ ( k ) produced by any input u ~ ( k ) to the model M. This means finding two mappings fc and h c such that the compensated unperturbed plant satisfies the property given before in some neighbourhoods VI of (x:, xO, u;) in Xc x X x UM and Vz of u0 in U.

A way to solve the above problem is to define an output error yE(k) = y(k) - y ~ ( k ) , and to choose a control u(k) such that yE(k) is decoupled from the model input u ~ ( k ) , for all k 2 0, and converges asymptotically to zero. The first point is equivalent to transforming the DAMMP to a disturbance-decoupling problem with disturbance measurements of an auxiliary system. Such an approach also allows us to treat the second point in such a way that the output error depends only on the initial conditions x(0) and x ~ ( 0 ) . and not on the model input u ~ ( k ) . To this end, let us define the auxiliary system

with state XE = col(x, XM), input vectors u E = u and WE = UM, and

Note that WE is considered as a disturbance signal acting on the auxiliary system. Given this system, together with an equilibrium point x! = (xO, x k ) it is known (see e.g. Griz- zle (1985). Kotta (1995), and Nijmeijer (1990)) that, if the disturbance-decoupling problem with measurement disturbance WE associated with this system has a solution on a:, an open and dense subset of X x XM x U x UM, defined around the equilibrium point (xO. x&, uO, u;), then there exists an analytic mapping defined on with the property that the control law


decouples the output y ~ ( k ) of the closed-loop system (3.6)-(3.7) from the disturbance w ~ ( k ) for every initial state x ~ ( 0 ) in an open and dense subset of X x XM contained in 0;.

In the sequel the DAMMP will be treated in terms of a relative degree associated with each output components yi and YM;. Thus the following definitions are introduced. Let So, f,wo, and f~~ be the undriven state dynamics f(., O), f~ ( a , O), and fE (., 0, O), respec-

-. . lively, and f i , fho, and fi0 the j-times iterated compositions of h, fMo, and fEo, with -0 fo (XI = X, f i 0 ( x ~ ) = XM, and f i o ( x ~ ) = XE.

DEFINITION 3.2 (Monaco & Normand-Cyrot 1987) The output yi of the unperturbed plant (3.2) is said to have a relative degree di in an open and dense subset Oi of X x U containing the equilibrium point (xO, uO), if

a - au [hi 0 $ (.f(x, u))]

fo ra l lO<l <di - 1,forall < j <m,fora l l (x ,u) € O i , and

a - au j [hi 0 f;p, ( i (x, n)] # o

for some j E { I , . . . , m} and for all (x, u) E Oi.

A similar definition is given for the relative degree of the model (3.3). dMi , in an open and dense subset OMi, of XM x UM containing the equilibrium point (xL, uL). If none of

the iterated functions hi o ft f (x, U) or h~~ o fko( ~ M ( X M , UM)) depends on u or u ~ , ( - ) respectively, we set di = ca or dMi = ca. From these concepts and the definition of the auxiliary system (3.6), it immediately follows that, if the output yi of the nominal plant and the output y~~ of the model have relative degrees di and dMi, respectively defined on open and dense subsets Oi and OM;, for i = 1, a , m, then the relative degrees of the output y ~ ~ , associated to the input vectors U E = u and WE = U M are exactly di and dMi, respectively.

Now suppose that each number di (i = 1, . - . , m) is finite and set O = OI n . . . n 0,. We define the input-output decoupling matrix of the nominal plant (3.2) as the m x m matrix

1

and consider the following assumption.

(Dl) For all 1 < i < m and for all xE = (x, XM) E X x XM and uM E UM,

where im {(p}denotes the image of (p.

The following theorem gives necessary and sufficient conditions for the local solvability of the DAMMP.

THEOREM 3.3 Consider the unperturbed plant (3.2) and the model (3.3) around, respectively, their equilibria (xO, uO) and (x;, uL). Suppose that the output yi of the unperturbed plant and the output y~~ of the model have finite relative degrees di and dMi, respectively defined on Oi and OMi, for i = 1, . - . , m. Assume that assumption A1 holds. Suppose also that the input-output decoupling matrix A(x, u) in (3.10) is nonsingular for all (x, u) E X x U. Then the DAMMP is locally solvable on $2; if, and only if,

di < d ~ ~ (1 Q i < m ) . (3.12)

Proof: Sufficiency. If di < d~~ (i = 1, , m) holds, then it follows from the definitions ofdi anddMi (i = 1 , - . . ,m) that

where

Sj ( ~ ( k ) , u (k)) = hi o $ (f(~(k), ~ ( k ) ) ) - hi $ + ' ( ~ ( k ) ) ,

with Si(x, 0) = 0 for all i = 1,

a 1 , - .- .. (1 < i, j < m), (3.13)

This argument can be repeated at each iteration. Thus, each output of the auxiliary system (3.6) is decoupled from u~ on a:, so

yi (k) = ei,o(k) + Y M ~ (k) (i = mh (3.16)

STABILITY OF PERTURBED DISCRETE NONLINEAR SYSTEMS

where

Note that the term eiVo(k) depends only on an arbitrary signal vi (k) (i = 1, . , m) and on the initial values of x and XM, and not on the model input vector u ~ ( k ) . If we choose a signal vi (k) such that limk+,vi (k) = 0 (i = 1, . . , m) then limk,,yi (k) = y~~ (k). In particular, if we put vi (k) = 0, then ei.o(k) will be zero after k = di SO y (k) = y ~ ( k ) for k > max{di .

Necessity. The necessity follows directly from the assumptions of the theorem since the DAMMP is solvable on s*. 0

REMARK 3.2 Notice that, from the analyticity of the function y : X x XM x UM + U, equality (3.14) holds as long as (x(k), x ~ ( k ) , u(k), u ~ ( k ) ) E L?:. This equality is no longer satisfied if we leave this neighbourhood, which may happen for some k = k ~ .

REMARK 3.3 From (3.16) we notice that, if x i = 0 is an equilibrium point for the auxiliary system (3.6) with hi(xO) = 0 and h~~ (xL) = 0 for i = 1, . a * , m then ei,o(k) = 0, for all k 2 0.

REMARK 3.4 From (3. IS), the control law u = (xE, UM) can be chosen so that y~~ (k) satisfy the linear difference equation

where ai.0, - . . , ai,di are constant real coefficients, thus allowing us to make the outputs yi (k) converge to y~~ (k). Si (x, y E ( x ~ , UM)) then takes the form

and a proper location of the roots of the polynomials

entails the desired asymptotic behaviour for y~~ (k) (i.e. vi (k) + 0 as k + 00).

REMARK 3.5 Recently, the discrete-time nonlinear model-matching problem has been addressed by Kotta (1995), giving the same conditions (3.12), though the asymptotic convergence of the output tracking error, y ~ , is not considered in that work. Also, the approach used there to find a solution to the model-matching problem is based on the time-shift right invertibility of the unperturbed plant (3.2).

Next we show the representation of the auxiliary system (3.6) fed back by control law (3.7) in terms of the unperturbed plant (3.2) and of the model (3.3) in a different coordinate frame. Suppose that the nominal plant is fully linearizable, i.e. Cr=l(di + 1) = n. The definition of di shows that hi(x), - . . , hi o f$(x) are independent functions (Nijmeijer

T 1987) and can be chosen as new coordinates t ( x ) = ( x ) , . , gm ( x ) l , with & ( x ) =

t. J T

[gi,I(X),.-.I&,di+l(X)] = [h i (x ) . . . . , h i o $ ( ~ ) ] fori = 1,--- ,m,definedonthe

subset O = O 1 n . . n O m locally around xO, as follows:

for all j = 1, - . , di + 1 and for all i = 1 , . . , m. Let us consider the auxiliary system (3.6) and the new coordinate functions

where now ~ ( X E ) = [ t l ( x ~ ) , r t&E)lT with t i (xE) = [R,I (xE) , , 6 i , d i + l f ~ ~ ) ] ~ and

t i . j(xE) = hEi 0 f ! i1(xE) = k . j ( x ) 0 fL i l (xM) . (3.20)

for all j = 1 , . . . , di + 1, and for all i = 1, . . , m. This is indeed an admissible choice because the Jacobian matrix ~ [ ~ ( x E ) ] / ~ x E is nonsingular. By choosing u = E ( x ~ , u ~ ) in accordance with (3.18). the representation of the systems (3.3) and (3.6) in the new coordinates takes the form

From these last expressions we see that the outputs yi ( k ) of the plant differ from the outputs y~~ ( k ) of the model by a signal y~~ ( k ) obeying the linear difference equation (3.17). We can also identify two subsystems in the closed-loop system (3.21). namely:

1. The subsystem described by the equation

which represents the dynamics of the model, and 2. The subsystem described by the equations

with 0 0 0 0 0 0

At = 0 0 ... 1

-ai,0 -ai,l -ai.2 - . . -ai,di

which represents the dynamics of the signals y~~ (k ) .

Since the dynamics of M can be made stable by assumption, if we choose the control law from (3.18) such that all the eigenvalues of matrix Af have magnitude strictly less than 1 , then the closed-loop system will be exponentially stable.


3.2 Stability analysis of the model-matching problem

Let us consider, in addition to the unperturbed fully linearizable plant P and the model M, the perturbed plant P(E) described by equation (3.1), and suppose that xo = 0 and x k = 0. We will also assume that Dl holds so that the origin is an equilibrium point for the dynamics (3.23). The perturbed auxiliary system now takes the form

with state XE, input vectors U E and WE, mapping h~ defined as before, and

In order to have the representation of the perturbed auxiliary system in a coordinate setting which is suitable to study its stability properties, the following assumption is introduced.

(D2) There exist positive integers dCi (i = 1, . . , m), such that

a [hi o f;(f(x. U; E))] 1 0 au j

fo ra l l l = 0, . ,dCi - , forall j = l , - . . , m , for all (x,u) E Oi, and forall E E R4, and

a - [hi 0 (f (x, U; c))] # 0 au j

forsome j E { 1 , ~ ~ ~ , m ) , f o r a l l ( x , u ) E 0i,andforal16 E Wq.

Note that, if do, = di, i.e. when E = 0, these numbers correspond to the relative degrees of the unperturbed plant. Notice also that each dCi may be a constant for all E Rq; however, it also may happen that these numbers change abruptly when one of the components of E changes to zero. The case of major interest is when dCi < di, since, when d,, 2 di, both the perturbed and the unperturbed plant have the same input-output behaviour. We thus consider the former case and denote this change by ri = di - dCi. Under this consid- eration, the representation of the perturbed auxiliary system E ( c ) when it is fed back with (3.7) in accordance to (3.12) becomes, in the coordinates (3.19)-(3.20),

where A* = diag[ Af ly! and

with the matrices Af defined as before and

The components of the vector function (3.28) are given by

where f,o represents the undriven state dynamics f (., 0; 6).

We now introduce an assumption on the feedback that solves the DAMMP for the unperturbed plant.

(D3) The feedback (3.7), selected in accordance with (3.18), is such that all the eigenvalues of the matrix A* have magnitude strictly less than 1.

Notice that, whenever (D3) is satisfied, the origin 6 = 0 of the unperturbed dynamics (3.23) is an (exponentially) asymptotic equilibrium point. Thus, a converse Lyapunov theorem (Khalil 1992; Vidyasagar 1993) guarantees the existence of a Lyapunov function V(6) = .!jTp!f, where P = pT > 0 is a solution of the discrete-time Lyapunov matrix equation A * ~ P A * - P = -Q, with Q > 0. that satisfies

where cl = )cmin(P), c2 = )cm,(P), and c3 = )cmin(Q), and 11.11 denotes the Euclidean norm.

Further properties are necessary to study the stability of the closed-loop perturbed system (3.26). These properties are stated in the following assumptions.


(D4) Locally around the equilibrium point, the dynamics of the model M is bounded- state-stable, i.e. for any initial condition x ~ ( 0 ) in a neighbourhood X: of x: = 0 , and for any input sequence ( u ~ ( k ) E U: : k 3 0 ) with IluM(k)ll < 11 and 11 > 0 , there exists a constant 12 > 0 such that I l x ~ ( k ) I1 < 12 for all k 3 0 .

(D5) The perturbation term + ( t , X M , U M ; r ) is C' in r and satisfies

for all (6, u M ) in a domain 2) = ((6, X M ) : II((, X M ) 1 1 < r } with r > 0 , for U M E U M and 6.1 > 0 .

Next we show that a feedback (3.7) which satisfies (D3) can withstand the effects of the perturbation terms to some extent in achieving overall system stability. This is summarized in the following theorem.

THEOREM 3.4 Consider the perturbed plant (3.1), the fully linearizable unperturbed plant (3.2), and the model (3.3). Suppose that the DAMMP is solvable for the nominal plant under the hypothesis of the previous theorem, and there exists a feedback (3.7) that fulfills D3. Suppose that assumptions D2 and D5 hold and that the model dynamics satisfies D4. Then there exist two constants 6 , l > 0 such that, for any initial state {(O), the solution trajectories ( ( k ) of the closed-loop perturbed system (3.26) are ultimately bounded.

Proof: Similarly to the proof of Theorem 2.8, one can show now in this particular case that, if 11((0)11 < J(CIICZ)r, then the solution trajectories of the closed-loop perturbed system (3.26) are ultimately bounded, for all k 2 0 , with an ultimate bound

where a , = c3 - 8c4 - i2c5 > 0 , a2 = c4 + k s 8 , 0 < 8 < 1 , c4 = I I ~ A * ~ P I I , and cs = 1 1 P 1 1 . Furthermore,

It is interesting to note that, if we restrict our control purpose to output regulation alone at the equilibrium of the model M , then an assumption which governs the dynamics of the model is needed. So, let us consider the following assumption.

(D6) x: = 0 is an asymptotically stable equilibrium point with respect to the model dynamics (3.3).

Suppose also that the perturbation +((, X M , U M ; r ) satisfies @(O, 0 , u k ; r ) = 0 , i.e. that @ is a 'space-vanishing' perturbation, and that assumption D5 holds with 1 = 0 . Then we have the following corollary.

COROLLARY 3.5 Assume that the DAMMP is solvable for the nominal plant P under the assumptions D2 and D3. Suppose that assumption D6 holds and that the perturbation +((, X M , U M ; r ) satisfies D5 with 1 = 0. If S is small enough to satisfy the bound 6 6

d < (-c4 + 4-) / 2 q in (3.32). then the closed-loop perturbed system (3.26) \ I

has the origin as an exponentially stable equilibrium point.

REMARK 3.6 The constant d can be considered as a measure of the effect of the perturbation term to the tracking-error dynamics. Also, condition (3.32) with 1 = 0 implies that this term increases at a rate no faster than a linear function of the state norm (I) ( 11).

4. Conclusions

Using Lyapunov theory, we have presented an analysis of the stability of non-autonomous discrete-time nonlinear systems under nonvanishing perturbations. We have shown that the state trajectories are ultimately bounded when the equilibrium point of the unperturbed system is uniformly asymptotically or exponentially stable, and if some conditions on the perturbations are satisfied, complementing the results already obtained for continuous-time nonlinear systems. An application of these results to the nonlinear model-matching problem for discrete nonlinear systems when a mismatch exists between parameters of the plant and those of the so-called unperturbed plant has exhibited that the state trajectories of the perturbed closed-loop system are ultimately bounded when the equilibrium point of the unperturbed closed-loop system is asymptotically stable, and if some conditions on the perturbation are satisfied. Moreover, for this particular case, we have determined explicitly the ultimate bound.

Acknowledgement

This work was partly supported by The National Council for Science and Technology (CONACYT), Mexico.

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