On analysis of discrete singularly perturbed non-linear systems: Application to the study of stability properties

~ ) Pergamon

PII:

J. FranklinInst. Vol. 334B, No. 2, pp. 199 212, 1997 Copyright ~, 1997 The Franklin Institute Published by Elsevier Science Ltd Printed in Great Britain 0016~032/97 $17.00+0.00

S001641032(96)00076--2

On Ana sis of Discrete Singularly Perturbed Non-linear Systems."Application to the Study of Stability Properties b.}' R . B O U Y E K H F

and

A. EL MOUDNI

Laboratoire de MOcanique et Productique, Ecole Nationale d'Ingknieurs de Belfort, Espace Bartholdi, Beljort Technop6le, B P 525, 90016 Belfort, France (Received 10 M a r c h 1996; accepted 18 June 1996)

: Recently we have introduced a model of singular perturbation for discrete-time nonlinear systems. This paper is aimed at validating the proposed model. In fact, a discrete version of the well-known Tikhonov 3' theorem on singular perturbation o f continuous-time systems is established. The second aim is to study stability problems o f such systems. SuJficient conditions for both asymptotic and exponential stability are obtained. As a result, significant order reduction o f stability problems is achieved. This" is achieved by allowing a small parameter whose upper bound is estimated. Finally a simple example is given to illustrate the applications of the results. Copyright © 1997 ABSTRACT

Published by Elsevier Science Ltd

Notations With some obvious exceptions, lower-case R o m a n letters denote vectors, capital script R o m a n letters denote sets and Greek letters denote scalars. The meaning of the notations is explained in what follows.

A p × p constant matrix C ( ~ ) the set of all functions of x continuous on 5e C('J)(T × 5:) the family of all functions /-times differentiable on T and j-times differentiable on 5P D c7_ ~ n + m an open connected neighbourhood of (x, y)T = 0 D x )< D r the Cartesian product of D, ~_ ~ and D,. _c ~m f : T x ~R" x ~Rm--+ ~" a given non-linear vector function g : T x ~}~nx ~ m - + ~ m a given non-linear vector function k ~ T discrete values of time at which the behaviour of the system can be or is observed m , n , p ~ { 1 , 2 . . . . } the dimensions of the systems the set of all real numbers 2 + = [ 0 , + ~ [ the set of all non-negative numbers r ( a ) the spectral radius of matrix A 5~ c ~Rp a neighbourhood of x = 0

199

200

R. B o u y e k h f and A. E l M o u d n i

T = [0,1,2 .... + ~ [ the largest discrete-time interval To = [k0,k0+ 1.... + ~ [ the largest time interval beginning with the initial instant k0 v: T x ~ " x ~ " - - * ~ + a positive definite function Av: T x ~ " x 9 ~ ' ~ 9{ the total first difference of v, Av = v(k + 1, x ( k + 1; ko, Xo)) - v(k, x)

V: T× ~R"~ ~R+ a positive definite function W: T× ~'~ ~ ~R+ a positive definite function a positive definite function a positive definite function te : T ~ ffl" x 9t '~ m x rn matrix function ~,fl, 7,.-. Greek letters denote scalars unless otherwise specified llxll ~ = xTx Euclidean norm of vector x [,1 denotes a closed interval [ , [ denotes an open interval X T denotes transpose of vector x ;t,,(.), ,~M(') the minimum and the maximum eigenvalues of matrix (.), respectively

L Introduction The singular perturbation approach for discrete-time non-linear systems has been recently introduced, defined and studied in Ref. (1). It has been successfully used to advantage for model-order reduction and separation of time scales. This paper is aimed at exploiting this approach in an effective new way for analysis of discrete-time nonstationary non-linear systems. Stability properties of singularly perturbed continuous-time non-linear systems have been investigated by several authors (2-9). They showed the advantage of the singular perturbation approach to stability investigations by reducing the order of the stability problems and separating the time scale. The purpose of this paper is to extend the concept of singular perturbation to discrete-time non-stationary non-linear systems and derive a discrete-time counterpart to the stability theory of singularly perturbed continuous-time systems. This objective first requires a discrete version of the wellknown Tikhonov's theorem (10), on singular perturbation of continuous-time systems and, secondly, solves the stability problems of such systems. The solution will be carried out on the Lyapunov second method. It will be shown how the Lyapunov function can be generated by the form of the corresponding subsystems of the whole system whose stability is investigated. It will also be shown that maximal allowable values of the small parameter can be estimated. An illustrative example is given.

I1. S y s t e m Description and Decomposition

Let us develop first a discrete version of Tikhonov's Theorem (10), which validates the model and the mode decoupling approach proposed in Ref. (1). Let us consider a discrete-time non-linear system possessing the two-time-scale properties in the form x ( k + 1) = f ( k , x ( k ) , l ~ y ( k ) ) ,

x ( k = O) = Xo

(la)

Discrete Singularly Perturbed Non-linear Systems y(k + l) = g(k,x(k),12y(k)),

y(k = O) = Yo

201 (lb)

where (x,y)rE ~ × ~ " is the state vector of system (1) at the instant k, k~ T, (f,g) are assumed to be smooth functions of their arguments (k, x,y)./~ is the small parameter. Furthermore, it is assumed that (x,y) r = 0 is the unique isolated equilibrium state of Eqns (1) for all k E T. We first formulate a procedure to decompose Eqns (1) into two lower-order slow and fast subsystems. A slow subsystem is defined by neglecting the fast modes, which is equivalent to setting/~ formally equal to zero. Then the dimension of the system (1) reduces from n + m to n because the system (1) degenerates into the n-dimensional system (2).

xs(k+ l) = f ( k , xs(k), 0)

(2a)

y~(k+ 1) = g(k, xs(k), 0)

(2b)

where the subscript (s) signifies the slow part of the variables.

Assumption l Assume that, the Jacobian matrix (Of/Ox) is non-singular in the region of interest. Thus, by the implicit function theorem (11), Eq. (2a) has a unique root

xs(k) = h(k, xs(k+ 1)).

(3)

To obtain the reduced model or Slow model we substitute Eq. (3) into (2b)

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

(4a)

ys(k) = gs(k, xs(k))

(4b)

where def

f(k, xs(k), O) = fs(k, xs(k)) def

g(k, h(k, xs(k + 1)), 0) = gs(k, xs(k + 1)). The only information we can obtain about Ys by solving Eqns (4) is to compute

ys(k) = gs(k, xs(k))

(5)

which describes the quasi-steady-state behaviour o f y when x = xs. It is straightforward to see that Xs starts from the same initial condition as x. By contrast, y~ is not free to start from Y0, hence the approximation of x by Xs may be uniform for all k = [0, + oo [, that is

x(k) = x~(k) + O(/0.

(6)

y(k) = ys(k) + O(/0

(7)

However, the approximation

holds only on an interval excluding 0, that is, for k ~ [ k ~ , + ~ [ , where kl > 0. Thus,

202

R. Bouyekhf and A. El Moudni

following the nomenclature of the singular perturbation literature (12), system (1) is said to be in the singularly perturbed form, and a boundary layer occurs at k -- 0. To investigate this initial value problem we assume that the variables x and y can be written in the interval [0, k d in the form of a/x-power series as

Ixk--I xf(k)/ixkyf(k)

(8)

where the subscript (f) signifies the fast part of the variables. This assumption is quite reasonable because x, y are smooth functions of k, due t o f and 9 which are smooth functions of their arguments, besides, the series (8) converges to zero when/x --+ 0, excluding k = 0 for Yr, which corresponds to hypotheses (6) and (5). Substituting series (8) into (1), expanding the Taylor series expansion about (~, 0, 0), and setting again IX -- 0 we obtain after simple computation y r ( k + l ) =~t'(k)y6k)

(9)

where

y,.(o) = yo - y ~ ( O ) .

The details for obtaining the fast or boundary layer system (9) are exactly as in Ref. (1) and will not be presented here. The solution yf(k) of Eq. (9) is used as a boundary layer correction of Eq. (7) for a possibly uniform approximation of y for all k e [0, + oo[

y(k) = ys(k) +yr(k) + O(#x).

(10)

It is important to note that the solution yr(k) holds only on the interval [0,kd and converges rapidly during this interval to an O(#) quantity. In order for the approximations (6), (7) and (10) to hold, the following two assumptions are useful.

Assumption 2 The equilibrium y = 0 of Eq. (9) is exponential stable, that is, there exist positive constants e, 0, p such that Hyf(k;O, yf(O))ll ~ lTIlyf(O)He x p ( - O k ) ,

Vke T,

Vliyf(O)l] ~ p

where yf(0) -- Yo-Ys(0) - O(#).

Assumption 3 The functions f, 9 are Lipschitz in (k, x,y) on T x 91n x 01m. Comments on Assumptions 1 3 Assumption 1 plays two distinct roles: it ensures that Xs and Ys are uniquely determined as smooth functions of k (13) and it determines the solvability of the solutions of the boundary layer system (9). In the framework of Assumption 2, it is quite

Discrete Sinyularly Perturbed Non-linear Systems

203

reasonable to assume that the boundary-layer model has an exponentially stable origin. In fact, if the dynamics associated with the fast part were unstable, we should not have neglected them in the first place. Besides, the fact of assuming exponential stability instead of only asymptotic stability is also reasonable because the boundary-layer system (9) is linear and hence, the assumption is automatically held. Finally, the Lipschitz property o f f , g in Assumption 3 is basically a smoothness requirement. It is implied by continuous differentiability. It is reasonable to expect models of physical systems to have a Lipschitz function. Examples of continuous functions which are not Lipschitz are quite exceptional and rarely arise in practice. With these assumptions, we can state our first result which establishes the validity of the approximations (6), (7) and (10) and extends Tikhonov's Theorem to the discretetime case.

Theorem I If Assumptions 1-3 are satisfied, then the approximations (6) and (10) are valid for all k e [0,+ ~ [ , and there exists kt > 0 such that the approximation (7) is valid for all kE [kl,+ ~ [ .

Proof'. Define z(k) = x(k)-x~(k),

Vke T.

Then

z(k + 1) = f(k, x(k), fry(k)) - f ( k , xs(k), 0).

(11)

We show now that Vx(O) = xs(O) = x0 the solution z(k) must hold : for each k ~ T there exists ~ > 0 such that (Definition l, Appendix)

IIz(k)ll ~< ¢~.

(12)

The claim can be proved by induction. Taking k -- 0 in Eq. (11), one has

IIz(1) li -- Ill(0, x(0), ~y(0)) --f(0, x~(0), 0)tl ~< L~II Y0 II where we have used the fact t h a t f i s Lipschitz on T x 9~nx 9t m and L is the Lipschitz constant. Hence the inequality (12) holds for j = 1 with ~ = ZllY0H. Suppose (12) is satisfied for a positive integer k = j. Then to prove for j + 1, we have IIz(j+ 1)II ~< [If(j, x(j), try(j)) - f ( j , x~(j), 0)l] ~< L[rlx(j)-- x~(j)II + ~NY(J)II1

<<,Z[llz(j) l[+#ily(J) II] t p [ ~ + Hy(J)II].

(13)

Since g is Lipschitz on T x 9~~ x N" and f(k, 0, 0) = #(k, 0, 0) = 0, Vk e T, then y(k) is bounded by a constant depending only upon (x0, Y0), i.e. there is a positive real number ct = ct(Xo,Yo) such that

Ily(k) II ~ =,

Vk e T.

R. Bouyekhf and A. E1 Moudni

204

Thus, it follows from (13) that IIz(j+l)ll ~< r/~,

~t = L(~+c0.

This shows that (12) holds for j + 1, so (12) holds for all k. We conclude for sufficiently small/~ that

0(~) = x(k)-x~(k),

Vk6 T

which proves the approximation (6). To prove the approximation (10), we set

v(k) = y ( k ) - y + ( k ) - y f ( k ) ,

Vk6 T,

(14)

then [[Iv(k)II ~< Ilg(k- 1, x(k - 1), fry(k- 1)) - g ( k -

1, x ( k - 1), 0)[I + IIyf(k)ll]

<~L~I£~ + [lYe(k)11 where we have used the fact that 9 is Lipschitz and x(k) = xs(k) + 0(#). For sufficiently small # we can select ~ > 1 such that [Iv(k)ll ~< ~llYs(k)ll.

(15)

By Assumption 2 we obtain IIv(k; O, Vo)]] ~< ~e IIv0 II exp( - Ok) provided that IIv011 ~< p. Indeed, by Assumption 2, we have v0 = O(#) (i.e. Ilvoll ~< ,~), then we conclude for sufficiently small # that I1%11 ~< p. Therefore qlv(k;0, vo)l[ <~ Y/~, V k ~ T where 7 = ~s2, we conclude that

v(k) = 0(~),

Vks T

which implies Eq. (10) for sufficiently small #. Finally, since yf satisfies Assumption 2, and exp(-0k)~<~t,

VOk>~InC~ ),

the term yf will be O(/0 uniformly in k ~ [kl, + ~ [ if/~ is small enough to satisfy

The proof of Theorem I is now completed.

•

When the system (1) is time-invariant, then ud is a constant matrix, and the proof of approximation (10) takes another form. Indeed, from Eq. (14) we have

v(k+ 1) = Udv(k)+AG where

(16)

Discrete Singularly Perturbed Non-linear Systems

205

AG = g ( x ( k ) , I~y(k) ) - g ( x ( k ) , O) - tP(y(k) - ys(k) ).

Using approximation (6), the fact that g is Lipschitz and y ( k ) is bounded, we derive

IIAGrl ~< L , # ~ + L , ~ l l ~ l r ~ < n/~, VxcgY,

(17)

where n = L t ~ ( 1 + It'll). Therefore AG = O(/x) (Definition 1, Appendix). Now we shall prove that the equilibrium v -- 0 of Eq. (16) is exponentially stable. By assumption 2, • is stable, then there is a positive definite symmetric matrix P satisfying (A3) in the Appendix. Let W(v) = vrPv as a Lyapunov function candidate for system (16), Obviously W satisfies condition 1 of Lemma 1 (Appendix). Employing the properties of the positive definite matrix we then have 2m(P)llvll 2 ~< W(v) <~ ;.~(P)llvll:,

Vvm~".

(18)

Furthermore, we have, relative to (16), W ( v ( k + 1)) = v TqjT pUdv + 2VTtP Tp AG + W(AG) = vTpv -- vTQv + 2vrudTP AG + W(AG) = W(v(k)) - v TQv + 2vT~ r p AG + W(AG).

(19)

Each one of the terms in Eq. (19) may be maximized separately, which leads to the following W ( v ( k + 1)) ~< W(v(k)) -)~m(Q)nvn 2 +2M(P)g,u(2ltvH ]rtIJ]r+ ~,u).

(20)

it is easy to see that for sufficiently small/t, and for 0 < 6 < 1, we can select fl > 0 such that Eq. (20) can be written in the form

vlrvlt ~ 3.

W(v(k+ 1)) <<.6W(v(k)),

(21)

Therefore, together, Eqns (18) and (21) prove that all conditions of Lemma 1 (Appendix) are satisfied, then we may conclude that the origin of Eq. (16) is exponentially stable and there exist real numbers a ~> 1, ~ > 0 and Q0 > 0 such that IIv(k;O, vo)lt <<.allVoll exp(-c~k),

Vnv0ll ~< 0o.

By Assumption 2, we have IIv0rl ~< 2/~, therefore

IIv(k;O, vo)H ~ ~;~,

VkET

which implies for all k ~ [0, + oo [ and for sufficiently small/1 that y ( k ) = y~(k) + y,,(lc) + 0 ( ~ ) ,

which proves the approximation (10). Remark I

In the proof of Theorem I, stability requirements were imposed only on the fast subsystem (9). By contrast, no requirement is imposed on the stability properties of the slow subsystems (4). Hence, the reduced solution x d k ) was permitted to be unstable.

206

R. Bouyekhf and A. El Moudni

Under the conditions specified above, the response of system (1) can be approximated by x(k) = x~(k) + 0(~),

y(k) = ys(k) + yf(k) + 0(~).

Thus the properties of system (1) can be investigated by examining the subsystems (4) and (9).

III. Application to the Stability Analysis In the following (x, y)r = 0 is accepted to be the unique isolated equilibrium state of system (1) in D = Dx×Dy for all k s T; where Dx ~- 91n is an open connected neighbourhood of x = 0 and Dy ~ 91m is also the open connected neighbourhood of y = 0. Separation of time-scale is advantageous from the stability viewpoint owing to the possibility of a separate analysis of the degenerate system (4) and the fast system (9), and for the order reduction of the stability problems. This means that the following problem can be solved. Problem Assuming that the associated slow and fast subsystem are each of asymptotically stable origin, what additional conditions will guarantee the asymptotic stability of the origin of the system (1) for sufficiently small parameter p? The problem solution to be established is based on the following three assumptions that relate some positive definite function V and W to Eqs (4) and (9), respectively. Assumption 4 There exists V: T x 91n ~ 91+ that is positive definite on Dx and, in addition, radially unbounded as soon as Dx = 91", and there exists ~p: 91n ~ 91+, that is positive definite on Dx, such that V(k + 1,f(k, x(k), 0)) - V(k, x(k)) <~ - ~o(x),

V(k, x) s T x Dx.

Assumption 5 There exists W: T x 91m ~ 91+ that is positive definite on D~. and, in addition, radially unbounded as soon as D~. = 91% and there exists q/: 91m ~ 91+, that is positive definite on D , such that W ( k + 1,yf(k+ l ) ) - W(k,y~(k)) <~ --~9(y),

Y(k, yf) e T x D e.

Assumption 4 guarantees that x = 0 of the reduced system (4) is asymptotically stable, which is global as soon as Dx = 91". Assumption 5 plays the same role for the boundary-layer system (9) which has an equilibrium point y = 0. Remark 2 Assumptions 4 and 5 do not require a special form of ~0 and ~b, they are defined in terms of x and y of arbitrary form.

Discrete Sin#Marly Perturbed Non-linear Systems

207

Assumption 6 There exist real n u m b e r s 2i, i = l 6, such that (a) V(k + 1,f(k, x(k), ]Ay(k)) ) - V(k + 1,f(k, x(k), 0)) ~ 2.q~(x) +]A22~h(y) (b) W(k + 1,g(k, x(k), ]Ay(k) )) - W(k + 1, yf(k + 1)) ~< 23q)(x) +]A24~h(y) (c) W(k, yr(k)) - W(k, y(k)) <. 2Vp(x) + 26if(y)

V(k,x, y) ~ T × D~X Dy. The n u m b e r s 2~, i = 1-6 o f A s s u m p t i o n 6 should be chosen to be as small as possible. Inequalities (a) and (b) determine the permissible interaction between the slow and fast variables. They are the required qualitative properties o f f , g on D = D, × Dy. The term ]A22t//(y) in (a) is added to allow for general dependence o f f on ]A. Similarly in (b) the term #24~(y) is added to allow for general dependence o f g on ]A. We are n o w ready to state our stability criterion. Let 1 --26 ]Am = ~2 "~'-/~4 "

(22)

It will be shown that ]Amis a lower estimate of the u p p e r b o u n d o f allowable ]A that can be provided by the stability analysis.

Theorem H Suppose that (i) A s s u m p t i o n s 4 - 6 are satisfied and, (ii) 2 1 + 2 3 + 2 5 ~< 1. Then the equilibrium state (x, y ) r = 0 of the system (1) is asymptotically stable for all ]A~ [0, ]Am], where ]Am is given by Eq. (22). If in addition D = 9~" x 9t% then the equilibrium state (x, y)a- = 0 is globally asymptotically stable for all ]Ae [0, #m]-

Proof'. Let v : T x ~R" x 9t" -~ 9~+ be a tentative L y a p u n o v function of the system (1) defined by v = V + W. Then v is positive definite on D = Dx × Dy due to A s s u m p t i o n s 4 and 5, and radially u n b o u n d e d as soon as D = ~N"+m. Its forward difference Av(k, x(k), y(k ) ) -= v[k + 1,f(k, x(k), ]Ay(k) ), g(k, x(k ), ]Ay(k))] - v(k, x(k), y(k ) ) along motions o f system (1) is obtained in the following f o r m after simple c o m p u t a t i o n s and by employing A s s u m p t i o n s 4 - 6 [Av(k, x(k), y(k)) <~ - (1 - )h - 23 - 25)q~(x) - (1 - 26 - ]A(22 + 24))O(y)] [V(k, x, y) ~ T x D~ x D)..] Condition (ii) o f T h e o r e m II n o w proves negative definiteness o f Av on D for all ]Ae [0,/~m]. This result, negative definiteness o f Av on D proves a s y m p t o t i c stability of (x,y) r = 0 o f system (1) (14). If D = 9~"+m then v is radially u n b o u n d e d , hence (x, y) r = 0 is then globally asymptotically stable for every ]A~ [0, #m]. •

Remark 3 The p r o o f o f T h e o r e m II delineates a procedure for constructing a L y a p u n o v function candidate for system (1). One starts by studying the equilibrium o f the slow and fast

208

R. Bouyekhf and A. El Moudni

subsystems, searching for Lyapunov functions V and W that satisfy Assumptions 4-6 and condition (ii) of Theorem II. The Theorem also provides information about upper bounds of the small parameter/~, which is important for engineering applications. At this point, the asymptotic stability of the equilibrium state (x, y)T = 0 is guaranteed for sufficiently small/a. However, for a given value of #, say g0, one can improve the bound #m by exploiting the freedom that usually exists in verifying inequalities (a), (b) and (c) of Assumption 6. The Lyapunov function v is known to be valid when P0 ~< Pm. Referring to the Theorem on uniform asymptotic stability of the equilibrium of discrete systems which is proved by Kalman and Bertram (14), and from the proof of Theorem II, the following corollary is straightforward.

Corollary l Let the assumptions of Theorem II hold and let V and W be positive definite and decreseents, that is h,(llxtl) ~< V(k,x) <~hz(ltxll),

h2(ItYll) <~ W(k,y) <~h4(llyll)

where hi, i = 1 4 belong to class ~# (15). Then, the equilibrium state (x,y) T = 0 of system (1) is uniformly asymptotically stable (globally uniformly asymptotically stable as soon as D = 9in+m) for all/re [0, #m]. In the following, we consider the case of exponential stability and use the well-known Gruji6 and Siljak Lemma 1 (16), (Appendix) on exponential stability to prove a result of conceptual importance.

Theorem III If x = 0 is an exponentially stable equilibrium of the reduced system (4) and y = 0 is an exponentially stable equilibrium of the boundary layer system (9), then (x, y)T = 0 is an exponentially stable equilibrium of the system (1) for sufficiently small #. Proof'. By Lemma 1 (Appendix) and hypothesis, there are two Lyapunov functions V: T x 91n _. 9t+ and W: T x 91m ~ 91+ for the reduced system (4) and the boundarylayer system (9), respectively, which satisfy (d) V(k,x)eC°'°(T, Dx), W(k,y)eC°'°(T,D~.) (e) t/,llxll 2 ~ V(k,x) <~t/211x[I2 t/*llyfl]2 ~< W(k, yf) <~t/*llydl2 (f) V(k+ l , x ( k + 1)) ~ t/3V(k,x(k)) W(k + 1,yr(k+ 1)) ~< t/*W(k, yr(k) ) V(k, x, y) ~ T x D x x Dy where (t/i, t/*) and (t/R, t/*) > (0, 0), 0 < t/3 < 1,0 < t/* < 1 are real numbers. We are going to use

v= V+W as a Lyapunov function candidate for the system (4)-(9). Clearly v satisfied condition (1) of Lemma 1 (Appendix). From (e) and (f), we find

Discrete Singularly Perturbed Non-linear Systems

209

(i)

O~yXf 2 ~ v(k, x(k), yf(k)) ~< fiX Yf (ii)

v(k + 1), x(k + 1), yf(k q- 1)) <~ vv(k, x(k), yf(k)),

V(k, x, y) ~ T x D, x D,.

where c~= min(rh, q*), fl = max(r/2, q*), 7 = max(~/3, r/*). According to Lemma 1 (Appendix), inequalities (i) and (ii) imply exponential stability of the equilibrium (x,y) T = 0 of the system (4)-(9), that is, there exist e > 1, p > 0, ~o > 0 such that x

~
exp(-pk),

Vk6T, V

<~(o.

From y f = y - g s ( k , x ) and Ilgs(k,x)ll ~ Lrlxll (Assumption 3, Section II), we obtain after simple computations Y <~ a y~ e x p ( - p k ) ,

Vk e T, V y~ <<.~Oo

where a = e max(2, 1 + 2L 2) and eg0 = (0(max(2, 1 + 2L2))1/2, which implies exponential stability of the equilibrium (x, y) r = 0 of the system (i) and completes the proof of the theorem. • If D = ~n+m, then Theorem III justifies the following corollary.

Corollary 2 If x = 0 is a globally exponentially stable equilibrium of the reduced system (4) and y = 0 is a globally exponentially stable equilibrium of the boundary layer system (9), then ( x , y ) T = 0 is a globally exponential stable equilibrium of the system (1) for sufficiently small p. Remark 4 Theorem III is conceptually important because it establishes robustness of exponential stability to fast dynamic (i.e. to small parameter #). Indeed, in the singular perturbation theory, we often use the model order reduction (4) in the analysis when we neglect the small parameter (i.e. neglect fast dynamics). Hence, suppose that we have analysed stability of the origin of the system (4) and conclude that it is exponentially stable, then Theorem III guarantees that the origin of the system (1) is exponentially stable provided the neglected fast dynamics are sufficiently fast, which explains why the small parameter remains free. Example Let us consider the non-linear stationary system possessing the two-time-scale properties

210

R. B o u y e k h f and A. El Moudni x ( k + 1) = ax(k) - ]Ay(k) = f i x ( k ) , ]Ay(k)) y ( k + 1) = - cx 2 (k) + b]Ay(k) = # ( x ( k ) , ]Ay(k)),

(23)

where lal < l, a ~ O, Ibl < l, b ¢ O, c e 9t. Clearly system (23) has a unique equilibrium state ( x , y ) r = O. Let D = {(x,y)E912: Ix] < 1, lY[ < 1}. We shall investigate the stability of the equilibrium in the three steps. Step 1 : Reduced system. Setting ]A = 0 in system (23), we get x ( k + 1) = ax(k).

(24)

Taking V(x) = x 2, we have A V(x) = (a 2 - 1)x 2. Hence Assumption 4 is satisfied with ~p(x) = (1 - aZ)x 2. Step 2: Boundary layer system. Employing the procedure established in Section II we obtain the boundary layer system as follows (25)

yr(k + 1) = byr(k).

We take W ( y ) = ]y] so that Assumption 5 is satisfied with ~k(y) = (1 -[bl)]Aly[. Step 3: It remains now to verify Assumption 6. Indeed, after simple computation we have (1) V ( f i x ( k ) , ]Ay(k))) - V ( f ( x ( k ) , 0)) ~< a2/(l --a2)(p(x) +2]A/(1 -Ibl)&(y) (2) W ( g ( x ( k ) , ]Ay(k))) - W(yr(k + 1)) ~< c/(1 -a2)~p(x) (3) W ( y r ( k ) ) - W ( y ( k ) ) <~ c/a2(1 - a2)~o(x) V(x,y) ~D. Hence, Assumption 5 is satisfied with a2 21

-- -

-

(l--a2) '

2 •2

--

( 1 - Ibl) '

c 23 --

-

-

(1--a2) '

e 1~4 =

O,

25

--

a2(1-a2) '

26 :

0

and e should obey c ~< a2(1 -2a2)/(1 + a 2) in order to ensure condition (ii) of Theorem II. Hence all conditions of Theorem II are satisfied then the equilibrium state (x, y ) r = 0 is asymptotically stable for all ]A ~ ]Am = ( l - - [bl)/2. We can conclude the consideration of this example by noting that the analysis for the non-linear system can be achieved by two first-order linear systems instead of the straightforward analysis of the second-order non-linear system, which clarifies the advantage of the separation of time scales. IV. Conclusion

The links between the singular perturbation theory, the second Lyapunov method and discrete-time non-linear systems have been made in this investigation. First, the well-known Tikhonov's result on singular perturbation of continuous systems has been extended to discrete systems. Slightly weaker conditions have been formulated to prove validity of the model and the approximations established. Second, the scalar Lyapunov function approach to the stability analysis of such systems has been developed. It has been shown how the stability conditions can be refined and relaxed in general. The main advantage of the proposed approach to the stability analysis of the singu-

Discrete Singularly Perturbed Non-linear S y s t e m s

211

larly perturbed systems is the essential reduction of the order of the stability problems. It also provides the opportunity for the easier construction of the overall system scalar Lyapunov function, which was achieved on the second hierarchical level. The results have been established in such a form that information about the upper bounds of the small parameter is obtained. Another advantage of the approach developed in this paper is the robustness of the exponential stability to fast dynamics, which has permitted the small parameter to be free. The analysis developed in this paper imposes the problem of determination of an estimate of the domain of asymptotic stability of the appropriate set for the given singularly perturbed system which can be a topic for future research.

Acknowledgement The authors wish to thank the reviewer for his careful reviews and for constructive suggestions to improve this paper. References

(1) R. Bouyekhf, A. El Moudni, A. E1 Hami, N. Zerhouni and M. Ferney, "Reduced order modelling of two-time-scale discrete non-linear systems", J. Franklin Inst., Vol. 333(B), No. 4, pp. 499-512, 1996. (2) J. H. Chow, "Asymptotic stability of a class of nonlinear singularly perturbed systems", J. Franklin Inst., Vol. 306, pp. 275-281, 1978. (3) L. T. Gruji6, "Uniform asymptotic stability of non-linear singularly perturbed general and large-scale systems", Int. J. Control, Vol. 33, No. 3, pp. 481-504, 1981. (4) B. S. Chen and C. L. Lin, "On the stability bounds of singularly perturbed systems", IEEE. Trans. Automatic Control, Vol. AC-35, pp. 1265-1270, 1990. (5) T. H. S. Li and J. H. Li, "Stability bounds of discrete two-time-scale systems", Syst. Control Lett., Vol. 18, pp. 476-489, 1992. (6) S. Sen and K. B. Datta, "Stability bounds of singularly perturbed systems", IEEE. Trans. Autom. Control, Vol. AC-38, pp. 302-304, 1993. (7) Z. H. Shao and M. E. Sawan, "Robust stability of singularly perturbed systems", Int. J. Control, Vol. 58, pp. 1469-1476, 1993. (8) A. Saberi and H. Khalil, "Quadratic-type Lyapunov function for singular perturbed systems", IEEE Trans. Autom. Control, Vol. AC-29, pp. 542-550, 1984. (9) P. V. Kokotovic, H. Khalil and J. O'Reilly, "Singular Perturbations Methods in Control: Analysis and Design", Academic Press, London, 1986. (10) A. N. Tichonov, "Systems of differential equations containing a small parameter multiplying the derivative", Math. Sbor. Vol. 31, No. 73, pp. 575-586, 1952. (11) F. F. Wu and C. A. Desoer, "Global inverse function theorem", IEEE Trans. Circuit Theory, Vol. CT-20, pp. 193-202, 1973. (12) P. V. Kokotovic, R. E. O'Mally and P. Sannuti, "Singular perturbations and order reduction in control theory--An overview", Automatica, Vol. 12, pp. 123 132, 1976. (13) K. Ikeda and N. Kodama, "Large-scale dynamical systems: state equations, Lipschitz conditions, and linearization", IEEE Trans. Circuit Theory, Vol. CT-20, pp. 193-202, 1973. (14) R. E. Kalman and J. E. Bertram, "Control system analysis and design via the second method of Lyapunov. Part II: Discrete-time systems", Trans. A S M E , J. Bas. Engng, Vol. 82, pp. 394-400, 1960.

212

R. B o u y e k h f and A. El M o u d n i

(15) W. Hahn, "Stability of Motion", Springer-Verlag, New York, 1967. (16) L. T. Gruji6 and D. D. Siljak, " O n stability of discrete composite systems", IEEE Trans. Autom. Control, Vol. 10, No. 5, pp. 522-524, 1973. (17) H. K. Khalil, "Nonlinear Systems", Macmillan, New York, 1992.

Appendix Let a discrete-time system S described by the vector difference equation

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

(A 1)

where x ( k ) e ~r is the state of the system at discrete time k s T, f : T × fflp ~ ~P is continuous function of its arguments (k, x).

Definition 1 (17) Let x ~ ~P, we write x = O(p) if there exist positive constants g~ and g2 such that Iixll < ~ , l a l ,

Vltd < ~ .

For sufficiently small # the norm will be small. Lemma 1 is obtained from Gruji6 and Siljak (16).

Lemma 1 The equilibrium state x = 0 of the system S is exponentially stable on 5e m ~Rp (globally exponentially stable as soon as ,5e = ~P), that is, there exist real numbers e >/ 1, 0 > 0, 6 > 0 such that Ilx(k;ko,xo)ll <<.~11xolle x p ( - O ( k - k o ) ) V k e T0,Vllx0ll ~ 6,

(Vllx0ll egtp),

if and only if there exists a scalar function V: T x 9~p ~ 9~+ with properties (i) V(k, x) ~ C°'°(T, 5P), (eC°,°(T, 9tP)) (ii) thllxll 2 ~< V(k,x) <~ .211xll ~ (iii) V(k+ l , x ( k + 1)) ~< rl3V(k,x(k)) V(ko, k , x ) e Tox TxSP, ( e T 0 x T x g ~ p) where (q~, ~/2) > (0,0), 0 < r/3 < 1 are real numbers and I1"11 is the Euclidean norm. For the linear system

x ( k + 1) = Ax(k)

(A2)

the term exponential stability implies that the spectral radius r(A) of A is less than one, and there exists a positive definite symmetric matrix P such that v = xTPx is a Lyapunov function candidate of the system, and given a positive definite symmetric matrix Q, the following Lyapunov equation is satisfied

A r e A -- P = - Q.

(A3)