On Stability Criteria for Time-Delay Linear Systems: A Matrix Pencil Approach

E Cn ,~ : II 11,< v}, where v is a positive real number.

stabilizing solution, that is, there exists a pair

(X, F) with X symmetric for which (3) is fulfilled and, in addition, A + BF is Hurwitz stable .

\1\1\1

The next result is a slight modification of Theorem 1. Lemma 1 Let E be a Popov triplet. Then the following assertions are t.quivalent:

1. Th e associated EHP is non-negativt! disconjugate. 2. Th e LUT'e sy.,tem (9) associated to E has a soluXT ~ O. tion paiT (X, F) with X

=

PToof. Obviously.

Notation.. The following notations will be used throughout the paper. W denotes the set of real numbers, !R+ is the set of non-negative real numbers, ~n denotes the n dimensional Euclidean space, and !?" xm denotes the set of all n x m real matrices. The nota~ tion X ~ Y (r""p"ctively, X > where X and Y are symmetric matrices, means that the matrix X - Y is positive semi-definite (respectively, positive definite). Cn , 7 = CO-To Oj , !Rn) denotes the Banach space oC con-

n,

.,(t) = Ax(!) + AdX(t - r)

(4)

with the initial condition

x(to

+ 0)

= (0), V 0 E [-T,OI;

(to, 4» E W+ )(

C::,T

(5)

where ,,(t) E !Rn ia the state, T > 0 is the delay and A and Ad are constant matrices of appropriate dimension. A first problem consists in finding a SUfficient condition, expressed in finite dimensional terms, for the asymptotic stability of the system (4)-(5) which ia independent of the size of the delay as well. Due to this last requierment we have to assume throughout this paper that A is Hurwitz stable (aee Niculescu et al., 1995b). Following Niculescu et al. (1995b) the Hurwitz stability of the matrices A, A + Ad are necessary conditions for delay-independent asymptotic stability property.

It i. worthwhile to point out that the approach adopted here combines the matrix pencil technique , briefly presented in Section I, with a Lyapunov-Krasovskii functional technique. As is well known, there are two ways in treating the question of stability of functional differential equations using a time-domain approach based on the Lypaunov's second method. Such ways essentially depend on the framework in which the solution is seen: as an element of a function space (Lyapunov-Krasovskii functional approach; see Hale and Lunel, 1991) or as an element of an Euclidian space (Lyapunov-Razumikhin function approach ; see Hale and Lune! , 1991). In this note our option was for the first kind of treatment cited above.

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Concerning the Lyapunov-Krasovskii functional a.p-

proach, a basic result belongs to Hale (see Hale and Lunel, 1991 and the references therein) . In the cited reference the following Lyapunov-Krasovskii functional is

introduced: V(x,) = x(t)T P,x(t)

+

1°,

x(t

+ of P2x(t + O)dO,

(6)

where Pt is the symmetric positive definite solution to the Lyapunov equation

(7)

disconjugacy (see Definition 1) of an adequate EHP. Theorem 2. Associate to (4}-(5) the Papa v triplet E = (A, Ad; Q, 0, R), where R ~ 0 and Q + R > O. If the EH P associat~d to E is non· negative disconjugate, then the linear delay system (4)-(5) is delay-independent asymptotically stable. Proof. From the non-negative disconjugacy of the EHP associated to E , it follows via Lemma 1 that there exists a matrix pair (X, F), with X symmetric and positive

definite, such that:

with D > 0, and P2 is a symmetric positive definite matri2; such that the following lintlar matrix inequality

(LMI) holds D - P, [ -P,Ad

After simple algebraic manipu1atioDs, one obtains:

.)T] 2

-(Pp,A

> O.

Thus, the asymptotic stability property reduces in fact to

aD

ATp,+P,A+P, AI P,

(9)

V(Xt)

x(t) ] ] [ X(t-T) . (14)

By substituting (12) and (13) in (14) it follows, after some algebraic calculations, that:

V(x,J ~ -x(tf(Q + R )x(t)

(9) transformed via Schur complement into an algebraic

(15)

Since Q + R is a symmetric and positive definite matrix,

Riccati inequation (or equation):

+ P,A + P, AdP2-1 AI PI + P, <

=

=

r(t) ]T[ATX+XA_R [ x(1 - T) AI X

Similar results, but with the linear matrix inequality

AT Pj

(12) (13)

The derivative of the Lyapunov functional candidate V(r,) given in (6) with Pj X ~ 0 and P, -R 2: 0, along the solutions of (4)-([,) is:

=

LMI condition.

A simila.r but more elaborated result , from a numerically point of view, is given in Boyd et al. (1994). Using the same Lyapunov-Krasovskii functional the asymptotic stability condition is converf.led into an LMI feasibility problem , that is, the existence of PI > 0, P'l > 0 such that [

+ X A - FT Ri' + Q = 0 AIx + RF =0

AT X

(8)

0

(10)

were propooed in Lee et al. (1994) (stability for the closed-loop system), in Shen et al. (1991) (robust stability for the closed-loop system) or in Niculescu et al. (lgg5a) (robust stability; the inequality sign is replaced by an equality one). In thi s paper we are interested in finding delayindependent stability criteria, via the same LyapunovKrasovskii functional (6L but under relaxed conditions:

P, > 0 and P, > 0, and avoidmq LaSalle type princip/;$ for junctio;;al differential equations (for example Matro80V criterion; see Kolmanovskii and Nosov , 1986).

Notice that if P2 > 0 then the inequation (10) has no sense and the line;r matrix inequalities (8)-(9) become non-strict. The technique proposed here combines tbe

Lyapunov-Krasovskii functional approach (6) with the

(15) combined with the Krasovskii stability theorem (see Appendix) allow us to conclude that the asymptotic stability of the trivial solution of the linear time-delay system (4)-(5) holds. Notice also that Q + R > 0 and the Lur'e system form (3) imply the positiveness of the Lyapunov-Krasovskii VVV functional (6) . Remark 1. The conditions of the Theorem 2 are easy checkable via numerically reliable procedures based on the generalized Kronecker canonical form (see Beelen and Van Doren, 1988; Oarli, 1994 and the references 0 tberein). Suppose now that we are interested in giving sufficient conditions to guarantee the a-stability (i.e, the exponential stability with a given decay rate; see Bourles,

1987 and the references tberein). We have the following result:

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Corollary 1. Let be er a given positive ntimber. Associate to (4)-{5) the Popov triplet

If the EHP associate to E is non.negative disconjugate, then {eo} is st.bi/i •• ble via memoryless state-space feedback and the stability of the closed-loop system is in depende •.t of the size of delay.

where R :5 0 and Q + R > O. If the EH P associated to E is non-11egative dtsconjugate, then the linear system (4)-{5} is o-stable.

Proof. Denote by the triplet (X , K , F) the solution of the Lur'e system associated to E, Then the proof runs similarly to the one given in Theorem 1 with A replaced by A+ BIC VVV

Proof. The o-stability of x(t) is equivalent to the stability of yet) = x(t)exp{o(t -to)}. Thus, (4) becomes in y coordinates:

yet) = (A + al)y(t) + AdeaT y(f -

r).

(16)

Then, the result directly follows by applying Theorem 2 to the system (16). VVV Suppose now that th e delay r(t) is a continuous differentiable function satisfying:

o :5 r(t) :5 f < 00

(17)

0:5 Tit) :5 P < 1

(18)

where f and j3 a.re given non-negative real numbers (see also Niculescu et al., 1995a).

Corollary 2. Associate to (4)-{5) with the time-varying de/ay (17}-{18) the Popov triplet E = (A,Ad;Q,O,R), where R:5 0 and Q + Rj(l - fJ) > O. Ifthe EHP associated to l: is non-negative disconjugate, then the linear syst em (4)-(5) with the time-varying delay (17}-(18) is asymptotically stable.

Proof. The result is a. dired consequ ence of Theorem 2 with the Lyapunov··Krasovskii functi onal candidate:

Vex,)

=x(lf P,x{!) + -1 -1-fJ /.'

Remark 2. Memoryless stabilization of (20) for the cases of a time-varying delay or of a prescribed decay ra.te in the closed-loop sy9tem can be easily derived combining Theorem 3 with Corollaries 2 and 1! respectively. 0

3. CONCLUSIONS In this paper an attempt has been made in order to rei axe sufficient Btability and memoryless stabilization conditions along the lineB of Lyapunov-Krasovskii approach . By introducing an adequate matrix pencil, termed as Enended Hamilton Pencil, the above mentioned sufficient condition are easily expressed in terms of non-diconjugacy of the considered EHP. An open problem is still that of deriving simple conditions for the positiveness of the first entry X of tbe Lur'e system solution pair (X, F) .

4. APPENDIX: KRASOVSKIl STABILITY THEOREM

Consider the functional differential equation of retarded type

,,(Bf P,x(B)dB(19)

x(1)

'-T(' )

=

with P2 -R Theorem 2).

?: 0 and P,

=X

x •• {B)

> 0 (see the proof of VVV

where x,(-), for a given t

(20)

where u(t) E lR m is tbe control input, tbe following theorem holds: Theore m 3. Associate to (20}-(5) the Popov triplet:

E

=

with R

(A, [B A"j; Q, 0,

:5 0, Q + n > o.

[b

~]).

~ tOI

(21) (22)

denotes the restriction of

x(·) to the interval [t - T,tj translated to [-r, OJ, i.e.

If additionally the system (4)-(5) is to be controlled, that is, (4) becomes

x{t) = Ax(t) + AdX(t - r) + Bu{t)

= f{l,x.), t > to = .p{O) , It B E [--r, OJ

x.(B)

= z(t + B),

It B E [-T,O) .

It i. assumed that .p E C~ T and the map t(t,.p) : ~+ x C~ r -+ ~n is contjnu~us and Lipschitzian in QJ and fit, 0) o.

=

Let us denote by x(to, c,I) the solution of the functional differential equation (21) with the initial condition (to , q,) E !R+ x C~.T'

=

Definition A.1 The trivi,,1 solution ,,(t) 0 of (21)(!!2) is said to b. 'uniformly asymptotically stable' if: (a) for' every" > 0 and for every to ~ 0 there exist a

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5 = 5(~) independent of to such that for any 4> E C~ T the solution x(t o, 4» of(21}-{22} satisfies x,(to,4» E C~',T for

REFERENCES Beelen, T. and P. Van Doren, (1988). An improved algorithm for the computation of Kronecker's canonical form of a singular pencil. Linear Alg. fj Appl., 105, 9-65. Bourles, H. (1987). a-Stability of systems governed by a functional differential equation - extension of results concerning linear delay systems. Int. J. Contr., 45, 2233-2234. Boyd, S., L. El Ghaoui, E. Feron and V. Balakrishnan, (1994). Linear matrix inequalities in system and control theory. SIAM Studies in Applied Mathematics, 15. Hale, J. K. and S. M. Verduyn Lunel, (1991). Introduction to Functional Differential Equations. Applied Math. Sciences~ 99, Springer-Verlag, New York. Ionescu, V. and M. Weiss, (1993). Continuous and discrete-time Riccati theory: a Popov function approach. Linear Alg. g Appl., 193, 173-209. Ionescu, V. and C. Oara, (1995). Generalized continuous-time Riccati theory, to appear in Lin. Alg. g Appl.. Kolmanovskii, V. B. and V. R. Nosov, (1986). Stability of Functional Differential Equations, Academic Press, New York. Lee, J. H., S. W. Kim and W. H. Kwon, (1994). Memoryless Hoc, controllerR for state delayed systems. IEEE Trans. Automat. Contr., 39, 159-162. Niculescu, S. I., C. E. de Souza, 1. Dugard and J. -M. DioD, (1995a). Robust Hoo memoryless control for uncertain linear systems with time-varying delay. in Proc. 9rd European Contr. Conf., Roma, Italy, pp. 1814-1818. Niculescu, S. I., Dugard, L. and J. -M. Dion, (1995h). StabiliU et stabilisation robustes des systemes a retard. (in french) Invited Paper, in Proc. "Journees Robustesse", Toulouse, France, Febr. 8-9. Oara, C., (1994). Proper deflating subspaces: properties, algorithmes and applications. Numerical algorithms, pp. 355-373. Shen, J. C., B. S. Chen and F. C. Kung, (1991). Memoryless stabilization of uncertain dynamic delay systems: Riccati equation approa.ch. IEEE Trans. Automat. Contr., AC-36, 638-640.

allt::>: to; {b} for every 'I > 0 and for every to ::>: 0 ther< exist a T(1J) independent of to and an Vo > 0 independent of1J and to such that for any 4> E Cn ,,, 11 4> 11,< Vo implies that 11 x,(to, q,) 11,< 'I, \I t::>: to + T(1J).

We recall that condition (a) implies uniform stability whereas (b) is required for the asymptotic stability property. In the stability analysis of a system of the form (21)(22), we shall make use of a Lyapunov functional that is continuous but not necessarily differentiable (in the usual sense of derivative). In this situation, the derivative used for a Lyapunov functional candidate, W(t, x~)j is the largest Dini's time-derivative of W(t, x,), defined by · ( ) W t,x,

I' = c-+o+ lm

sup

W(t

+ c, x,+,) -

W(t, Xt)

e

.

The above Dini's derivative is referred to as the Dini's time-derivative of W(t, x,). Next, let V;.(.), i = 1,2,3 denote scalar, continuous and nondecreasing functions such that: V,(r) > 0, \I r > 0;

V,(O)

= O.

Theorem A.1 Krasovskii Stability Theorem Consider the functional differential equation {21}-{22}. Let there exists a continuous functional W(t, q,) such that

{a} V, (II q,(0)

11) S W(t,.p) S V,(lI4>(O) 11) {b} W(t, x,) S - \,:,(11 x(t) III where Wet, X1) denotes tht: Dim' '5 time-derivative

of W(t, x,) along Ihe ,olutions of (21}-{22). Then, the trivial solutiun of (21 }-{22} is 'uniformly asymptotically .Iable. Note that condition (a) means that the functional W(t,q,) is positive definite and has an infinitesimal upper limit.

1619