Delay-Dependent Stability of Linear Systems With Delayed State: An L.M.I. Approach

Copyright © IFAC Linear Time Delay Systems, Grenoble, France, 1998

MIXED DELAY-INDEPENDENT / DELAY-DEPENDENT STABILITY OF LINEAR SYSTEMS WITH DELAYED STATE: AN L.M.I. APPROACH Silviu-Iulian Niculescu 1 • Jean-Michel Dion 2 Luc Dugard 2 1 HEUDIASYC (UMR CNRS 6599), Universite de Technologie de Compiegne, Centre de Recherche de Royallieu, BP. 20529, 60205, Compiegne, France. E-mail: silviu
·On leave from the Department of Automatic Control, University "Politehnica" Bucharest, Romania and from Dept. of Applied Mathematics, ENS TA , Paris, France.

Abstract. This paper focuses on some delay-independent / delay-dependent asymptotic stability problems of a class of linear systems described by delay-differential equations involving several constant (or time-varying) delays. We give sufficient conditions (expressed in terms of LMIs) for characterizing unbounded stability regions in the delays parameter space. Note that the presented results allow to recover previous sufficient delay-independent or delay-dependent conditions from the control literature. Key Words. Delay; stability; Liapunov-Razumikhin function; LMIs

In the time-domain framework we can include the matrix measure approach combined with a comparison theorem for functional differential equations (Niculescu et al., 1995 and the references therein) and respectively the Liapunov second method (Hale and Lunel, 1991; Kolmanovskii and Nosov, 1986) combined with Liapunov equations (Su, 1994), Riccati equations (Niculescu et al., 1994a) or linear matrix inequalities (LMIs) (Boyd et al., 1994). Thus, two different techniques can be used: the Liapunov-Krasovskii functional (Niculescu et al., 1994a) or the LiapunovRazumikhin function approach (Niculescu et al., 1994; Su, 1994; Li and de Souza, 1995, 1997). Further remarks and comments on the methods can be found in Niculescu et al. (1997a).

1. INTRODUCTION

Stability analysis of linear systems including delays in their physical model is a problem of recuring interest since the existence of a delay induces often instability (see Kolmanovskii et Nosov, 1986; Malek-Zavarei and Jamshidi, 1987). A lot of work has been done concerning the stability analysis of systems with a single delay (see, e.g. Kolmanovskii et Nosov, 1986 and the references therein). The main interest for asymptotic stability of linear systems with a single delay has been focused on: delay-independent stability criteria (no information on the delay size; see, e.g. Chen and Latchman, 1994; Lewis and Andersson, 1980), and delay-dependent stability criteria (including this information: see, e.g. Chiasson, 1988; Niculescu et al., 1994; Chen et al., 1994), respectively.

Algorithms for computing optimal or suboptimal bounds on the delay are proposed in Chiasson, Chen et al. (1994) (a frequency-based approach), or in Su (1994), Li and de Souza (1995, 1997), Niculescu et al. (1995) (a Lypaunov-Razumikhin function approach).

The corresponding methods can be classified in two classes: frequency-domain (two or several polynomial variables, small gain or matrix pencil techniques, etc.) and time-domain based approaches.

Delay-independent or delay-dependent stability regions in the parameter space have been considered by Boese (1994) (scalar case, frequency-

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domain approach) or by Niculescu (1997) (multivariable case, frequency-domain and time-domain techniques) .

al. (1997).

This paper focuses on the problem of asymptotic stability of linear systems involving several delays. Sufficient delay-independent / delaydependent stability conditions are given in terms of finite dimensional linear matrix inequalities. These conditions allow to give some characterizations for the unbounded stability regions in the delays parameter space. The "limit" cases (i.e. delays-independent or delays-dependent type results) can be easily derived. For the asymptotic stability analysis, a Razumikhin technique is used, combined with an LMI method.

Real systems cannot always be described by mathematical models involving a single delay (see, for example, biological or population dynamics models: Stepan, 1989 and the references therein). For linear systems with several delays, asymptotic stability criteria have been proposed in Niculescu et al. (1994b) (Liapunov-Krasovskii functional approach), in Niculescu et al. (1995) (matrix measure approach) or in Hale et al. (1985) (frequencydomain approach). We will use the following terminology: a "delaysindependent" stability condition means an asymptotic stability condition which is independent of each delay and respectively a "delays-dependent" stability condition means an asymptotic stability condition which is dependent of each involved delay. With this terminology, delays-independent stability criteria were proposed in Hale et al. (1985) (for a special class of time-delay systems), Boyd et al. (1994), Chen and Latchman (1994) and delays-dependent criteria were proposed in Niculescu et al. (1994b), (1995), which are restrictive and do not allow an easy optimization.

The paper is organized as follows: Section 2 briefly presents the problem statement and the main result; Section 3 addresses the corresponding delaysindependent and delays-dependent cases. Some final remarks end the paper. Notations. The following notations will be used throughout the paper. R denotes the set of real numbers, R+ is the set of non-negative real numbers, Rn denotes the n dimensional Euclidean space, and R nxm denotes the set of all n x m real matrices. The notation X 2': Y (respectively, X > Y), where X and Y are symmetric matrices, means that the matrix X - Y is positive semi-definite (respectively, positive definite). Cn,T = C([-7",O], Rn) denotes the Banach space of continuous vector functions mapping the interval [-7",0] into Rn with the topology of uniform convergence. The following norms will be used: 11 . 11 refers to either the Euclidean vector norm or the induced matrix 2-norm; 11 4> Ile= SUP_T(t) 11 stands for the norm of a function 4> E en,T' Moreover, we denote by C~ T the set defined by C~,T = {4> E Cn,T 11 4> Ife< v}, where v is a positive real number.

In this paper we consider a class of linear systems described by delayed differential equations involving several constant time-delays, seen as parameters. We are interested in: giving bounds on the delays such that the asymptotic stability of the considered system is still preserved for delays smaller than these bounds.

The stable region is defined as a maximal connected set 1) included in R~ which contains the origin 0 E R r of the delays-parameter space (7"1, ... 7"r) such that for each r-uplet (7"1, ... 7"r) E 1) the considered system is asymptotically stable (see Hale and Huang, 1993). For linear systems containing multiple delays, when the stability region in the delays parameter space is bounded then the corresponding asymptotic stability criteria is always of delay-dependent type. Notice that an unbounded stability region does not guarantee that the stability is delay-independent. Hale and Huang (1993), for the tW
2. PROBLEM STATEMENT AND PRELIMINARY RESULTS Consider the following linear system with delayed state: r

x(t)

Ax(t)

+

L

AiX(t - 7"d

(1)

i=l

with the initial condition x(to

+ B)

f --

To the best authors' knowledge, there does not exist any result devoted to the geometry of the stable regions in the delays parameter space for a non-scalar linear system involving multiple delays. Note that bounded stability regions for the nonscalar case have been considered in Niculescu et

= 4>(8), \:/ B E [-7",0]; max{7":} _ 1, (to, 4» E R+ X C~,T (2)

i=l ,r

where x(t) E Rn is the state, Ti (i = G) are the delays of the system supposed to be constant, but unknown, A, Ai (i = G) are known real constant matrices of appropriate dimensions (Ai :I 0).

70

Let us consider a decomposition of the delay vector:

lays Ti (i

o <

= G) such that: Ti ~ Tt, i = n,

if there exist (k + l)r + 1 symmetric and positivej = 0, r), R; definite matrices X, R;j (i = (i = k + 1, r) such that the following matrix inequalities hold:

n,

III

where Tdep contains, without loss of generality 1 , the first k delays {Tl' ... , Tr }.

N,AXA T N,T ~ R,o,

1

N,A)XA;N,T ~ R,),

The stability problem cam be formulated in this case as: find bounds on the delays Tdep such that the asymptotic stability of the delay system (1)-(2) is preserved for any positive delays Tind and for all delays Tdep smaller than the corresponding bounds.

+

MiNi, .Mi E R nxm "

with Mi and Ni full-rank matrices (mi

~

= (k+ l),r.

MiR;MT

+ (r -

k)X.

An 'alternative' equivalent form of Theorem 1 based on a different majoration technique is: Theorem 2 The linear system (1)-(2) satisfying (3) is asymptotically stable for any positive delays Ti (i = G) such that:

o <

Ti ~ Tt,

i=

n,

if there exist (k + l)r + 1 symmetric and positivedefinite matrices X, R;j (i = 1, k, j = ~), R; (i = k + 1, r) such that the following matrix inequalities hold: QI (X, Tdep)

X [ NiAX

<

°

XATN T RiO' ]

~ 0,

i=G (5)

(3)

X [ NiX

n).

XNt] Ri ~ 0,

i=(k+1),r.

Using a Razumikhin-Liapunov approach, we have the following result:

where

Theorem 1 The linear system (1 )-(2) satisfying (3) is asymptotically stable for any positive de-

I

.E

have a de-

Ni E Rm,xn,

(4)

i=k+l

Notice also that the scalar case involving two delays Tl and T2 as parameters was treated by Hale and Huang (1993), and the particular commensurate delays case T2 = 2Tl was considered by Schoen and Geering (1993). In fact, Schoen and Geering (1993) have pointed out some similarities between the single delay case (Boese, 1994) and the particular considered time-delay system. Let us remark that this similarity is due to the particular form of the associated characteristic equation for the commensurate delays case.

Ai

t

=G = G, j = l,T i

r

It is clear that if Tdep or Tind are empty sets, one recovers the delays-independent and delaysdependent cases, respectively. Thus, if Tind contains, at least, one delay, then the corresponding stability region is of unbounded type.

=G

N,XN,T ~ R"

t

°

where

This problem is the so-called mixed delayindependent / delay-dependent stability problem since we are interested in giving stability conditions which are delay-independent in some of delay variables and delay-dependent in the others. Notice that this is possible only for the case when the system has at least two delays.

Assume that the matrices Ai, i composition of the form:

<

QI (X, Tdep)

via an appropriate index pennutation

71

r

+

L

MiR;Mt

+ (r -

k)X.

i=k+1 r

+

L

MiR;Mt

+ (r -

k)X.

Theorems 1 and 2 are also valid if one assumes that the delays Ti (i = 'l,r) are continuous timevarying functions belonging to the set:

i=k+1

The proof idea is that we can easily interpret the retarded functional differential equation (RFDE) (1) on Cn ,T as a differential equation on Cn ,2T' With this change, we take into account in the 'new' RFDE an integration of the 'old' RFDE over one interval f = max {T;} (i = "G) obtained by the considered partition of the T vector. Applying the Razumikhin stability theorem to the 'new' RFDE, allows to conclude the uniform asymptotic stability of the trivial solution in terms of the algebraic delays-dependent inequalities (4), (for the complete proof see the full version of the paper, N iculescu et al., 1997b). V''V''v

TiC)

V(h)

NiXNF ~ R"

i

Furthermore, note that if A is unstable, then the system can not be stable independently of the delays size, but the corresponding stability regions can be unbounded function of the existence of some matrices A j (1 ~ j ~ r), which allows simultaneously:

=

• the Hurwitz stability of the real matrix (Tj == 0, j E :1) A + Aj (Denote :1 the corre-

L j

sponding set of indices, which is a subset of {l, ... ,r}), • the delay-independent stability of the system:

(6)

~(t) = (A + L

= ..,..,(k:-+-l-:-),-r.

iE:J

t

Q(X, Td"l = (A + A;) X+X (A+ t,Ay

(t,

M; R;MT )

Ai)

~(t) + L Ai~(t -

Ti)'

ii:J

Remark 3 In Niculescu (1997), some delaydependent necessary conditions for the single delay case are related to instability of the matrix A - A 1

where

(A;

Ai. Thus,

• if k = r, one recovers the asymptotic stability of the systems without delays (Ti == O. 'Vi = 'l,r), which is a necessary condition for delay-dependent stability ('first' delayintervals of the form [0, Tt); for more remarks, see Niculescu, 1997), • if k = 0, one has the stability of the matrix A, which is a necessary condition for delayindependent stability (see, e.g. Hale et al., 1985).

Ql(X,Tdep) < 0

1

+L i=1

1):

= l,r

(7)

'VtER}

k

=

j

E CO(R, R+)

stability of the matrix A

Furthermore, if one assumes that the matrix A has a similar decomposition A MoNo, the condition (4) can be written into a different form as (following the same steps of the proof of Theorem

NjXNJ ~ R J ,

{T

A necessary condition for the existence of a solution to the matrix inequalities (4) is the Hurwitz

obtain different 'equivalent' forms of the matrix inequality (4) (the same holds for Theorem 2).

NaXNl' ~ Ra,

= 'l,r,

2.1. Some necessary conditions

Remark 2 It is clear that if some Ai (i = 'l,r) are full-rank, one may set Mi Ai and Ni In, or Mi In and Ni Ai, and thus, one may

=

i

O~T(t)~h,

Note also that such kind of equivalence allows to 'mix' majorants with minorants as another way to express such sufficient stability conditions, etc.

=

hi E R+,

where

Remark 1 We may interpret the inequalities (4)(5) in the variable X as "majorants" , and respectively "minorants" . Note that via an appropriate change of variables, we may prove that matrix inequalites (4) have solutions if and only if (5) have solutions. For the brevity of the paper, we do not develop such kind of arguments here.

=

E V(h i ),

=

(r 1), results which can be summarized as follows: if A - A 1 has at least one strictly unstable eigenvalue (the real part is positive), but A + Al is a Hurwitz stable matrix, then the stability is always of delay-dependent type. Note that such kind of arguments can be also developed for the multiple delays case. Some comments are presented in

AT +X)

72

Niculescu et al. (1997a).

j

= ~ such that the following matrix inequalities

hold: Q.(X,r) < 0

2.2. Some remarks on 'sub-optimal'Tdep

T

Following how are considered the variables in the matrix inequalities (4), we may have several ways to analyze the corresponding delay system.

= G,

j

(9)

= G.

X

+X

4. CONCLUDING REMARKS In this paper we have derived sufficient delayindependent / delay-dependent conditions that guarantee the asymptotic stability of a class of linear systems involving several delays using a Liapunov-Razumikhin function approach combined with LMI techniques. The results presented allow to recover as "limit cases" previous delaysindependent or delays-dependent conditions. Furthermore, the result can be easily extended to structured uncertain linear systems with several delays (see also Niculescu et al., 1994b; Li and de Souza, 1995).

3. DELAYS-INDEPENDENT / DELAYS-DEPENDENT CASES Let us consider now the 'limit' cases, i.e. k = 0 and k r, respectively. Thus, if k 0, one obtains the following corollary. For the sake of simplicity, we have directly considered the case of continuous time-varying delays and we have developed the results only for Theorem 1.

=

REFERENCES Boese, F. G., (1994). Stability in a special class of retarded difference-differential equations with interval-valued parameters. J. Math. Anal. Appl., 181, 367-386. Boyd, S., 1. El Ghaoui, E. Feron and V. Balakrishnan, (1994). Linear matrix inequalities in system and control theory. SIAM Studies in Applied Mathematics, 15. Chen, J. and H. A. Latchman, (1994). Asymptotic stability independent of delays: simple necessary and sufficient conditions. Proc. 1994 A mer. Contr. Conf., Baltimore,

Corollary 1 (Delays-independent) The linear system (1 )-(2) satisfying (3) is asymptotically stable for any delays TiC) E V(h i ), with arbitrary bounds hi (i = G), if there exist r + 1 symmetric and positive-definite matrices X, Ri (i = G) such that the following matrix inequalities hold: (8)

where

U.5.A., 1027-1031. r

Q3(X) =AX+XAT

i

T

Remark 4 If one considers the single delay case, the delay-dependent result obtained from Theorem 2 (by setting M i = Ai and Ni = In) allows to completely recover the sufficient conditions proposed in Li and de Souza (1997).

• fixing (k -1) delays in T dep and searching the 'optimum' in the corresponding "last" (positive) direction, which turns out to be a generalized eigenvalue problem (see Boyd et al., 1994); • finding maximal ellipsoids in the same manner as in Niculescu et al. (1997b). For the sake of brevity, we do not develop such kind of arguments here.

Ni;~:~'<~ = G.

t ),

= 1,

(A + t,A,) (A+ t,Ay + t,h' (M'~~jMr +X) .

Q.(X,T) =

Secondly, if we want to find a 'maximal' set Tdep , by considering Tt (i = l,"k) as parameters, then the inequality Q1(X, Tdep ) < 0 is not an LMI due to the dependences "TiRi/' or TiX (for appropriate indices i and j). In such a case, we may have some alternative ways:

{ NiX

NtA]XA;N tT ~ R

i

where

Firstly, if we 'impose' some delay set 7 dep ' we may verify if the asymptotic stability of the corresponding delay system is ensured, problem which turns out to be a feasibility standard problem in terms of linear matrix inequalities (see Boyd et al., 1994).

=

N.AXA N,T $ RiO,

{

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+ LMiRiMl +rx. i=l

Corollary 2 The linear system (1 )-(2) satisfying (3) is asymptotically stable for any delays Ti(') E V(h i ) (i = G) if there exist (r+1)r+1 symmetric and positive-definite matrices X, R;j (i = G,

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