Stability of Linear Systems with Delayed State: A Guided Tour

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

STABILITY OF LINEAR SYSTEMS WITH DELAYED STATE: A GUIDED TOUR Silviu-Iulian Niculescu 1 , Erik I. Verriest 2 , Luc Dugard3 and Jean-Michel Dion3 1 HEUDIASYC, UTCompiegne, BP 20529, Royallieu, 60205, Compiegne, France. 2School of Electrical and Computer Eng., GeorgiaTech, Atlanta, GA 30332-0250, USA. 3LAGrenoble, ENSIEG, BP 46, 38402 Saint-Martin-d'Heres, France.

Abstract. In this paper, some recent stability results on linear time-delay systems are outlined. The goal is to give an overview of the state of the art of the techniques used in delay system stability analysis. In particular, two specific problems (delay-independent / delay-dependent) are considered and some references where the reader can find more details and proofs are pointed out. This paper is based on Niculescu et al. (1997). Copyright © J998 IFAC Key Words. Delay, stability, delay-independent, delay-dependent

1. Introduction

are not always easy to apply for specific problems. • Functional differential equations We may have: evolutions in a finite-dimensional space or in a function space (see, e.g. Hale and Lunel, 1993).

In the mathematical description of physical processes, one generally assumes that the corresponding behaviour depends only on the present l state, asumption which is verified for a large class of dynamical systems. However, there exist situations2 , where this assumption is not satisfied and it is better to consider that the system's behaviour includes also information on the former states. These systems are called delay systems. The existence of a delay in a system can have several causes (Malek-Zavarei and Jamshidi, 1987), as, for example: the measure of a system variable, the physical nature of a system's composant or a signal transmission; thus, we may encounter technological, transmission or information delay.

Some remarks on the effect of a delay on the stability or oscillations can be found in Burton (1985). Thus, one can use classical concepts specific to "finite-dimensional" systems, or introduce "new" concepts more appropriate to a function space interpretation (see, e.g. Manitius and Triggiani, 1978; Salamon, 1984). One of the possible advantages lies in its facility to treat such problems using "finite-dimensional" tools, with a trade-off to be paid on the conservatism of the results. • Differential equations over rings or modules We have interesting "structural" properties, as stabilizability and observability: Morse (1976), Sontag (1976), Kamen (1978), Fliess and Mounier (1994). In our opinion, some of these interpretations are better adapted for the cases when no explicit information on the delay is needed.

1.1. Delay systems representations

There are mainly three ways to model such systems, each way having some advantages / inconvenients depending on the handled problem: • Evolutions in abstract spaces The system class is embedded in a larger class of linear systems for which the evolution is described by appropriate (bounded or not bounded) operators in infinite dimensional spaces (see Curtain and Zwart, 1995). From a system theory point of view, this approach needs the introduction of appropriate concepts of stabilizability, observability, detectability, etc. (see also Bensoussan et al., 1993). Although this way is very general, the corresponding methods

1.2. A "short" historical perspective

The study of FDE3 started long before 1900 (see the works of Bernoulli, Euler or Volterra), but the basics were developed in the 20th century. Thus, the notion of a FDE was introduced by Myshkis (1949) as a differential equation involving the function "x(t) " and its derivatives not only in the argument "t" but in several values of "t. " Without being exhaustive, we cite some of the books which have marked the study of such sys-

1

in the usual sense

3

functional differential equations

2

material or information transport

4

called time

31


terns: Bellman and Cooke (1966) (entire functions), Krasovskii (1963) (on Lyapunov method), Halanay (1966) (on Popov theory), Rasvan (1977) (absolute stability), Lakshmikantam and Leela (1969) (comparison theorems), Burton (1985) (on Krasovskii theory) or Diekman et al. (1995) (operator theory). Some recent comprehensive introductions are G6recki et al. (1989), Kolmanovskii and Myshkis (1992) and Hale and Lunel (1993).

£to = {t ER: t =

1.5.1. Single delay. Following Mori (1985), we have two different kinds of asymptotic stability for E, depending on the information on the delay size in the property 7 : Delay-independent: the property holds for all positive (and finite) values of the delays. Delay-dependent: the stability is preserved for some values of delays and the system is unstable for other values. If the delay-independent notion is clear, the delaydependent case has to be better specified. For the sake of simplicity and in order to have no ambiguity, we introduce the following: Assumption 1: The triplet E free of delay %s asymptotically stable. Thus, the problem to be considered is: Problem 1: Determine if the triplet E satisfying Assumption 1 is delay-independent asymptotically stable or not. If not, find an optimal (sub-optimal) bound on the delay size which still ensures the stability property.

nd

+L

Adix(t - T;),

(1)

i=1 with an initial conditionS:

x(to

Note that we only consider here, in the delaydependent case, the interval containing T = 0, i.e. of the form [0, T*), independently of possible other intervals (for "delay-intervals" of the form (1:, r), with 1: > 0 see also Niculescu, 1997; Chen, 1995).

+ 8) =


(3)

1.5. Asymptotic stability

1.4. Linear delay systems class Consider a system with "point" delays of the form:

Ax(t)

T(1]) ~ to, 1] ~ O}.

and continuous with bounded derivatives delay function: +(t) ~ {3 < 1, etc.

1.3. Delay as a parameter In the sequel, we consider a system class described by linear delay-differential equations with "point delays". The delays are seen as parameters, and we are interested in analyzing the stability property with respect to them. The idea is to give characterizations of the corresponding stability regions in terms of delays. To the best authors' knowledge, this problem is still open (see Diekmann et al., 1995), but in some cases, complete characterizations can be given. In this sense, two notions are introduced: delay-independent and delay-dependent stability. The time-varying, the multiple delays cases can be also considered.

x(t)

1] -

(2)

For such systems, one may associate a triplet E of the form E = (A, Ad, T), where: Ad = [Adl, ... , Adnd] (A an d A d correspond { T = [Tl,'" Tnd ] to the present and former state). Throughout this paper, E asymptotically stable means that the system (1)-(2) is asymptotically stable.

1.5.2. Multiple delays. The considered stability notions can be extended to this case, by taking into account the behaviour with respect to each delay. We have a particular "mixed" case8 : delaydependent stability in one delay (or several) and delay-independent in others (at least one). If the problem is posed in the delay-parameter space, we may have two different delays sets: unbounded (delays-independent and "mixed") and bounded sets (delays-dependent case).

The case nd = 1 is known as the single delay case. For nd ~ 2, we also consider the delay parameter space (Tl' ... Tnd ) E Rnd. For multiple delays case, we may have a particular situation - the commensurable case (i.e. there exists a delay value T, such that all Ti are rational "multipliers" of T).

Problem 2: Determine if a triplet E satisfying Assumption 1 is delays-independent asymptotically stable or not. If not, find an optimal (sub-optimafJ) region in the delay-parameter space which still ensures the stability property.

Further specifications are given when the delays are bounded time-varying functions 6 : (piecewise) continuous delay function T : R+ 1-+ R, T(t) ~ l' for any t E R +; in this case, the initial condition (2) becomes: x(to + 8) =
If the delays are commensurate, we have the same stability notions as in the single delay case. 7

The time-varying case can be defined by analogy.

5

delays supposed constant

8

called also mixed delay-independent / delay-dependent

6

the single delay case developed here

9

convex or not

32

2. Frequency-Domain In this class of methods, one has: • analytical tests, including all the criteria that generalize Hurtwitz method to delay systems: Pontryagin, Chebotarev or Yesupovisch-Svirskii (Stepan, 1989; Kolmanovskii and Nosov, 1986), • root locus method: the V-decomposition (Neimark, 1949) or T-decomposition methods (Lee and Hsu, 1969; only single delay) and its derived forms: Walton and Marshall (1987), Cooke and van den Driessche (1986), which can be applied to the considered stability problems: The idea can be summarized as follows: the characteristic equation can be written as eH = Do(s), with Do a ratio of 2 polynomials. Thus, the problem is reduced to analyze the behaviour of the contour Do(jw) with respect to the unit circle eiWT . In the delay-independent case, ejwT and jw are independent variables, so no intersection means delay-independent stability, etc. • argument principle method: including the Nyquist criterion, the Satche diagram or the Michailov criterion; these principles can be applied to linear delay systems since the number of the unstable roots in the complex plane is finite. All these methods (except T-decomposition) seem difficult to be applied for the considered stability problem. Further comments can be found in Niculescu et al. (1997).

A is Hurwitz stable and

X n4

... bnJIe"J : ble E C, I ble

1$

I} ,

n4

L k; = nnd· ;=1

Using similar ideas combined with matrix measures properties, one can obtain various delayindependent (Chen et al., 1994a) or delaydependent conditions. Thus, for the single delay case, E is delay-independent stable if:

where P is the solution of: AT P + PA = -2In . Thus, we recover the result due to Mori et al. (1982) (comparison principle) p.(A) + IIA d ll < 0, where p.(A) is the matrix measure of A.

Mori and Kokame criterion (1989). The idea is based on the maximum principle combined with the property: if there exist unstable roots of the characteristic equation, then they are located in a compact domain in C+. Thus, the problem is reduced to the computation of a given function on the boundary of a compact. Methods to restrict the compact are proposed in Wang (1992), Su et al. (1994), Wang and Wang (1995).

2.1.2. Polynomial criteria. We have included: • One variable: the Tsypkin criterion (Els'golts' and Norkin, 1973), the Thowsen lemma (Thowsen, 1981) with its matrix derived condition (Su, 1995) and the matrix pencil techniques (Chen et al., 1994b, Niculescu, 1997). • Two (Kamen, 1981) or several variables (Hertz et al., 1984; Hale et al., 1985).

< 1, then it folweR+ lows via the maximum principle that 1,

= {diag (b1 lie"

and

If H xu satisfies sup IIH xu (jw)1I

H

w > 0,

Here, X n4 is the corresponding family of block diagonal matrices, defined by:

2.1.1. Maximum Principle Based Criteria. Small gain theorem criteria. Consider the delayindependent problem in the single delay case nd = 1 (see also Chen et al., 1994a). Suppose that A is Hurwitz stable 1o . Introduce now the following system:

IIH xu (s)e- II <

1,

where M(s) is defined as follows:

2.1. Special criteria

sup sec-

<

P.X"4 (M(jw))

"IT E R+.

This condition leads to:

Two variables polynomial criteria. The basic idea can be summarized as follows: First, the characteristic equation on jR for commensurate delays:

(for all positive delay T), which allows to conclude the delay-independent stability property. Related criteria can be found in Verriest et al. (1993) (resulting from the "Strict bounded real lemma" and an appropriate Riccati equation), etc. For the general case, we have delay-independent stability (with respect to each delay) if the matrix

det (jW1n - A -

f

Adk e - iWh ) = 0, w E R

k=1

can be interpreted as a two independent variables equation: • one on the imaginary axis "jw E jR" and • the other one on the unit circle "z = e- jwT ", since T is a "free" parameter.

10 a necessary condition for delay-independent stability, see also Hale et al. (1985), etc.

33

Second, due to the continuity properties l l the delay-independent stability can be reduced to check (Hale et al., 1985) Assumption 1 and if the characteristic equation has no roots on jR (see also Cooke and Ferreira, 1982). Other two variables methods can be found in Chiasson (1988) (delay-dependent using two polynomials) or in Repin (1960) (single delay only).

Niculescu et al., 1997) is:

Proposition 1: The triplet E is delay-independent asymptotically stable if there exists P,5 > 0 satisfying the Riccati inequality:

Matrix pencils techniques. One of the major inconvenient of the two variables polynomial based results consists in the difficulty to check the condition on numerical example. Thus, we need to simplify it, and one of the way is to reduce the variables number from two to one; the "reduced" one can be, for example, the imaginary axis variable, idea exploited by Chen et al. (1994b). The stability problem is reduced to analyze the generalized eigenvalue distribution with respect to the unit circle for two constant, and finite dimensional matrix pencils with larger size: one associated to finite (Chen et al., 1994b) and respectively infinite delays (Niculescu, 1997, 1998). Note that such matrix pencils are the linearization of some appropriate matrix polynomials. Furthermore, this technique can be easily extended to the delay-interval stability analysis, etc.

For details, see Verriest and Ivanov (1993) and Verriest et al. (1993) (frequency interpretations). Using the Schur complement, (5) can be transformed in (Boyd et al., 1994): AT P

[

+PA AJP

+5

0

PAd]

(6)

<.

-5

Thus the stability problem is reduced to the feasibility of an LMI: finding if there exist P, 5 > 0 which satisfy the constraint (6). Time-varying delay: if T(t) satisfies +(t) ~ f3 < 1, then we choose (4) as: V(xr)

= x(tfPx(t) + 1 _1 f3

jl l-r x(l:1f Sx(l:1)dl:1, (7)

and Proposition 1 can be easily rewritten. Multiple delays (not commensurable). Ways to ex-

tend the Krasovskii functional (4) are:

3. Time-Domain 3.1. Lyapunov's Second Method There are two ways to develop the Lyapunov method: Lyapunov-Krasovskii functionals and

i=1

jl

.x(I:1)T Si x (l:1)dl:1,

(-T.

or:

Lyapunov-Razumikhin functions.

Whereas the Lyapunov functional notion may seem like an obvious choice to extend the "classical" Lyapunov stability analysis for ODE 12 to the FDE case, the Lyapunov-Razumikhin notion is not so clear. In the latter, one uses a "finitedimensional" tool for an "infinite-dimensional" problem. The main idea of the stability result can be summarized as follows: in the case of a Lyapunov-Krasovskii functional, V, a sufficient condition for stability is that the derivative, V, of the candidate be negative along all the system's trajectories. In the Razumikhin based approach the negativity of the derivative of the function V : Rn ........ R is only required for the trajectories which leave at t+ a certain set, defined by the system evolution on the interval [t - T, t].

V(xt)

= x(t)T Px(t) +

f: _=1

1

jl-r._ X(I:1)T Si x (l:1)dl:1, t-'1",

if we assume Ti ~ Ti+l, TO = 0 and 51 > 0, 5i +1 - 5i > 0 (this restriction turns out to be in fact immaterial for the resulting criterion). Delay-dependent type criteria One usually considers the model transformation (single delay):

obtained using the Leibniz-Newton formula, etc. The associated Lyapunov-Krasovskii functional IS:

3.1.1. Lyapunov-Krasovskii approach. Delayindependent type criteria. The "classical" Lyapunov-Krasovskii functional used for such stability analysis (see Hale and Lunel, 1993;

V(Xt)

=

sup

e 68 x(t

+ B)Px(t + B)

8E[-2r,0]

{ P> 0,

1 {) = - 109(1 T

+ ,): ,

.

E R+

Although this form is quite complex, it depends only on one matrix P > O. Connections between this functional and the Lyapunov-Razumikhin function V(x(t)) = x(t)T Px(t) have been considered in Kato (1980) via a comparison principle.

11 the maximal negative (minimal positive) real part is continuously dependent on the delay parameter; see also Datko (1978)

12

= x(t)T Px(t) + L nd

V(Xl)

ordinary differential equation

34

3.1.2. Lyapunov-Razumikhin approach. Delayindependent type results. Consider ~ (single delay) with a continuous delay function r(t) S r

3.2. Comparison Principle

The idea is to find an ODE, or a FDE, called (B), with known behaviour such that its (asymptotic) stability implies the (asymptotic) stability for the initial delay system, called (A). Then (B) is called a comparison system for (A). The first comparison principles have been estabilished by Halanay (1966), Lakshmikhantam and Leela (1969) and Driver (1962). The tool which seems best adapted is the vector Lyapunov functions (Bellman, 1962; Matrosov, 1968). Using such ideas combined with some matrix techniques, we may have:

(r > 0). The 'usual' Razumikhin function is: V(x(t» = x(tf Px(t).

(8)

and one of the form of Proposition 1 can be: Proposition 2: The triplet ~ is delay-independent asymptotically stable (any r, r(t) S r) if there exists 5 > 0, P and 13 E R+ satisfying 5 S P and:

Matrix measures One of the first results in this framework is due to Mori et al. (1981), where the delay-independent stability of a triplet ~ (single delay case) is reduced to check:

For details and further comments, see Niculescu et al. (1997). Note that (9) can not be reduced to an LMI, but if 13 = lone obtains the LMIs:

{

[

AT P

P;::: S,

+ PA + P

A~P

PAd]

-s

<

0

(12)

.

S> 0

In a Lyapunov vector function approach, one needs V(x) = !lxll, etc.

All the ideas proposed in the Krasovskii framework may be developed along the same lines. For the sake of brevity, we do not develop them here.

Time-varying delays For the sake of simplicity, we consider here only the single delay 1s case. Multiple delays case are considered in Niculescu et al., (1995a) or Lehmann and Shujaee (1994) (different technique,delay-independent only). For constant delays, see also Wang et al. (1987). Introduce:

Delay-dependent type results. On the considered model transformation, one has: Proposition 3: The triplet ~ is asymptotically stable for any r(t) S r· if there exists Q51 , 52 > 0, 131,132 E R+, satisfying 13;lQ S 5i , i =

D:

Q(A

iJ(t)

+ Ad)T + (A + Ad)Q + T" (131 + 132)Q

+T" Ad (AS1A T + AdS2A~) Ar < 0,

q(t) = (~- T~t))

or other convex regions

(13)

exp

(-(I l~T(t) T~:)) ,

M -matrices The basic ideas are: • First, to introduce a comparison system which

can be, for example (single delay case, with a constant delay), of the form: iJ(t)

=

My(t)

+ Ny(t

- r),

with M and N computed from ~ . • Second, to test if -(M + N) is an M-matrix 16 . In terms of Lyapunov vector functions, one uses (Dambrine and Richard, 1993):

v (x) = [J X 1 I ... I x n I], x E Rn. 15

14

T(t»,

(11)

Multiple delays (not commensurate). The idea is to compute sub-optimal ellipsoids14 in the delay parameter space using apropriate model transformations obtained via an integration over one Ti (i index) delay interval (Niculescu et al., 1996). For scalar case, a complete characterization is given in Hale and Huang (1993) (two delays). Further comments are in Niculescu et al. (1997), etc.

in the sense m=imal allowable

+ q(t)y(t -

which is a comparison system for ~. Depending on the relationships between TJ, A and A + Ad, one obtains "delay-independent / delay-dependent" results (see Niculescu et al., 1997).

which is a standard LMI generalized eigenvalue problem (quasiconvex: Boyd et al., 1994).

13

-TJy(t)

where

(10)

Further comments can be found in Niculescu et al. (1997) or Li and de Souza (1995, 1997) (the variable Q = p- 1 , etc.). The computation of a suboptimal delay bound 13 can be reduced (directly) to an LMI optimization problem (if /3i = 1): max r· such that Q=:;QT>O { (10) holds,

=

continuous with bounded derivative

16 A matrix D is called an M-m atrix if the elements on the diagonal are non-positive, the matrix D is not singular, and furthermore, all the elements of D-l are non-negative.

35

4.2. Other remarks

4. Remarks

4.2.1. Complexity in multiple delays case. We have seen in the single or commensurate delays case that it is possible to solve the considered stability problems in polynomial time. In the general multiple delays case, Toker and Ozbay (1996) have proved that such problems are NP-hard using the NP-hardness of complex bilinear programming over the unit polydisk. In conclusion, it is rather unlikely to find efficient procedures (of polynomial-time type) for such problems in the general case. However, we should point out that better approximations schemes can be thought of to improve the "sufficient" (relatively simple) "delay-independent / delay-dependent" conditions presented above.

4.1. Various interpretations of delay systems 4.1.1. Differential Equations over Rings. The basic idea is to rewrite the system (1) using the

translation operator, and to interpret it as a differential equation on the ring R[z]. For exemplification, let us consider the single delay case E. The translation operator V T , defined by: Vd(t) = f(t - T), allows to rewrite the considered system as: x(t) = F(V T )x(t), where F(V T ) = A + V T is an operator acting on the evolution "x(O)" for 0 E [-T,O] of the system. Thus, we can associate the differential equation on the ring R[z] (see also Kamen, 1978) given by: x(t)

F(z)x(t).

(14)

4.2.2. Ill-possedeness. An interesting problem is to study the stability property for commensurate delays in the hypothesis when at least one of the commensurate delays does not satisfy this condition, but it (they) is (are) perturbed into an (some) interval(s)l7. Generally, it is clear that, if the system involving the delays as "independent" parameter is asymptotically stable, then also the commensurate delays stability case, or the perturbed delays stability problem is guaranteed. The question is if this condition is also necessary. The answer is positive in some cases.

Some connections between the characteristic equation associated to (14) and E are given in Kamen (1978, 1981). The delay-independent criteria given in Hale et al. (1985) or Brierely et al. (1982) can be seen also in this framework.

4.1.2. 2 - D Equations. The basic idea is to rewrite the FDE as a 2 - D equation. Thus, for the scalar case, x(t) = -ax(t) - bx(t - T), can be rewritten as:

4.2.3. Existence of several stability regions. Another interesting problem is to analyze if in the delay-dependent stability easelS, the stability region thus obtained is the only existing one. The examples given in Niculescu (1997) prove that one can have a sequence stability / instability / stability for non scalar systems. Some remarks have been considered in Niculescu et al. (1997).

(15) which is combines an "ordinary" differential equation and a "functional equation". Sufficient delay-independent stability conditions expressed in terms of 2-D Lyapunov equations have been given in Agathoklis and Foda (1989). For other criteria see Chiasson et al. (1985), etc.

4.2.4. Discretizing Lyapunov-Krasovskii functionalso In the previous paragraphs, we have considered stability issues using particular forms for the Krasovskii functionals. Other methods for constructing such functionals have been proposed in Infante and Castellan (1978) or Huang (1989). If one of the major advantages of such methods is that the corresponding conditions are closed to "necessary and sufficient" conditions, the major inconvenient lies in their complexity to handle numerical examples. Discretization techniques have been proposed in Gu (1997).

4.1.3. Matrix Characteristic Equation Approach The basic idea is to transform the FDE as an ODE via an appropriate linear transformation. For the single delay case, the transformation is:

where A mc satisfies:

(16) The corresponding linear system is i(t) Amcz(t). Note that (16) is a transcendental matrix equation and it is hard to use it for "delay-independent / delay-dependent" type results. However, we can mention the algorithms for computing such A mc matrices are given in Fiagbedzi and Pearson (1986) or Zheng et al. (1993).

5. Conclusions In this paper, some topics on delay systems stability have been considered. A specific problem

36

17

so-called ill-poucdcncu problem

18

always for commensurate delays

has been considered throughout the paper: the delay effects on the asymptotic stability property, i.e. delay-independent or delay-dependent. The intention of the authors was not only to classify existing results and methods, but also to present some trends in this field. For further comments, see also Kharitonov (1998) and Richard (1998).

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