Necessary and sufficient stability criteria for linear systems with delays

IFAC

Copyright © IFAC Time Delay Systems, New Mexico, USA, 2001

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NECESSARY AND SUFFICIENT STABILITY CRITERIA FOR LINEAR SYSTEMS WITH DELAYS

Bugong Xu

Department 0/ Automatic Control Engineering, South China University o/Technology, Guangzhou 510641, People's Republic a/China

Abstract: New necessary and sufficient stability criteria for linear systems with multiple uncertain delays are established by using the time-domain and the frequency-domain methods, respectively. The results are first established based on a new-type time-domain stability theorem for retarded dynamical systems and a new technique for estimating the derivative of Lyapunov function along the solution of system at some specific instants. Then, the same criteria are derived by the frequency-domain techniques. The established stability criteria are not only the least conservative but also involve the least tuning parameters. Copyright © 2001 IFAC Keywords: method.

stability criteria, delay, linear systems, time-domain method. frequency-domain

I. INTRODUCTION

some free tuning scalar and/or matrix parameters. Therefore, the numerical schemes for determining these parameters so that less conservative or optimal results can be obtained have attracted many researchers, see Chen and Latchman (1994), Li and de Souza (1995), Trinh and Aldeen (1995), Niculescu et at. (1995), Luo and Bosch (1997), Kolmanovskii et at. (1999), Xu (1997b, 1999, 2000a, 200 I), Xu and Lam (1999), Kolmanovskii and Ricard (1999), Gu (1999) and also see the recent results in Mahmoud (2000) and the book edited by Dugard and Verriest (1998). However, a more meaningful work for stability analysis of linear time-delay systems is to establish as least conservative stability conditions as possible and at the same time to reduce the number of tuning parameters as least as possible.

It is well known that the main time-domain stability

analysis methods for linear time-delay systems are Lyapunov methods including Lyapunov functional method and Lyapunov function method together with Razumikhin-type techniques (see for example, Kolmanovskii, and Nosov, 1986; Hale, and Lunel, 1993). For linear systems with time-varying delays, Lyapunov functional method needs generally the derivative conditions on delays, see for example, Trinh and Aldeen (1995), Jeung et at. (1996), Kim (1996), and Xu (1997b). But the derivative conditions on delays may introduce some conservative factors into the results, see the remarks given by Xu (1999, 2000a, 2001). On the other hand, Lyapunov function method together with Razumikhin-type techniques can be used to deal easily with the case of time-varying delays without the derivative conditions on delays. Unfortunately, the existing Razumikhin-type techniques in the literature also provides generally conservative results, see some results and remarks given in Xu and Liu (1994) and Xu (1994, 1995, I 997a, 1999, 2000a, 2001). Generally speaking, both Lyapunov functional method and Lyapunov function method provide sufficient stability conditions that include

Consider the following linear system with multiple uncertain time-varying delays: m

x(t)

= Aox(t) + I

Akx(t - Tk (t))

hI

X/

O

«()) = X(t + ()) = ifJ«())

T to +()E G{t-Tk(t)1 t- k(>t)5,t o'}U{t o } k=1 t - to

123

(1)

(1) or (2), the fol1owing stability definitions (Xu, 2001, 2000b) are used.

and its special case: m

X(t) = Aox(t) + L Akx(t - 'k)'

t~0

(2)

Definition 1. System (1) or (2) is said to be exponential1y (uniformly asymptotical1y) stable if the

k=l

equilibrium x· = 0 of system (1) or (2) is global1y exponential1y (uniformly asymptotical1y) stable and

with multiple unknown constant delays 'k < 00 for nxn n k=I,2, ... ,m,where t/JEC n , xER , AkER

k = 0, 1, ... , m

for

are

constant

said to be uniformly stable if the equilibrium x· of system (1) or (2) is global1y uniformly stable.

matrices,

'k(t):5,'kM <00 and 'k :5, 'kM <00 for k = 1, 2, ... , m with known constant 'kM > 0 for k=I,2, ... ,m

al1

'k(t):5,,
or

k = 1, 2, ... , m with unknown constant

2.

Definition

for

,> 0

a >0

Let

~R+

TI: Rn

be

continuous,

be

=0

constant,

11xI1:5, TI(x)

for

x ERn, and TIt = SUP-T$O$O {TI(t/J«(}))} for given

are the uncertain delays. In this paper, new necessary and sufficient delay-dependent and delay-independent uniformly asymptotical1y (or uniformly) stability conditions for linear systems with multiple uncertain delays (1) and (2) are fIrst established based on the new-type stability theorem for retarded dynamical systems and the new technique for estimating the derivative of Lyapunov function along the solution of system at some specific instants (Xu, 1999, 2000a, 2001). Then, the same criteria are derived by the frequency-domain techniques.

t/J E C n

System (1) is exponential1y stable with

.

respect to the constant decay degree a > 0 if along the solution x(to, t/J)(t) of system (1) through any

(to, t/J)ERxC n , one has

Definition 3. System (2) is said to be a-stable 10cal1y in the delays if there are positive 'kM > 0 for k = 1, 2, ... , m such that al1 the roots of the characteristic equation:

Rn nxn is the real vector space of dimension n; R is the real matrix space of dimension n x n; R+ The notations used in this paper are as fol1ows.

denotes the set of nonnegative real numbers; C n is the complex vector space of dimension n; Cn denotes the Banach space of continuous functions mapping [-,,0] into Rn, where 0 ;

lie in the region of Re S :5, -a < 0 in the complex plane for arbitrary values of 'k E [0, 'kM] ,

,>

k = 1,2, ... , m.

y, (0) E Rn denotes y(t + 0) E Rn for t E Rand o

E

R

so that

Euclidean norm in Rn; with t/J«(})

E

Rn

11·11

y(t) = Yt (0); 11

Definition 4. System (2) is said to be e-stable independently of the delays if for any positive 'kM > 0 for k = I, 2, ... , m, there is a sufficiently

denotes the

t/J liT = sup-T$O$oll t/J(O) 11

for given t/J

E

C n ; AT

smal1 positive number &= &(, kM ) > 0 such that al1 the roots of the characteristic equation (4) lie in the region of Re S < -& < 0 in the complex plane for arbitrary values of 'k E [0, 'kM)' k = 1,2, ... , m ,

is the

transpose of A; A' is the conjugate transpose of A, i.e. AT; A(') denotes any eigenvalue of a A max (A)

and

Amin (A)

the real part of SEC; Z

E

Iz I

&~

where

are the maximum nxn , and minimum eigenvalues of AT = A E R respectively; A> 0 (or < 0) denotes a positive defmite (or negative definite) matrix; A:5, B means that A - B :5, 0 is negative semi-defmite; Re s is matrix;

k

E

0

as

Let P > 0 E R

any

for

any

7J =

pe-j(O/2)

nxn and DE R nxn

Then, for any x, y

constant matrices.

C or the absolute value of z ER; Jn is the

~ 00

{I, 2, ... , m} .

Lemma 1.

is the modulus of

'kM

EC

with

p >0

E

and

C

n

be and

(} ER,

one has

n x n identity matrix; and finally, j =..r:I. x· PDye- jO + e JO y" D T Px 1. -1 T • :5,-x PDP D PX+PY Py P

2. PRELIMINARIES Since al1 the stability properties discussed in this paper are global ones in the solution space of system

and

124

the

equality

holds

if

and

only

(5)

if

e}(1J/2) D T Px = 1]py . Proof It is 1] = pe- }(1J/2) E e

rnax {_I_x' PDp-1 D T Px+ PM/ pY} PM

easy to see and any

x Px=K y'Py=K

that for any x, y E en

. PDP -I D T Px+ PMx .Px} rnax { -I -x PM

(e}(1J/2) D T Px -1]py)' p-I (e J (IJ/2) D T Px -7]py) ~ 0 and the equality e}(1J/2) D T Px = 7]py .

holds

if

and

only if Q.E.D.

= 2P M K = rnax {2X' PDy} x Px=K y'Py=K

Let P > 0 E R nxn and DE R nxn be

Lemma 2:

x, y E en , and

constant matrices,

K > O.

(10)

x Px=K

By (6) and Lemma I, (7) is obvious.

Q.E.D.

Then,

Remark 1: Obviously, all the results established in Lemma I and Lemma 2 still hold for the case of x, y E Rn and 7] = P > 0 E R+ .

there exists a positive PM> 0 such that

(6)

3. TIME-DOMAIN METHOD In this section, the necessary and sufficient stability criteria for system (I) are derived by using the time-domain method (Xu, 200 I).

and

Theorem 1. Let 'kM > 0 for k = 1,2, ... , m and r > O. System (I) is exponentially stable with

I. -I T .} { - - x PDP D Px+ PMx Px

rna~

PM

x Px-K

$

rnax x Px=K

respect to the constant decay degree a = r /2 > 0 if and only if there is a positive definite matrix P > 0 E R nxn and m positive scalars Pk > 0 for k = I, 2, ... , m such that

(7)

{~x· PDP- DTpx+ px' px} I

P

for any P >0. (11)

Since K > 0, there must exist a posItIve PM > 0 such that pit K = rnax {x' PDP- I D T Px}.

Proof

x Px=K

Let X(K)

where To = PAo +A6 P.

= {x E en Ix' Px = K}. For any given

x E X(K) , it is not difficult to see that there exists a positive

p>0

and a vector

ZE

nxn

Proof Let V(x) = x T Px , where P > 0 E R . Then, by Lemma I, Lemma 2, Remark I, and Theorem A.I in Appendix, one obtains

X(K) satisfying

T

D Px = jje}1J pz with a () E R such that

-2K =x -'PDP-1D T rx=pz re: -2'pz=px -2'pX P for any x

E

X(K).

Note that PM pI/2 z depends

linearly on pI/2 y when y together with (8) implies that

=z E

X(K)

PMK = rnax {(y' pyx' PDP-1D T Px x Px=K y'Py=K

= rnax

(8)

This

r}

{x' PDy}

(9)

x Px=K y'Py=K

From (8) and (9), it is easy to obtain (6), Le.

(12)

125

if and whenever (Le. V(x,(O))=V;o exp{-y(t-t o )} on where

Sas,a(L,(B))

is

defmed

only if) t~to ER as

(24)

lY(Tk(t)+OXk )

Appendix, Y,(-,dt))=y,(Ok)e2 V(y, (Ok))

=Y; (Ok )Py, (Ok) = Vex, (0))

3. Let a > 0 and 'kM > 0 for k = 1, 2, ... , m. System (2) is a-stable locally in the delays if and only if there is a positive defmite IJXIJ and m positive scalars matrix P > 0 E R P k > 0 for k = 1, 2, ... , m such that Theorem

in and with

BXk E (-00,0] for all k = I, 2, ... , m . By Theorem in Appendix and Defmition 2 with A.I T T Amin (P)x X ~ Vex) ~ Amax (P)x x and I1(x)

= V(x), the proof is completed.

To + Le aTkM m

hI

Q.E.D.

[PA p-IAT k k Pk

p

J

+Pk P ~-2aP

(15)

where To = PA O + A6 P .

Theorem 2. System (I) is uniformly asymptotically stable if and only if there is a positive defmite matrix

Let s=a+jOJEC

Proof

P > 0 E R IJXIJ and m positive scalars Pk > 0 for k = 1,2, ... , m such that

[0, 'kM] for k = 1,2, ... , m , with aER and OJER

'k E

m

W('b s)

= Aa + Le-rk"Ak =W('k' a, OJ)

and

k=1

y := y(,k ,

a, iiJ) E C n

satisfy

a, w)y = A(W('k' a, iiJ))y s = s = a+ jw C, where

at

W('k'

E

and uniformly stable if and only if there is a positive definite matrix P > 0 E R IJxn and m positive scalars Pk > 0 for k = I, 2, ... , m such that

a = Re A(W('b a, w)) _ y' (PW('k' a, w) + W' ('k' a, w)p)y -

2y' py y' (PAo + A6 P)Y

1

[ + Ie-rkO'y' (PAke-JTkW + A{PeJrkW)'y k=l

Proof According to the standard definitions of uniform stability (Hale, and Lunel, 1993, Xu, 2001) and Defmition 2, it is not difficult to see that the uniform asymptotical stability (or uniform stability) for the zero solution of system (I) is equivalent to the corresponding exponential stability with respect to a unknown positive constant decay degree (or zero decay degree). Therefore, this theorem can be proved similarly to Theorem 1 based on Theorem A.l in Appendix so that the proof is omitted here. Q.E.D.

(16) Let us prove the sufficiency • for any P > 0 E R by contradiction. Assume that condition (15) holds. If there IS a root S = a + jw E C of the corresponding characteristic equation (4) for 'kE[O"kM]' k=I,2, ... ,m, such that a>-a, then by Lemma I, Lemma 2, and equation (16), one has IJXIJ

Remark 2. According to Lemma I, Lemma 2, Remark 1, Defmition I, Definition 2, and the proof of Theorem 1, it is easy to see that the established stability criteria above are not only the least conservative but also involve the least tuning parameters.

-a
4. FREQUENCY-DOMAIN METHOD It should be pointed out that the results established in

Section 3 are also suitable to system (2). However, in order to compare the time-domain method with the frequency-domain method, let us derive the corresponding stability criteria by using the frequency-domain method in this section. (17) which is a contradiction.

126

The proof for the

ACKNOWLEGEMENTS

sufficiency is completed. For the necessity, since ii~-a for all 'kE[O"kM], k=I,2, ... ,m, by Lemma 2 and equation (16), it is easy to see that nxn and m positive there must exist a P > 0 E R scalars Pk > 0 for k = I, 2, ... , m such that

This work was supported by NSFC Project 60074026 and Guangdong Province Natural Science Foundation of China Project 000409.

APPENDIX: A NEW-TYPE STABILITY THEOREM FOR RETARDED DYNAMICAL SYSTEMS Consider a retarded dynamical system

x(t) = f(t,

=-a. (18)

where "." denotes the right-hand derivative, f: RxC n ~Rn, and f(t,f/J) is continuous and

Otherwise, by Lemma 2, equation (16) and the last equality in (18), - a> max rkE[O, rkM 1{ii} which is a

Lipschitzian in f/J so that for an initial function

x lO =f/JECn at t=t o ER, system (20) has a

Q.E.D.

contradiction.

unique solution x(to, f/J)(t) on [to - " 00) . Suppose that f(t, 0) == 0 for all t E R so that

Theorem 4. System (2) is t::-stable independently of the delays if and only if there is a constant matrix P > 0 E R nxn and m positive scalars Pk > 0 for

x· == 0 is the equilibrium of system (20). For simplicity, we denote the value of the solution

k = I, 2, ... , m such that (13) holds.

x(to, f/J)(t)E Rn by x(t) and the solution segment x(t + ()) = XI «()) E Rn

Proof Since the eigenvalues of a matrix depend continuously on its elements, condition (13) implies that for any positive 'kM> 0, k = 1,2, ... , m, there is a sufficiently small number & > 0 such that

XI

Note that

() E

0]

[-"

by

Let V(x) = x T Px, where X ERn R nxn . Then, (i) The equilibrium

P>0E

x· == 0 of system (20) is globally unifonnly stable if along the solution x(to, f/J)(t) of system (20) through any (to, f/J) E R x C n ,

~ 0 as 'kM ~ 00 for any k E {I, 2, ... , m}. By Lemma 2 and Theorem 3, it is easy to see that all the roots of equation (4) lie in the region of Re s < -& < 0 in the complex plane for arbitrary values of 'k E [0, 'kM) for k = I, 2, ... , m if and only if inequality (19) (Le. (13)) holds. According to Definition 4, the proof is completed. Q.E.D.

= PAo + Ab P.

all

EC n atagiven te-to'

and

To

for

Theorem A.I.

(19)

where

(20)

XI)

&

whenever

V(x l (0)) = V;o

t e- to

on

'

where

S 510. (V;o) is defined as

S 510

(v;o)

YI EC n

5. CONCLUSION

V(YI (0))

The new necessary and sufficient stability criteria for linear systems with multiple uncertain time-varying/constant delays have been established by using both time-domain and frequency-domain methods. The results are derived based on the new stability theorems and the new technique established and used in Xu (2001, 2000). It has been remarked that the established stability criteria for the systems under consideration are not only the least conservative but also involve the least tuning parameters.

=

= VIO

IIpl/2 y, (B~12 = lip 1/2 y, (0), l' x () E [-"

0],

()X E

r

(22)

(-00,0]

(ii) The equilibrium x· == 0 of system (20) is globally exponentially (unifonnly asymptotically) stable with respect to the constant decay degree y /2 > 0 if along the solution x(t 0' f/J )(t) of system

(20) through any (to, f/J)

127

E

R x Cn ,

V(y, (0» ~ -yV(y, (0»,

'ify,

E

Sas,a. (L, (B»)

uncertain linear state space models. Automatica, 33, 171-179. Mahmoud, M.S. (2000). Robust Control and Marcel Filtering for Time-Delay Systems. Dekker, Inc., New York. Niculescu, S.I., A.T. Neto, J.M. Dion and L. Dugard (1995). Delay-dependent stability of linear systems with delayed state: a LMI approach. In Proceedings of the 34th Conference on Decision and Control, New Orleans, Louisiana, USA, 2, 1495-1496. Trinh, H. and M. Aldeen (1995). Stability robustness bounds for linear systems with delayed perturbations. lEE Proceedings Control Theory and Applications, 142,345-350. Xu, B. (1994). Comments on 'Robust stability of delay dependence for linear uncertain systems'. IEEE Transactions on Automatic Control, 39, 2365. Xu, B. (1995). On delay-independent stability of large-scale systems with time delays. IEEE Transactions on Automatic Control, 40, 930-933. Xu, B. (\ 997a). An improved Razumikhin-type theorem and its applications--Author's reply. IEEE Transactions on Automatic Control, 42, 430. Xu, B. (1997b). Stability robustness bounds for linear systems with multiple time-varying delayed perturbations. International Journal of Systems Science, 28, 13 11-13 17. Xu, B. (1999). Decay estimates for retarded dynamic systems. International Journal ofSystems Science, 30,427-439; Xu, B. (2000a). Decentralized stabilization of large-scale linear continuous systems with N xN time-varying delays. International Journal of Systems Science, 31,489-496. Xu, B. (2000b). Stability criteria for linear time-invariant systems with multiple delays. Journal of Mathematical Analysis and Applications, 252, 484-494. Xu, B. (2001). Stability of Retarded Dynamical Systems: A Lyapunov Function Approach. Journal of Mathematical Analysis and Applications, 253, 590-615. Xu, B., Y. Fu and L. Bai (1996). Further results on robust bounds for large-scale time-delay systems with structured and unstructured uncertainties. International Journal of Systems Science, 27, 1491-1495. Xu, B. and l Lam (1999). Decentralized stabilization of large-scale interconnected time-delay systems. Journal ofOptimization Theory and Applications, 103,231-240. Xu, B. and Y. Uu (\994). An improved Razumikhin-type theorem and its applications. IEEE Transactions on Automatic Control, 39, 839-841.

(23)

whenever V(x,(O»=V;oexp{-yU-t o )} on t"2t o , where y > 0 and S a.s'a. (L, (B») is defined as

L, (B)

= L(t + B) = V;o e -y(t+B-,o)

V(y, (0» = L, (0)

lip '/2 y, (Of = lip '/2 y, (O)e +
Proof

(24)

Bx E (-00, 0]

For the proof, see Xu (2001).

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