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Robust stability analysis of time delay systems: A survey
Annual Reviews in Control
PERGAMON
Annual Reviews in Control 23 (1999) 185-196
ROBUST
STABILITY ANALYSIS OF TIME DELAY SYSTEMS: A SURVEY
Vladimir L. Kharitonov
Automatic Control Section CINVESTAV-IPN P.O. BOX 14-7~0 07300 Mexico, D.F.
Abstract: In this survey some recent contributions to stability and robust stability analysis of linear time delay systems with parameter uncertainty are discussed. The main aim of the paper is to discuss some basic techniques used for deriving tractable stability and robust stability conditions. The reference list presents rather a small portion of the large number of publications in this field. Rdsume: On discute quelques contributions rdcentes a l'analyse de la estabilitd et de la estabilitd robuste des systemes lindares a retards a paramdtres incertaius. L'espirit de article est de prdsenter l'essential des techniques de base que sout utilisdes pour obtenir des conditions vdrifiables de stabilitd et de stabilitd robuste. Les rdfdrences listdes ne reprdsenteut qu'une petite pattie du gran nombre des publications dans ce domaine. Keywords: delay system; stability; robust stability
1: BASIC DEFINITIONS
mapping the delay interval [ - h , 0] into R '~ with the standard uniform norm
1.1 Systems and solutions In this section some classes of time delay systems will be introduced as well as necessary notations and basic definitions will be given. The main attention is paid to the case of linear time invariant systems. An example of such systems is the single delay system of the form
dz(t) _ Ax(t) + Alx(t - h), dt
(1)
I1¢(.)11= ee[-h,o] max {11¢(0)11}. For given initial conditions of the form (2) the system (1) admits the unique solution x(t) = x(t, to, ¢(.)), which is defined on [to - h, c¢). A natural extension of the class of single delay systems is the class of systems with multiple delays of the form N
where A and A1 are constant n x n matrices, h is a nonnegative delay. In order to define a particular solution of the system one has first to set initial conditions which include a time instant to and an initial vector function ¢(.)
(to + 0) = ¢(0), fo . all 0 e I-h, 0].
(2)
Here ¢(.) is an element of the Banach space C ( [ - h , 0 ] , R n) of contimlous vector functions
dx(t) = Ax(t) + ~ dt
Akx(t - h~),
(3)
k=l
where all delays are nonnegative real values~ In order to define a particular solution of the system (3) one has to specify an initial vector function ¢(.) on the segment [ - h , 0], where h = maxl
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V. L. Kharitonov / A n n u a l Reviews in Control 23 (1999) 185-196
186
0
dx(t) dt = f [dF(O)]x(t + 0).
(4)
-h
It is assumed here that all elements of the matrix valued function F(.) have bounded variations on [ - h , 0]
of delay free systems is a description of delay elements, scalar h in the case of systems (1), (4), or delays hi, h2, ..., hN in the case of system (3). Assume that a time delay system is given along with particular uncertainty descriptions :D in delay elements, and S in system matrices, then one can define the following notion of robust stability.
0
f IlaF(0)ll< -h
The single delay system and the multiple delay system are particular cases of the distributed delay system of the form (4) and can be obtained from it with a special choice of piecewise constant system matrix F(-). Initial vector function ¢(.) for a particular solution of the system (4) must be specified on the whole integration segment [ - h , 0].
1.2 Stability There is no difference in stability definitions for systems (1)-(4), so for definiteness they will be given for the system (4) only.
Definition 1. The system (4) is said to be Lyapunov stable if for every positive e there exists a positive 5 such that if I1¢(')11 < 5, then IIz(t,t0,¢(.))ll < c for all t > to.
Remark 2. If the value 5 from the above definition may be chosen such that in addition x(t, to, ¢(.)) --~ 0 when t --~ c~, then the system (4) is said to be Lyapunov asymptotic stable. Definition 3. The system (4) is said to be exponential stable if there exist positive constants #, a such that every solution of the system satisfy the following inequality
IIz(t, to,¢(.))ll to. It is worth to be mentioned now that for the system (4) the exponential stability and the Lyapunov asymptotic stability are equivalent. In the sequel the word "stability" will be used instead of "exponential stability".
1.3 Robust stability
Definition 4. Given a time delay system with uncertainty descriptions :D, S. The system is said to be robustly stable if it is stable for all delay elements from 79 and for all system matrices from S. The case when each of the sets :D and ~g contains just one element corresponds to uncertainty free system. It should be mentioned here that the stability analysis of systems with uncertainty in delay elements is, in general, a more complicated task than that of systems with uncertainty in system matrices (Toker and Ozbay, 1996).
2. STABILITY CONDITIONS In this section some basic stability conditions will be presented.
2.1 Frequency domain conditions Having in mind the fact that the main concern of the paper is about the class of linear time invariant systems one has to address first to stability conditions based on the zero analysis of the system characteristic function,see (Hale, 1977).
2.1.1. Distributed delay case Theorem 5. The system (4) is stable if and only if its characteristic function f(s) = det
(]) sE-
e°'dF(O)
-h
has no zeroes with nonnegative real parts. The characteristic function may be written in the polynomial form r*--I
There are two different places where uncertainty may affect a time delay system. First, as in the case of delay free systems, uncertainty may appear in a description of the system matrices, A, A1,..., AN for the cases of the systems (1), (3) or in that of the matrix valued function F(-), in the case of the system (4). The second source of uncertainty which does not exist in the case
f(s) = s" +
k(s)s k, k---0
where coefficient functions are bounded in the closed right half plane of the complex plane. Applying the argument principle to f(s) one may arrive to the following result which is an extension of the Mikhailov polynomial stability criterion (Kabakov, 1946).
V. L. Kharitonov / A n n u a l Reviews in Control 23 (1999) 185-196
Theorem 6. Let f(s) have no zeros on the imaginary axis. It has no zeros with positive real parts if and only if the net change of the argument function satisfies the condition ~z.
•
2.1.2. Pontryagin criterion In the case of systems (1), (3) this characteristic function is a finite sum of power and exponent terms n--1 rn
~2 . . , a k ~ s k=Oj=O
~ e ~"
,
where the exponent coefficients 0 = 7"0 > 1"1 > •.. > Vra > - c o are linear combinations of delays with nonnegative entire coefficients. Function f(s) may be written as a sum of polynomials multiplied by exponents
y(s) = p0(s)e ~o' + .
direct method. There are two different ideas how one can apply this method for time delay systems. In the first one the state of a delay system is defined as the time depending trajectory segment
7r
A a r g ( y ( ~ w ) )oo =
f ( s ) = s" + ~
187
+ p~(s)~ ~-',
(~)
where deg (P0) = n > deg (pj) for j = 1, 2, ..., m. Any function of the form (5) is called quasipolynomial. The zero set of f(s) is not changed after multiplication of f(s) by an arbitrary exponent er" . T h a t means that in the zero analysis one can ignore assumption that all exponent coefficients are nonnegative. Let us define two real functions u(w) and v(w) as the real and the imaginary parts
of f(i~), y(i~) = u(~) + ~v(~).
• , = { ~(t + o) I 0 e I-h, 0] }. Here instead of the classical Lyapunov functions it is proposed to use Lyapunov functionals. This approach has been developed in (Krasovskii, 1956).
The second idea is based on classical Lyapunov functions and uses a special estimation procedure which allows exclude delay states in the derivative of the Lyapunov functions. This procedure has been proposed in (Ra~umikhin, 1960). 2.2.1• Lyapunov-Krasovskii approach
Theorem 9. The time delay system (4) is exponential stable if and only if there exists a continuous functional v(xt) such that (1) 4111~(t)ll ~ < v(~,) < 4~ II~tll ~ , for some positive 41, 42; (2) ~v(xt)l(4 ) _< - 3 IIx(t)ll 2, for a positive constant 3. The direct application of this theorem is seriously complicated by difficulties arising in construction of such functionals. In general one has to look for a functional of the form
v(~t) = ~T(t)P~(t)+
The following theorem has been proven first in (Pontryagin, 1942) for the case of entire exponent coefficients, and then in (Tchebotarev, N.G. and N.N. Meiman, 1949) for the general case.
Theorem 7. Given a quasipolynomial (5) with r0+vm _> 0. All zeros of the function have negative real parts if and only if functions u(w) and v(w) have only real, simple, interlacing zeros, and at least for one real
(7)
0 -t-2xT (t)
/ Q(O)x(t + e)dO+ -h
0
+ / xT(, + O)n(o)x(~ + O)dO+ -h 0
0
+ f f T(t + el)S(01,02)( +o,)dOldO2, -h -h
where weight matrices P, Q(0), R(0) and S(01, e2)
Remark 8. If f(s) has no zeros with nonnegative real parts and T0 + rm >__ 0 then the above inequality (6) holds for all real w. It means that arg(f(iw)) is a strictly increasing function of w. For brevity f(s) is said to be stable if it has no zeros in the closed right half plane of the complex plane.
2.2 Lyapunov direct method A more general approach to stability analysis of time delay systems is based on the Lyapunov's
satisfy a system of algebraic, ordinary and partial differential equations• A solution construction for this system and the following analysis of its positivity is a very difficult task. Therefore the theorem is usually used for deriving sufficient conditions when one tries different quadratic functionals of particular types. This approach is known as the Lyapunov-Krasovskii functional approach. 2.2.2. Lyapunov-Razumikhin approach An attempt to rescue the idea to use classical Lyapunov functions for stability analysis of time delay systems has been successfully launched in (Razumikhin, 1960). The main problem here is that
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V. L. Kharitonov / A n n u a l Reviews in Control 23 (1999) 185-196
the time derivative of a Lyapunov function depends not only on the present state x(t) of the system but also on past states. The key point of the Lyapunov-Razumikhin approach is a special procedure how one can obtain an upper bound of the time derivative which does not depend of these past states.
where P is a positive definite matrix, P > 0. Then
Theorem 10. The system (3) is exponential stable if there exists a positive definite function v(x) such that
The integral terms may be estimated as follows
-d- v I = ~T (t) [(A + A1)TP + P(A + A1)] ~ ( t ) dt (8) o
-2~T (t)PA1 f [A~(t + 0) + Ax~(t - h + 0)] dO. -h
o
-2 T(t)PA1 f A (t + 0)d0 _<
(1) Ul(llxll) _< v(z) _< u2(llxll) for some continuous nondecreasing functions, ul(s) > O, u2(s) > 0, for s > 0 and ul(0) = u2(0) = 0; (2) _< -w(llxll) for a continuous nondecreasing function w(s), where the derivative is evaluated for all vector functions satisfying the condition
-h 0
< ~T (t)PAIA / Y - 1(O)dOATA~e~(t)+ -h 0
+
v(x(t + 0)) < p(v(x(t))), for all 0 E [-h, 0]. Here p(s) is a continuous function such that p(s) > s, for s > 0.
f
~T(t q- 0 ) Y ( 0 ) ~ ( t -~- 0)d0,
-h
and o
-2~T (t)PA1 / Al~(t - h + O)da < 2.2.3. System transformation For simplicity the single delay system (1) is treated. It may be written as
0
< ~T(t)PA~ f Y~'X(O)dO(A~)T P~(t)+
0
-h 0
-h
If now one changes differencial under the integral using the right hand side of the equation (1) and replaces the state variable x with the new one, ~, then one comes to the system (8)
+ / ~T(t -- h + O)YI(O)~(t - h + O)dO, -h
where Y(O) > 0 and YI(O) > 0 for all 0 E [-h,0] are arbitrary positive definite matrices with continuous elements. Now the derivative can be estimated as
--v Jt
0
/ [A~(t + e) + Al~(t - h + 0)1 dO. ,J
-h
It is easy to observe that the system (8) belongs to the class distributed delay system and that the time lag for the system is twice larger than that for the original system (1). But on the Other hand every solution of the original system is also a solution of the latter one. It means that stability of the system (8) implies that of the system (1). There are also other possibilities how one may transform the original system in order to include explicitly delay elements in the system description, see (Kolmanovskii, 1995). 2.2.4. Lyapunov-Razumikhin approach Consider a Lyapunov-Razumikhin function candidate
v = ~T(t)P~(t),
(10)
-h
d (t) = (A + A1)x(t) - A1 / dx(t + 0). dt
d~(t) = (A + A1)~(t)dt
(9)
(s)
< ~T(t)[(A + A1)Tp + P(A + A1)+ o
+PAIA f Y-I(O)dOAT ATP+ -h 0
+PA~ / Y{'l(O)dO (A~)w PlY(t)+ -h 0
+ f [ r(t +
+ o)+
-h
+ ~T(t -- h + O)Yl(O)~(t - h + 0)] dO. If one sets that in the R ~ u m i k h i n theorem q(s) = (1 + e)s, with e > 0, then terms with delay states may be estimated as follows
~T (t q- O)P~(t +0) <_(1 + e)~T (t)P~(t). The form of this estimation makes clear why in practice there is no big choice in selection of the weight matrices Y(O) and YI(O). T h e simplest
V. L. K h a r i t o n o v / A n n u a l R e v i e w s in C o n t r o l 23 (1999) 1 8 5 - 1 9 6
case is just to assume that Y(O) = tiP, and YI(0) = fliP for some positive constants fl, fil. In this case the derivative admits the following quadratic estimation
-~v
If in addition S(-h) = &(0), then for arbitrary positive definite Sl (-h) = R 0
& (o) = R + f ~'~(O)dO,
<_~T(t) [(A + AoTp + P(A + A1)+ (s)
189
-h
and
+
g
0
IAr,+
S(O) =.R + f (Y(O) + Y1(0)) dO.
+h(1 + ¢)(fl + ill)P] ~(t).
Theorem I1. Assume that the matrix (A + A1) has no eigenvalues in the closed right half plane of the complex plane. The system (8), and henceforth the system (1) is stable if there exist matrix P > 0 and positive values fl, j31 such that
0 > (A + A1)TP + P(A + A1) + h(fl + flOP+ 1 -1 A1)A T T1 P. +hPAl( AP-1A T + fl-~A1P
-h
After these calculation one arrives to the following upper bound for the time derivative
< ~T(0 [(A + A O r e + P(A + AO+
-~
dt
(8)-
0
0
+R + J ,(O)dO + J (O)dO+ -h
-h
+PA1A f Y-I(O)dOAT AT1P+ 2.2.5. Lyapunov-Krasovskii approach Try now a Lyapunov-Krasovskii functional candidate
-h 0
+PA~ f Y~'l(O)dO (A~)T P]~(t)-
o
J
-h
v(~t) = {T (t)P~(t) + f ~T (t + e)s(e)~(t + e)dO
_~T (t
-h 0
+ f ~T(t -- h + O)&(O)~(t - h + O)dO, -h
where P > O, S(O) and SI(0) are positive definite matrices with continuously differentiable elements. The time derivative of the functional along the trajectories of the system (8)
--v = ~T(t) [(A + A1)TP + P(A + A1)+ dt (s) + S(O)] ~(t)+ C(t - h)[&(O) - S(-h)] ~(t - h ) o
_f
+ O)@0
+ 0)d0--
-h
-
2h)/~(t
-
2h).
Theorem 12. Let the matrix (A + A1) have no eigenvalues with nonnegative real parts. The system (8), and therefore the system (1), is stable if there exist positive definite matrix P and positive definite matrix valued functions S(0), $1 (0) such that 0
0 > (A + AoTp + P(A+ A1) + f Yl(0)d0 -h 0
0
+ J Y(O)dO+ PA1A J Y-'(O)dOAT ArP -h
-h 0
+PA~ f Yi-l(O)dO (A~)T P.
0
-2(T(t)PA1 f [A~(t + O) + Al~(t - h + 0)] de-h
-~r(t - 2h)&(-h)~(t - 2h)0
-h
Remark 13. It is clear that the last condition coincides with that of the previous theorem if one sets Y(0) = fiR and Y1(0) = /~IP. But it is no longer the compulsory choice here.
-h
The mixed integral terms admit estimation (9)(10), where Y(O) > 0 and Yt(0) > 0 are two matrix valued functions with continuous elements. In order to cancel all integral terms one has to assume that dS(O) = Y(O) and d&(O) - I"1(0).
dO
dO
3. STABILITY INDEPENDENT OF DELAY In this section some basic results on a specific robust stability notion for time delay systems, namely stability independent of delay are discussed. Probably the first step in analysis of this type of stability has been made in (Tsypkin, 1946)
V. L. Kharitonov / A n n u a l Reviews in Control 23 (1999) 185-196
190
where it was proved that a quasipolynomial p(s) + q(s)e -~s, deg(p) > deg(q) has no zeros in the closed right h a l f plane of the complex plane for all nonnegative values of r if and only if p(s) is a stable polynomial and the inequality Ip(iw)l > Iq(iw)l, holds for all w E R.
See (Kamen, 1982).
Assume that for the time delay system (3) delays are nonnegative functions of some parameters
3.2 Norm based sufficient conditions
hk = hk(ql,q2,...,qt) k = 1, 2 , . . . , N where (ql, q2, ..., q~) E Q c R I. Definition 14. The system (3) is said to be stable independent of these parameters if it is stable for all (ql, q2, ..., ql) from Q. There are two special cases which attract a special attention. The first one is the case when delays depend on one parameter, l = 1, and hk = rkq,
k = 1,2,...,N.
Here rh > 0, are given and q is a positive free parameter. The change of the time scale t I = qt allows reduce the problem to the case of fixed delay. In the second case delays are independent parameters by themselves, i.e., l = N, and
• det(sE - A - zA1) # < 1.
0
for all s E C+and
For arbitrary induced matrix norm the spectral radius of a square matrix A does not exceed the norm of the matrix
p(A) <
IIAII.
Based on this fact one can immediately obtain the following sufficient condition, see for example (Chen, et all, 1995). Theorem 17. The time delay system (1) is stable independent of delay if the system matrix A is stable and [ [ ( i w E - A ) - I A I [ [ < 1 forallw_>O. (11) The last theorem is main source of a number of sufficient stability independent of delay conditions. Now we observe some of them.
Here Q = R ~ = [0, oo) × [0, oo) ×... × [0, o¢). This case corresponds to stability independent of delay.
3.2.1. Stability radius interpretation The theorem condition may be interpreted in the terms of the structural complex stability radius of a stable matrix A. It has been shown in (Hinrichsen and Pritchard, 1992) that a system
These two cases coincide for the single delay system (1).
dz - - = (A + B Z C ) x , dt
hk = qk k= l,2,...,N "
3.1 Stability independent of delay criterion The following statement is a slightly modified result from (Chen, et all, 1995) and (Chen and Latchman, 1995). Theorem 15. The system (1) is stable independent of delay if and only if • matrices A and (A + A1) are stable; • the spectral radius of the frequency dependent matrix (iwE - A ) - I A 1 is less than one for all positive w: p((iwE - A ) - I A 1 ) < 1,
for all w > 0.
Remark 16. Let matrix A be stable. The following two sufficient conditions for stability independent of delay are equivalent •
p((iwE - A)-IA1) < 1,
for all w _> 0;
with given matrices A, B, C and unknown norm bounded complex matrix Z, IIZI] < r, is stable if and only if
r < r ( A I B , C) =
i m a ~ e R (flC(i~E - A ) - ' B I D
(assuming by definition that r(A, B , C ) = oo when C (iw E - A )- I B = 0). The value r( A, B, C) is known as the structural complex stability radius of the matrix A. Assuming B = Ax and C = E one can detect that the theorem conditions is equivalent to the statement that r(A, A1, E) > 1. 3.2.2. H ~ approach For the case of g2 norm the theorem condition may be also interpreted as that the H ~ - n o r m of the stable transfer matrix G(s) = (sE - A ) - I A x is less than one. This observation opens a door for application of the H¢~ theory methods for stability analysis of time delay systems. A good example of a such application is (Verriest, et all, 1993).
V. L. Kharitonov / A n n u a l Reviews in Control 23 (1999) 185-196
3.2.3. Time domain approach If one looks at the condition (11) from the time domain point of view then, having in mind stability of the system matrix A, one can use the following identity co
(iwE - A)-IA1 =
f e-~%AtAldt. 0
From here, using the classical exponential estimation IIeA'Alll <- ke-"', where k and t, are positive constants, one can immediately conclude that the time delay system (1) is stable if k < ~.
191
3.4 Lyapunov.Krasovskii approach An application of the Lyapunov-Krasovskii functionals also provides some sufficient stability independent of delay conditions in terms of positive definite solutions of specific matrix inequalities. To see that one can start with the following functional 0
u = xT(t)Px(t) + / xr(t + O)Qz(t + O)dO, -h
where weight matrices P and Q are positive definite. Direct calculations show that the derivative of the functional along the trajectories of the system (1) is the quadratic form:
3.3 Lyapunov-Razumikhin approach Some sufficient conditions for stability independent of delay may be obtained by means of the Lyapunov-l=tazumikhin approach. Let
v=
02)
be a positive definite quadratic form, P > 0. Then the derivative of the function along the trajectories of the system (1) has the following form
dd
--v
= xT(t)(ATp + PA)x(t)
(1)
+2xT(t)PAlx(t - h). The mixed term of the derivative may be estimated as follows
d ] (ATP+PAPtl) --Vdt (1) = zT(t) A~P -
z(t),
where zT (t) -- (xT (t), xT (t -- h) ). It is easy to see that the derivative is negative definite if
_Q
< 0.
Having in mind that Q > 0 one can easily transform this condition to the matrix Riccati inequality for P
AT P + P A + PA1Q-1AT p + Q < O. It is worth to be mentioned that the last matrix inequality coincides with that obtained via Lyapunov-Razumikhin approach if one sets Q = P. 4. DELAY DEPENDENT ROBUST
2xT (t)PAlx(t - h) <_xT (t -- h)Sx(t - h) +z T (t)PAx S - 1A[ Px(t). This inequality holds for arbitrary positive definite matrix S, and in particular for S : P. Applying the Razumikhin condition with q(s) : (l+s)s, where ~ > 0, one obtains
xT (t - h)Px(t - h) < (1 + ~)xT (t)Px(t),
STABILITY CONDITIONS 4.1 Distributed delay systems 4.1.1. Stability radius approach The stability radius concept may be extended to the case of a distributed delay system
d (t) dt
=
/
0
[d (f(O) + BA(O)) C] x(t + 0),
-h
SO
dI
--v < xT (t)(AT p + P A + ( 1 T ~)P dt (1) +PA1P-1ATp)x(t). Therefore the derivative is negative definite if
A T p + P A + P + P A 1 P - 1 A T p < O. Pre and post multiplying the le.st inequality by R = P-1 one may transform it to the equivalent linear one
RA T + A R + R + A1RA T <0.
where F(O) is a given matrix function whose elements have bounded variations on [-h, 0], B, C are known matrices, and A(0) is an unknown perturbation matrix satisfying condition
/
0
lidA(O)ll <_ r.
-h
Assume that the unperturbed system (4) is stable. Then the perturbed system remains stable if 1 r<
V. L. Kharitonov /Annual Reviews in Control 23 (1999) 185-196
192
4.1.2. Lyapunov-I(rasovskii approach An application of the Lyapunov-Krasovskii approach for a special class of distributed delay systems has been done in (Verriest, 1994). Consider the following time delay system o
dx(t)dt - Ax(t) + / A ( O ) x ( t + O)dO, (13)
+p f
+ a(o)n-'(o)ar(0)] dOP,
-h
then the system (13) is stable.
Theorem 18. The perturbed system (13) is ~robustly stable if AT p + PA + E + P@P < O.
-h
where A is a stable matrix and A(0) is an unknown matrix valued function with square integrable on I-h, 0] elements. In the following it is assumed that A(0) is such that there is a positive scalar function/?(.) satisfying the condition 0
f 3(0)d0 = 1, -h
such that 0
/ Z-l(O)/,(O)Ar(O)dO<_¢,
o
(14)
4.2 Fixed delay systems 4.2.1. Quasipolynomial families In this section we consider stability analysis of a parametrically perturbed time delay system. Consider the system (3) whose matrices, A, A1, ..., AN, and delays, hi, h2,..., hN, depend continuously on uncertain parameters q = (ql,q2, ...,qt). It is clear that the coefficients akj and exponent coefficients 1-j of the characteristic quasipolynomial (5) of this system are continuous real valued functions of these parameters n-1
m
f(s, q) = she rocq)' + E E akj(q)sk er~Cq)'"
-h
k=Oj=O
where ¢ is a given positive definite matrix. The system (13) is called (I)-robustly stable if it is stable for all A(0) satisfying condition (14) The Lyapunov-Krasovskii functional 0
xT (t)Px(t) + [ xT (t + O)Q(O)x(t + 0)d0,
u
. 2
-h
has been proposed. It is assumed here that P is a positive definite matrix. Let the weight matrix Q(O) is defined as 0
-h
where R(~) > 0 for all ~ E I-h, 0]. By construction
dQ(O) dO = R(O) > 0, and Q(-h) = O.
That means that the Lyapunov-Krasovskii functional is positive definite. Its derivative along the trajectories of (13) satisfies the inequality --v
dd
(131
~_ x T (t) [ A T p + P A +
The following statement is known as the zero exclusion principle.
Proposition 19. All quasipolynomialsfrom F have no zeros with nonnegative real parts if and only if (1) At least one quasipolynomial from F is stable; (2) The value set of the family does not include the origin
O ~ VF(iw) = { f(iw) I f e F } for all w > 0.
o
+
v0(q) _> vj(q) for all q E Q and for all j = 1,2, ...,m.
Q(e) = /
Q(O) > O,
Assume that q E Q c R :, where the uncertainty bounding set Q is a given compact pathwise connected set, one obtains the family of quasipolynomials F = { f(s,q) I qeQ}. Stability of the uncertain system is now equivalent to the statement that all the zeros of quasipolynomials from the family F have negative real parts. In order to prevent a possible lost of the principal term the following necessary condition should be imposed
[ [R(e)+ PA(O)R-I(e)AT(o)P] de]x(t). -h
Hence, when there are a positive definite matrix P > 0 and a positive definite matrix valued function R(0) > 0, for all 0 E [-h, 0], such that 0 > ATp + PA+
(15)
Remark 20. One can replace the second condition by a slightly weaker one • the condition 2 holds at least for one we >_ 0, and the boundary of the value set does not contain the origin:
0 ~ OVF(iw) for all w > O.
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1I. L. Kharitonov / A n n u a l Reviews in Control 23 (1999) 185-196
Assume that there is no uncertainty in delays and that the coefficient uncertainty is specified by a convex polytope
jCk
where aj = 1,2,3,4; and s =
E"5"
. j > 0,
j=l
= 1
,
where points aj, j = 1,2, ...,/, correspond to the vertices of the polytope. The polytope defines the corresponding polytopic family of quasipolynomials
j=l
The following theorem has been proven in (Fu, et all, 1989). Theorem 21. All quasipolynomials of the polytopic family are stable (have no zeros in the closed right half plane of the complex plane) if and only if all quasipolynomials from the edge subfamilies are stable. This theorem shows that for the polytopic class of quasipolynomial families stability analysis is equivalent to stability check of a finite number of simple one parameter subfamilies corresponding to edges. But it is also true that the number of such subfamilies growth exponentially with n and m. In order to get additional reduction one may try to apply the following statement from (Kharitonov and Zhabko, 1994). Theorem 22. A s s n m e that /5(s) and fk(~) are stable, then if the difference g(s) = fj(s) - fk(s) satisfies the condition
d
d~arg(g(i~))---
vo + Vm _
_
2
°
+
sin(arg(g(iw))-- (r0 + r,,)w) + 2w ' for all w > O, where 9(iw) 7£ O, then all convex combinations ttfj(s) + (1 - # ) f k ( s ) # E [0, 1], are also stable quasipolynomials. Let a quasipolynomial family has an interval structure i = P0(s)~ T0s + P , ( s ) ~ , ~
Theorem 23. The interval quasipolynomial I is stable if and only if those of the (m + 1)4m+l one parametric subfamilies for which v k > - -r o + r m 2 are stable.
j=l
Here quasipolynomial J'j(s) corresponds to the vertex coefficient vector aj.
-
(~,/3) = (1,2), (2, 3), (3, 4), (4, 1).
j=l
+ ... + Pm(~)~ ~=~,
where Pj(s), j = O, 1, ..., m are interval polynomials. With every one of them one may associate four polynomials pll)(s), p~2)(s), p~3)(s), p~4)(s) as in (Kharitonov, 1978). The first reduction is based on the value set analysis: There are (m + 1)4 m+l one parameter families of the form
See (Kharitonov and Zhabko, 1994). Another examples of robust stability conditions for quasipolynomial families may be found in (Hoeherman, at all, 1995). 4.2.2. Lyapunov-Krasovskii approach As it was already discussed above by means of LyapunovKrasovskii or Lyapunov-Ra~umikhin approaches one can obtain a number of sufficient stability conditions. These conditions may be e x p r e ~ d in two closely related forms: In the form of nonlinear matrix inequalities or in the form of linear (quasilinear) matrix inequalities. Let us start with the first form. This approach has been actively developed in (Li and Souza, 1995;1997). They have treated the single delay case, but all results can be easily extended to the general multiple delay case. Consider the following uncertain system dr(t) = (A + AA) z(t)+ dt
(16)
+ (A1 + AA1) z ( t - h), where AA and AA1 are unknown norm bounded matrices of the form AA = DFB,
AA1 = DIF1B1.
(17)
Here D, B, D1 and B1 are given constant matrices which describe uncertainty structure, while the uncertain matrices F and F1 satisfy the following constrains
IIFII _< 1,
IIF~II _< 1.
(18)
Remark 24. It was assumed in the original papers (Li and Souza, 1995;1997) that F and F1 are time varying uncertain matrices, but this does not affect the following stability condition. When one is looking for stability conditions which depend on delay the standard step is to replace the system (16) by the following one
194
V. L. Kharitonov / A n n u a l Reviews in Control 23 (1999) 185-196
d((t) = ( A + A A + A ~ + A A J ( ( t ) dt
(19)
0
f[(A
- (A~ + AA~)
+ A A ) ( ( t + 0)+
Theorem 26. Let matrix (A + A1) be stable. For a given delay value h, the uncertain system (16)(18) is stable if there exist X > 0 and vector = ( a l , ..., as) with positive elements solving the following LMI Q(X) + 5 J 1 D X/~ T X M
-h
(A~ + AA~)~(t
BX MT x
h + e)]dO.
-
Let = C(t)P
(0 +
NT
where 0
g
-J1 ,,0 0 -1.12 0
0
)
0 0
<0
- h J3
where
0
Q(X) = X (A + A1) 7c+ (A + A1) X, -h 0
D = (D, D1), ~7" __ (BT,B~), N = (A1,DJ, M T = (AT,BT,A~,BT),
0
0
J1 = diag (alE, a~E), - h O--h
J2 = diag(aaE- a4DD T,
The term w(~) is introduced in order to be exploited later for cancellation of integral terms in an estimation of the time derivative of the functional. After such a cancellation
J3 = diag(E - a6B~B1, a6E), Slightly different conditions have been obtained in (Niculescu, et all, 1994).
dv(&) (tg) <- ~T(t)S(P' e, h)~(t), where
Theorem 27. Let the matrix (A + A J be stable and D1 = 0, B1 = 0 (no uncertainty on the matrix A1). For a given value h > 0 the uncertain system (16)-(18) is stable if there exist matrix Y > 0 and positive values e, fl, f l satisfying the following LMI
S(P,e,h) = (A + A~) r P + P (A + A J + +P [¢~DDT + h A t ( E - grBT B~)-~A[
¢3¢7
a4E, (1 - aa)E - ahD1D~, othE).
J
Y
hg3g4 4T(E _ e~DDT)_~A+
h~TATY
E2
+hs3(1 + &,)AT (E - e~D1DT)-~AI+
+
hg3"~4
BTB
(20)
E > e~DDT,E > cGD1D~',E > eTBTBI(21) 4.2.3. Sufficient conditions in the LMI form the new variables X = ~3 P - l ,
In
g4
1+e4' C4E5
~4
--
1 + c4
~6 ,
0'5
--
-
-
~
1 + ~..~
(~6
=
0 I"
<0,
O= (A + A1)Ty + Y(A + A1) + BT B h + -gD O T + h(f + fit)Y,
Theorem 25. Let matrix (A + A1) be stable. For a given delay vaJue h, the uncertain system (16)(18) is sta.ble if there exist P > 0 and vector e = (el, ...,~7) with positive elements satisfying the inequalities S(P, e,h) > 0
O0
0 - h0~0
where
+ h¢3(1 -I- 64) BTB~" e~
(1 + ~4) e~
-E
C7~
the theorem conditions may be rewritten in the LMI form.
t~ =
flY 0 0 Y - ¢B TB
,
and (I) = (A, A1, DA),
F = eBTB - Y.
Another employment of the LMI approach has been proposed in (Gu, 1997). Here the LyapunovKrasovskii functional (7) is considered. The principal idea of the procedure proposed in (Gu, 1997) is based on the observation that if the matrix valued functions Q(O), R(8), S(01,0~) are assumed to be continuous and piecewise linear, then both d the functional v(xt) and its derivative 77v1(16) may be expressed as some quadratic forms whose
V. L. Kharitonov / A n n u a l Reviews in Control 23 (1999) 185-196
matrices depend linearly on the values of the matrices Q(O), R(O), S(01,02) at corresponding griding points. This observation opens a door to exploiting powerful interior point algorithms. 4.3 Robust stability independent of delay 4.3.1. Interval quasipolynomial Robust stability of an interval quasipolynomial
I = s" + ~
tiff
~k,-dk] s k +
[bkj,-bkj] ske ~j~,
j=l k=0
k=0
with rj < 0, j = 1, 2, ..., m, independent of exponential coefficients was addressed first in (Kim and Bose, 1990) where some sufficient conditions were obtained. Later this problem has been considered in (Olbrot and Igwe, 1995), where the following necessary and sufficient were proposed.
Theorem 28. The interval quasipolynomial I is stable independent of exponential coefficients if and only if
(1) (2)
max(Ib-. l, Ib-Jl) < 1;
max(lb-0j[, I 0jl) -< min ([a-0[, I~01) ;
(3) All 4 "~+1 extreme quasipolynomials.
195
5. CONCLUDING REMARKS It should be mentioned explicitly that this survey covers only a small part of the large number of contributions in this area. For more complete reference lists the readers are advised to see recently published survey papers (Niculescu, et all, 1997) and (Richard, 1998). In order to make this survey self-contained the author deliberately concentrates on the class of linear time invariant delay systems. On the other hand this allows to study principle features of the basic methods for stability and robust stability analysis with minimum technical details.
6. ACKNOWLEDGEMENTS The author would like to thank CNRS and IRCyN France for the priveledge to be invited for a working visit to the Ecole Centrale de Nantes. This paper has been prepared during this visit. The author would also like to thank Dr. Silviuhlian Niculescu for helpful discussion of the subject.
77~
j=0
where (~j -- 1, 2, 3, 4, are stable independent of exponent coefficients U < 0. 4.3.2. Lyapunov-Krasovskii approach Consider again the system (16)-(17), where the uncertain matrices F, Ft satisfy the constrains
F F T < E and F1F1T < E.
(22)
Some conditions for stability independent of delay have been obtained in (Lisong, 1996) with the following Lyapunov-Krasovskii functional 0
v(xt) = x'r(t)Pz(t) +
/
xT(t + O)Sx(t + O)dO.
-- h
Here it is assumed that P > 0 and S > O.
Theorem 29. Let matrix A be stable. The system (16)-(17) with uncertainty constrains (22) is stable independent of delay if there exist matrices P > 0, S > 0 and positive constants e, el such that 0 > A T p + P A + S + 1-BTB + e P D D T P 6
+ P A l ( S - - ¢ I B T B1) - 1 A T p + 1pDIDT1 £1
and
0 < S - E1BTB1.
7. REFERENCES [1]Bellman, R.E. and K.L. Cooke, (1963), Differential-Difference Equations, Academic Press, New York. [2] Chen, J., D. Xu and B. Shafai, (1995), On sufficient conditions for stability independent of delay, IEEE Trans. on Automatic Control,
40, 1676-1680. [3] Chen J., and H.A. Latchman, (1995), Frequency sweeping tests for stability independent of delay, IEEE Trans. on Automatic Control, 40, 1640-1645 [4] Fu, M., A.W. Olbrot and M.P. Polls, (1989), Robust stability of time delay systems: The edge theorem and graphical tests, IEEE Trans. on Automatic Control, 34, 813-820. [5] Gu K., (1997), Discretized LMI set in the stability problem of linear uncertain timedelay systems, International Journal of Control, 68,923-934. [6] Hale, J., (1977), Theory of Functional Differential Equations. Springer-Verlag, New York. [7] Hinrichsen, D. and A.J. Pritchard, (1992), Destabilization by output feedback, Differential and Integral Equations, 5, 357-386. [8] Hocherman, J., J. Kogan and E. Zeheb, (1995), On exponential stability of linear systems and Hurwitz stability of characteristic quasipolynomials, Systems and Control Letters, 25, 1-7.
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V. L. Kharitonov /Annual Reviews in Control 23 (1999) 185-196
[9] Kabakov, I.P., (1946), Concerning the control process for the steam preasure, Inzhenernii Sbornik, 2, 27-76. [10] Kamen, E.W., (1982), Lineal" systems with commensurate time-delays: Stability and stabilization independent of delay, IEEE Trans. on Automatic Control, 27, 367-375. [11] Kharitonov, V.L., (1978), Asymptotic stability of an equilibrium position of a family of systems of linear differential equations, Differentsial 'nie Uravneniya, 14, 2086-2088. [12] Kharitonov, V.L. and A.P. Zhabko, (1994), Robust stability of time-delay systems, IEEE Trans. on Automatic Control, 39, 2388-2397. [13] Kim K.D. and N.K. Bose, (1990), Vertex implications of stabilityfor a class of delaydifferential interval system, IEEE Trans. on Circuits and Systems, 37, 969-972. [14] golmanovskii, V.B., (1995), Stability of some nonlinear functional differential equations, Journal of Nonlinear Differential Equations and Applications, no. 2, 185-198. [15]Kolmanovskii, V.B. and A.D. Myshkis, (1992), Applied Theory of Functional Differential Equations. Kluwer, Dordrecht. [16] Krasovskii, N.N., (1956), On the applications of the second Lyapunov method for equations with Delay, Journal of Appl. Math. Mech., 20, 315-327. [17] Li, X. and C.E. de Souza, (1995), LMI approach to delay-dependent robust stability of uncertain linear systems, Proc. of the 34th CDC, New Orleans, 3614-3619. [18] Li, X. and C.E. de Souza, (1997), Delay dependent robust stability and stabilization of uncertain linear delay systems: A linear Matrix Inequality Approach, IEEE Trans. on Automatic Control,42, 1144-1148. [19] Lisong Yuan, (1996), Robust analysis and synthesis of linear time-delay systems with norm-bounded time-varying uncertainty, Systems and Control Letters, 28,281-289. [20] Mori, T., and H. Kokame, (1989), Stability of d~(t) = Ax(t) + Bx(t - r), IEEE Trans. on dt Automatic Control, 34, 460-462. [21] Niculescu, S.-I., C.E. de Souza, L. Dugard and J.-M. Dion, (1994), Robust stability and stabilization of uncertain linear systems with state delay: Single delay case (1), Proc. of the IFA C Syrup. on Robust ContTvl Design, Rio de Janeiro, Brasil, 469-474. [22] Niculescu, S.-I., E.I. Verriest, L. Dugard and J.-M. Dion, (1997), Stability of linear systems with delayed state: A guided tour, in Stability and Control of Time-Delay Systems (L. Dugard and E.I. Verriest, Eds), Lecture Notes in Control and Information Sciences, Springer-Verlag, London, 228, 1-71. [23] Oil)rot A.W.a.nd C.IJ.T. lgwe, (1995), Necessary and sufficient conditions for robust sta-
bility independent of delays and coefficient perturbations, Proc. of the 34th CDC, New Orleans, 392-394. [24] Pontryagin, L.S., (1942), On the zeros of some elementary transcendental functions, Izvestiya Akademii Nauk SSSR, 6, no. 3, 115134. [25] Richard, J.-P., (1998), Some trends and tools for the study of time delay systems. Proc. of the IMA CS-IEEE Multiconference, CESA '98, Hammamet, Tunisia, 27-43. [26] Razumikhin B.S., (1960), Application of Lyapunov method to problems in stability of systems with delay, Automaticai Telemehanika, 21,740-749. [27] Tchebotareff, N.G. and N.N. Meiman, 1949, The Routh-Hurwitz Problem for Polynomials and Entire Functions, In Proc. of Mathematical Institute of the Acad. Sci. USSR. [28] Townley, S. and A.J. Pritchard,-(1991), On problems of robust stability for uncertain systems with time-delay, Proc. of the 1st ECC, Grenoble, France, 2078-2083. [29] Toker, O. and H. Ozbay, (1996), Complexity issues in robust stability of linear delaydifferential systems, Math. in Contr., and Sygnals, Syst., 9,386-400. [30] Tsypkin, Ya.Z., (1946), Stability of systems with delayed feedback, Automation and Remote Control, 7, no. 2, 107-129. [31] Verriest E.I., (1994), Robust stability of time varying systems with unknown bounded delays, Proe. of the 33rd CDC, Lake Buena Vista, Florida, 417-422. [32] Verriest E.I., M.K.H. Fan and J. Kullstam, (1993), Frecuency domain robust stability criteria for linear delay systems, Proc. of the 32nd CDC, San Antonio, Texas, 3473-3478.
PERGAMON
Annual Reviews in Control 23 (1999) 185-196
ROBUST
STABILITY ANALYSIS OF TIME DELAY SYSTEMS: A SURVEY
Vladimir L. Kharitonov
Automatic Control Section CINVESTAV-IPN P.O. BOX 14-7~0 07300 Mexico, D.F.
Abstract: In this survey some recent contributions to stability and robust stability analysis of linear time delay systems with parameter uncertainty are discussed. The main aim of the paper is to discuss some basic techniques used for deriving tractable stability and robust stability conditions. The reference list presents rather a small portion of the large number of publications in this field. Rdsume: On discute quelques contributions rdcentes a l'analyse de la estabilitd et de la estabilitd robuste des systemes lindares a retards a paramdtres incertaius. L'espirit de article est de prdsenter l'essential des techniques de base que sout utilisdes pour obtenir des conditions vdrifiables de stabilitd et de stabilitd robuste. Les rdfdrences listdes ne reprdsenteut qu'une petite pattie du gran nombre des publications dans ce domaine. Keywords: delay system; stability; robust stability
1: BASIC DEFINITIONS
mapping the delay interval [ - h , 0] into R '~ with the standard uniform norm
1.1 Systems and solutions In this section some classes of time delay systems will be introduced as well as necessary notations and basic definitions will be given. The main attention is paid to the case of linear time invariant systems. An example of such systems is the single delay system of the form
dz(t) _ Ax(t) + Alx(t - h), dt
(1)
I1¢(.)11= ee[-h,o] max {11¢(0)11}. For given initial conditions of the form (2) the system (1) admits the unique solution x(t) = x(t, to, ¢(.)), which is defined on [to - h, c¢). A natural extension of the class of single delay systems is the class of systems with multiple delays of the form N
where A and A1 are constant n x n matrices, h is a nonnegative delay. In order to define a particular solution of the system one has first to set initial conditions which include a time instant to and an initial vector function ¢(.)
(to + 0) = ¢(0), fo . all 0 e I-h, 0].
(2)
Here ¢(.) is an element of the Banach space C ( [ - h , 0 ] , R n) of contimlous vector functions
dx(t) = Ax(t) + ~ dt
Akx(t - h~),
(3)
k=l
where all delays are nonnegative real values~ In order to define a particular solution of the system (3) one has to specify an initial vector function ¢(.) on the segment [ - h , 0], where h = maxl
1367-5788/99/$20© 1999Publishedby ElsevierScienceLtd on behalfof the InternationalFederationof AutomaticControl.All rights reserved. PII: S1367-5788(99)00021-8
V. L. Kharitonov / A n n u a l Reviews in Control 23 (1999) 185-196
186
0
dx(t) dt = f [dF(O)]x(t + 0).
(4)
-h
It is assumed here that all elements of the matrix valued function F(.) have bounded variations on [ - h , 0]
of delay free systems is a description of delay elements, scalar h in the case of systems (1), (4), or delays hi, h2, ..., hN in the case of system (3). Assume that a time delay system is given along with particular uncertainty descriptions :D in delay elements, and S in system matrices, then one can define the following notion of robust stability.
0
f IlaF(0)ll< -h
The single delay system and the multiple delay system are particular cases of the distributed delay system of the form (4) and can be obtained from it with a special choice of piecewise constant system matrix F(-). Initial vector function ¢(.) for a particular solution of the system (4) must be specified on the whole integration segment [ - h , 0].
1.2 Stability There is no difference in stability definitions for systems (1)-(4), so for definiteness they will be given for the system (4) only.
Definition 1. The system (4) is said to be Lyapunov stable if for every positive e there exists a positive 5 such that if I1¢(')11 < 5, then IIz(t,t0,¢(.))ll < c for all t > to.
Remark 2. If the value 5 from the above definition may be chosen such that in addition x(t, to, ¢(.)) --~ 0 when t --~ c~, then the system (4) is said to be Lyapunov asymptotic stable. Definition 3. The system (4) is said to be exponential stable if there exist positive constants #, a such that every solution of the system satisfy the following inequality
IIz(t, to,¢(.))ll to. It is worth to be mentioned now that for the system (4) the exponential stability and the Lyapunov asymptotic stability are equivalent. In the sequel the word "stability" will be used instead of "exponential stability".
1.3 Robust stability
Definition 4. Given a time delay system with uncertainty descriptions :D, S. The system is said to be robustly stable if it is stable for all delay elements from 79 and for all system matrices from S. The case when each of the sets :D and ~g contains just one element corresponds to uncertainty free system. It should be mentioned here that the stability analysis of systems with uncertainty in delay elements is, in general, a more complicated task than that of systems with uncertainty in system matrices (Toker and Ozbay, 1996).
2. STABILITY CONDITIONS In this section some basic stability conditions will be presented.
2.1 Frequency domain conditions Having in mind the fact that the main concern of the paper is about the class of linear time invariant systems one has to address first to stability conditions based on the zero analysis of the system characteristic function,see (Hale, 1977).
2.1.1. Distributed delay case Theorem 5. The system (4) is stable if and only if its characteristic function f(s) = det
(]) sE-
e°'dF(O)
-h
has no zeroes with nonnegative real parts. The characteristic function may be written in the polynomial form r*--I
There are two different places where uncertainty may affect a time delay system. First, as in the case of delay free systems, uncertainty may appear in a description of the system matrices, A, A1,..., AN for the cases of the systems (1), (3) or in that of the matrix valued function F(-), in the case of the system (4). The second source of uncertainty which does not exist in the case
f(s) = s" +
k(s)s k, k---0
where coefficient functions are bounded in the closed right half plane of the complex plane. Applying the argument principle to f(s) one may arrive to the following result which is an extension of the Mikhailov polynomial stability criterion (Kabakov, 1946).
V. L. Kharitonov / A n n u a l Reviews in Control 23 (1999) 185-196
Theorem 6. Let f(s) have no zeros on the imaginary axis. It has no zeros with positive real parts if and only if the net change of the argument function satisfies the condition ~z.
•
2.1.2. Pontryagin criterion In the case of systems (1), (3) this characteristic function is a finite sum of power and exponent terms n--1 rn
~2 . . , a k ~ s k=Oj=O
~ e ~"
,
where the exponent coefficients 0 = 7"0 > 1"1 > •.. > Vra > - c o are linear combinations of delays with nonnegative entire coefficients. Function f(s) may be written as a sum of polynomials multiplied by exponents
y(s) = p0(s)e ~o' + .
direct method. There are two different ideas how one can apply this method for time delay systems. In the first one the state of a delay system is defined as the time depending trajectory segment
7r
A a r g ( y ( ~ w ) )oo =
f ( s ) = s" + ~
187
+ p~(s)~ ~-',
(~)
where deg (P0) = n > deg (pj) for j = 1, 2, ..., m. Any function of the form (5) is called quasipolynomial. The zero set of f(s) is not changed after multiplication of f(s) by an arbitrary exponent er" . T h a t means that in the zero analysis one can ignore assumption that all exponent coefficients are nonnegative. Let us define two real functions u(w) and v(w) as the real and the imaginary parts
of f(i~), y(i~) = u(~) + ~v(~).
• , = { ~(t + o) I 0 e I-h, 0] }. Here instead of the classical Lyapunov functions it is proposed to use Lyapunov functionals. This approach has been developed in (Krasovskii, 1956).
The second idea is based on classical Lyapunov functions and uses a special estimation procedure which allows exclude delay states in the derivative of the Lyapunov functions. This procedure has been proposed in (Ra~umikhin, 1960). 2.2.1• Lyapunov-Krasovskii approach
Theorem 9. The time delay system (4) is exponential stable if and only if there exists a continuous functional v(xt) such that (1) 4111~(t)ll ~ < v(~,) < 4~ II~tll ~ , for some positive 41, 42; (2) ~v(xt)l(4 ) _< - 3 IIx(t)ll 2, for a positive constant 3. The direct application of this theorem is seriously complicated by difficulties arising in construction of such functionals. In general one has to look for a functional of the form
v(~t) = ~T(t)P~(t)+
The following theorem has been proven first in (Pontryagin, 1942) for the case of entire exponent coefficients, and then in (Tchebotarev, N.G. and N.N. Meiman, 1949) for the general case.
Theorem 7. Given a quasipolynomial (5) with r0+vm _> 0. All zeros of the function have negative real parts if and only if functions u(w) and v(w) have only real, simple, interlacing zeros, and at least for one real
(7)
0 -t-2xT (t)
/ Q(O)x(t + e)dO+ -h
0
+ / xT(, + O)n(o)x(~ + O)dO+ -h 0
0
+ f f T(t + el)S(01,02)( +o,)dOldO2, -h -h
where weight matrices P, Q(0), R(0) and S(01, e2)
Remark 8. If f(s) has no zeros with nonnegative real parts and T0 + rm >__ 0 then the above inequality (6) holds for all real w. It means that arg(f(iw)) is a strictly increasing function of w. For brevity f(s) is said to be stable if it has no zeros in the closed right half plane of the complex plane.
2.2 Lyapunov direct method A more general approach to stability analysis of time delay systems is based on the Lyapunov's
satisfy a system of algebraic, ordinary and partial differential equations• A solution construction for this system and the following analysis of its positivity is a very difficult task. Therefore the theorem is usually used for deriving sufficient conditions when one tries different quadratic functionals of particular types. This approach is known as the Lyapunov-Krasovskii functional approach. 2.2.2. Lyapunov-Razumikhin approach An attempt to rescue the idea to use classical Lyapunov functions for stability analysis of time delay systems has been successfully launched in (Razumikhin, 1960). The main problem here is that
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V. L. Kharitonov / A n n u a l Reviews in Control 23 (1999) 185-196
the time derivative of a Lyapunov function depends not only on the present state x(t) of the system but also on past states. The key point of the Lyapunov-Razumikhin approach is a special procedure how one can obtain an upper bound of the time derivative which does not depend of these past states.
where P is a positive definite matrix, P > 0. Then
Theorem 10. The system (3) is exponential stable if there exists a positive definite function v(x) such that
The integral terms may be estimated as follows
-d- v I = ~T (t) [(A + A1)TP + P(A + A1)] ~ ( t ) dt (8) o
-2~T (t)PA1 f [A~(t + 0) + Ax~(t - h + 0)] dO. -h
o
-2 T(t)PA1 f A (t + 0)d0 _<
(1) Ul(llxll) _< v(z) _< u2(llxll) for some continuous nondecreasing functions, ul(s) > O, u2(s) > 0, for s > 0 and ul(0) = u2(0) = 0; (2) _< -w(llxll) for a continuous nondecreasing function w(s), where the derivative is evaluated for all vector functions satisfying the condition
-h 0
< ~T (t)PAIA / Y - 1(O)dOATA~e~(t)+ -h 0
+
v(x(t + 0)) < p(v(x(t))), for all 0 E [-h, 0]. Here p(s) is a continuous function such that p(s) > s, for s > 0.
f
~T(t q- 0 ) Y ( 0 ) ~ ( t -~- 0)d0,
-h
and o
-2~T (t)PA1 / Al~(t - h + O)da < 2.2.3. System transformation For simplicity the single delay system (1) is treated. It may be written as
0
< ~T(t)PA~ f Y~'X(O)dO(A~)T P~(t)+
0
-h 0
-h
If now one changes differencial under the integral using the right hand side of the equation (1) and replaces the state variable x with the new one, ~, then one comes to the system (8)
+ / ~T(t -- h + O)YI(O)~(t - h + O)dO, -h
where Y(O) > 0 and YI(O) > 0 for all 0 E [-h,0] are arbitrary positive definite matrices with continuous elements. Now the derivative can be estimated as
--v Jt
0
/ [A~(t + e) + Al~(t - h + 0)1 dO. ,J
-h
It is easy to observe that the system (8) belongs to the class distributed delay system and that the time lag for the system is twice larger than that for the original system (1). But on the Other hand every solution of the original system is also a solution of the latter one. It means that stability of the system (8) implies that of the system (1). There are also other possibilities how one may transform the original system in order to include explicitly delay elements in the system description, see (Kolmanovskii, 1995). 2.2.4. Lyapunov-Razumikhin approach Consider a Lyapunov-Razumikhin function candidate
v = ~T(t)P~(t),
(10)
-h
d (t) = (A + A1)x(t) - A1 / dx(t + 0). dt
d~(t) = (A + A1)~(t)dt
(9)
(s)
< ~T(t)[(A + A1)Tp + P(A + A1)+ o
+PAIA f Y-I(O)dOAT ATP+ -h 0
+PA~ / Y{'l(O)dO (A~)w PlY(t)+ -h 0
+ f [ r(t +
+ o)+
-h
+ ~T(t -- h + O)Yl(O)~(t - h + 0)] dO. If one sets that in the R ~ u m i k h i n theorem q(s) = (1 + e)s, with e > 0, then terms with delay states may be estimated as follows
~T (t q- O)P~(t +0) <_(1 + e)~T (t)P~(t). The form of this estimation makes clear why in practice there is no big choice in selection of the weight matrices Y(O) and YI(O). T h e simplest
V. L. K h a r i t o n o v / A n n u a l R e v i e w s in C o n t r o l 23 (1999) 1 8 5 - 1 9 6
case is just to assume that Y(O) = tiP, and YI(0) = fliP for some positive constants fl, fil. In this case the derivative admits the following quadratic estimation
-~v
If in addition S(-h) = &(0), then for arbitrary positive definite Sl (-h) = R 0
& (o) = R + f ~'~(O)dO,
<_~T(t) [(A + AoTp + P(A + A1)+ (s)
189
-h
and
+
g
0
IAr,+
S(O) =.R + f (Y(O) + Y1(0)) dO.
+h(1 + ¢)(fl + ill)P] ~(t).
Theorem I1. Assume that the matrix (A + A1) has no eigenvalues in the closed right half plane of the complex plane. The system (8), and henceforth the system (1) is stable if there exist matrix P > 0 and positive values fl, j31 such that
0 > (A + A1)TP + P(A + A1) + h(fl + flOP+ 1 -1 A1)A T T1 P. +hPAl( AP-1A T + fl-~A1P
-h
After these calculation one arrives to the following upper bound for the time derivative
< ~T(0 [(A + A O r e + P(A + AO+
-~
dt
(8)-
0
0
+R + J ,(O)dO + J (O)dO+ -h
-h
+PA1A f Y-I(O)dOAT AT1P+ 2.2.5. Lyapunov-Krasovskii approach Try now a Lyapunov-Krasovskii functional candidate
-h 0
+PA~ f Y~'l(O)dO (A~)T P]~(t)-
o
J
-h
v(~t) = {T (t)P~(t) + f ~T (t + e)s(e)~(t + e)dO
_~T (t
-h 0
+ f ~T(t -- h + O)&(O)~(t - h + O)dO, -h
where P > O, S(O) and SI(0) are positive definite matrices with continuously differentiable elements. The time derivative of the functional along the trajectories of the system (8)
--v = ~T(t) [(A + A1)TP + P(A + A1)+ dt (s) + S(O)] ~(t)+ C(t - h)[&(O) - S(-h)] ~(t - h ) o
_f
+ O)@0
+ 0)d0--
-h
-
2h)/~(t
-
2h).
Theorem 12. Let the matrix (A + A1) have no eigenvalues with nonnegative real parts. The system (8), and therefore the system (1), is stable if there exist positive definite matrix P and positive definite matrix valued functions S(0), $1 (0) such that 0
0 > (A + AoTp + P(A+ A1) + f Yl(0)d0 -h 0
0
+ J Y(O)dO+ PA1A J Y-'(O)dOAT ArP -h
-h 0
+PA~ f Yi-l(O)dO (A~)T P.
0
-2(T(t)PA1 f [A~(t + O) + Al~(t - h + 0)] de-h
-~r(t - 2h)&(-h)~(t - 2h)0
-h
Remark 13. It is clear that the last condition coincides with that of the previous theorem if one sets Y(0) = fiR and Y1(0) = /~IP. But it is no longer the compulsory choice here.
-h
The mixed integral terms admit estimation (9)(10), where Y(O) > 0 and Yt(0) > 0 are two matrix valued functions with continuous elements. In order to cancel all integral terms one has to assume that dS(O) = Y(O) and d&(O) - I"1(0).
dO
dO
3. STABILITY INDEPENDENT OF DELAY In this section some basic results on a specific robust stability notion for time delay systems, namely stability independent of delay are discussed. Probably the first step in analysis of this type of stability has been made in (Tsypkin, 1946)
V. L. Kharitonov / A n n u a l Reviews in Control 23 (1999) 185-196
190
where it was proved that a quasipolynomial p(s) + q(s)e -~s, deg(p) > deg(q) has no zeros in the closed right h a l f plane of the complex plane for all nonnegative values of r if and only if p(s) is a stable polynomial and the inequality Ip(iw)l > Iq(iw)l, holds for all w E R.
See (Kamen, 1982).
Assume that for the time delay system (3) delays are nonnegative functions of some parameters
3.2 Norm based sufficient conditions
hk = hk(ql,q2,...,qt) k = 1, 2 , . . . , N where (ql, q2, ..., q~) E Q c R I. Definition 14. The system (3) is said to be stable independent of these parameters if it is stable for all (ql, q2, ..., ql) from Q. There are two special cases which attract a special attention. The first one is the case when delays depend on one parameter, l = 1, and hk = rkq,
k = 1,2,...,N.
Here rh > 0, are given and q is a positive free parameter. The change of the time scale t I = qt allows reduce the problem to the case of fixed delay. In the second case delays are independent parameters by themselves, i.e., l = N, and
• det(sE - A - zA1) # < 1.
0
for all s E C+and
For arbitrary induced matrix norm the spectral radius of a square matrix A does not exceed the norm of the matrix
p(A) <
IIAII.
Based on this fact one can immediately obtain the following sufficient condition, see for example (Chen, et all, 1995). Theorem 17. The time delay system (1) is stable independent of delay if the system matrix A is stable and [ [ ( i w E - A ) - I A I [ [ < 1 forallw_>O. (11) The last theorem is main source of a number of sufficient stability independent of delay conditions. Now we observe some of them.
Here Q = R ~ = [0, oo) × [0, oo) ×... × [0, o¢). This case corresponds to stability independent of delay.
3.2.1. Stability radius interpretation The theorem condition may be interpreted in the terms of the structural complex stability radius of a stable matrix A. It has been shown in (Hinrichsen and Pritchard, 1992) that a system
These two cases coincide for the single delay system (1).
dz - - = (A + B Z C ) x , dt
hk = qk k= l,2,...,N "
3.1 Stability independent of delay criterion The following statement is a slightly modified result from (Chen, et all, 1995) and (Chen and Latchman, 1995). Theorem 15. The system (1) is stable independent of delay if and only if • matrices A and (A + A1) are stable; • the spectral radius of the frequency dependent matrix (iwE - A ) - I A 1 is less than one for all positive w: p((iwE - A ) - I A 1 ) < 1,
for all w > 0.
Remark 16. Let matrix A be stable. The following two sufficient conditions for stability independent of delay are equivalent •
p((iwE - A)-IA1) < 1,
for all w _> 0;
with given matrices A, B, C and unknown norm bounded complex matrix Z, IIZI] < r, is stable if and only if
r < r ( A I B , C) =
i m a ~ e R (flC(i~E - A ) - ' B I D
(assuming by definition that r(A, B , C ) = oo when C (iw E - A )- I B = 0). The value r( A, B, C) is known as the structural complex stability radius of the matrix A. Assuming B = Ax and C = E one can detect that the theorem conditions is equivalent to the statement that r(A, A1, E) > 1. 3.2.2. H ~ approach For the case of g2 norm the theorem condition may be also interpreted as that the H ~ - n o r m of the stable transfer matrix G(s) = (sE - A ) - I A x is less than one. This observation opens a door for application of the H¢~ theory methods for stability analysis of time delay systems. A good example of a such application is (Verriest, et all, 1993).
V. L. Kharitonov / A n n u a l Reviews in Control 23 (1999) 185-196
3.2.3. Time domain approach If one looks at the condition (11) from the time domain point of view then, having in mind stability of the system matrix A, one can use the following identity co
(iwE - A)-IA1 =
f e-~%AtAldt. 0
From here, using the classical exponential estimation IIeA'Alll <- ke-"', where k and t, are positive constants, one can immediately conclude that the time delay system (1) is stable if k < ~.
191
3.4 Lyapunov.Krasovskii approach An application of the Lyapunov-Krasovskii functionals also provides some sufficient stability independent of delay conditions in terms of positive definite solutions of specific matrix inequalities. To see that one can start with the following functional 0
u = xT(t)Px(t) + / xr(t + O)Qz(t + O)dO, -h
where weight matrices P and Q are positive definite. Direct calculations show that the derivative of the functional along the trajectories of the system (1) is the quadratic form:
3.3 Lyapunov-Razumikhin approach Some sufficient conditions for stability independent of delay may be obtained by means of the Lyapunov-l=tazumikhin approach. Let
v=
02)
be a positive definite quadratic form, P > 0. Then the derivative of the function along the trajectories of the system (1) has the following form
dd
--v
= xT(t)(ATp + PA)x(t)
(1)
+2xT(t)PAlx(t - h). The mixed term of the derivative may be estimated as follows
d ] (ATP+PAPtl) --Vdt (1) = zT(t) A~P -
z(t),
where zT (t) -- (xT (t), xT (t -- h) ). It is easy to see that the derivative is negative definite if
_Q
< 0.
Having in mind that Q > 0 one can easily transform this condition to the matrix Riccati inequality for P
AT P + P A + PA1Q-1AT p + Q < O. It is worth to be mentioned that the last matrix inequality coincides with that obtained via Lyapunov-Razumikhin approach if one sets Q = P. 4. DELAY DEPENDENT ROBUST
2xT (t)PAlx(t - h) <_xT (t -- h)Sx(t - h) +z T (t)PAx S - 1A[ Px(t). This inequality holds for arbitrary positive definite matrix S, and in particular for S : P. Applying the Razumikhin condition with q(s) : (l+s)s, where ~ > 0, one obtains
xT (t - h)Px(t - h) < (1 + ~)xT (t)Px(t),
STABILITY CONDITIONS 4.1 Distributed delay systems 4.1.1. Stability radius approach The stability radius concept may be extended to the case of a distributed delay system
d (t) dt
=
/
0
[d (f(O) + BA(O)) C] x(t + 0),
-h
SO
dI
--v < xT (t)(AT p + P A + ( 1 T ~)P dt (1) +PA1P-1ATp)x(t). Therefore the derivative is negative definite if
A T p + P A + P + P A 1 P - 1 A T p < O. Pre and post multiplying the le.st inequality by R = P-1 one may transform it to the equivalent linear one
RA T + A R + R + A1RA T <0.
where F(O) is a given matrix function whose elements have bounded variations on [-h, 0], B, C are known matrices, and A(0) is an unknown perturbation matrix satisfying condition
/
0
lidA(O)ll <_ r.
-h
Assume that the unperturbed system (4) is stable. Then the perturbed system remains stable if 1 r<
V. L. Kharitonov /Annual Reviews in Control 23 (1999) 185-196
192
4.1.2. Lyapunov-I(rasovskii approach An application of the Lyapunov-Krasovskii approach for a special class of distributed delay systems has been done in (Verriest, 1994). Consider the following time delay system o
dx(t)dt - Ax(t) + / A ( O ) x ( t + O)dO, (13)
+p f
+ a(o)n-'(o)ar(0)] dOP,
-h
then the system (13) is stable.
Theorem 18. The perturbed system (13) is ~robustly stable if AT p + PA + E + P@P < O.
-h
where A is a stable matrix and A(0) is an unknown matrix valued function with square integrable on I-h, 0] elements. In the following it is assumed that A(0) is such that there is a positive scalar function/?(.) satisfying the condition 0
f 3(0)d0 = 1, -h
such that 0
/ Z-l(O)/,(O)Ar(O)dO<_¢,
o
(14)
4.2 Fixed delay systems 4.2.1. Quasipolynomial families In this section we consider stability analysis of a parametrically perturbed time delay system. Consider the system (3) whose matrices, A, A1, ..., AN, and delays, hi, h2,..., hN, depend continuously on uncertain parameters q = (ql,q2, ...,qt). It is clear that the coefficients akj and exponent coefficients 1-j of the characteristic quasipolynomial (5) of this system are continuous real valued functions of these parameters n-1
m
f(s, q) = she rocq)' + E E akj(q)sk er~Cq)'"
-h
k=Oj=O
where ¢ is a given positive definite matrix. The system (13) is called (I)-robustly stable if it is stable for all A(0) satisfying condition (14) The Lyapunov-Krasovskii functional 0
xT (t)Px(t) + [ xT (t + O)Q(O)x(t + 0)d0,
u
. 2
-h
has been proposed. It is assumed here that P is a positive definite matrix. Let the weight matrix Q(O) is defined as 0
-h
where R(~) > 0 for all ~ E I-h, 0]. By construction
dQ(O) dO = R(O) > 0, and Q(-h) = O.
That means that the Lyapunov-Krasovskii functional is positive definite. Its derivative along the trajectories of (13) satisfies the inequality --v
dd
(131
~_ x T (t) [ A T p + P A +
The following statement is known as the zero exclusion principle.
Proposition 19. All quasipolynomialsfrom F have no zeros with nonnegative real parts if and only if (1) At least one quasipolynomial from F is stable; (2) The value set of the family does not include the origin
O ~ VF(iw) = { f(iw) I f e F } for all w > 0.
o
+
v0(q) _> vj(q) for all q E Q and for all j = 1,2, ...,m.
Q(e) = /
Q(O) > O,
Assume that q E Q c R :, where the uncertainty bounding set Q is a given compact pathwise connected set, one obtains the family of quasipolynomials F = { f(s,q) I qeQ}. Stability of the uncertain system is now equivalent to the statement that all the zeros of quasipolynomials from the family F have negative real parts. In order to prevent a possible lost of the principal term the following necessary condition should be imposed
[ [R(e)+ PA(O)R-I(e)AT(o)P] de]x(t). -h
Hence, when there are a positive definite matrix P > 0 and a positive definite matrix valued function R(0) > 0, for all 0 E [-h, 0], such that 0 > ATp + PA+
(15)
Remark 20. One can replace the second condition by a slightly weaker one • the condition 2 holds at least for one we >_ 0, and the boundary of the value set does not contain the origin:
0 ~ OVF(iw) for all w > O.
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1I. L. Kharitonov / A n n u a l Reviews in Control 23 (1999) 185-196
Assume that there is no uncertainty in delays and that the coefficient uncertainty is specified by a convex polytope
jCk
where aj = 1,2,3,4; and s =
E"5"
. j > 0,
j=l
= 1
,
where points aj, j = 1,2, ...,/, correspond to the vertices of the polytope. The polytope defines the corresponding polytopic family of quasipolynomials
j=l
The following theorem has been proven in (Fu, et all, 1989). Theorem 21. All quasipolynomials of the polytopic family are stable (have no zeros in the closed right half plane of the complex plane) if and only if all quasipolynomials from the edge subfamilies are stable. This theorem shows that for the polytopic class of quasipolynomial families stability analysis is equivalent to stability check of a finite number of simple one parameter subfamilies corresponding to edges. But it is also true that the number of such subfamilies growth exponentially with n and m. In order to get additional reduction one may try to apply the following statement from (Kharitonov and Zhabko, 1994). Theorem 22. A s s n m e that /5(s) and fk(~) are stable, then if the difference g(s) = fj(s) - fk(s) satisfies the condition
d
d~arg(g(i~))---
vo + Vm _
_
2
°
+
sin(arg(g(iw))-- (r0 + r,,)w) + 2w ' for all w > O, where 9(iw) 7£ O, then all convex combinations ttfj(s) + (1 - # ) f k ( s ) # E [0, 1], are also stable quasipolynomials. Let a quasipolynomial family has an interval structure i = P0(s)~ T0s + P , ( s ) ~ , ~
Theorem 23. The interval quasipolynomial I is stable if and only if those of the (m + 1)4m+l one parametric subfamilies for which v k > - -r o + r m 2 are stable.
j=l
Here quasipolynomial J'j(s) corresponds to the vertex coefficient vector aj.
-
(~,/3) = (1,2), (2, 3), (3, 4), (4, 1).
j=l
+ ... + Pm(~)~ ~=~,
where Pj(s), j = O, 1, ..., m are interval polynomials. With every one of them one may associate four polynomials pll)(s), p~2)(s), p~3)(s), p~4)(s) as in (Kharitonov, 1978). The first reduction is based on the value set analysis: There are (m + 1)4 m+l one parameter families of the form
See (Kharitonov and Zhabko, 1994). Another examples of robust stability conditions for quasipolynomial families may be found in (Hoeherman, at all, 1995). 4.2.2. Lyapunov-Krasovskii approach As it was already discussed above by means of LyapunovKrasovskii or Lyapunov-Ra~umikhin approaches one can obtain a number of sufficient stability conditions. These conditions may be e x p r e ~ d in two closely related forms: In the form of nonlinear matrix inequalities or in the form of linear (quasilinear) matrix inequalities. Let us start with the first form. This approach has been actively developed in (Li and Souza, 1995;1997). They have treated the single delay case, but all results can be easily extended to the general multiple delay case. Consider the following uncertain system dr(t) = (A + AA) z(t)+ dt
(16)
+ (A1 + AA1) z ( t - h), where AA and AA1 are unknown norm bounded matrices of the form AA = DFB,
AA1 = DIF1B1.
(17)
Here D, B, D1 and B1 are given constant matrices which describe uncertainty structure, while the uncertain matrices F and F1 satisfy the following constrains
IIFII _< 1,
IIF~II _< 1.
(18)
Remark 24. It was assumed in the original papers (Li and Souza, 1995;1997) that F and F1 are time varying uncertain matrices, but this does not affect the following stability condition. When one is looking for stability conditions which depend on delay the standard step is to replace the system (16) by the following one
194
V. L. Kharitonov / A n n u a l Reviews in Control 23 (1999) 185-196
d((t) = ( A + A A + A ~ + A A J ( ( t ) dt
(19)
0
f[(A
- (A~ + AA~)
+ A A ) ( ( t + 0)+
Theorem 26. Let matrix (A + A1) be stable. For a given delay value h, the uncertain system (16)(18) is stable if there exist X > 0 and vector = ( a l , ..., as) with positive elements solving the following LMI Q(X) + 5 J 1 D X/~ T X M
-h
(A~ + AA~)~(t
BX MT x
h + e)]dO.
-
Let = C(t)P
(0 +
NT
where 0
g
-J1 ,,0 0 -1.12 0
0
)
0 0
<0
- h J3
where
0
Q(X) = X (A + A1) 7c+ (A + A1) X, -h 0
D = (D, D1), ~7" __ (BT,B~), N = (A1,DJ, M T = (AT,BT,A~,BT),
0
0
J1 = diag (alE, a~E), - h O--h
J2 = diag(aaE- a4DD T,
The term w(~) is introduced in order to be exploited later for cancellation of integral terms in an estimation of the time derivative of the functional. After such a cancellation
J3 = diag(E - a6B~B1, a6E), Slightly different conditions have been obtained in (Niculescu, et all, 1994).
dv(&) (tg) <- ~T(t)S(P' e, h)~(t), where
Theorem 27. Let the matrix (A + A J be stable and D1 = 0, B1 = 0 (no uncertainty on the matrix A1). For a given value h > 0 the uncertain system (16)-(18) is stable if there exist matrix Y > 0 and positive values e, fl, f l satisfying the following LMI
S(P,e,h) = (A + A~) r P + P (A + A J + +P [¢~DDT + h A t ( E - grBT B~)-~A[
¢3¢7
a4E, (1 - aa)E - ahD1D~, othE).
J
Y
hg3g4 4T(E _ e~DDT)_~A+
h~TATY
E2
+hs3(1 + &,)AT (E - e~D1DT)-~AI+
+
hg3"~4
BTB
(20)
E > e~DDT,E > cGD1D~',E > eTBTBI(21) 4.2.3. Sufficient conditions in the LMI form the new variables X = ~3 P - l ,
In
g4
1+e4' C4E5
~4
--
1 + c4
~6 ,
0'5
--
-
-
~
1 + ~..~
(~6
=
0 I"
<0,
O= (A + A1)Ty + Y(A + A1) + BT B h + -gD O T + h(f + fit)Y,
Theorem 25. Let matrix (A + A1) be stable. For a given delay vaJue h, the uncertain system (16)(18) is sta.ble if there exist P > 0 and vector e = (el, ...,~7) with positive elements satisfying the inequalities S(P, e,h) > 0
O0
0 - h0~0
where
+ h¢3(1 -I- 64) BTB~" e~
(1 + ~4) e~
-E
C7~
the theorem conditions may be rewritten in the LMI form.
t~ =
flY 0 0 Y - ¢B TB
,
and (I) = (A, A1, DA),
F = eBTB - Y.
Another employment of the LMI approach has been proposed in (Gu, 1997). Here the LyapunovKrasovskii functional (7) is considered. The principal idea of the procedure proposed in (Gu, 1997) is based on the observation that if the matrix valued functions Q(O), R(8), S(01,0~) are assumed to be continuous and piecewise linear, then both d the functional v(xt) and its derivative 77v1(16) may be expressed as some quadratic forms whose
V. L. Kharitonov / A n n u a l Reviews in Control 23 (1999) 185-196
matrices depend linearly on the values of the matrices Q(O), R(O), S(01,02) at corresponding griding points. This observation opens a door to exploiting powerful interior point algorithms. 4.3 Robust stability independent of delay 4.3.1. Interval quasipolynomial Robust stability of an interval quasipolynomial
I = s" + ~
tiff
~k,-dk] s k +
[bkj,-bkj] ske ~j~,
j=l k=0
k=0
with rj < 0, j = 1, 2, ..., m, independent of exponential coefficients was addressed first in (Kim and Bose, 1990) where some sufficient conditions were obtained. Later this problem has been considered in (Olbrot and Igwe, 1995), where the following necessary and sufficient were proposed.
Theorem 28. The interval quasipolynomial I is stable independent of exponential coefficients if and only if
(1) (2)
max(Ib-. l, Ib-Jl) < 1;
max(lb-0j[, I 0jl) -< min ([a-0[, I~01) ;
(3) All 4 "~+1 extreme quasipolynomials.
195
5. CONCLUDING REMARKS It should be mentioned explicitly that this survey covers only a small part of the large number of contributions in this area. For more complete reference lists the readers are advised to see recently published survey papers (Niculescu, et all, 1997) and (Richard, 1998). In order to make this survey self-contained the author deliberately concentrates on the class of linear time invariant delay systems. On the other hand this allows to study principle features of the basic methods for stability and robust stability analysis with minimum technical details.
6. ACKNOWLEDGEMENTS The author would like to thank CNRS and IRCyN France for the priveledge to be invited for a working visit to the Ecole Centrale de Nantes. This paper has been prepared during this visit. The author would also like to thank Dr. Silviuhlian Niculescu for helpful discussion of the subject.
77~
j=0
where (~j -- 1, 2, 3, 4, are stable independent of exponent coefficients U < 0. 4.3.2. Lyapunov-Krasovskii approach Consider again the system (16)-(17), where the uncertain matrices F, Ft satisfy the constrains
F F T < E and F1F1T < E.
(22)
Some conditions for stability independent of delay have been obtained in (Lisong, 1996) with the following Lyapunov-Krasovskii functional 0
v(xt) = x'r(t)Pz(t) +
/
xT(t + O)Sx(t + O)dO.
-- h
Here it is assumed that P > 0 and S > O.
Theorem 29. Let matrix A be stable. The system (16)-(17) with uncertainty constrains (22) is stable independent of delay if there exist matrices P > 0, S > 0 and positive constants e, el such that 0 > A T p + P A + S + 1-BTB + e P D D T P 6
+ P A l ( S - - ¢ I B T B1) - 1 A T p + 1pDIDT1 £1
and
0 < S - E1BTB1.
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