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On the Delay Independent Stabilization of Linear Neutral Systems with Limited Measurable State Variables
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
213, 601]619 Ž1997.
AY975564
On the Delay Independent Stabilization of Linear Neutral Systems with Limited Measurable State Variables Wanbiao Ma* and Norihiko Adachi Department of Applied Systems Science, Faculty of Engineering, Kyoto Uni¨ ersity, Yoshida Hon-machi, Sakyo-ku, Kyoto, 606-01, Japan
and Takashi Amemiya Tokyo Metropolitan Institute of Technology, 6-6, Asahigaoka, Hino, Tokyo, 191, Japan Communicated by Harold L. Stalford Received January 19, 1995
The delay-independent stabilizability of linear neutral systems with limited measurable state variables is considered. A sufficient condition for the delay-independent stabilizability of linear retarded systems with limited measurable state variables and linear neutral systems with measurable state variables which had been obtained by Amemiya and Ma et al. is proved to be also valid for linear Q 1997 Academic Press neutral systems with limited measurable state variables.
1. INTRODUCTION It is well known that the robust stabilization of uncertain dynamical systems has been one of the major research subjects in the field of control systems due to the existence of uncertainty in the large number of practical control systems. Various kinds of necessary and sufficient condi* Present address: Department of Applied Mathematics, Faculty of Engineering, Shizuoka University, Hamamatsu, 432, Japan. 601 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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MA, ADACHI, AND AMEMIYA
tions have been derived on the existence of robust stabilizing controllers heretofore after the work of Leitmann et al. w17, 18x. For example, in w7x, Lyapunov function theory is used to design a class of stabilizing controllers for linear and nonlinear uncertain dynamical systems. In w6x, the state feedback controllers guaranteeing uniform ultimate boundedness of the trajectories of a class of uncertain linear systems in the absence of matching conditions has been presented. In w8, 9, 24x, the quadratic stabilizability via linear control for uncertain linear systems in the absence of matching conditions has been considered and some necessary and sufficient conditions have been obtained. During recent years, the stabilization problem for systems with delay has also received considerable attention and many papers dealing with this problem have appeared because of the existence of delays in various practical control problems and also because of the fact that the delay is frequently a source of instability of systems. For example, in w14, 22x, the state feedback controllers which guaranteed the uniform ultimate boundedness of the solutions of two classes of delayed systems with matching conditions have been constructed based on the Lyapunov functions and Razumikin theorem w15x. In particular, in w1]5x, the delay independent stabilization of several classes of uncertain closed loop and open loop delay systems without the matching assumptions have been considered using the differential-difference inequality in w23x and the transformation technique; and some sufficient conditions for stabilization have been obtained. Recently, w19x considered the delay independent stabilization of a class of uncertain closed loop linear neutral systems without the matching assumptions, and also obtained geometric patterns similar to Wei’s and Amemiya’s. This paper is an extension of the previous work w1]5, 19x. We consider the delay independent stabilization of uncertain open loop linear neutral systems by the methods proposed in w1]5, 19x, with the hope of obtaining the same geometric patterns as w3, 5, 19, 24x.
2. LEMMAS In this section, we first present the definition of M-matrices and a simple property. DEFINITION 1. An n = n real matrix A s Ž a i j . n=n is said to be an M-matrix, if all the off-diagonal elements of A are nonpositive Ži.e., a i j F 0, i / j, i, j s 1, 2, . . . , n., and all the principal minors of A are positive.
603
LINEAR NEUTRAL SYSTEMS
There are some other equivalent definitions of M-matrices. We can find them in Berman and Plemmons w10x. From Definition 1 we can easily obtain the following LEMMA 1. Let P be an n = n lower-triangular M-matrix, and Q be an n = n real matrix such that all the off-diagonal elements of Q and PQ are nonpositi¨ e. Then the matrix PQ is an M-matrix iff the matrix Q is an M-matrix. Proof. In fact, if we let Pk , Q k , and Sk be the kth principal minors of the matrices P, Q, and PQ, respectively, k s 1, 2, . . . , n, then we have det Ž Sk . s det Ž Pk . det Ž Q k . . Clearly, the above equality implies that Lemma 1 is true. The following Lemma 2 can be found in w20x. LEMMA 2. Assume the function pŽ t . s colŽ p1Ž t ., . . . , pnŽ t .. : w t 0 y t , n q`. ª Rq is continuous and satisfies the following ¨ ector differential-difference inequality KDy p Ž t . F Ap Ž t . q Bp Ž t , t . ,
for t G t 0 ,
Ž 1.
where K s diagŽ k 1 , . . . , k n . is a non-negati¨ e constant matrix and D y pŽ t . is a Dini left deri¨ ati¨ e of pŽ t .. t is a non-negati¨ e constant, A s Ž a i j . n=n is a constant matrix with all non-negati¨ e off-diagonal elements, B s Ž bi j . n=n is a non-negati¨ e matrix such that yŽ A q B . is an M-matrix, and p Ž t , t . s col Ž p1 Ž t , t . , . . . , pn Ž t , t . .
and
pi Ž t , t . s maxy t F s F 0 pi Ž t q s . , i s 1, 2, . . . , n. Then there exist constants M G 1 and a ) 0 such that n
Ý pi Ž t . F M 5 p0 5 eya Ž tyt . 0
is1
for t G t 0 , where 5 p 0 5 s Ý nis1 pi Ž t 0 , t .. Throughout this paper, for any matrix D s Ž d i j . n=m , the matrix Ž< d i j <. n=m is denoted by < D <.
604
MA, ADACHI, AND AMEMIYA
3. SYSTEMS AND ASSUMPTIONS Let n be a fixed positive integer. The system considered in this paper is described by the following linear delay differential equation of neutral-type, d dt
Ž x Ž t . y A 0 Ž t . x Ž t y t Ž t . . . s AŽ t . x Ž t . q B Ž t . x Ž t y t Ž t . . q bu Ž t . , Ž 2.
for t G t 0 , where x g R n ; AŽ t ., B Ž t ., and A 0 Ž t . are n = n real, continuous, and bounded matrix functions, uŽ t . g C Žw t 0 , q`., R . is a control variable, b g R n is a known constant vector, and the delay function t Ž t . : w t 0 , q`. ª Rq is continuous and bounded, e.g., 0 F t Ž t . F t , for t G t 0 , where the upper bound t is not necessarily assumed to be known. The initial condition of Ž2. is x Ž t0 q s . s w Ž s . ,
for yt F s F 0,
Ž 3.
where w Ž s . g C Žwyt , 0x, R n .. The vector x Ž t . g R n is called the instantaneous system state at time t. It is well known that under the above assumptions, for each control uŽ t . g C Žw t 0 , q`., R ., there exists a unique solution to Ž2. for t G t 0 y t w11, 15x. The output function y Ž t . g R 2 of the system Ž2. is given by yŽ t . s CT x Ž t . , where C is a known n = 2 real constant matrix. Our first assumption is that for every t g R, the pair Ž AŽ t ., b . is represented as a controllable cannonical form. Assumption I. The matrix function AŽ t . and vector b can be written in the form A s A1 q A 2 Ž t . , 0 0 . A1 s ..
1 0 .. .
0 1 .. .
??? ??? .. .
0 0 .. . ,
0 0
0 0
??? 0
0 ???
1 0
0
605
LINEAR NEUTRAL SYSTEMS
A2 Ž t . s
0 0 .. .
0 0 .. .
0 0 .. .
??? ??? .. .
0 0 .. .
0
0 a2 Ž t .
0 ???
0
a1 Ž t .
??? a3 Ž t .
an Ž t .
0
,
0 0 . b s .. ,
0 0 1
where a i Ž t . Ž i s 1, 2, . . . , n. are real, continuous, and bounded functions. Our next assumption is that two elements of the instantaneous state vector of the system Ž2. can be measured. Namely: Assumption II. The output matrix C s Ž c1 , c 2 . can be written as 1 ¡¦
0 ¡¦
0 .. .
c1 s
0 , 0 0 .. .
¢§ 0
0 .. .
c2 s
0 , 1 0 .. .
¢§ 0
where c 2 has only one nonzero element at the Ž n y k q 1.st element. Here k is an integer satisfying 0 F k F n. Remark 1. Assumption II is first introduced in w5x and contains the previous case w3x as special ones. In fact, if k s 0, then c 2 s 0; if k s n, then c1 s c 2 . Furthermore, we can easily show that the pair Ž A1 , C T . is an observable pair. Our third assumption reflects the fact that the elements of the matrix functions A 0 Ž t ., B Ž t ., and A 2 Ž t . can enter into the system in a way to form a particular configuration. We first introduce a set of matrices with regards to the matrix functions A 0 Ž t ., B Ž t ., and A 2 Ž t ., which is also introduced in w4, 5, 19x. Clearly, from the boundedness of the matrix functions A 0 Ž t ., B Ž t ., A 2 Ž t ., and Assumption I, there are non-negative n = n constant matrices A 0 , B, and A 2
606
MA, ADACHI, AND AMEMIYA
such that for t G t 0 < A 0 Ž t . < F Ž aŽ0. ij .
n=n
< B Ž t . < F Ž bi j .
' A0 ,
< A 2 Ž t . < F Ž aŽ2. ij .
n=n
n=n
' B,
' A2 ,
where aŽ2. i j s 0 for i s 1, 2, . . . , n y 1; j s 1, 2, . . . , n. DEFINITION 2. Let the integer k be the same as in Assumption II. For this k, let V Ž k . s G s Ž g i j . n=n g R n= n4 and QŽ k . s W s Ž h i j . n=n g R n= n4 be two sets of matrices which satisfy Ž1. and Ž2. below: Ž1. If 1 F i F n y k; then gi j s 0
for j F 1 q i or j G 2 Ž n y k . y i q 1;
hi j s 0
for j F i or j G 2 Ž n y k . y i q 2.
Ž2. If n y k q 1 F i F n, then gi j s 0
for j G 1 q i or j F 2 Ž n y k . y i q 1;
hi j s 0
for j G i or j F 2 Ž n y k . y i q 2.
Remark 2. The forms of the matrices belonging to the matrix sets V Ž k . and QŽ k . can be shown clearly below, where, for the matrix set V Ž k ., ) and ^ denote its entries which may not be equal to zero, and e s 0; for the matrix set QŽ k ., ) and e denote its entries which may not be equal to zero, and ^ s 0. Assumption III.
The constant matrices B, A 0 , and A 2 satisfy B q A2 g V Ž k . ,
A0 g Q Ž k . .
For the case of the retarded system with limited measurable state variables, i.e., the case of A 0 Ž t . s 0Ž t G t 0 . in the system Ž2., it has been shown that if the Assumptions I]III are satisfied, then the system Ž2. can be made globally delay independently stable w5x. As for the case of the neutral system with all the state variables being measurable, it has been shown that the system Ž2. can be also made globally delay independently stable by a linear state variables feedback uŽ t . s g T x Ž t . ,
for t G t 0 , g g R n
as long as Assumption I and Assumption III are satisfied w19x.
LINEAR NEUTRAL SYSTEMS
607
In the next section, we construct a linear feedback control law to make the system Ž2. with limited measurable state variables also globally delay independently stable.
4. MAIN RESULT AND CONSTRUCTION OF A CONTROLLER We first consider the construction of a controller for Ž2.. Following w1]5x, we define z Ž t . g R n which satisfies the equation
˙z Ž t . s Ž A1 y DC T . z Ž t . q Dy Ž t . q bu Ž t .
Ž 4.
for t G t 0 , where D is an n = 2 constant matrix to be chosen later. Also we define the error function eŽ t . of the systems Ž2. and Ž4. as e Ž t . s z Ž t . y x Ž t . q A0 Ž t . x Ž t y t Ž t . .
Ž 5.
608
MA, ADACHI, AND AMEMIYA
and the control uŽ t . g R is defined as u Ž t . s g T z Ž t . s g T e Ž t . q g T x Ž t . y g TA 0 Ž t . x Ž t y t Ž t . .
Ž 6.
for t G t 0 , where g g R n is a constant vector to be chosen in the following discussion. Then from Ž2., Ž4., Ž5., and Ž6. we have for t G t 0
˙e Ž t . s Ž A1 y DC T . e Ž t . y A 2 Ž t . x Ž t . y Ž A1 y DC T . A 0 Ž t . q B Ž t . 4 x Ž t y t Ž t . .
Ž 7.
and d dt
Ž x Ž t . y A 0 Ž t . x Ž t y t Ž t . . . s Ž A1 q bg T . x Ž t . q A 2 Ž t . x Ž t . q B Ž t . x Ž t y t Ž t . . q bg T e Ž t . . Ž 8 .
Next, let us consider the stability of the 2n-dimensional system Ž7. ] Ž8.. The concept of delay independent stabilization is defined as follows w3, 5x. DEFINITION 3. The system Ž2. is called delay independently stabilizable, if there exist a vector g g R n and an n = 2 matrix D such that the 2 n-dimensional systems Ž7. ] Ž8. are globally asymptotically stable, independently of delay t Ž t .. Here we introduce the coordinate transformation defined by x s Ty1 x,
e s Se
for t G t 0 y t ,
Ž 9.
where T and S are n = n constant matrices to be chosen in the sequel. Then the systems Ž7. and Ž8. can be written as d dt
Ž x Ž t . y Ty1A 0 Ž t . T x Ž t y t Ž t . . s Ty1 Ž A1 q bg T . T x Ž t . q Ty1A 2 Ž t . T x Ž t . q Ty1 B Ž t . Tx Ž t y t Ž t . . q Ty1 bg T Sy1 e Ž t .
Ž 10 .
˙e Ž t . s S Ž A1 y DC T . Sy1 e Ž t . y SA 2 Ž t . T x Ž t . y S Ž A1 y DC T . A 0 Ž t . q B Ž t . 4 T x Ž t y t Ž t . .
Ž 11 .
for t G t 0 . Let y Ž t . s x Ž t . y Ty1A 0 Ž t . T x Ž t y t Ž t . . ,
for t G t 0 ;
Ž 12 .
609
LINEAR NEUTRAL SYSTEMS
then it follows from Ž10. that
˙y Ž t . s Ty1 Ž A1 q bg T . T y Ž t . q Ty1 Ž A1 q bg T . A 0 Ž t . T x Ž t y t Ž t . . q Ty1A 2 Ž t . Tx Ž t . q Ty1 BT x Ž t y t Ž t . . q Ty1 bg T Sy1 e Ž t . .
Ž 13 . Owing to the controllability of Ž A1 , b . and the observability of Ž A1 , C T ., it is possible to make all the roots of the characteristic equations detŽ l I y Ž A1 q bg T .. s 0 and detŽ s I y Ž A1 q DC T .. s 0, denoted by l1 , . . . , l n and s 1 , . . . , sn , respectively, be real and distinct by suitable choices of g and D. In fact, from the forms of the matrices A1 and C, it is easy to see that the matrix D can take the simple form
¡
d1 d2 .. .
d ny ky1 d ny k Ds 0 0 .. .
¢
0 0 .. . 0 y1
,
d ny kq1 d ny kq2 .. . dn
0
¦
§
where d i Ž i s 1, 2, . . . , n. can be chosen according to si Ž i s 1, 2, . . . , n.. Now we let the matrices T and S be the Vandermonde matrices defined by
Ts
1 l1 .. .
1 l2 .. .
??? ??? .. .
1 ln .. .
l1ny 1
l2ny1
???
l ny1 n
S1 0
0 , S2
Ss
ž
/
0
,
610
MA, ADACHI, AND AMEMIYA
where S 1 and S 2 are Ž n y k . = Ž n y k . and k = k matrices given as
S1 s
and
S2 s
s 1ny ky1
s 1ny ky2
???
s1
1
s 2nyky1
s 2nyky2 .. .
??? .. .
s2 .. .
1 .. , .
ny ky1 sny k
nyky2 snyk
???
snyk
1
.. .
0 0
ky 1 sny kq1
ky 2 snykq1
???
snykq1
1
ky1 sny kq2
ky2 snykq2
.. .
??? .. .
snykq2 .. .
1 .. , .
snky 1
snky2
???
sn
.. .
1
respectively. Define L, S, pŽ t ., and q Ž t . as L s diag Ž l1 , . . . , l n . ,
S s diag Ž s 1 , . . . , sn .
and p Ž t . ' < y Ž t . < s Ž p1 Ž t . , . . . , pn Ž t . .
T
T
s Ž < y 1 Ž t . < , . . . , < yn Ž t . < . , q Ž t . ' < e Ž t . < s Ž q1 Ž t . , . . . , q n Ž t . .
T
T
s Ž < e1 Ž t . < , . . . , < e n Ž t . < . , r Ž t . ' < x Ž t . < s Ž r 1 Ž t . , . . . , rn Ž t . . s Ž < x1Ž t . < , . . . , < x n Ž t . <.
T
T
for t G t 0 . Then from Ž11., Ž12., and Ž13. we have for t G t 0 Dy p Ž t . s Dy < y Ž t . < s diag Ž sgn y 1 Ž t . , . . . , sgn yn Ž t . . ˙ yŽ t. s diag Ž sgn y 1 Ž t . , . . . , sgn yn Ž t . . Ž T .
y1
Ž A1 q bg T . T y Ž t .
611
LINEAR NEUTRAL SYSTEMS
q Ty1 Ž A1 q bg T . A 0 Ž t . T x Ž t y t Ž t . . q Ty1A 2 Ž t . Tx Ž t . q. Ty1 B Ž t . T x Ž t y t Ž t . . q Ty1 bg T Sy1 e Ž t . F L p Ž t . q Ž < L < < Ty1 < A 0 < T < q < Ty1 < A 2 < T < q< Ty1 < B < T < . r Ž t , t . q < Ty1 bg T Sy1 < q Ž t . F L p Ž t . q Ž < L < < Ty1 < A 0 < T < q < Ty1 < A 2 < T < q< Ty1 < B < T < . r Ž t , t . q < Ty1 bg T Sy1 < q Ž t , t . ,
Ž 14 .
Dy q Ž t . s D y < e Ž t . < s diag Ž sgn e1 Ž t . , . . . , sgn e n Ž t . . ˙ eŽ t . s diag Ž sgn e1 Ž t . , . . . , sgn e n Ž t . . Ž S Ž A1 y DC T . Sy1 e Ž t . ySA 2 Ž t . T x Ž t . y S Ž A1 y DC T . A 0 Ž t . q B Ž t . 4 T x Ž t yt Ž t . . . F S q Ž t . q Ž < S < < S < A0 < T < q < S < A2 < T < q < S < B < T <. r Ž t , t . ,
Ž 15 .
and 0 F yr Ž t . q p Ž t . q < Ty1A 0 Ž t . T < r Ž t , t . F yr Ž t . q p Ž t , t . q < Ty1 < A 0 < T < r Ž t , t . .
Ž 16 .
where pŽ t, t . s Ž p1Ž t, t ., . . . , pnŽ t, t ..T , r Ž t, t . s Ž r 1Ž t, t ., . . . , rnŽ t, t ..T , pi Ž t, t . s maxy t F s F 0 pi Ž t q s . s maxy t F s F 0 < yi Ž t q s .<, ri Ž t, t . s maxyt F s F 0 ri Ž t q s . s maxy t F s F 0 < x i Ž t q s .<, for i s 1, 2, . . . , n. For the sake of convenience, we also define q Ž t . ' < e Ž t . < s Ž q1 Ž t . , . . . , q n Ž t . .
T
T
s Ž < e1 Ž t 0 . < , . . . , < e n Ž t 0 . < . , for t 0 y t F t F t 0 , and q Ž t, t . s Ž q1Ž t, t ., . . . , qnŽ t, t ..T for t G t 0 ; here qi Ž t, t . s maxy t F s F 0 qi Ž t q s . s maxy t F s F 0 < e i Ž t q s .<. Let
R1 s
ž
L 0 0
0 S 0
0 0 yE
/
612
R2 s
MA, ADACHI, AND AMEMIYA
0
< Ty1 bg T Sy1 <
< L < < Ty1 < A 0 < T < q < Ty1 < A 2 < T < q < Ty1 < B < T <
0
0
< S < < S < A0 < T < q < S < A2 < T < q < S < B < T <
E
0
y1 <
A0 < T <
0
,
K s diag Ž E, E, 0 . , T
w Ž t . s Ž pT Ž t . , q T Ž t . , r T Ž t . . g R 3 n , T
w Ž t , t . s Ž pT Ž t , t . , q T Ž t , t . , r T Ž t , t . . , where E and 0 are an n = n unit matrix and an n = n 0 matrix, respectively. Then, with these notations, Ž14., Ž15., and Ž16. can be written in the 3n-dimensional vector form KDy w Ž t . F R1w Ž t . q R 2 w Ž t , t .
Ž 17 .
for t G t 0 . Let < Ty1 < A 0 < T < s Ž a i j . n=n ; then it follows from Lemma 2 that if the matrix yŽ R1 q R 2 . is an M-matrix, there exist positive constants a and M G 1 such that for t G t 0 3n
3n
is1
is1
Ý wi Ž t . F M Ý wi Ž t 0 , t . eya Ž tyt . 0
n
F M Ý Ž pi Ž t 0 , t . q qi Ž t 0 , t . q ri Ž t 0 , t . . eya Ž tyt 0 . is1 n
FM Ý is1
ž
max < yi Ž t 0 q s . < q < e i Ž t 0 . <
ytFsF0
q max < x i Ž t 0 q s . < eya Ž tyt 0 .
/
ytFsF0
n
FM Ý is1
ž
max < x i Ž t 0 q s . < q
ytFsF0
n
Ý js1
q< e i Ž t 0 . < q
ža
ij
max < x j Ž t 0 q s . <
y t FsF0
max < x i Ž t 0 q s . < eya Ž tyt 0 .
ytFsF0
n
s M Ý 2 M1 max < x i Ž t 0 q s . < q < e i Ž t 0 . < eya Ž tyt 0 . , is1
ž
ytFsF0
/
/
/
613
LINEAR NEUTRAL SYSTEMS
where n
M1 s 2 q max
1FiFn
žÝ /
a i j G 1.
js1
Thus n
Ý
n
Ž < x i Ž t . < q < ei Ž t . < . F M Ý 2 M1 max < x i Ž t 0 q s . <
is1
is1
ž
ytFsF0
q< e i Ž t 0 . < eya Ž tyt 0 . Ž 18 .
/
for t G t 0 , which implies that the trivial solution of Ž10. and Ž11. is delay independently exponentially stable. Hence, from the non-singularity of T and S we have that the trivial solution of Ž7. and Ž8. is delay independently exponentially stable. We choose l i - 0, si - 0 Ž i s 1, 2, . . . , n. and set Ns
E 0 yLy1
0 E Ly1 < Ty1 bg T Sy1 < Sy1
0 0 . E
0
Then N Ž R1 q R 2 . s
Ly1
< Ty1 bg T Sy1 <
< L < < Ty1 < A 0 < T < q < Ty1 < A 2 < T < q < Ty1 < B < T <
0 0
Sy1 0
< S < < S < A0 < T < q < S < A2 < T < q < S < B < T < yD
0
,
where D s E y < Ty1 < A 0 < T < y Ly1 < Ty1 bg T Sy1 < Sy1 = Ž < S < < S < A0 < T < q < S < A2 < T < q < S < B < T <. q Ly1 Ž < L < < Ty1 < A 0 < T < q < Ty1 < A 2 < T < q < Ty1 < B < T < . . Thus, from Lemma 1 we have that yŽ R1 q R 2 . is an M-matrix iff D is an M-matrix which is also equivalent to the matrix Q s < L < y 2 < L < < Ty1 < A 0 < T < y < Ty1 bg T Sy1 < = Ž < S < A 0 < T < q < Sy1 < Ž < S < A 2 < T < q < S < B < T < . . y < Ty1 < A 2 < T < y < Ty1 < B < T < being an M-matrix. Therefore, in the following, we consider the matrix Q.
614
MA, ADACHI, AND AMEMIYA
We choose l1 , . . . , l n and s 1 , . . . , sn such that
l1 , . . . , l nyk ª 0,
s 1 , . . . , snyk ª 0,
l ny kq1 , . . . , l n ª y`,
snykq1 , . . . , sn ª y`.
Hence, we can express l1 , . . . , l n and s 1 , . . . , sn as
li s o
1
ž / l
si s o
,
1
ž / l
for i s 1, 2, . . . , n y k, and
li s O Ž l. ,
si s O Ž l .
for i s n y k q 1, . . . , n, where l is a positive number which is sufficiently large. Following the papers by Amemiya w1]5x, the following notations are again introduced here. Let j Ž l. g R be a continuous function; if there exists a constant m such that j Ž l.rl m < - q` as < l < ª q` and < j Ž l.rl my a < ª q` as < l < ª q` for any positive constant a, then j is called a function of order m denoted by OrdŽ j . s m. Let m denote a matrix, a column vector, or row vector every element of which is a function of order m. Thus, with these notions, the matrices T, S, Ty1 , and Sy1 can be expressed in the forms
Ts
ž
T11 T21
T12 , T22
Ty1 s
/
ž
Tˆ11
Tˆ12
Tˆ21
Tˆ22
¡
0
/
where
¡ T11 s
0 y1 .. .
¦ ,
¢ yn q k q 1 §
T12 s
¦
1 .. .
¢ nyky1 §
615
LINEAR NEUTRAL SYSTEMS
¡
yn q k y 1
T21 s
Tˆ21 s
ž
ž
nyky2
ž
y2 n q 2 k
Tˆ22 s
.. .
§
ny1
/
,
n y k y 3 ??? n y 2 k y 1
/
,
y2 n q 2 k q 1 ??? yn q k y 1
yn q k
ž
¢
1 ??? n y k y 1
0
¦
nyk nykq1
T22 s
§
yn q 1 Tˆ11 s
¡
,
.. .
¢ Tˆ12 s
¦
yn q k
yn q k y 1 ??? yn q 1
/
,
/
and S1 s
1ynqk
ž
S2 s
ž
2 y n q k ??? 0
y1 q k
y2 q k ??? 0
¡ nyky1 ¦ Ž S1 .
y1
nyky2
s
¢
.. .
,
,
,
¡ 1yk ¦ Ž S2 .
§
0
/
/
y1
2yk
s
¢
.. .
.
§
0
By Assumptions I]III and papers w1]5, 19x, the matrices < Ty1 bg T Sy1 <, < S < A 2 < T < q < S < B < T <, < Ty1 < A 2 < T < q < Ty1 < B < T <, < Ty1 < A 0 < T <, and < Ty1 < B < T < have the forms
< Ty1 bg T Sy1 < s
nyky2
n y 2k
yn q k
2yn
0
,
616
MA, ADACHI, AND AMEMIYA
< S < A2 < T < q < S < B < T < s
y1 <
A2 < T < q < T
y1 <
< Ty1 < A 0 < T < s
y1 <
B
nyky1
2k y n y 1
ny1
2n y 2k y 2
2k y 2n
0 2n y 2k y 1
2k y 2n y 1
0
y2
y1
B
yn q k y 1
0
y1
y2
2n y 2k y 2
2k y 2n
0
0
,
0
,
,
.
Furthermore, by Assumption III and the form of the matrices T and S, we can also find that the matrix < S < A 0 < T < has the form
< S < A0 < T < s
kyn
nyk
2k y n y 2
ny2
0
.
Therefore, the matrices < L < < Ty1 < A 0 < T <, < Ty1 bg T Sy1 < < S < A 0 < T <, and < Ty1 bg T Sy1 < < Sy1 <Ž< S < A 2 < T < q < S < B < T <. have the same structure as the matrix < Ty1 < A 2 < T < q < Ty1 < B < T <. Thus, by the same arguments as in w1x, we can find that the matrix Q becomes an M-matrix as long as l is large enough. Thus, we have our main result. THEOREM. If Assumptions I]III are satisfied, then Ž2. is delay independently stabilizable. Remark 3. It is easy to see that if A 0 Ž t . s 0 for t G t 0 in Ž2., then the system Ž2. reduces to the retarded system considered in Refs. w3, 5x. Therefore, the above Theorem generalizes the results in w3, 5x.
LINEAR NEUTRAL SYSTEMS
617
5. ALGORITHMS Now we can construct the stabilizing controller the existence of which is guaranteed under Assumptions I]III by the Theorem. The system considered must be transformed into the one whose nominal system is in controllable canonical form. It must also be checked whether Assumption II is satisfied. From Assumption II we can determine the value of k. Step 1. After transforming the system, check whether the matrix functions AŽ t ., B Ž t ., and A 0 Ž t . of the system satisfy Assumption III. Step 2. Select positive number l to be sufficiently large so that the matrix Q is an M-matrix. Step 3. Select the roots, l i s oŽ1rl., si s oŽ1rl. for i s 1, 2, . . . , n y k, and l i s O Ž l., si s O Ž l. for i s n y k q 1, . . . , n, of the characteristic equations detŽ l I y Ž A1 q bg T .. s 0 and detŽ s I y Ž A1 q DC T .. s 0 as functions of a parameter l. Step 4. Vector g can be decided from the relation between eigenvalues and coefficients of the characteristic equation detŽ l I y Ž A1 q bg T .. s 0. The matrix D with the form of
¡
d1 d2 .. .
d ny ky1 d ny k Ds 0 0 .. .
¢
0
0 0 .. .
¦
0 y1 d ny kq1 d ny kq2 .. . dn
§
can also be found from the relation between eigenvalues and coefficients of the characteristic equation detŽ s I y Ž A1 q DC T .. s 0. Step 5. From the value of g and D, the controller can be constructed.
6. CONCLUDING REMARKS In this paper, the delay-independent stabilizability of the linear neutral system Ž2. has been considered using the differential-difference inequali-
618
MA, ADACHI, AND AMEMIYA
ties and the transformation techniques developed in w1]5, 19]21, 23x. A sufficient condition for the delay-independent stabilizability of linear retarded systems with limited measurable state variables and linear neutral systems with measurable state variables which had been obtained by Amemiya and Ma et al. has been proved also to be valid for the linear neutral system Ž2. with limited measurable state variables. Our sufficient condition is also independent of the construction of Lyapunov functions and Lyapunov functionals and has a geometric pattern similar to w3, 5, 19, 24x too. Finally, a procedure to design the state feedback controller which can guarantee the delay-independent stabilizability of the system considered has been presented.
ACKNOWLEDGMENTS The authors are very grateful to the referee for his valuable comments. The first author also thanks Professor Y. Takeuchi for his valuable suggestions.
REFERENCES 1. K. Akazawa, T. Amemiya, and H. Tokumaru, Further condition for the delay independent stabilization of linear systems, Internat. J. Control 46 Ž1987., 1195]1202. 2. T. Amemiya, Delay-independent stabilization of linear systems, Internat. J. Control 37 Ž1983., 1071]1079. 3. T. Amemiya, K. Akazawa, and H. Tokumaru, Delay independent stabilization and decay rate assignability of linear systems with limited measurable state variables, Internat. J. Control 47 Ž1988., 65]73. 4. T. Amemiya and G. Leitmann, A method for designing a stabilizing control for a class of uncertain linear delay systems, Dynam. Control 4 Ž1994., 147]167. 5. T. Amemiya, A method of delay independent stabilization of a class of linear delay systems with limited measurable state variables, submitted for publication. 6. B. R. Barmish and G. Leitmann, On ultimate boundedness control of uncertain systems in the absence of matching assumptions, IEEE Trans. Automat. Control 27 Ž1982., 153]158. 7. B. R. Barmish, M. Corless, and G. Leitmann, A new class of stabilizing controllers for uncertain dynamical systems, SIAM J. Control Optim. 21 Ž1983., 246]255. 8. B. R. Barmish, Stabilization of uncertain systems via linear control, IEEE Trans. Automat. Control 28 Ž1983., 848]805. 9. B. R. Barmish, Necessary and sufficient conditions for quadratic stabilizability of an uncertain systems, J. Optim. Theory Appl. 46 Ž1985., 399]408. 10. A. Berman and R. J. Plemmons, ‘‘Nonnegative Matrices in the Mathematical Sciences,’’ Academic Press, New York, 1979. 11. E. N. Chekwu, ‘‘Stability and Time-Optimal Control of Hereditary Systems,’’ Academic Press, New York, 1992. 12. D. A. O. Connor and T. J. Tarn, On the function space controllablity of linear neutral systems, SIAM J. Control Optim. 21 Ž1983., 306]329.
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619
13. D. A. O. Connor and T. J. Tarn, On stabilization by state feedback for neutral differential difference equations, IEEE Trans. Automat. Control 28 Ž1983., 615]618. 14. E. Cheres, S. Gutman, and Z. J. Plamor, Stabilization of uncertain dynamic systems including state delay, IEEE Trans. Automat. Control 34 Ž1989., 1199]1203. 15. J. Hale, ‘‘Theory of Functional Differential Equations,’’ Springer-Verlag, New York, 1977. 16. M. Q. Jacobs and C. E. Langenhop, Criteria for the function space controllability of linear neutral systems, SIAM J. Control Optim. 14 Ž1976., 1009]1048. 17. G. Leitmann, Guaranteed asymptotic stability for some linear systems with bounded uncertainties, J. Dynam. Systems Measurement Control 101 Ž1979., 212]216. 18. G. Leitmann, On the efficacy of nonlinear control in uncertain linear systems, J. Dynam. Systems Measurement Control 102 Ž1981., 94]102. 19. Wanbiao Ma, N. Adachi, and T. Amemiya, On the delay independent stabilization of uncertain linear systems of neutral-type, J. Optim. Theory Appl. 84 Ž1995., 393]405. 20. Ligeng Si, ‘‘Stability of Neutral Differential Difference Equations with Delays,’’ Inner Mongolia Education Press, Huhhot, 1994. 21. Ligeng Si and Wanbiao Ma, Differential-difference inequalities with unbounded delays and their applications, J. Nonlinear Anal. 17 Ž1991., 787]801. 22. A. Thowsen, Uniform ultimate boundedness of the solutions of uncertain dynamic delay systems with state-dependent and memoryless feedback control, Internat. J. Control 37 Ž1983., 1135]1143. 23. H. Tokumaru, N. Adachi, and T. Amemiya, Macroscopic stability of interconnected systems, in ‘‘6th IFAC ID,’’ Vol. 44, No. 4, 1975. 24. Kehui Wei, Quadratic stabilizability of linear systems with structural independent timevarying uncertainties, IEEE Trans. Automat. Control 35 Ž1990., 268]277.
213, 601]619 Ž1997.
AY975564
On the Delay Independent Stabilization of Linear Neutral Systems with Limited Measurable State Variables Wanbiao Ma* and Norihiko Adachi Department of Applied Systems Science, Faculty of Engineering, Kyoto Uni¨ ersity, Yoshida Hon-machi, Sakyo-ku, Kyoto, 606-01, Japan
and Takashi Amemiya Tokyo Metropolitan Institute of Technology, 6-6, Asahigaoka, Hino, Tokyo, 191, Japan Communicated by Harold L. Stalford Received January 19, 1995
The delay-independent stabilizability of linear neutral systems with limited measurable state variables is considered. A sufficient condition for the delay-independent stabilizability of linear retarded systems with limited measurable state variables and linear neutral systems with measurable state variables which had been obtained by Amemiya and Ma et al. is proved to be also valid for linear Q 1997 Academic Press neutral systems with limited measurable state variables.
1. INTRODUCTION It is well known that the robust stabilization of uncertain dynamical systems has been one of the major research subjects in the field of control systems due to the existence of uncertainty in the large number of practical control systems. Various kinds of necessary and sufficient condi* Present address: Department of Applied Mathematics, Faculty of Engineering, Shizuoka University, Hamamatsu, 432, Japan. 601 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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MA, ADACHI, AND AMEMIYA
tions have been derived on the existence of robust stabilizing controllers heretofore after the work of Leitmann et al. w17, 18x. For example, in w7x, Lyapunov function theory is used to design a class of stabilizing controllers for linear and nonlinear uncertain dynamical systems. In w6x, the state feedback controllers guaranteeing uniform ultimate boundedness of the trajectories of a class of uncertain linear systems in the absence of matching conditions has been presented. In w8, 9, 24x, the quadratic stabilizability via linear control for uncertain linear systems in the absence of matching conditions has been considered and some necessary and sufficient conditions have been obtained. During recent years, the stabilization problem for systems with delay has also received considerable attention and many papers dealing with this problem have appeared because of the existence of delays in various practical control problems and also because of the fact that the delay is frequently a source of instability of systems. For example, in w14, 22x, the state feedback controllers which guaranteed the uniform ultimate boundedness of the solutions of two classes of delayed systems with matching conditions have been constructed based on the Lyapunov functions and Razumikin theorem w15x. In particular, in w1]5x, the delay independent stabilization of several classes of uncertain closed loop and open loop delay systems without the matching assumptions have been considered using the differential-difference inequality in w23x and the transformation technique; and some sufficient conditions for stabilization have been obtained. Recently, w19x considered the delay independent stabilization of a class of uncertain closed loop linear neutral systems without the matching assumptions, and also obtained geometric patterns similar to Wei’s and Amemiya’s. This paper is an extension of the previous work w1]5, 19x. We consider the delay independent stabilization of uncertain open loop linear neutral systems by the methods proposed in w1]5, 19x, with the hope of obtaining the same geometric patterns as w3, 5, 19, 24x.
2. LEMMAS In this section, we first present the definition of M-matrices and a simple property. DEFINITION 1. An n = n real matrix A s Ž a i j . n=n is said to be an M-matrix, if all the off-diagonal elements of A are nonpositive Ži.e., a i j F 0, i / j, i, j s 1, 2, . . . , n., and all the principal minors of A are positive.
603
LINEAR NEUTRAL SYSTEMS
There are some other equivalent definitions of M-matrices. We can find them in Berman and Plemmons w10x. From Definition 1 we can easily obtain the following LEMMA 1. Let P be an n = n lower-triangular M-matrix, and Q be an n = n real matrix such that all the off-diagonal elements of Q and PQ are nonpositi¨ e. Then the matrix PQ is an M-matrix iff the matrix Q is an M-matrix. Proof. In fact, if we let Pk , Q k , and Sk be the kth principal minors of the matrices P, Q, and PQ, respectively, k s 1, 2, . . . , n, then we have det Ž Sk . s det Ž Pk . det Ž Q k . . Clearly, the above equality implies that Lemma 1 is true. The following Lemma 2 can be found in w20x. LEMMA 2. Assume the function pŽ t . s colŽ p1Ž t ., . . . , pnŽ t .. : w t 0 y t , n q`. ª Rq is continuous and satisfies the following ¨ ector differential-difference inequality KDy p Ž t . F Ap Ž t . q Bp Ž t , t . ,
for t G t 0 ,
Ž 1.
where K s diagŽ k 1 , . . . , k n . is a non-negati¨ e constant matrix and D y pŽ t . is a Dini left deri¨ ati¨ e of pŽ t .. t is a non-negati¨ e constant, A s Ž a i j . n=n is a constant matrix with all non-negati¨ e off-diagonal elements, B s Ž bi j . n=n is a non-negati¨ e matrix such that yŽ A q B . is an M-matrix, and p Ž t , t . s col Ž p1 Ž t , t . , . . . , pn Ž t , t . .
and
pi Ž t , t . s maxy t F s F 0 pi Ž t q s . , i s 1, 2, . . . , n. Then there exist constants M G 1 and a ) 0 such that n
Ý pi Ž t . F M 5 p0 5 eya Ž tyt . 0
is1
for t G t 0 , where 5 p 0 5 s Ý nis1 pi Ž t 0 , t .. Throughout this paper, for any matrix D s Ž d i j . n=m , the matrix Ž< d i j <. n=m is denoted by < D <.
604
MA, ADACHI, AND AMEMIYA
3. SYSTEMS AND ASSUMPTIONS Let n be a fixed positive integer. The system considered in this paper is described by the following linear delay differential equation of neutral-type, d dt
Ž x Ž t . y A 0 Ž t . x Ž t y t Ž t . . . s AŽ t . x Ž t . q B Ž t . x Ž t y t Ž t . . q bu Ž t . , Ž 2.
for t G t 0 , where x g R n ; AŽ t ., B Ž t ., and A 0 Ž t . are n = n real, continuous, and bounded matrix functions, uŽ t . g C Žw t 0 , q`., R . is a control variable, b g R n is a known constant vector, and the delay function t Ž t . : w t 0 , q`. ª Rq is continuous and bounded, e.g., 0 F t Ž t . F t , for t G t 0 , where the upper bound t is not necessarily assumed to be known. The initial condition of Ž2. is x Ž t0 q s . s w Ž s . ,
for yt F s F 0,
Ž 3.
where w Ž s . g C Žwyt , 0x, R n .. The vector x Ž t . g R n is called the instantaneous system state at time t. It is well known that under the above assumptions, for each control uŽ t . g C Žw t 0 , q`., R ., there exists a unique solution to Ž2. for t G t 0 y t w11, 15x. The output function y Ž t . g R 2 of the system Ž2. is given by yŽ t . s CT x Ž t . , where C is a known n = 2 real constant matrix. Our first assumption is that for every t g R, the pair Ž AŽ t ., b . is represented as a controllable cannonical form. Assumption I. The matrix function AŽ t . and vector b can be written in the form A s A1 q A 2 Ž t . , 0 0 . A1 s ..
1 0 .. .
0 1 .. .
??? ??? .. .
0 0 .. . ,
0 0
0 0
??? 0
0 ???
1 0
0
605
LINEAR NEUTRAL SYSTEMS
A2 Ž t . s
0 0 .. .
0 0 .. .
0 0 .. .
??? ??? .. .
0 0 .. .
0
0 a2 Ž t .
0 ???
0
a1 Ž t .
??? a3 Ž t .
an Ž t .
0
,
0 0 . b s .. ,
0 0 1
where a i Ž t . Ž i s 1, 2, . . . , n. are real, continuous, and bounded functions. Our next assumption is that two elements of the instantaneous state vector of the system Ž2. can be measured. Namely: Assumption II. The output matrix C s Ž c1 , c 2 . can be written as 1 ¡¦
0 ¡¦
0 .. .
c1 s
0 , 0 0 .. .
¢§ 0
0 .. .
c2 s
0 , 1 0 .. .
¢§ 0
where c 2 has only one nonzero element at the Ž n y k q 1.st element. Here k is an integer satisfying 0 F k F n. Remark 1. Assumption II is first introduced in w5x and contains the previous case w3x as special ones. In fact, if k s 0, then c 2 s 0; if k s n, then c1 s c 2 . Furthermore, we can easily show that the pair Ž A1 , C T . is an observable pair. Our third assumption reflects the fact that the elements of the matrix functions A 0 Ž t ., B Ž t ., and A 2 Ž t . can enter into the system in a way to form a particular configuration. We first introduce a set of matrices with regards to the matrix functions A 0 Ž t ., B Ž t ., and A 2 Ž t ., which is also introduced in w4, 5, 19x. Clearly, from the boundedness of the matrix functions A 0 Ž t ., B Ž t ., A 2 Ž t ., and Assumption I, there are non-negative n = n constant matrices A 0 , B, and A 2
606
MA, ADACHI, AND AMEMIYA
such that for t G t 0 < A 0 Ž t . < F Ž aŽ0. ij .
n=n
< B Ž t . < F Ž bi j .
' A0 ,
< A 2 Ž t . < F Ž aŽ2. ij .
n=n
n=n
' B,
' A2 ,
where aŽ2. i j s 0 for i s 1, 2, . . . , n y 1; j s 1, 2, . . . , n. DEFINITION 2. Let the integer k be the same as in Assumption II. For this k, let V Ž k . s G s Ž g i j . n=n g R n= n4 and QŽ k . s W s Ž h i j . n=n g R n= n4 be two sets of matrices which satisfy Ž1. and Ž2. below: Ž1. If 1 F i F n y k; then gi j s 0
for j F 1 q i or j G 2 Ž n y k . y i q 1;
hi j s 0
for j F i or j G 2 Ž n y k . y i q 2.
Ž2. If n y k q 1 F i F n, then gi j s 0
for j G 1 q i or j F 2 Ž n y k . y i q 1;
hi j s 0
for j G i or j F 2 Ž n y k . y i q 2.
Remark 2. The forms of the matrices belonging to the matrix sets V Ž k . and QŽ k . can be shown clearly below, where, for the matrix set V Ž k ., ) and ^ denote its entries which may not be equal to zero, and e s 0; for the matrix set QŽ k ., ) and e denote its entries which may not be equal to zero, and ^ s 0. Assumption III.
The constant matrices B, A 0 , and A 2 satisfy B q A2 g V Ž k . ,
A0 g Q Ž k . .
For the case of the retarded system with limited measurable state variables, i.e., the case of A 0 Ž t . s 0Ž t G t 0 . in the system Ž2., it has been shown that if the Assumptions I]III are satisfied, then the system Ž2. can be made globally delay independently stable w5x. As for the case of the neutral system with all the state variables being measurable, it has been shown that the system Ž2. can be also made globally delay independently stable by a linear state variables feedback uŽ t . s g T x Ž t . ,
for t G t 0 , g g R n
as long as Assumption I and Assumption III are satisfied w19x.
LINEAR NEUTRAL SYSTEMS
607
In the next section, we construct a linear feedback control law to make the system Ž2. with limited measurable state variables also globally delay independently stable.
4. MAIN RESULT AND CONSTRUCTION OF A CONTROLLER We first consider the construction of a controller for Ž2.. Following w1]5x, we define z Ž t . g R n which satisfies the equation
˙z Ž t . s Ž A1 y DC T . z Ž t . q Dy Ž t . q bu Ž t .
Ž 4.
for t G t 0 , where D is an n = 2 constant matrix to be chosen later. Also we define the error function eŽ t . of the systems Ž2. and Ž4. as e Ž t . s z Ž t . y x Ž t . q A0 Ž t . x Ž t y t Ž t . .
Ž 5.
608
MA, ADACHI, AND AMEMIYA
and the control uŽ t . g R is defined as u Ž t . s g T z Ž t . s g T e Ž t . q g T x Ž t . y g TA 0 Ž t . x Ž t y t Ž t . .
Ž 6.
for t G t 0 , where g g R n is a constant vector to be chosen in the following discussion. Then from Ž2., Ž4., Ž5., and Ž6. we have for t G t 0
˙e Ž t . s Ž A1 y DC T . e Ž t . y A 2 Ž t . x Ž t . y Ž A1 y DC T . A 0 Ž t . q B Ž t . 4 x Ž t y t Ž t . .
Ž 7.
and d dt
Ž x Ž t . y A 0 Ž t . x Ž t y t Ž t . . . s Ž A1 q bg T . x Ž t . q A 2 Ž t . x Ž t . q B Ž t . x Ž t y t Ž t . . q bg T e Ž t . . Ž 8 .
Next, let us consider the stability of the 2n-dimensional system Ž7. ] Ž8.. The concept of delay independent stabilization is defined as follows w3, 5x. DEFINITION 3. The system Ž2. is called delay independently stabilizable, if there exist a vector g g R n and an n = 2 matrix D such that the 2 n-dimensional systems Ž7. ] Ž8. are globally asymptotically stable, independently of delay t Ž t .. Here we introduce the coordinate transformation defined by x s Ty1 x,
e s Se
for t G t 0 y t ,
Ž 9.
where T and S are n = n constant matrices to be chosen in the sequel. Then the systems Ž7. and Ž8. can be written as d dt
Ž x Ž t . y Ty1A 0 Ž t . T x Ž t y t Ž t . . s Ty1 Ž A1 q bg T . T x Ž t . q Ty1A 2 Ž t . T x Ž t . q Ty1 B Ž t . Tx Ž t y t Ž t . . q Ty1 bg T Sy1 e Ž t .
Ž 10 .
˙e Ž t . s S Ž A1 y DC T . Sy1 e Ž t . y SA 2 Ž t . T x Ž t . y S Ž A1 y DC T . A 0 Ž t . q B Ž t . 4 T x Ž t y t Ž t . .
Ž 11 .
for t G t 0 . Let y Ž t . s x Ž t . y Ty1A 0 Ž t . T x Ž t y t Ž t . . ,
for t G t 0 ;
Ž 12 .
609
LINEAR NEUTRAL SYSTEMS
then it follows from Ž10. that
˙y Ž t . s Ty1 Ž A1 q bg T . T y Ž t . q Ty1 Ž A1 q bg T . A 0 Ž t . T x Ž t y t Ž t . . q Ty1A 2 Ž t . Tx Ž t . q Ty1 BT x Ž t y t Ž t . . q Ty1 bg T Sy1 e Ž t . .
Ž 13 . Owing to the controllability of Ž A1 , b . and the observability of Ž A1 , C T ., it is possible to make all the roots of the characteristic equations detŽ l I y Ž A1 q bg T .. s 0 and detŽ s I y Ž A1 q DC T .. s 0, denoted by l1 , . . . , l n and s 1 , . . . , sn , respectively, be real and distinct by suitable choices of g and D. In fact, from the forms of the matrices A1 and C, it is easy to see that the matrix D can take the simple form
¡
d1 d2 .. .
d ny ky1 d ny k Ds 0 0 .. .
¢
0 0 .. . 0 y1
,
d ny kq1 d ny kq2 .. . dn
0
¦
§
where d i Ž i s 1, 2, . . . , n. can be chosen according to si Ž i s 1, 2, . . . , n.. Now we let the matrices T and S be the Vandermonde matrices defined by
Ts
1 l1 .. .
1 l2 .. .
??? ??? .. .
1 ln .. .
l1ny 1
l2ny1
???
l ny1 n
S1 0
0 , S2
Ss
ž
/
0
,
610
MA, ADACHI, AND AMEMIYA
where S 1 and S 2 are Ž n y k . = Ž n y k . and k = k matrices given as
S1 s
and
S2 s
s 1ny ky1
s 1ny ky2
???
s1
1
s 2nyky1
s 2nyky2 .. .
??? .. .
s2 .. .
1 .. , .
ny ky1 sny k
nyky2 snyk
???
snyk
1
.. .
0 0
ky 1 sny kq1
ky 2 snykq1
???
snykq1
1
ky1 sny kq2
ky2 snykq2
.. .
??? .. .
snykq2 .. .
1 .. , .
snky 1
snky2
???
sn
.. .
1
respectively. Define L, S, pŽ t ., and q Ž t . as L s diag Ž l1 , . . . , l n . ,
S s diag Ž s 1 , . . . , sn .
and p Ž t . ' < y Ž t . < s Ž p1 Ž t . , . . . , pn Ž t . .
T
T
s Ž < y 1 Ž t . < , . . . , < yn Ž t . < . , q Ž t . ' < e Ž t . < s Ž q1 Ž t . , . . . , q n Ž t . .
T
T
s Ž < e1 Ž t . < , . . . , < e n Ž t . < . , r Ž t . ' < x Ž t . < s Ž r 1 Ž t . , . . . , rn Ž t . . s Ž < x1Ž t . < , . . . , < x n Ž t . <.
T
T
for t G t 0 . Then from Ž11., Ž12., and Ž13. we have for t G t 0 Dy p Ž t . s Dy < y Ž t . < s diag Ž sgn y 1 Ž t . , . . . , sgn yn Ž t . . ˙ yŽ t. s diag Ž sgn y 1 Ž t . , . . . , sgn yn Ž t . . Ž T .
y1
Ž A1 q bg T . T y Ž t .
611
LINEAR NEUTRAL SYSTEMS
q Ty1 Ž A1 q bg T . A 0 Ž t . T x Ž t y t Ž t . . q Ty1A 2 Ž t . Tx Ž t . q. Ty1 B Ž t . T x Ž t y t Ž t . . q Ty1 bg T Sy1 e Ž t . F L p Ž t . q Ž < L < < Ty1 < A 0 < T < q < Ty1 < A 2 < T < q< Ty1 < B < T < . r Ž t , t . q < Ty1 bg T Sy1 < q Ž t . F L p Ž t . q Ž < L < < Ty1 < A 0 < T < q < Ty1 < A 2 < T < q< Ty1 < B < T < . r Ž t , t . q < Ty1 bg T Sy1 < q Ž t , t . ,
Ž 14 .
Dy q Ž t . s D y < e Ž t . < s diag Ž sgn e1 Ž t . , . . . , sgn e n Ž t . . ˙ eŽ t . s diag Ž sgn e1 Ž t . , . . . , sgn e n Ž t . . Ž S Ž A1 y DC T . Sy1 e Ž t . ySA 2 Ž t . T x Ž t . y S Ž A1 y DC T . A 0 Ž t . q B Ž t . 4 T x Ž t yt Ž t . . . F S q Ž t . q Ž < S < < S < A0 < T < q < S < A2 < T < q < S < B < T <. r Ž t , t . ,
Ž 15 .
and 0 F yr Ž t . q p Ž t . q < Ty1A 0 Ž t . T < r Ž t , t . F yr Ž t . q p Ž t , t . q < Ty1 < A 0 < T < r Ž t , t . .
Ž 16 .
where pŽ t, t . s Ž p1Ž t, t ., . . . , pnŽ t, t ..T , r Ž t, t . s Ž r 1Ž t, t ., . . . , rnŽ t, t ..T , pi Ž t, t . s maxy t F s F 0 pi Ž t q s . s maxy t F s F 0 < yi Ž t q s .<, ri Ž t, t . s maxyt F s F 0 ri Ž t q s . s maxy t F s F 0 < x i Ž t q s .<, for i s 1, 2, . . . , n. For the sake of convenience, we also define q Ž t . ' < e Ž t . < s Ž q1 Ž t . , . . . , q n Ž t . .
T
T
s Ž < e1 Ž t 0 . < , . . . , < e n Ž t 0 . < . , for t 0 y t F t F t 0 , and q Ž t, t . s Ž q1Ž t, t ., . . . , qnŽ t, t ..T for t G t 0 ; here qi Ž t, t . s maxy t F s F 0 qi Ž t q s . s maxy t F s F 0 < e i Ž t q s .<. Let
R1 s
ž
L 0 0
0 S 0
0 0 yE
/
612
R2 s
MA, ADACHI, AND AMEMIYA
0
< Ty1 bg T Sy1 <
< L < < Ty1 < A 0 < T < q < Ty1 < A 2 < T < q < Ty1 < B < T <
0
0
< S < < S < A0 < T < q < S < A2 < T < q < S < B < T <
E
0
y1 <
A0 < T <
0
,
K s diag Ž E, E, 0 . , T
w Ž t . s Ž pT Ž t . , q T Ž t . , r T Ž t . . g R 3 n , T
w Ž t , t . s Ž pT Ž t , t . , q T Ž t , t . , r T Ž t , t . . , where E and 0 are an n = n unit matrix and an n = n 0 matrix, respectively. Then, with these notations, Ž14., Ž15., and Ž16. can be written in the 3n-dimensional vector form KDy w Ž t . F R1w Ž t . q R 2 w Ž t , t .
Ž 17 .
for t G t 0 . Let < Ty1 < A 0 < T < s Ž a i j . n=n ; then it follows from Lemma 2 that if the matrix yŽ R1 q R 2 . is an M-matrix, there exist positive constants a and M G 1 such that for t G t 0 3n
3n
is1
is1
Ý wi Ž t . F M Ý wi Ž t 0 , t . eya Ž tyt . 0
n
F M Ý Ž pi Ž t 0 , t . q qi Ž t 0 , t . q ri Ž t 0 , t . . eya Ž tyt 0 . is1 n
FM Ý is1
ž
max < yi Ž t 0 q s . < q < e i Ž t 0 . <
ytFsF0
q max < x i Ž t 0 q s . < eya Ž tyt 0 .
/
ytFsF0
n
FM Ý is1
ž
max < x i Ž t 0 q s . < q
ytFsF0
n
Ý js1
q< e i Ž t 0 . < q
ža
ij
max < x j Ž t 0 q s . <
y t FsF0
max < x i Ž t 0 q s . < eya Ž tyt 0 .
ytFsF0
n
s M Ý 2 M1 max < x i Ž t 0 q s . < q < e i Ž t 0 . < eya Ž tyt 0 . , is1
ž
ytFsF0
/
/
/
613
LINEAR NEUTRAL SYSTEMS
where n
M1 s 2 q max
1FiFn
žÝ /
a i j G 1.
js1
Thus n
Ý
n
Ž < x i Ž t . < q < ei Ž t . < . F M Ý 2 M1 max < x i Ž t 0 q s . <
is1
is1
ž
ytFsF0
q< e i Ž t 0 . < eya Ž tyt 0 . Ž 18 .
/
for t G t 0 , which implies that the trivial solution of Ž10. and Ž11. is delay independently exponentially stable. Hence, from the non-singularity of T and S we have that the trivial solution of Ž7. and Ž8. is delay independently exponentially stable. We choose l i - 0, si - 0 Ž i s 1, 2, . . . , n. and set Ns
E 0 yLy1
0 E Ly1 < Ty1 bg T Sy1 < Sy1
0 0 . E
0
Then N Ž R1 q R 2 . s
Ly1
< Ty1 bg T Sy1 <
< L < < Ty1 < A 0 < T < q < Ty1 < A 2 < T < q < Ty1 < B < T <
0 0
Sy1 0
< S < < S < A0 < T < q < S < A2 < T < q < S < B < T < yD
0
,
where D s E y < Ty1 < A 0 < T < y Ly1 < Ty1 bg T Sy1 < Sy1 = Ž < S < < S < A0 < T < q < S < A2 < T < q < S < B < T <. q Ly1 Ž < L < < Ty1 < A 0 < T < q < Ty1 < A 2 < T < q < Ty1 < B < T < . . Thus, from Lemma 1 we have that yŽ R1 q R 2 . is an M-matrix iff D is an M-matrix which is also equivalent to the matrix Q s < L < y 2 < L < < Ty1 < A 0 < T < y < Ty1 bg T Sy1 < = Ž < S < A 0 < T < q < Sy1 < Ž < S < A 2 < T < q < S < B < T < . . y < Ty1 < A 2 < T < y < Ty1 < B < T < being an M-matrix. Therefore, in the following, we consider the matrix Q.
614
MA, ADACHI, AND AMEMIYA
We choose l1 , . . . , l n and s 1 , . . . , sn such that
l1 , . . . , l nyk ª 0,
s 1 , . . . , snyk ª 0,
l ny kq1 , . . . , l n ª y`,
snykq1 , . . . , sn ª y`.
Hence, we can express l1 , . . . , l n and s 1 , . . . , sn as
li s o
1
ž / l
si s o
,
1
ž / l
for i s 1, 2, . . . , n y k, and
li s O Ž l. ,
si s O Ž l .
for i s n y k q 1, . . . , n, where l is a positive number which is sufficiently large. Following the papers by Amemiya w1]5x, the following notations are again introduced here. Let j Ž l. g R be a continuous function; if there exists a constant m such that j Ž l.rl m < - q` as < l < ª q` and < j Ž l.rl my a < ª q` as < l < ª q` for any positive constant a, then j is called a function of order m denoted by OrdŽ j . s m. Let m denote a matrix, a column vector, or row vector every element of which is a function of order m. Thus, with these notions, the matrices T, S, Ty1 , and Sy1 can be expressed in the forms
Ts
ž
T11 T21
T12 , T22
Ty1 s
/
ž
Tˆ11
Tˆ12
Tˆ21
Tˆ22
¡
0
/
where
¡ T11 s
0 y1 .. .
¦ ,
¢ yn q k q 1 §
T12 s
¦
1 .. .
¢ nyky1 §
615
LINEAR NEUTRAL SYSTEMS
¡
yn q k y 1
T21 s
Tˆ21 s
ž
ž
nyky2
ž
y2 n q 2 k
Tˆ22 s
.. .
§
ny1
/
,
n y k y 3 ??? n y 2 k y 1
/
,
y2 n q 2 k q 1 ??? yn q k y 1
yn q k
ž
¢
1 ??? n y k y 1
0
¦
nyk nykq1
T22 s
§
yn q 1 Tˆ11 s
¡
,
.. .
¢ Tˆ12 s
¦
yn q k
yn q k y 1 ??? yn q 1
/
,
/
and S1 s
1ynqk
ž
S2 s
ž
2 y n q k ??? 0
y1 q k
y2 q k ??? 0
¡ nyky1 ¦ Ž S1 .
y1
nyky2
s
¢
.. .
,
,
,
¡ 1yk ¦ Ž S2 .
§
0
/
/
y1
2yk
s
¢
.. .
.
§
0
By Assumptions I]III and papers w1]5, 19x, the matrices < Ty1 bg T Sy1 <, < S < A 2 < T < q < S < B < T <, < Ty1 < A 2 < T < q < Ty1 < B < T <, < Ty1 < A 0 < T <, and < Ty1 < B < T < have the forms
< Ty1 bg T Sy1 < s
nyky2
n y 2k
yn q k
2yn
0
,
616
MA, ADACHI, AND AMEMIYA
< S < A2 < T < q < S < B < T < s
y1 <
A2 < T < q < T
y1 <
< Ty1 < A 0 < T < s
y1 <
B
nyky1
2k y n y 1
ny1
2n y 2k y 2
2k y 2n
0 2n y 2k y 1
2k y 2n y 1
0
y2
y1
B
yn q k y 1
0
y1
y2
2n y 2k y 2
2k y 2n
0
0
,
0
,
,
.
Furthermore, by Assumption III and the form of the matrices T and S, we can also find that the matrix < S < A 0 < T < has the form
< S < A0 < T < s
kyn
nyk
2k y n y 2
ny2
0
.
Therefore, the matrices < L < < Ty1 < A 0 < T <, < Ty1 bg T Sy1 < < S < A 0 < T <, and < Ty1 bg T Sy1 < < Sy1 <Ž< S < A 2 < T < q < S < B < T <. have the same structure as the matrix < Ty1 < A 2 < T < q < Ty1 < B < T <. Thus, by the same arguments as in w1x, we can find that the matrix Q becomes an M-matrix as long as l is large enough. Thus, we have our main result. THEOREM. If Assumptions I]III are satisfied, then Ž2. is delay independently stabilizable. Remark 3. It is easy to see that if A 0 Ž t . s 0 for t G t 0 in Ž2., then the system Ž2. reduces to the retarded system considered in Refs. w3, 5x. Therefore, the above Theorem generalizes the results in w3, 5x.
LINEAR NEUTRAL SYSTEMS
617
5. ALGORITHMS Now we can construct the stabilizing controller the existence of which is guaranteed under Assumptions I]III by the Theorem. The system considered must be transformed into the one whose nominal system is in controllable canonical form. It must also be checked whether Assumption II is satisfied. From Assumption II we can determine the value of k. Step 1. After transforming the system, check whether the matrix functions AŽ t ., B Ž t ., and A 0 Ž t . of the system satisfy Assumption III. Step 2. Select positive number l to be sufficiently large so that the matrix Q is an M-matrix. Step 3. Select the roots, l i s oŽ1rl., si s oŽ1rl. for i s 1, 2, . . . , n y k, and l i s O Ž l., si s O Ž l. for i s n y k q 1, . . . , n, of the characteristic equations detŽ l I y Ž A1 q bg T .. s 0 and detŽ s I y Ž A1 q DC T .. s 0 as functions of a parameter l. Step 4. Vector g can be decided from the relation between eigenvalues and coefficients of the characteristic equation detŽ l I y Ž A1 q bg T .. s 0. The matrix D with the form of
¡
d1 d2 .. .
d ny ky1 d ny k Ds 0 0 .. .
¢
0
0 0 .. .
¦
0 y1 d ny kq1 d ny kq2 .. . dn
§
can also be found from the relation between eigenvalues and coefficients of the characteristic equation detŽ s I y Ž A1 q DC T .. s 0. Step 5. From the value of g and D, the controller can be constructed.
6. CONCLUDING REMARKS In this paper, the delay-independent stabilizability of the linear neutral system Ž2. has been considered using the differential-difference inequali-
618
MA, ADACHI, AND AMEMIYA
ties and the transformation techniques developed in w1]5, 19]21, 23x. A sufficient condition for the delay-independent stabilizability of linear retarded systems with limited measurable state variables and linear neutral systems with measurable state variables which had been obtained by Amemiya and Ma et al. has been proved also to be valid for the linear neutral system Ž2. with limited measurable state variables. Our sufficient condition is also independent of the construction of Lyapunov functions and Lyapunov functionals and has a geometric pattern similar to w3, 5, 19, 24x too. Finally, a procedure to design the state feedback controller which can guarantee the delay-independent stabilizability of the system considered has been presented.
ACKNOWLEDGMENTS The authors are very grateful to the referee for his valuable comments. The first author also thanks Professor Y. Takeuchi for his valuable suggestions.
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13. D. A. O. Connor and T. J. Tarn, On stabilization by state feedback for neutral differential difference equations, IEEE Trans. Automat. Control 28 Ž1983., 615]618. 14. E. Cheres, S. Gutman, and Z. J. Plamor, Stabilization of uncertain dynamic systems including state delay, IEEE Trans. Automat. Control 34 Ž1989., 1199]1203. 15. J. Hale, ‘‘Theory of Functional Differential Equations,’’ Springer-Verlag, New York, 1977. 16. M. Q. Jacobs and C. E. Langenhop, Criteria for the function space controllability of linear neutral systems, SIAM J. Control Optim. 14 Ž1976., 1009]1048. 17. G. Leitmann, Guaranteed asymptotic stability for some linear systems with bounded uncertainties, J. Dynam. Systems Measurement Control 101 Ž1979., 212]216. 18. G. Leitmann, On the efficacy of nonlinear control in uncertain linear systems, J. Dynam. Systems Measurement Control 102 Ž1981., 94]102. 19. Wanbiao Ma, N. Adachi, and T. Amemiya, On the delay independent stabilization of uncertain linear systems of neutral-type, J. Optim. Theory Appl. 84 Ž1995., 393]405. 20. Ligeng Si, ‘‘Stability of Neutral Differential Difference Equations with Delays,’’ Inner Mongolia Education Press, Huhhot, 1994. 21. Ligeng Si and Wanbiao Ma, Differential-difference inequalities with unbounded delays and their applications, J. Nonlinear Anal. 17 Ž1991., 787]801. 22. A. Thowsen, Uniform ultimate boundedness of the solutions of uncertain dynamic delay systems with state-dependent and memoryless feedback control, Internat. J. Control 37 Ž1983., 1135]1143. 23. H. Tokumaru, N. Adachi, and T. Amemiya, Macroscopic stability of interconnected systems, in ‘‘6th IFAC ID,’’ Vol. 44, No. 4, 1975. 24. Kehui Wei, Quadratic stabilizability of linear systems with structural independent timevarying uncertainties, IEEE Trans. Automat. Control 35 Ž1990., 268]277.