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An improved criterion for exponential stability of linear systems with multiple time delays
Applied Mathematics and Computation 202 (2008) 870–876
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
An improved criterion for exponential stability of linear systems with multiple time delays Phan T. Nam Department of Mathematics, Quynhon University, 170 An Duong Vuong Road, Binhdinh 84, Viet Nam
a r t i c l e
i n f o
a b s t r a c t
Keywords: Exponential stability Time delays Lyapunov functional Linear matrix inequalities
Exponential stability of linear systems with multiple time delays is studied in this paper. By using an improved Lyapunov functional, we have obtained an improved criterion, which is strictly less conservative than the very recent criterion in Ren and Cao [Fengli Ren, Jinde Cao, Novel a-stability of linear systems with multiple time delays, Appl. Math. Comput. 181 (2006), 282–290], and an estimation of Lyapunov factor. A numerical example is also given to shown the superiority of our result to those in the literature. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction Consider the linear systems with multiple time delays of the form: 8 m P < xðtÞ _ ¼ A0 xðtÞ þ Ak xðt sk Þ; k¼1 : xðtÞ ¼ /ðtÞ; t 2 ½s; 0;
t 2 Rþ
ð1:1Þ
where 0 6 sk 6 s; k ¼ 1; 2; . . . ; m, are positive delays and s is a positive constant, xðtÞ 2 Rn is the state, Ak ; k ¼ 0; 1; 2; . . . ; m are given matrices, and initial condition is /ðhÞ 2 Cð½s; 0; Rn Þ. Definition 1.1. The system (1.1) is a-stable, with a > 0, if there is a positive number N such that for each /ð:Þ, the solution xðt; /Þ of the system (1.1) satisfies kxðt; /Þk 6 Neat k/ks ;
8t P 0;
where k/ks ¼ maxfk/ðtÞk : t 2 ½s; 0g. N is called Lyapunov factor. Exponential stability with convergence rate a of linear systems with multiple delays has been an interesting topic in recent years (see [2,4,6–10] and references therein). Based on the characteristic function of linear time-delay systems, in Cao et al. [2] have given criteria for a-stability of uncertain systems with multiple delays. By using both the time-domain and the frequency-domain methods, in Xu [10] has proposed other a-stability criteria for linear systems with multiple uncertain delays. By combining the Lyapunov method and the linear matrix inequality technique, in Mondie and Kharitonov [6] not only give a a-stability criterion for system (1.1) but also give an estimation of Lyapunov factor. Based on the Hamiltonian matrix
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P.T. Nam / Applied Mathematics and Computation 202 (2008) 870–876
871
and a solvable algebraic Riccati equation, a delay-dependent criterion for robust stability of uncertain system with time delay has proposed in [3]. Exponential stability for linear non-autonomous systems with multiple delays or for linear uncertain polytopic time-delay systems have been studied in Ref. [7,8]. Recently, by combining the Lyapunov–Krasovski stability theory and parameterized neutral model transformation, in Ren and Cao [4] have just given a novel criterion for exponential stability of system (1.1). Although this criterion is less conservative than those in the published literature, this criterion has still depended on the stability of operators Dk ðxt Þ ¼ xðtÞþ Rt G xðsÞ ds; k ¼ 1; 2; . . . ; m and Lyapunov factor has not been estimated. Therefore, the main purpose of this paper is to tsk k find a new criterion, which does not depend on the stability of operators Dk ðxt Þ and can give an estimation of Lyapunov factor. By using an improved Lyapunov functional, we avoid the use of the stability assumption on the main operators and derive an improved a-stability criterion given in term of LMIs. Hence, our criterion will be less conservative than that given by Ren and Cao [4]. Moreover, by simple computations we have given an estimation of Lyapunov factor. The following lemmas are needed for our main result. Lemma 1.1. ½1; 7. Assume that S 2 Rnn is a symmetric positive definite matrix. Then for every Q 2 Rnn , 2hQy; xi hSy; yi 6 hQS1 Q T x; xi;
8x; y 2 Rn :
If we take S ¼ I then we have j2hQy; xij 6 kyk2 þ kQxk2 . Lemma 1.2 [4]. For any constant matrix M 2 Rnn ; M ¼ M T > 0, scalar r > 0, vector function w : ½0; r ! Rn such that the integrations concerned are well defined, then Z r T Z r Z r wðsÞ ds M wðsÞ ds 6 r wT ðsÞMwðsÞ ds: 0
0
0
Q ðxÞ Lemma 1.3 [4]. The following linear matrix inequality T S ðxÞ affinely on x, is equivalent to
SðxÞ RðxÞ
< 0, where Q ðxÞ ¼ Q T ðxÞ; RðxÞ ¼ RT ðxÞ and SðxÞ depends
(i) Q ðxÞ < 0; RðxÞ ST ðxÞQ 1 ðxÞSðxÞ < 0, (ii) RðxÞ < 0; Q ðxÞ SðxÞR1 ðxÞST ðxÞ < 0.
Lemma 1.4 [4]. The operator Dk ðxt Þ ¼ xðtÞ þ metric matrix Mk such that ! dk sk GTk Mk < 0: I Mk
Rt
tsk
Gk xðsÞ ds is stable if there exist a scalar 0 < dk < 1 and a positive definite sym-
We also need the result of Fengli Ren and Jinde Cao in [4] for our main result. So, we recall their theorem and proof as follows. Theorem 1.5 [4]. For given sk > 0 and b > 1, system (1.1) is a-stable if there exist positive definite symmetric matrices Rt X; W; F 11 ; F 22 ; F 33 and any matrices Y k , k ¼ 1; 2; . . . ; m, F 12 ; F 13 ; F 23 , such that the operators Dk ðyt Þ ¼ xðtÞ þ tsk Y k X 1 yðsÞ ds, k ¼ 1; 2; . . . ; m are stable and the following LMIs hold 0
X B BI B B B B N ¼ BI B B . B . B . @ I
M1
M2
...
Mm
bs1 X
0
...
0
I
..
.
..
.
.. .
.. .
..
.
..
.
0
I
...
I
1 C C C C C C C < 0; C C C C A
bsm X
X þ F 22 < 0; 0 1 F 11 F 12 F 13 B C B I F 22 F 23 C > 0; @ A I
I
ð1:2Þ
F 33
where B0 ¼ aI þ A0 ; Bk ¼ eask Ak ; k ¼ 1; 2; . . . ; m and
ð1:3Þ
ð1:4Þ
872
P.T. Nam / Applied Mathematics and Computation 202 (2008) 870–876
0
R1
U1
U2
...
Um
W1
W2
...
Wm1
Wm
I
K1
I
I .. . .. . .. . .. . .. . .. .
0 .. . .. .
... .. .
0 .. . .. .
C1 .. . .. .
B2 X .. . .. .
... .. . .. .
Bm1 X .. . .. .
Bm X .. . .. .
.. . .. . .. . .. .
I .. . .. . .. . .. .
Km
B1 X
B2 X
...
Bm1 X
Cm
I .. . .. . .. .
W þ s1 F 33
0 .. .
... .. . .. . I
0 .. . .. . .. .
0 .. . .. . .. .
I
I
W þ sm F 33
B B B B B B B B B B B B B X¼B B B B B B B B B B B B B @
I I I I I I
0
I .. . .. .
I I I I I I m X ðB0 X þ Y k þ XBT0 þ Y Tk þ W þ sk F 11 Þ; R1 ¼
I .. . I
1 C C C C C C C C C C C C C C; C C C C C C C C C C C C A
ð1:5Þ
Uk ¼ XBT0 þ Y Tk þ F 12 ;
k¼1
Wk ¼ mBk X Y k þ sk F 13 ; Mk ¼
Ck ¼ Bk X Y k þ F 23 ;
ðbsk Y k ; 0; . . . ; 0ÞT2mþ1;1 ;
Kk ¼ sk ðb 1ÞX;
k ¼ 1; 2; . . . ; m:
Proof. We take the following change of the state variable t 2 Rþ ;
yðtÞ ¼ eat xðtÞ;
then the system (1.1) is transformed to the delay system 8 m P < yðtÞ _ Bk yðt sk Þ ¼ B0 yðtÞ þ k¼1 : at yðtÞ ¼ e /ðtÞ; t 2 ½s; 0
ð1:6Þ
where B0 ¼ aI þ A0 ; Bk ¼ eask Ak ; k ¼ 1; 2; . . . ; m. Consider the following Lyapunov functional V¼
m X ðV 1k þ V 2k þ V 3k þ V 4k Þ
ð1:7Þ
k¼1
where V 1k ¼ DTk ðyt ÞPDk ðyt Þ; Z t Z t V 2k ¼ b yT ðuÞGTk PGk yðuÞ duds; V 3k ¼
Z
tsk
s
t
yT ðsÞTyðsÞ ds;
tsk
1T 0 1 1 0 F 11 F 12 F 13 yðsÞ yðsÞ B C B C C B V 4k ¼ @ Gk yðuÞ A P @ I F 22 F 23 AP@ Gk yðuÞ A 0 ssk I I F 33 yT ðs sk Þ yT ðs sk Þ Rt 1 and Dk ðyt Þ ¼ yðtÞ þ tsk Gk yðsÞ ds; Gk ¼ Y k X ; T > 0; P > 0; P ¼ diagfP; P; Pg. By Fengli Ren’s proof, if (1.2)–(1.4) hold then we have Z
t
Z
0
s
V P0 and 0
1T 0 1 yðtÞ yðtÞ B R t G yðsÞ ds C B R t G yðsÞ ds C B ts1 1 C B ts1 1 C B C B C B C B C . . .. .. B C B C m Z t BR C BR C X B B C C yT ðsÞGTk ðP þ PF 22 PÞGk yðsÞ ds V_ 6 B t Gm yðsÞ ds C XB t Gm yðsÞ ds C þ ts ts m m B C B C k¼1 tsk B yðt s Þ C B yðt s Þ C 1 1 B C B C B C B C B C B C .. .. @ A @ A . . yðt sm Þ
yðt sm Þ
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P.T. Nam / Applied Mathematics and Computation 202 (2008) 870–876
with P þ PF 22 P < 0 and X < 0: This implies that there exists a positive number d such that V_ < dkyðtÞk2 : Combining with the stability assumption on the main operators Dk ðyt Þ, we have the system (1.6) is asymptotically stable. This implies that the system (1.1) is a-stable. The proof is completed. Note that the stability of the operators Dk ðyt Þ can check by using Lemma 1.4 . h Remark 1. If P þ PF 22 P < 0 then by Lemma 1.2, we have 0
1T 0 1 yðtÞ yðtÞ R R B t G yðsÞ ds C B t G yðsÞ ds C B ts1 1 C B ts1 1 C B C B C B C B C . . B C B C . . B B C C . . Z m t X BRt C BRt C T T B B C C y ðsÞGk ðP þ PF 22 PÞGk yðsÞ ds 6 B DB G yðsÞ ds G yðsÞ ds m m C C tsm tsm ts B B C C k k¼1 B yðt s1 Þ C B yðt s1 Þ C B C B C B C B C .. .. B C B C @ @ A A . . yðt sm Þ
yðt sm Þ
where 0
0 B B0 B B0 B B B B0 B B B0 D¼B B B0 B B .. B. B B B0 B B @0
1
0
0
...
0
0
0
...
0
0
s1 1 ðP þ PF 22 PÞ
0
...
0
0
0
...
0
0
s1 2 ðP þ PF 22 PÞ
0
0
0
0
...
0
0
0
..
0
.. .
.. .
.. .
.. .
0
0
0
s1 m ðP þ PF 22 PÞ
0
0
...
0
0 .. .
0 .. .
0 .. .
0 .... ..
0 .. .
0 .. .
... .. .
0 .. .
0
0
0
0
0
0
...
0
0
0
0
0
0
0
...
0
C 0C C 0C C C .. C .C C C 0C C 6 0: C 0C C C C C C 0C C C 0A
0
0
0
0
0
0
...
0
0
0
.
ð1:8Þ
Hence, we have 0
0 1T 1 yðtÞ yðtÞ B Rt B Rt C C B ts G1 yðsÞ ds C B ts G1 yðsÞ ds C B B C C 1 1 B B C C B B C C .. .. B B C C . . B B C C R R B t C C t _V 6 B B ts Gm yðsÞ ds C ðX þ DÞB ts Gm yðsÞ ds C: m m B B C C B B C C B yðt s1 Þ C B yðt s1 Þ C B B C C B B C C .. .. B B C C . . @ @ A A yðt sm Þ
yðt sm Þ
Let X ¼ P 1 ; W ¼ XTX; Y k ¼ Gk X; k ¼ 1; 2; . . . m and multiplying both sides of inequality X þ D < 0 by diagfX; X; . . . ; Xg and using Lemma 1.3, then X þ D < 0 is equivalent to 0 B B B B N1 ¼ B B B @
X1
M1
M2
...
Mm
I
bs1 X
I .. .
I .. .
0 .. . .. .
... .. . .. .
0 .. .
I
I
...
I
bsm X
where X1 ¼ X þ D1 and
0
1 C C C C C<0 C C A
874
P.T. Nam / Applied Mathematics and Computation 202 (2008) 870–876
0
0
B0 B B B0 B B B0 B B B0 D1 ¼ B B B0 B. B. B. B B0 B B @0
0
0
...
0
0
0
s1 1 ðX þ F 22 Þ
0
...
0
0
0
0
s1 2 ðX þ F 22 Þ
0
0 0
0 0
0 .. . 0
0 s1 m ðX þ F 22 Þ
0 .. . 0
0 .. . 0
0 .. .
0 .. .
0 .. .
0 .... ..
0 .. .
0 .. .
... .. .
0 .. .
0
0
0
0
0
0
...
0
0
0
0
0
0
0
...
0
0
0
0
0
0
...
0
1
0
0
...
0
... .. . ...
0 .. . 0
0
0C C C 0C C .. C .C C C 0C C 6 0: C 0C C C C C 0C C C 0A
0
0
...
Thus, if X þ F 22 < 0 and N1 < 0 then there is a positive number k such that V_ 6 kðkyðtÞk2 þ kyðt s1 Þk2 þ þ kyðt sm Þk2 Þ: Remark 2. Since N1 6 N, the condition N1 < 0 is less conservative than the condition N < 0. 2. Main result Combining Remark 1 and the following Lyapunov functional, we get an improved a-stability criterion of the system (1.1) as follows: Theorem 2.1. The system (1.1) is a-stable if the exist b > 1, positive definite symmetric matrices X; W; F 11 ; F 22 ; F 33 and any matrices Y k ; F 12 ; F 13 ; F 23 ,k ¼ 1; 2; . . . ; m, which which satisfy the LMIs (1.3), (1.4) and N1 < 0. Proof. Consider the following Lyapunov functional: V ¼ V þ V5 where V 5 ¼ gkyðtÞk2 ; and ( ) 1 k ; k > 0: g ¼ min P 2 2 2kB0 k þ m k¼1 kBk k Since V P 0, we have V P gkyðtÞk2 : Taking the derivative of V in t and using Lemma 1.1, we have _ T yðtÞ þ V_ V_ ¼ 2gyðtÞ 6 2g yT ðtÞBT0 yðtÞ þ
m X
! yT ðt sk ÞBTk yðtÞ
k¼1
6g
2kB0 k þ
m X
!
kBk k2 kyðtÞk2 þ
k¼1
m X
k kyðtÞk2 þ kyðt s1 Þk2 þ þ kyðt sm Þk2 !
kyðt sk Þk2
kðkyðtÞk2 þ kyðt s1 Þk2 þ þ kyðt sm Þk2 Þ 6 0:
k¼1
This follows that V ðtÞ V ð0Þ 6 0;
8t 2 Rþ :
Since V ðtÞ P gkyðtÞk2 , we have sffiffiffiffiffiffiffiffiffiffiffiffi V ð0Þ : kyðtÞk 6 g It means yðtÞ is bounded. This implies that the system (1.1) is a-stable. The proof is completed.
h
Remark 3. To reduce the conservatism of our stability criterion, we have used the function V 5 , which allows our criterion Rt does not depend on the stability of operators Dk ðyt Þ ¼ xðtÞ þ tsk Gk yðsÞ ds; k ¼ 1; . . . ; m. In addition, the condition N1 < 0 is less conservative than the condition N < 0. Hence, our criterion is strictly less conservative than the criterion in Ref. [4]. On the other hand, Fengli Ren’s criterion is less conservative than those in the literature. Therefore, our criterion will be less conservative than those in the literature. We can shown directly the superiority of our result to those in the literature by considering the following numerical example, which is considered in Ref. [4].
875
P.T. Nam / Applied Mathematics and Computation 202 (2008) 870–876
Example 2.2. Consider the following linear system ( pffiffiffi _ xðtÞ ¼ A0 xðtÞ þ A1 xðt 1Þ þ A2 xðt 3Þ; t P 0 pffiffiffi xðtÞ ¼ /ðtÞ; t 2 ½ 3; 0; 7 1 10:1 14:2 5:2 ; A1 ¼ ; A2 ¼ where A0 ¼ 0:5 5:5 6:6 10:2 3:6 Matlab [5], the LMIs solutions of Theorem 2.1 are found 3:5370
X ¼ 107
2:3266
!
2:1428
W ¼ 108
;
ð2:1Þ 6:2 . Taking b ¼ 12; a ¼ 0:1. Using the LMI Toolbox in 5:1 !
1:4004
;
1:7142 1:4004 1:0252 ! ! 25:2902 4:4060 37:4902 2:2060 Y1 ¼ ; Y2 ¼ ; 4:2402 20:7282 4:0402 25:1282 ! ! 4:6381 2:3249 2:9990 0:0500 ; F 12 ¼ ; F 11 ¼ 106 2:3249 1:1687 0:0510 0:0193 ! ! 1:3047 0:0007 2:2534 1:4797 7 F 13 ¼ ; F 22 ¼ 10 ; 0:0021 1:7505 1:4797 1:0941 ! ! 5:0555 5:5001 4:9181 2:4645 ; F 33 ¼ 106 : F 23 ¼ 4:0050 5:0506 2:4645 1:2386 2:3266
See Table 1. Remark 4. To get the bound more precisely, integrating both sides of the above inequality V_ 6 0 from 0 to t, we have V ðtÞ V ð0Þ 6 0;
8t 2 Rþ :
Since V ðtÞ P gkyðtÞk2 , we have m X
gkyð0Þk2 6 V ð0Þ ¼ gkzð0Þk2 þ
V 1k ð0Þ þ V 2k ð0Þ þ V 3k ð0Þ þ V 4k ð0Þ:
k¼1
On the other hand, we have yð0Þ þ
V 1k ð0Þ ¼
Z
!T
0
Gk yðsÞ ds
P yð0Þ þ
Z
sk
yð0Þ þ
¼
Z
!
0
Gk yðsÞ ds sk
!T
0
Y k X 1 yðsÞ ds
X 1 yð0Þ þ
sk
Z
!
0
Y k X 1 yðsÞ ds
sk
¼ yT ð0ÞX 1 yð0Þ þ yT ð0ÞX 1 Y k X 1
Z
0
Y k X 1
yðsÞ ds þ
Z
sk
Y kX
þ
1
Z
!T
0
yðsÞ ds
X
1
YkX
sk
and
R0 sk
yðsÞ ds ¼
V 1k ð0Þ 6
R0 sk
kX
k þ 2kX
!T yðsÞ ds
X 1 yð0Þ
sk
Z
!
0
yðsÞ ds
sk
xðsÞeas ds 6 k/ks
1
1
0
R0 sk
ask
eas ds ¼ 1ea
k/ks . Hence,
! ask 2 1 eask Þ 2 ð1 e 1 3 þ kX k kY k k k kY k k k/k2s : a a2
1 2
Table 1 Comparison between results of our result and recent ones Method
Year
Convergence rate a
Y.J. Sun [9] D.Q. Cao [2] Bugong Xu [10] Fengli Ren [4] Ours
1998 2003 2003 2006 2008
0.04 0.09 Not applicable 0.095 0.1
876
P.T. Nam / Applied Mathematics and Computation 202 (2008) 870–876
By the same computations, we have Z 0 Z 0 V 2k ð0Þ ¼ be2as yT ðuÞX 1 Y Tk X 1 Y k X 1 yðuÞ duds sk
6 bkX
s
1 3
k kY k k2
Z
0 sk
Z
0
yT ðuÞyðuÞ duds s
2ask 1 þ e2ask k/k2s ; 4a2 Z 0 yT ðuÞyðuÞ du 6 k/k2s kTk e2au du
6 bkX 1 k3 kY k k2 V 3k ð0Þ ¼
Z
0 sk
sk
1 e2ask 1 2 kX k kWkk/k2s 6 2a and V 4k ð0Þ ¼ 0. We denote 1 eask ð1 eask Þ2 N 1k ¼ kX 1 k þ 2kX 1 k2 kY k k þ kX 1 k3 kY k k2 ; a a2 2ask 2ask 2ask 1 þ e 1e kX 1 k2 kWk N 2k ¼ bkX 1 k3 kY k k2 ; N 3k ¼ 4a2 2a P gþ m k¼1 ðN 1k þ N 2k þ N 3k Þ : N2 ¼ g
and
Then, we have gkyðtÞk2 6 V ð0Þ 6 gN 2 k/k2s : Hence, kyðtÞk 6 Nk/ks . This implies kxðt; /Þk 6 Nk/ks eat ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P gþ m k¼1 ðN 1k þ N 2k þ N 3k Þ N¼ : g
8t 2 Rþ : Thus, the Lyapunov factor is
3. Conclusions In this paper, we present an improved a-stability criterion for linear systems with multiple time delays. This criterion is strictly less conservative than the very recent criterion in [4]. Hence, our criterion is less conservative than those in the literature. Moreover, an estimation the Lyapunov factor is also given. Acknowledgement The author would like to thank anonymous referees for valuable comments and suggestions, which have improved the paper. The author would like to thank Professor Fengli Ren for giving the example solving. References [1] [2] [3] [4] [5] [6] [7] [8]
S. Boyd, El. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix and Control Theory, SIAM Studies in Applied Mathematics, vol. 15, SIAM, PA, 1994. D.Q. Cao, P. He, K.Y. Yhang, Exponential stability criteria of uncertain systems with multiple time delays, J. Math. Anal. Appl. 283 (2003) 274–362. J. Cao, J. Wang, Delay-dependent robust stability of uncertain nonlinear systems with time delay, Appl. Math. Comput. 154 (1) (2004) 289–297. Fengli Ren, Jinde Cao, Novel a-stability of linear systems with multiple time delays, Appl. Math. Comput. 181 (2006) 282–290. P. Gahinet, A. Nemirovski, A. Laub, M. Chilali, LMI Control Toolbox User’s Guide, The Mathworks, Natick, Massachusetts, 1995. S. Mondie, V.L. Kharitonov, Exponential estimates for retarded time-delay systems: an LMI approach, IEEE Trans. Aut. Contr. 50 (2) (2005) 268–273. V.N. Phat, P.T. Nam, Exponential stability criteria of linear non-autonomous systems with multiple delays, Elect. J. Diff. Equations 58 (2005) 1–9. P.T. Nam, V.N. Phat, Robust exponential stability and stabilization of linear uncertain polytopic time-delay systems, J. Contr. Theory Appl. 5 (3) (2007) 123–131. [9] Y.J. Sun, J.G. Hsieh, On a-stability criteria of nonlinear systems with multiple time delays, J. Franklin Inst. 335B (1998) 695–705. [10] B. Xu, Stability criteria for linear systems with uncertain delays, J. Math. Anal. Appl. 284 (2003) 455–470.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
An improved criterion for exponential stability of linear systems with multiple time delays Phan T. Nam Department of Mathematics, Quynhon University, 170 An Duong Vuong Road, Binhdinh 84, Viet Nam
a r t i c l e
i n f o
a b s t r a c t
Keywords: Exponential stability Time delays Lyapunov functional Linear matrix inequalities
Exponential stability of linear systems with multiple time delays is studied in this paper. By using an improved Lyapunov functional, we have obtained an improved criterion, which is strictly less conservative than the very recent criterion in Ren and Cao [Fengli Ren, Jinde Cao, Novel a-stability of linear systems with multiple time delays, Appl. Math. Comput. 181 (2006), 282–290], and an estimation of Lyapunov factor. A numerical example is also given to shown the superiority of our result to those in the literature. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction Consider the linear systems with multiple time delays of the form: 8 m P < xðtÞ _ ¼ A0 xðtÞ þ Ak xðt sk Þ; k¼1 : xðtÞ ¼ /ðtÞ; t 2 ½s; 0;
t 2 Rþ
ð1:1Þ
where 0 6 sk 6 s; k ¼ 1; 2; . . . ; m, are positive delays and s is a positive constant, xðtÞ 2 Rn is the state, Ak ; k ¼ 0; 1; 2; . . . ; m are given matrices, and initial condition is /ðhÞ 2 Cð½s; 0; Rn Þ. Definition 1.1. The system (1.1) is a-stable, with a > 0, if there is a positive number N such that for each /ð:Þ, the solution xðt; /Þ of the system (1.1) satisfies kxðt; /Þk 6 Neat k/ks ;
8t P 0;
where k/ks ¼ maxfk/ðtÞk : t 2 ½s; 0g. N is called Lyapunov factor. Exponential stability with convergence rate a of linear systems with multiple delays has been an interesting topic in recent years (see [2,4,6–10] and references therein). Based on the characteristic function of linear time-delay systems, in Cao et al. [2] have given criteria for a-stability of uncertain systems with multiple delays. By using both the time-domain and the frequency-domain methods, in Xu [10] has proposed other a-stability criteria for linear systems with multiple uncertain delays. By combining the Lyapunov method and the linear matrix inequality technique, in Mondie and Kharitonov [6] not only give a a-stability criterion for system (1.1) but also give an estimation of Lyapunov factor. Based on the Hamiltonian matrix
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P.T. Nam / Applied Mathematics and Computation 202 (2008) 870–876
871
and a solvable algebraic Riccati equation, a delay-dependent criterion for robust stability of uncertain system with time delay has proposed in [3]. Exponential stability for linear non-autonomous systems with multiple delays or for linear uncertain polytopic time-delay systems have been studied in Ref. [7,8]. Recently, by combining the Lyapunov–Krasovski stability theory and parameterized neutral model transformation, in Ren and Cao [4] have just given a novel criterion for exponential stability of system (1.1). Although this criterion is less conservative than those in the published literature, this criterion has still depended on the stability of operators Dk ðxt Þ ¼ xðtÞþ Rt G xðsÞ ds; k ¼ 1; 2; . . . ; m and Lyapunov factor has not been estimated. Therefore, the main purpose of this paper is to tsk k find a new criterion, which does not depend on the stability of operators Dk ðxt Þ and can give an estimation of Lyapunov factor. By using an improved Lyapunov functional, we avoid the use of the stability assumption on the main operators and derive an improved a-stability criterion given in term of LMIs. Hence, our criterion will be less conservative than that given by Ren and Cao [4]. Moreover, by simple computations we have given an estimation of Lyapunov factor. The following lemmas are needed for our main result. Lemma 1.1. ½1; 7. Assume that S 2 Rnn is a symmetric positive definite matrix. Then for every Q 2 Rnn , 2hQy; xi hSy; yi 6 hQS1 Q T x; xi;
8x; y 2 Rn :
If we take S ¼ I then we have j2hQy; xij 6 kyk2 þ kQxk2 . Lemma 1.2 [4]. For any constant matrix M 2 Rnn ; M ¼ M T > 0, scalar r > 0, vector function w : ½0; r ! Rn such that the integrations concerned are well defined, then Z r T Z r Z r wðsÞ ds M wðsÞ ds 6 r wT ðsÞMwðsÞ ds: 0
0
0
Q ðxÞ Lemma 1.3 [4]. The following linear matrix inequality T S ðxÞ affinely on x, is equivalent to
SðxÞ RðxÞ
< 0, where Q ðxÞ ¼ Q T ðxÞ; RðxÞ ¼ RT ðxÞ and SðxÞ depends
(i) Q ðxÞ < 0; RðxÞ ST ðxÞQ 1 ðxÞSðxÞ < 0, (ii) RðxÞ < 0; Q ðxÞ SðxÞR1 ðxÞST ðxÞ < 0.
Lemma 1.4 [4]. The operator Dk ðxt Þ ¼ xðtÞ þ metric matrix Mk such that ! dk sk GTk Mk < 0: I Mk
Rt
tsk
Gk xðsÞ ds is stable if there exist a scalar 0 < dk < 1 and a positive definite sym-
We also need the result of Fengli Ren and Jinde Cao in [4] for our main result. So, we recall their theorem and proof as follows. Theorem 1.5 [4]. For given sk > 0 and b > 1, system (1.1) is a-stable if there exist positive definite symmetric matrices Rt X; W; F 11 ; F 22 ; F 33 and any matrices Y k , k ¼ 1; 2; . . . ; m, F 12 ; F 13 ; F 23 , such that the operators Dk ðyt Þ ¼ xðtÞ þ tsk Y k X 1 yðsÞ ds, k ¼ 1; 2; . . . ; m are stable and the following LMIs hold 0
X B BI B B B B N ¼ BI B B . B . B . @ I
M1
M2
...
Mm
bs1 X
0
...
0
I
..
.
..
.
.. .
.. .
..
.
..
.
0
I
...
I
1 C C C C C C C < 0; C C C C A
bsm X
X þ F 22 < 0; 0 1 F 11 F 12 F 13 B C B I F 22 F 23 C > 0; @ A I
I
ð1:2Þ
F 33
where B0 ¼ aI þ A0 ; Bk ¼ eask Ak ; k ¼ 1; 2; . . . ; m and
ð1:3Þ
ð1:4Þ
872
P.T. Nam / Applied Mathematics and Computation 202 (2008) 870–876
0
R1
U1
U2
...
Um
W1
W2
...
Wm1
Wm
I
K1
I
I .. . .. . .. . .. . .. . .. .
0 .. . .. .
... .. .
0 .. . .. .
C1 .. . .. .
B2 X .. . .. .
... .. . .. .
Bm1 X .. . .. .
Bm X .. . .. .
.. . .. . .. . .. .
I .. . .. . .. . .. .
Km
B1 X
B2 X
...
Bm1 X
Cm
I .. . .. . .. .
W þ s1 F 33
0 .. .
... .. . .. . I
0 .. . .. . .. .
0 .. . .. . .. .
I
I
W þ sm F 33
B B B B B B B B B B B B B X¼B B B B B B B B B B B B B @
I I I I I I
0
I .. . .. .
I I I I I I m X ðB0 X þ Y k þ XBT0 þ Y Tk þ W þ sk F 11 Þ; R1 ¼
I .. . I
1 C C C C C C C C C C C C C C; C C C C C C C C C C C C A
ð1:5Þ
Uk ¼ XBT0 þ Y Tk þ F 12 ;
k¼1
Wk ¼ mBk X Y k þ sk F 13 ; Mk ¼
Ck ¼ Bk X Y k þ F 23 ;
ðbsk Y k ; 0; . . . ; 0ÞT2mþ1;1 ;
Kk ¼ sk ðb 1ÞX;
k ¼ 1; 2; . . . ; m:
Proof. We take the following change of the state variable t 2 Rþ ;
yðtÞ ¼ eat xðtÞ;
then the system (1.1) is transformed to the delay system 8 m P < yðtÞ _ Bk yðt sk Þ ¼ B0 yðtÞ þ k¼1 : at yðtÞ ¼ e /ðtÞ; t 2 ½s; 0
ð1:6Þ
where B0 ¼ aI þ A0 ; Bk ¼ eask Ak ; k ¼ 1; 2; . . . ; m. Consider the following Lyapunov functional V¼
m X ðV 1k þ V 2k þ V 3k þ V 4k Þ
ð1:7Þ
k¼1
where V 1k ¼ DTk ðyt ÞPDk ðyt Þ; Z t Z t V 2k ¼ b yT ðuÞGTk PGk yðuÞ duds; V 3k ¼
Z
tsk
s
t
yT ðsÞTyðsÞ ds;
tsk
1T 0 1 1 0 F 11 F 12 F 13 yðsÞ yðsÞ B C B C C B V 4k ¼ @ Gk yðuÞ A P @ I F 22 F 23 AP@ Gk yðuÞ A 0 ssk I I F 33 yT ðs sk Þ yT ðs sk Þ Rt 1 and Dk ðyt Þ ¼ yðtÞ þ tsk Gk yðsÞ ds; Gk ¼ Y k X ; T > 0; P > 0; P ¼ diagfP; P; Pg. By Fengli Ren’s proof, if (1.2)–(1.4) hold then we have Z
t
Z
0
s
V P0 and 0
1T 0 1 yðtÞ yðtÞ B R t G yðsÞ ds C B R t G yðsÞ ds C B ts1 1 C B ts1 1 C B C B C B C B C . . .. .. B C B C m Z t BR C BR C X B B C C yT ðsÞGTk ðP þ PF 22 PÞGk yðsÞ ds V_ 6 B t Gm yðsÞ ds C XB t Gm yðsÞ ds C þ ts ts m m B C B C k¼1 tsk B yðt s Þ C B yðt s Þ C 1 1 B C B C B C B C B C B C .. .. @ A @ A . . yðt sm Þ
yðt sm Þ
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P.T. Nam / Applied Mathematics and Computation 202 (2008) 870–876
with P þ PF 22 P < 0 and X < 0: This implies that there exists a positive number d such that V_ < dkyðtÞk2 : Combining with the stability assumption on the main operators Dk ðyt Þ, we have the system (1.6) is asymptotically stable. This implies that the system (1.1) is a-stable. The proof is completed. Note that the stability of the operators Dk ðyt Þ can check by using Lemma 1.4 . h Remark 1. If P þ PF 22 P < 0 then by Lemma 1.2, we have 0
1T 0 1 yðtÞ yðtÞ R R B t G yðsÞ ds C B t G yðsÞ ds C B ts1 1 C B ts1 1 C B C B C B C B C . . B C B C . . B B C C . . Z m t X BRt C BRt C T T B B C C y ðsÞGk ðP þ PF 22 PÞGk yðsÞ ds 6 B DB G yðsÞ ds G yðsÞ ds m m C C tsm tsm ts B B C C k k¼1 B yðt s1 Þ C B yðt s1 Þ C B C B C B C B C .. .. B C B C @ @ A A . . yðt sm Þ
yðt sm Þ
where 0
0 B B0 B B0 B B B B0 B B B0 D¼B B B0 B B .. B. B B B0 B B @0
1
0
0
...
0
0
0
...
0
0
s1 1 ðP þ PF 22 PÞ
0
...
0
0
0
...
0
0
s1 2 ðP þ PF 22 PÞ
0
0
0
0
...
0
0
0
..
0
.. .
.. .
.. .
.. .
0
0
0
s1 m ðP þ PF 22 PÞ
0
0
...
0
0 .. .
0 .. .
0 .. .
0 .... ..
0 .. .
0 .. .
... .. .
0 .. .
0
0
0
0
0
0
...
0
0
0
0
0
0
0
...
0
C 0C C 0C C C .. C .C C C 0C C 6 0: C 0C C C C C C 0C C C 0A
0
0
0
0
0
0
...
0
0
0
.
ð1:8Þ
Hence, we have 0
0 1T 1 yðtÞ yðtÞ B Rt B Rt C C B ts G1 yðsÞ ds C B ts G1 yðsÞ ds C B B C C 1 1 B B C C B B C C .. .. B B C C . . B B C C R R B t C C t _V 6 B B ts Gm yðsÞ ds C ðX þ DÞB ts Gm yðsÞ ds C: m m B B C C B B C C B yðt s1 Þ C B yðt s1 Þ C B B C C B B C C .. .. B B C C . . @ @ A A yðt sm Þ
yðt sm Þ
Let X ¼ P 1 ; W ¼ XTX; Y k ¼ Gk X; k ¼ 1; 2; . . . m and multiplying both sides of inequality X þ D < 0 by diagfX; X; . . . ; Xg and using Lemma 1.3, then X þ D < 0 is equivalent to 0 B B B B N1 ¼ B B B @
X1
M1
M2
...
Mm
I
bs1 X
I .. .
I .. .
0 .. . .. .
... .. . .. .
0 .. .
I
I
...
I
bsm X
where X1 ¼ X þ D1 and
0
1 C C C C C<0 C C A
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P.T. Nam / Applied Mathematics and Computation 202 (2008) 870–876
0
0
B0 B B B0 B B B0 B B B0 D1 ¼ B B B0 B. B. B. B B0 B B @0
0
0
...
0
0
0
s1 1 ðX þ F 22 Þ
0
...
0
0
0
0
s1 2 ðX þ F 22 Þ
0
0 0
0 0
0 .. . 0
0 s1 m ðX þ F 22 Þ
0 .. . 0
0 .. . 0
0 .. .
0 .. .
0 .. .
0 .... ..
0 .. .
0 .. .
... .. .
0 .. .
0
0
0
0
0
0
...
0
0
0
0
0
0
0
...
0
0
0
0
0
0
...
0
1
0
0
...
0
... .. . ...
0 .. . 0
0
0C C C 0C C .. C .C C C 0C C 6 0: C 0C C C C C 0C C C 0A
0
0
...
Thus, if X þ F 22 < 0 and N1 < 0 then there is a positive number k such that V_ 6 kðkyðtÞk2 þ kyðt s1 Þk2 þ þ kyðt sm Þk2 Þ: Remark 2. Since N1 6 N, the condition N1 < 0 is less conservative than the condition N < 0. 2. Main result Combining Remark 1 and the following Lyapunov functional, we get an improved a-stability criterion of the system (1.1) as follows: Theorem 2.1. The system (1.1) is a-stable if the exist b > 1, positive definite symmetric matrices X; W; F 11 ; F 22 ; F 33 and any matrices Y k ; F 12 ; F 13 ; F 23 ,k ¼ 1; 2; . . . ; m, which which satisfy the LMIs (1.3), (1.4) and N1 < 0. Proof. Consider the following Lyapunov functional: V ¼ V þ V5 where V 5 ¼ gkyðtÞk2 ; and ( ) 1 k ; k > 0: g ¼ min P 2 2 2kB0 k þ m k¼1 kBk k Since V P 0, we have V P gkyðtÞk2 : Taking the derivative of V in t and using Lemma 1.1, we have _ T yðtÞ þ V_ V_ ¼ 2gyðtÞ 6 2g yT ðtÞBT0 yðtÞ þ
m X
! yT ðt sk ÞBTk yðtÞ
k¼1
6g
2kB0 k þ
m X
!
kBk k2 kyðtÞk2 þ
k¼1
m X
k kyðtÞk2 þ kyðt s1 Þk2 þ þ kyðt sm Þk2 !
kyðt sk Þk2
kðkyðtÞk2 þ kyðt s1 Þk2 þ þ kyðt sm Þk2 Þ 6 0:
k¼1
This follows that V ðtÞ V ð0Þ 6 0;
8t 2 Rþ :
Since V ðtÞ P gkyðtÞk2 , we have sffiffiffiffiffiffiffiffiffiffiffiffi V ð0Þ : kyðtÞk 6 g It means yðtÞ is bounded. This implies that the system (1.1) is a-stable. The proof is completed.
h
Remark 3. To reduce the conservatism of our stability criterion, we have used the function V 5 , which allows our criterion Rt does not depend on the stability of operators Dk ðyt Þ ¼ xðtÞ þ tsk Gk yðsÞ ds; k ¼ 1; . . . ; m. In addition, the condition N1 < 0 is less conservative than the condition N < 0. Hence, our criterion is strictly less conservative than the criterion in Ref. [4]. On the other hand, Fengli Ren’s criterion is less conservative than those in the literature. Therefore, our criterion will be less conservative than those in the literature. We can shown directly the superiority of our result to those in the literature by considering the following numerical example, which is considered in Ref. [4].
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P.T. Nam / Applied Mathematics and Computation 202 (2008) 870–876
Example 2.2. Consider the following linear system ( pffiffiffi _ xðtÞ ¼ A0 xðtÞ þ A1 xðt 1Þ þ A2 xðt 3Þ; t P 0 pffiffiffi xðtÞ ¼ /ðtÞ; t 2 ½ 3; 0; 7 1 10:1 14:2 5:2 ; A1 ¼ ; A2 ¼ where A0 ¼ 0:5 5:5 6:6 10:2 3:6 Matlab [5], the LMIs solutions of Theorem 2.1 are found 3:5370
X ¼ 107
2:3266
!
2:1428
W ¼ 108
;
ð2:1Þ 6:2 . Taking b ¼ 12; a ¼ 0:1. Using the LMI Toolbox in 5:1 !
1:4004
;
1:7142 1:4004 1:0252 ! ! 25:2902 4:4060 37:4902 2:2060 Y1 ¼ ; Y2 ¼ ; 4:2402 20:7282 4:0402 25:1282 ! ! 4:6381 2:3249 2:9990 0:0500 ; F 12 ¼ ; F 11 ¼ 106 2:3249 1:1687 0:0510 0:0193 ! ! 1:3047 0:0007 2:2534 1:4797 7 F 13 ¼ ; F 22 ¼ 10 ; 0:0021 1:7505 1:4797 1:0941 ! ! 5:0555 5:5001 4:9181 2:4645 ; F 33 ¼ 106 : F 23 ¼ 4:0050 5:0506 2:4645 1:2386 2:3266
See Table 1. Remark 4. To get the bound more precisely, integrating both sides of the above inequality V_ 6 0 from 0 to t, we have V ðtÞ V ð0Þ 6 0;
8t 2 Rþ :
Since V ðtÞ P gkyðtÞk2 , we have m X
gkyð0Þk2 6 V ð0Þ ¼ gkzð0Þk2 þ
V 1k ð0Þ þ V 2k ð0Þ þ V 3k ð0Þ þ V 4k ð0Þ:
k¼1
On the other hand, we have yð0Þ þ
V 1k ð0Þ ¼
Z
!T
0
Gk yðsÞ ds
P yð0Þ þ
Z
sk
yð0Þ þ
¼
Z
!
0
Gk yðsÞ ds sk
!T
0
Y k X 1 yðsÞ ds
X 1 yð0Þ þ
sk
Z
!
0
Y k X 1 yðsÞ ds
sk
¼ yT ð0ÞX 1 yð0Þ þ yT ð0ÞX 1 Y k X 1
Z
0
Y k X 1
yðsÞ ds þ
Z
sk
Y kX
þ
1
Z
!T
0
yðsÞ ds
X
1
YkX
sk
and
R0 sk
yðsÞ ds ¼
V 1k ð0Þ 6
R0 sk
kX
k þ 2kX
!T yðsÞ ds
X 1 yð0Þ
sk
Z
!
0
yðsÞ ds
sk
xðsÞeas ds 6 k/ks
1
1
0
R0 sk
ask
eas ds ¼ 1ea
k/ks . Hence,
! ask 2 1 eask Þ 2 ð1 e 1 3 þ kX k kY k k k kY k k k/k2s : a a2
1 2
Table 1 Comparison between results of our result and recent ones Method
Year
Convergence rate a
Y.J. Sun [9] D.Q. Cao [2] Bugong Xu [10] Fengli Ren [4] Ours
1998 2003 2003 2006 2008
0.04 0.09 Not applicable 0.095 0.1
876
P.T. Nam / Applied Mathematics and Computation 202 (2008) 870–876
By the same computations, we have Z 0 Z 0 V 2k ð0Þ ¼ be2as yT ðuÞX 1 Y Tk X 1 Y k X 1 yðuÞ duds sk
6 bkX
s
1 3
k kY k k2
Z
0 sk
Z
0
yT ðuÞyðuÞ duds s
2ask 1 þ e2ask k/k2s ; 4a2 Z 0 yT ðuÞyðuÞ du 6 k/k2s kTk e2au du
6 bkX 1 k3 kY k k2 V 3k ð0Þ ¼
Z
0 sk
sk
1 e2ask 1 2 kX k kWkk/k2s 6 2a and V 4k ð0Þ ¼ 0. We denote 1 eask ð1 eask Þ2 N 1k ¼ kX 1 k þ 2kX 1 k2 kY k k þ kX 1 k3 kY k k2 ; a a2 2ask 2ask 2ask 1 þ e 1e kX 1 k2 kWk N 2k ¼ bkX 1 k3 kY k k2 ; N 3k ¼ 4a2 2a P gþ m k¼1 ðN 1k þ N 2k þ N 3k Þ : N2 ¼ g
and
Then, we have gkyðtÞk2 6 V ð0Þ 6 gN 2 k/k2s : Hence, kyðtÞk 6 Nk/ks . This implies kxðt; /Þk 6 Nk/ks eat ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P gþ m k¼1 ðN 1k þ N 2k þ N 3k Þ N¼ : g
8t 2 Rþ : Thus, the Lyapunov factor is
3. Conclusions In this paper, we present an improved a-stability criterion for linear systems with multiple time delays. This criterion is strictly less conservative than the very recent criterion in [4]. Hence, our criterion is less conservative than those in the literature. Moreover, an estimation the Lyapunov factor is also given. Acknowledgement The author would like to thank anonymous referees for valuable comments and suggestions, which have improved the paper. The author would like to thank Professor Fengli Ren for giving the example solving. References [1] [2] [3] [4] [5] [6] [7] [8]
S. Boyd, El. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix and Control Theory, SIAM Studies in Applied Mathematics, vol. 15, SIAM, PA, 1994. D.Q. Cao, P. He, K.Y. Yhang, Exponential stability criteria of uncertain systems with multiple time delays, J. Math. Anal. Appl. 283 (2003) 274–362. J. Cao, J. Wang, Delay-dependent robust stability of uncertain nonlinear systems with time delay, Appl. Math. Comput. 154 (1) (2004) 289–297. Fengli Ren, Jinde Cao, Novel a-stability of linear systems with multiple time delays, Appl. Math. Comput. 181 (2006) 282–290. P. Gahinet, A. Nemirovski, A. Laub, M. Chilali, LMI Control Toolbox User’s Guide, The Mathworks, Natick, Massachusetts, 1995. S. Mondie, V.L. Kharitonov, Exponential estimates for retarded time-delay systems: an LMI approach, IEEE Trans. Aut. Contr. 50 (2) (2005) 268–273. V.N. Phat, P.T. Nam, Exponential stability criteria of linear non-autonomous systems with multiple delays, Elect. J. Diff. Equations 58 (2005) 1–9. P.T. Nam, V.N. Phat, Robust exponential stability and stabilization of linear uncertain polytopic time-delay systems, J. Contr. Theory Appl. 5 (3) (2007) 123–131. [9] Y.J. Sun, J.G. Hsieh, On a-stability criteria of nonlinear systems with multiple time delays, J. Franklin Inst. 335B (1998) 695–705. [10] B. Xu, Stability criteria for linear systems with uncertain delays, J. Math. Anal. Appl. 284 (2003) 455–470.