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New stability criteria for a class of neutral systems with discrete and distributed time-delays: an LMI approach
Applied Mathematics and Computation 150 (2004) 719–736 www.elsevier.com/locate/amc
New stability criteria for a class of neutral systems with discrete and distributed time-delays: an LMI approach Jenq-Der Chen
*
Institute of Engineering Science and Technology, National Kaohsiung First University of Science and Technology, Kaohsiung 824, Taiwan, ROC
Abstract In this paper, a problem of the asymptotic stability for a class of neutral systems with multiple discrete and distributed time-delays is considered. Lyapunov stability theory is applied to guarantee the stability for the systems. New discrete-delay-independent and discrete-delay-dependent stability conditions are derived in terms of the spectral radius and linear matrix inequality. By mathematical analysis, the stability criteria are proved to be less conservative than the ones reported in the current literatures. A numerical example is given to illustrate the availability of the proposed results. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Neutral systems; Discrete and distributed time-delays; Lyapunov stability theory; Linear matrix inequality
1. Introduction In the recent years, stability analysis of time-delay systems has received considerable attention and has been one of the most interesting topics in the control systems. This is due to theoretical interests as well as a powerful tool for practical system analysis and design, since delay phenomenon is often encountered in various mechanics, physics, biology, medicine, economy, and engineering systems, such as AIDS epidemic, aircraft stabilization, chemical * Address: Department of Electronic Engineering, Yung-Ta Institute of Technology & Commerce, 316 Chunsan Road, Lin-Lo, Ping-Tung 909, Taiwan, ROC. E-mail address: [email protected] (J.-D. Chen).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00302-3
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J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
Nomenclature C0 set of all continuous functions from ½H ; 0 to Rn Rþ set of all non-negative real numbers Rn n-dimensional real space Rm n set of all real m by n matrices AT (respectively xT ) transpose of matrix A (respectively, vector x) kxk Euclidean norm of vector x kAk spectral norm of matrix A P > 0 P is a positive definite symmetric matrix P < 0 P is a negative definite symmetric matrix qðAÞ spectral radius of real matrix A jAj ½jaij j, with A ¼ ½aij m f1; 2; . . . ; mg I unit matrix engineering systems, control of epidemics, distributed networks, infeed grinding and cutting model, manual control, microwave oscillator, models of lasers, neural network, nuclear reactor, population dynamic model, rolling mill, ship stabilization, and systems with lossless transmission lines [2–23]. Moreover, time-delay is frequently a source of instability and a source of generation of oscillation in many systems; for example, the trivial solution of ða 1Þ ½€xðtÞ þ x_ ðtÞ þ xðtÞ ¼ 0, with a > 1, is stable, but that of system €xðtÞ þ x_ ðtÞ þ xðtÞ ¼ a ½€xðt hÞ þ x_ ðt hÞ þ xðt hÞ is unstable for any h > 0 [13]. Consider the linear system " # m X x_ ðtÞ ¼ A þ Ei xðtÞ; t P 0;
ð1Þ
i¼1
The necessary and sufficient where xðtÞ 2 Rn , A, Ei 2 Rn n , i 2 m. Pm condition for asymptotic stability of system (1) is that the matrix A þ i¼1 Ei is Hurwitz. Recently, many reports are concentrated on systems with discrete delay [4,8– 11,14–23]. By increasing in the equation number of summands and simultaneously decreasing the differences between neighbouring argument values, one naturally arrives at equations with distributed (or continuous) and mixed (both distributed and discrete) delay arguments [2,3,6,7,13]. Now, we consider the neutral systems with discrete and distributed time-delays: Z t m X x_ ðtÞ ¼ AxðtÞ þ Bi xðt hi Þ þ Ci x_ ðt hi Þ þ Di xðsÞ ds ; t P 0; i¼1
tsi
ð2aÞ
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
xðtÞ ¼ /ðtÞ;
t 2 ½H ; 0;
721
ð2bÞ
where x 2 Rn , xt is the state at time t defined by xt ðsÞ :¼ xðt þ sÞ 8s 2 ½H ; 0, are H ¼ maxi2m fhi ; si g P 0, with kxt ks :¼ supH 6 r 6 0 kxðt þ rÞk, hi and si , i 2 m, both non-negative constants, which represent discrete delays and distributed are known, and delays, respectively. The matrices A, Bi , Ci , Di 2 Rn n , i 2 m, the initial vector / 2 C0 . play an important role In view of [2–6,13], the distributed delays si , i 2 m, about the stability of system (2), so our main results will also depend on those Depending on whether the stability criterion itself distributed delays si , i 2 m. contains the discrete delay argument as a parameter, stability criteria for neutral systems can be classified into two categories, namely discrete-delayindependent criteria [4,7,8,16,20,21] and discrete-delay-dependent criteria [2– 4,6,16,17,20,21]. Generally speaking, the latter ones are less conservative than the former ones, but the former ones are also important when the effect of time delay is small. In this paper, the discrete-delay-independent and discrete-delaydependent criteria will be proposed to guarantee the asymptotic stability for neutral systems with multiple time-delays. Many approaches have been used to searching sufficient conditions for the stability problem of time-delay systems. The stability problem for time-delay systems with distributed delay is considered using discretized Lyapunov functional [8]. A robust stability of time-delay systems was investigated by checking the Hamiltonian matrix and solving an algebraic Riccati equation [23]. Appropriate model transformations of original time-delay systems and Lyapunov theory are useful for the analysis of stability of systems [5,11–16,20–23]. Many sharp results are used the linear matrix inequality (LMI) approach to solve the stability problem of time-delay systems [5,9,14,16,20–22]. Furthermore, a model transformation and Lyapunov theory with LMI approach are used in this paper, less conservative criteria are proposed to guarantee the asymptotic stability for the neutral systems with discrete and distributed time-delays. A numerical example is given to illustrate the validity of the proposed results.
2. Problem formulation and main results By some model transformations, system (2) can be written as: " # Z t Z t m X d xðtÞ þ xðsÞ ds Ci xðt hi Þ þ Fi ðh t þ si ÞxðhÞ dh Ei dt thi tsi i¼1 Z t m X b ¼ AxðtÞ þ ðBi Ei Þxðt hi Þ þ ðDi Fi Þ xðhÞ dh ; t P 0; i¼1
tsi
ð3Þ
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J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
are some chosen matrices such that the matrix where Ei P , Fi 2 Rn n , i 2 m, m b A ¼ A þ i¼1 ðEi þ si Fi Þ is Hurwitz in view of (1). By this transformation, it will cause our obtained results are less conservative than other recent literatures. Remark 1. The system (2a) is a general representation for the description of neutral system discrete and distributed time-delays, for example the neutral system Z t x_ ðtÞ ¼ AxðtÞ þ Bxðt hÞ þ C x_ ðt gÞ þ D xðsÞ ds; ts
can be converted into the form of system (2a) with B1 ¼ B;
C1 ¼ D1 ¼ 0;
D3 ¼ D;
h1 ¼ h;
B2 ¼ D2 ¼ 0;
h2 ¼ g;
C2 ¼ C;
B3 ¼ C3 ¼ 0;
s3 ¼ s:
Lemma 1 (Chen et al., 2001 [3], Chen et al., 2002 [4]). For any matrices Ei , Ci , Fi 2 Rn n , if ! m X s2i q hi jEi j þ jCi j þ jFi j < 1; 2 i¼1 then the operator D : C0 ! Rn with Z t Z m X Dðxt Þ ¼ xðtÞ þ Ei xðsÞds Ci xðt hi Þ þ Fi i¼1
thi
t
ðh t þ si ÞxðhÞdh tsi
is stable. Lemma 2. For any vectors x, y 2 Rn and R > 0, we have 2xT y 6 xT Rx þ y T R1 y:
QðyÞ T SðyÞ 1 T QðyÞ SðyÞRðyÞ SðyÞ < 0;
Lemma 3 (Boyd et al., 1994 [1]). The LMI RðyÞ < 0;
SðyÞ < 0 is equivalent to RðyÞ
where QðyÞ ¼ QðyÞT , RðyÞ ¼ RðyÞT , and SðyÞ depend affinely on y. Now we present the main result for asymptotic stability of system (2). b is Theorem 1. System (2) is asymptotically stable provided that the matrix A Hurwitz and satisfying the condition
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
q
m X
hi jEi j þ jCi j þ
i¼1
s2i jFi j 2
723
! ð4aÞ
<1
and there exist some positive definite symmetric matrices P , Mi , Ni , Qi , Ri , Sij , Tij , such that the following LMI condition holds: Uij , Vi , Wij , Xij , and Yij , i, j 2 m, 3 2 U11 U12 U13 U14 U15 U16 U17 U18 6 UT12 U22 0 0 0 0 0 0 7 7 6 T 6U 0 U 0 0 0 0 0 7 33 7 6 13 6 UT 0 0 U44 0 0 0 0 7 7 6 14 U¼6 T 7 < 0; 6 U15 0 0 0 U55 0 0 0 7 7 6 T 6 U16 0 0 0 0 U66 0 0 7 7 6 T 4U 0 0 0 0 0 U77 0 5 17
UT18
0
0
0
0
0
0
U88 ð4bÞ
where bT P þ P A bþ U11 ¼ A
m X i¼1
þ
m X j¼1
U12 ¼
" hi Mi þ Ni þ
s2i Qi þ Ri þ si Vi 2
s2j s2j hj Sij þ Tij þ Uij þ si hj Wij þ si Xij þ si Yij 2 2
!# ;
h pffiffiffiffiffi pffiffiffiffiffi T b PEm ATPC b T PE1 hm A bT b h1 A 1 A PCm T b T PF1 sm A b T PFm ðB1 E1 Þ P ðBm Em ÞT P s1 A i pffiffiffiffi pffiffiffiffiffi T T s1 ðD1 F1 Þ P sm ðDm Fm Þ P ;
U22 ¼ diag ðM1 ; . . . ; Mm ; N1 ; . . . ; Nm ; 2 Q1 ; . . . ; 2 Qm ; R1 ; . . . ; Rm ; V1 ; . . . ; Vm Þ; pffiffiffiffiffi pffiffiffiffiffi T T h1 ðBi Ei Þ PE1 hm ðBi Ei Þ PEm ; U13 ¼ ½X1 Xm ; Xi ¼ b 1; . . . ; X b m Þ; X b i ¼ diag ðSi1 ; . . . ; Sim Þ; U33 ¼ diag ð X U14 ¼ ½R1 Rm ; Ri ¼ ½ðBi Ei ÞT PC1 ðBi Ei ÞT PCm ; b 1; . . . ; R b m Þ; R b i ¼ diag ðTi1 ; . . . ; Tim Þ; U44 ¼ diag ð R T
T
U15 ¼ ½N1 Nm ; Ni ¼ ½s1 ðBi Ei Þ PF1 sm ðBi Ei Þ PFm ; b 1; . . . ; N b m Þ; N b i ¼ diag ð2 Ui1 ; . . . ; 2 Uim Þ; U55 ¼ diag ð N U16 ¼ ½C1 Cm ; pffiffiffiffi pffiffiffiffiffi pffiffiffiffi pffiffiffiffiffi si h1 ðDi Fi ÞT PE1 si hm ðDi Fi ÞT PEm ; Ci ¼ b 1; . . . ; C b m Þ; C b i ¼ diag ðWi1 ; . . . ; Wim Þ; U66 ¼ diag ð C
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J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
U17 ¼ ½K1 Km ;
pffiffiffiffi pffiffiffiffi T T Ki ¼ ½ si ðDi Fi Þ PC1 si ðDi Fi Þ PCm ;
b 1; . . . ; K b m Þ; K b i ¼ diag ðXi1 ; . . . ; Xim Þ; U77 ¼ diag ð K pffiffiffiffi pffiffiffiffi T T U18 ¼ ½W1 Wm ; Wi ¼ ½ si s1 ðDi Fi Þ PF1 si sm ðDi Fi Þ PFm ; b 1; . . . ; W b m Þ; W b i ¼ diag ð2 Yi1 ; . . . ; 2 Yim Þ: U88 ¼ diag ð W Proof. By the schur complement of [1], the condition (4b) is equivalent to pðP ; Mi ; Ni ; Qi ; Ri ; Sij ; Tij ; Uij ; Vi ; Wij ; Xij ; Yij Þ b TP þ P A bþ ¼A
m h X
b T PEi M 1 ET P A b þ Mi Þ hi ð A i i
i
i¼1
þ
þ
m m 2 X X si b T 1 T b b T PCi N 1 C T P A b þ Ni Þ þ ð A ðA PF Q F P A þ Q Þ i i i i i i 2 i¼1 i¼1 m X
½ðBi Ei ÞT PR1 i P ðBi Ei Þ þ Ri
i¼1
þ
m X m X i¼1
þ
j¼1
m X m X i¼1
T
½hj ððBi Ei Þ PEj Sij1 EjT P ðBi Ei Þ þ Sij Þ ½ðBi Ei ÞT PCj Tij1 CjT P ðBi Ei Þ þ Tij
j¼1
# " m X m X s2j T 1 T þ ððBi Ei Þ PFj Uij Fj P ðBi Ei Þ þ Uij Þ 2 i¼1 j¼1 þ
m X T ½si ððDi Fi Þ PVi 1 P ðDi Fi Þ þ Vi Þ i¼1
þ
m X m X i¼1
þ
m X m X i¼1
þ
T
½si ððDi Fi Þ PCj Xij1 CjT P ðDi Fi Þ þ Xij Þ
j¼1
m X m X i¼1
T
½si hj ððDi Fi Þ PEj Wij1 EjT P ðDi Fi Þ þ Wij Þ
j¼1
j¼1
"
# s2j T 1 T si ððDi Fi Þ PFj Yij Fj P ðDi Fi Þ þ Yij Þ < 0: 2 ð5Þ
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
725
The functional given by V ðxt Þ ¼ V1 ðxt Þ þ V2 ðxt Þ þ V3 ðxt Þ þ V4 ðxt Þ þ V5 ðxt Þ;
ð6aÞ
where V1 ðxt Þ ¼ GT ðxt Þ P Gðxt Þ; m X
Gðxt Þ ¼ xðtÞ þ
Z
t
xðsÞds Ci xðt hi Þ þ Fi
Ei thi
i¼1
Z
ðh t þ si ÞxðhÞdh ;
t
tsi
ð6bÞ V2 ðxt Þ ¼
m Z X i¼1
(
t
xT ðsÞ ðs t þ hi Þ Mi þ Ni þ ðBi Ei Þ
T
thi
" R1 i
P
þ
m X
hj
Ej Sij1 EjT
þ
Cj Tij1 CjT
j¼1
s2j þ Fj Uij1 FjT 2
!#
)
P ðBi Ei Þ xðsÞ ds;
V3 ðxt Þ ¼
m Z X i¼1
(
t T
x ðhÞ
tsi
"
P Vi
1
þ
ð6cÞ
1 2 T ðh t þ si Þ Qi þ ðDi Fi Þ 2
m X
hj
Ej Wij1 EjT
j¼1
þ
Cj Xij1 CjT
s2j þ Fj Yij1 FjT 2
!#
)
P ðDi Fi Þ xðhÞ dh;
V4 ðxt Þ ¼
m X m Z X i¼1
j¼1
t
h xT ðsÞ ðs t þ hj Þ ðSij þ si Wij Þ:
thj
i þ ðTij þ si Xij Þ xðsÞ ds;
V5 ðxt Þ ¼
ð6dÞ
m X m Z t 1X 2 ðh t þ sj Þ xT ðhÞðUij þ si Yij ÞxðhÞ dh; 2 i¼1 j¼1 tsj
ð6eÞ
ð6fÞ
is a legitimate Lyapunov functional candidate [9]. The time derivative of Vi ðxt Þ, i¼ 5, along the trajectories of system (3) is given by
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J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
b T P þ P AÞxðtÞ b b TP V_1 ðxt Þ ¼ xT ðtÞð A þ 2xT ðtÞ A
m X
Z
xðsÞ ds thi
i¼1
b TP 2xT ðtÞ A b TP þ 2x ðtÞ A T
m X i¼1 m X
Ci xðt hi Þ Z
m X
t
ðh t þ si ÞxðhÞ dh
Fi tsi
i¼1
þ2
xT ðt hi ÞðBi Ei ÞT PxðtÞ
i¼1
þ2 2
m X m X i¼1 j¼1 m X m X i¼1
þ2 þ2 þ2 2
i¼1 j¼1 m Z t X
þ2
x ðt hi ÞðBi Ei Þ PEj
j¼1
j¼1
xðsÞ ds
T
xT ðt hi ÞðBi Ei Þ PCj xðt hj Þ T
xT ðt hi ÞðBi Ei Þ PFj
Z
t
ðh t þ sj ÞxðhÞ dh tsj
T
xT ðsÞ dsðDi Fi Þ PxðtÞ
Z
m X m Z X i¼1
t
thj
tsi i¼1 m X m Z t X i¼1 j¼1 m X m X
Z
T
T
j¼1
m X m X
i¼1
t
Ei
T
T
x ðhÞ dhðDi Fi Þ PEj
Z
t
xðsÞ ds
tsi
thj
t T
xT ðhÞ dhðDi Fi Þ PCj xT ðt hj Þ tsi t T
xT ðhÞ dhðDi Fi Þ PFj
Z
tsi
t
ðh t þ sj ÞxðhÞ dh:
tsj
By using the Lemma 2, it is ease to see b T P þP AÞxðtÞ b V_1 ðxt Þ 6 xT ðtÞð A " Z m X T T 1 T b b þ hi x ðtÞ A PEi M E P AxðtÞ þ i
i¼1
þ
#
t T
x ðsÞMi xðsÞds
i
thi
m X T b T PCi N 1 C T P AxðtÞþx b ½xT ðtÞ A ðt hi ÞNi xðt hi Þ i i i¼1
" # Z t m X s2i T b T 1 T b T x ðtÞ A PFi Qi Fi P AxðtÞ þ ðht þsi Þx ðhÞQi xðhÞdh þ 2 tsi i¼1 þ
m X T T ½xT ðt hi ÞðBi Ei Þ PR1 i P ðBi Ei Þxðt hi Þþx ðtÞRi xðtÞ i¼1
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
þ
m X m X i¼1
þ þ
Z
hj xT ðt hi ÞðBi Ei ÞT PEj Sij1 EjT P ðBi Ei Þxðt hi Þ #
j¼1 t
xT ðsÞSij xðsÞ ds
thj m X m X i¼1
727
"
T
½xT ðt hi ÞðBi Ei Þ PCj Tij1 CjT P ðBi Ei Þxðt hi Þ
j¼1
þ xT ðt "hj ÞTij xðt hj Þ m X m X s2j T x ðt hi ÞðBi Ei ÞT PFj Uij1 FjT P ðBi Ei Þxðt hi Þ þ 2 i¼1 j¼1 # Z t T ðh t þ sj Þ x ðhÞUij xðhÞ dh þ tsj m Z t X T þ xT ðhÞðDi Fi Þ PVi 1 P ðDi Fi ÞxðhÞ dh þ si xT ðtÞVi xðtÞ tsi i¼1 " Z t m X m X T þ hj xT ðhÞðDi Fi Þ PEj Wij1 EjT P ðDi Fi ÞxðhÞ dh ts i i¼1 j¼1 # Z t
xT ðsÞWij xðsÞ ds
þ si
thj
þ
"Z m X m X i¼1
j¼1
t T
xT ðhÞðDi Fi Þ PCj Xij1 CjT P ðDi Fi ÞxðhÞ dh tsi #
þ si xT ðt hj ÞXij xðt hj Þ " Z t m X m X s2j T þ xT ðhÞðDi Fi Þ PFj Yij1 FjT P ðDi Fi ÞxðhÞ dh 2 ts i i¼1 j¼1 # Z t T ðh t þ sj Þ x ðhÞYij xðhÞ dh ; þ si tsj
V_2 ðxt Þ ¼
m X
(
" T
x ðtÞ hi Mi þ Ni þ ðBi Ei Þ P R1 i þ T
m X
hj Ej Sij1 EjT
i¼1
) j¼1 !# 2 m Z t X s j xT ðsÞMi xðsÞ ds þ Cj Tij1 CjT þ Fj Uij1 FjT P ðBi Ei Þ xðtÞ 2 thi i¼1 ( " m m X X T T x ðt hi Þ Ni þ ðBi Ei Þ P R1 hj Ej Sij1 EjT i þ i¼1
s2j þ Cj Tij1 CjT þ Fj Uij1 FjT 2
!#
j¼1
)
P ðBi Ei Þ xðt hi Þ;
728
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736 m X
V_3 ðxt Þ ¼
i¼1
" s2i T Qi þ ðDi Fi Þ P Vi 1 x ðtÞ 2
þ
(
T
m X
hj
Ej Wij1 EjT
þ
Cj Xij1 CjT
j¼1
m Z X
!#
) P ðDi Fi Þ xðtÞ
t
xT ðhÞðh t þ si Þ Qi xðhÞ dh tsi
i¼1
s2j þ Fj Yij1 FjT 2
m X
(
" T
x ðt si Þ ðDi Fi Þ P Vi 1 T
i¼1
þ
m X
hj
Ej Wij1 EjT
þ
Cj Xij1 CjT
j¼1
s2j þ Fj Yij1 FjT 2
!#
)
P ðDi Fi Þ xðt si Þ;
V_4 ðxt Þ ¼
m X m X i¼1
xT ðtÞ½hj ðSij þ si Wij Þ þ ðTij þ si Xij ÞxðtÞ
j¼1
"Z m X m X i¼1
j¼1
t
xT ðsÞðSij þ si Wij ÞxðsÞ ds thj
# xT ðt hj ÞðTij þ si Xij Þxðt hj Þ ; " m X m X s2j T x ðtÞðUij þ si Yij ÞxðtÞ V_5 ðxt Þ ¼ 2 i¼1 j¼1
Z
#
t T
ðh t þ sj Þ x ðhÞðUij þ si Yij ÞxðhÞ dh :
tsj
Hence the derivative of V ðxt Þ is given by V_ ðxt Þ ¼ V_1 ðxt Þ þ V_2 ðxt Þ þ V_3 ðxt Þ þ V_4 ðxt Þ þ V_5 ðxt Þ 6 xT ðtÞ pðP ; Mi ; Ni ; Qi ; Ri ; Sij ; Tij ; Uij ; Vi ; Wij ; Xij ; Yij Þ xðtÞ: ð7Þ P s2i m In view of Lemma 1, the condition q <1 i¼1 hi jEi j þ jCi j þ 2 jFi j guarantees that the operator
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
Gðxt Þ ¼ xðtÞ þ þ Fi
m X i¼1 Z t
Z
729
t
xðsÞ ds Ci xðt hi Þ
Ei thi
ðh t þ si ÞxðhÞ dh ;
tsi
is stable. Thus, by Theorem 9.8.1 of [9] with (5)–(7), we conclude that systems (2) and (3) are both asymptotically stable. the entries of U in (4b) Remark 2. Notice that for given si , hi , Ei , and Fi , i 2 m, Hence, the are affine in P , Mi , Ni , Qi , Ri , Sij , Tij , Uij , Vi , Wij , Xij , and Yij , i, j 2 m. asymptotic stability problem for the neutral systems with discrete and distributed time-delays can be converted into a strictly feasible LMI problem [11]. it Remark 3. Notice that forR all chosen matrices R t Ei ¼ 0 (or Fi ¼ 0), i 2 m, t do represents that the term Ei thi xðsÞ ds (or Fi tsi ðhi t þ si ÞxðhÞ dh), i 2 m not convert to left-hand side of system (3). By the proof of Theorem 1, the LMI condition (4b) are reduced by deleting their corresponding elements. in Theorem 1, we may obtain the first Simply setting Ei ¼ 0, Fi ¼ 0, i 2 m, discrete-delay-independent criterion for asymptotic stability of system (2). Corollary 1. System (2) is asymptotically stable for any delays hi 2 Rþ provided Pm that A is Hurwitz, the condition q i¼1 jCi j < 1 is satisfied, and there exist some such that positive definite symmetric matrices P , Ni , Ri , Tij , Vi , and Xij , i, j 2 m, the following LMI condition holds: 2 3 b 12 b 13 b 14 b 11 U U U U T 6U b 22 0 0 7 6 b 12 U 7 ð8Þ 6 bT 7 < 0; b 33 4 U 13 0 U 0 5 b 44 bT 0 0 U U 14
where b 11 ¼ AT P þ PA þ U
m X
" N i þ Ri þ s i V i þ
i¼1
m X
# ðTij þ si Xij Þ ;
j¼1
pffiffiffiffiffi b 12 ¼ AT PC1 AT PCm BT P BT P pffiffiffiffi U s1 DT1 P sm DTm P ; 1 m b 22 ¼ diag ðN1 ; . . . ; Nm ; R1 ; . . . ; Rm ; V1 ; . . . ; Vm Þ; U b 13 ¼ ½R1 Rm ; Ri ¼ ½BT PC1 BT PCm ; U i
i
b 33 ¼ diag ð R b 1; . . . ; R b m Þ; R b i ¼ diag ðTi1 ; . . . ; Tim Þ; U p pffiffiffiffi ffi b 14 ¼ ½K1 Km ; Ki ¼ ½ ffiffiffi U si DTi PC1 si DTi PCm ; b 1; . . . ; K b m Þ; K b i ¼ diag ðXi1 ; . . . ; Xim Þ: b 44 ¼ diag ð K U
730
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
Remark 4. Corollary 1 coincides with the Theorem 1 of [3]. Theorem 1 of [3] could be seen as a special case of this result. in Theorem 1, we may obtain the Simply setting Ei ¼ 0, Fi ¼ Di , i 2 m, second discrete-delay-independent criterion for asymptotic stability of system (2). Corollary 2. System (2) is asymptotically stable for any delays hi 2 Rþ provided P P s2i m <1 that A ¼ A þ mi¼1 si Di is Hurwitz, the condition q i¼1 jCi j þ 2 jDi j is satisfied, and there exist some positive definite symmetric matrices P , Ni , Qi , Ri , such that the following LMI condition holds: Tij , and Uij , i; j 2 m, 2 3 U11 U12 U13 U14 6 T 7 0 0 7 6 U12 U22 6 T 7 < 0; ð9Þ 6U 0 U33 0 7 4 13 5 T U14 0 0 U44 where T
U11 ¼ A P þ P A þ
m X i¼1
"
h U12 ¼ AT PC1 AT PCm
m X s2j s2 N i þ i Q i þ Ri þ Tij þ Uij 2 2 j¼1 T
T
s1 A PD1 sm A PDm
!# ;
i BT1 P BTm P ;
U22 ¼ diag ðN1 ; . . . ; Nm ; 2 Q1 ; . . . ; 2 Qm ; R1 ; . . . ; Rm Þ; U13 ¼ ½R1 Rm ; Ri ¼ ½BTi PC1 BTi PCm ; b 1; . . . ; R b m Þ; R b i ¼ diag ðTi1 ; . . . ; Tim Þ; U33 ¼ diag ð R U14 ¼ ½N1 Nm ; Ni ¼ ½s1 BTi PD1 sm BTi PDm ; b 1; . . . ; N b m Þ; N b i ¼ diag ð2 Ui1 ; . . . ; 2 Uim Þ: U44 ¼ diag ð N in Theorem 1, we may obtain the first Simply setting Ei ¼ Bi , Fi ¼ 0, i 2 m, discrete-delay-dependent criterion for asymptotic stability of system (2). P e ¼ A þ m Bi Corollary 3. System (2) is asymptotically stable provided that A i¼1 Pm is Hurwitz, the condition q i¼1 ðhi jBi j þ jCi jÞ < 1 is satisfied, and there exist such some positive definite symmetric matrices P , Mi , Ni , Vi , Wij , and Xij , i; j 2 m, that the following LMI condition holds: 2 3 e 12 e 13 e 14 e 11 U U U U 6 eT 7 e 22 6 U 12 U 0 0 7 6 7 < 0; ð10Þ 6U T e 33 0 U 0 7 4 e 13 5 eT e 44 U 0 0 U 14
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
731
where e TP þ P A eþ e 11 ¼ A U
m X
"
# m X hi Mi þ Ni þ si Vi þ ðsi hj Wij þ si Xij Þ ;
i¼1
e 12 U
j¼1
pffiffiffiffiffi T pffiffiffiffiffi T e PB1 hm A e PBm ¼ h1 A pffiffiffiffi T pffiffiffiffiffi T s1 D1 P sm Dm P ;
e T PCm e T PC1 A A
e 22 ¼ diag ðM1 ; . . . ; Mm ; N1 ; . . . ; Nm ; V1 ; . . . ; Vm Þ; U h ffi pffiffiffiffiffi i pffiffiffiffi pffiffiffiffiffi e 13 ¼ ½C1 Cm ; Ci ¼ pffiffiffi si h1 DTi PB1 si hm DTi PBm ; U e 33 ¼ diag ð C b 1; . . . ; C b m Þ; C b i ¼ diag ðWi1 ; . . . ; Wim Þ; U p pffiffiffiffi ffi e 14 ¼ ½K1 Km ; Ki ¼ ½ ffiffiffi si DTi PC1 si DTi PCm ; U b 1; . . . ; K b m Þ; K b i ¼ diag ðXi1 ; . . . ; Xim Þ: e 44 ¼ diag ð K U Remark 5. By using the Lemma 3 and si ¼ hi , (10) is equivalent to eþ e TP þ P A A
m X
e T PBi M 1 BT P A e þ Mi Þ ½hi ð A i i
i¼1
þ
m X
e T PCi N 1 C T P A e þ Ni Þ þ ðA i i
i¼1
þ þ
½hi ðDTi PVi 1 PDi þ Vi Þ
i¼1
m X m X i¼1 j¼1 m X m X i¼1
m X
½hi hj
ðDTi PBj Wij1 BTj PDi
þ Wij Þ
½hi ðDTi PCj Xij1 CjT PDi þ Xij Þ < 0:
j¼1
By setting some matrices as Vi ¼ ðkPDi k þ eÞ I;
Wij ¼ ðkBTj PDi k þ eÞ I;
Xij ¼ ðkCjT PDi k þ eÞ I;
where e > 0, we have ( m X T e TP þ P A eþ e T PBi M 1 BT P A e þ Mi Þ x ðtÞ A ½hi ð A i
i
i¼1
þ
m m X X e T PCi N 1 C T P A e þ Ni Þ þ ðA ½hi ðDTi PVi 1 PDi þ Vi Þ i i i¼1
þ
i¼1
þ
i¼1
m X m X
m X m X i¼1
½hi hj ðDTi PBj Wij1 BTj PDi þ Wij Þ
j¼1
j¼1
) ½hi
ðDTi PCj Xij1 CjT PDi
þ Xij Þ xðtÞ
732
( J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736 m X e TP þ P A eþ e T PBi M 1 BT P A e þ Mi Þ 6 xT ðtÞ A ½hi ð A i
i
i¼1 2 m X e T PCi N 1 C T P A e þ Ni Þ þ hi kPDi k þ e þ kPDi k ðA þ i i kPDi k þ e i¼1 ! 2 T m X kBj PDi k hi hj kBTj PDi k þ e þ T I þ kBj PDi k þ e j¼1 " ! #) 2 T m X kC PD k i j hi kCjT PDi k þ e þ T I xðtÞ þ kCj PDi k þ e j¼1 ( m X e TP þ P A eþ e T PBi M 1 BT P A e þ Mi Þ 6 xT ðtÞ A ½hi ð A i
! I
i
i¼1
þ
m m X X e T PCi N 1 C T P A e þ Ni Þ þ ðA hi ð2 kPDi k þ eÞ I i i i¼1
þ
i¼1
)
i¼1
m X m X
hi ½hj ð2
kBTj PDi k
þ eÞ þ ð2
kCjT PDi k
þ eÞ I xðtÞ:
j¼1
Note that the parameter e > 0 could be chosen sufficiently small. Corollary 3 is less conservative than the result in [6]. in Theorem 1, we may obtain the Simply setting Ei ¼ Bi , Fi ¼ Di , i 2 m, second discrete-delay-dependent criterion for asymptotic stability of system (2). Corollary 4. System (2) is asymptotically provided that A¼ P stable Aþ Pm s2i m <1 i¼1 ðBi þ si Di Þ is Hurwitz, the condition q i¼1 hi jBi j þ jCi j þ 2 jDi j is satisfied, and there exist some positive definite symmetric matrices P , Mi , Ni , such that the following LMI condition holds: and Q"i , i 2 m, # 12 U11 U ð11Þ T U 22 < 0; U 12 where
m X s2 11 ¼ AT P þ P A þ hi Mi þ Ni þ i Qi ; U 2 i¼1 h pffiffiffiffiffi p ffiffiffiffiffi T T 12 ¼ U h1 A PB1 hm A PBm AT PC1 AT PCm i s1 AT PD1 sm AT PDm ; 22 ¼ diag ðM1 ; . . . ; Mm ; N1 ; . . . ; Nm ; 2 Q1 ; . . . ; 2 Qm Þ: U
Remark 6. Corollary 4 coincides with the Theorem 2 of [3]. Theorem 2 of [3] could be seen as a special case of this result. By Lemma 3 and m ¼ 1, (11) is equivalent to
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
733
AT P þ P A þ h ðAT PBM 1 BT P A þ MÞ þ ðAT PCN 1 C T P A þ N Þ þ
s2 T ðA PDQ1 DT P A þ QÞ < 0; 2
where A ¼ A þ B þ s D. Choose þ eÞ I; M ¼ ðkBT P Ak
þ eÞ I; N ¼ ðkC T P Ak
þ eÞ I; Q ¼ ðkDT P Ak
then, we have T x ðtÞ AT P þ P A þ h ðAT PBM 1 BT P A þ MÞ þ ðAT PCN 1 C T P A þ N Þ s2 þ ðAT PDQ1 DT P A þ QÞ xðtÞ " 2 ! T 2 kB P Ak þ h kBT P Ak þeþ 6 2lðP AÞ þe kBT P Ak ! T 2 þ e þ kC P Ak þ kC T P Ak þe kC T P Ak !# 2 s2 kDT P Ak T þ kD P Ak þ e þ kxðtÞk2 T 2 kD P Ak þ e s2 T T T 6 2 lðP AÞ þ h kB P Ak þ kC P Ak þ kD P Ak 2 e s2 2 þ h þ þ 1 kxðtÞk : 2 2 Note that the parameter e > 0 could be chosen sufficiently small. Corollary 4 is less conservative than the result in [2].
3. Illustrative example Consider the following neutral system: 2 0 1 0:2 x_ ðtÞ ¼ xðtÞ þ xðt 0:776Þ 0 2 0 1 0:1 0 0 0:1 þ xðt hÞ þ x_ ðt 0:776Þ 0 0:1 0:1 0 Z t 0:05 0 0:01 0 þ x_ ðt hÞ þ xðsÞ ds 0 0:05 0 0:01 t0:2 Z t 0:05 0 xðsÞ ds; þ 0 0:05 t0:1
ð12Þ
734
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
where h 2 Rþ is a finite constant. In view of (3) and (12), we have h1 ¼ 0:776, h2 ¼ h, s1 ¼ 0:2, and s2 ¼ 0:1. By Theorem 1 with 0:85 0 E1 ¼ 0 0:85 and E2 ¼ F 1 ¼ F 2 ¼
0 0 ; 0 0
we have q½h1 jE1 j þ jC1 j þ jC2 j < 1; 1:8162 0:0708 4:3979 0:1714 P¼ ; M1 ¼ ; 0:0708 3:4423 0:1714 8:3376 0:7077 0:0175 0:3030 0:2186 N2 ¼ ; N1 ¼ 0:0175 0:7136 0:2186 0:5743 0:4097 0:1244 0:1820 0:0068 R1 ¼ ; R2 ¼ ; 0:1244 0:3795 0:0068 0:3432 0:0217 0:0027 0:0960 0:0052 ; V2 ¼ ; V1 ¼ 0:0027 0:0389 0:0052 0:1750 0:3484 0:1057 0:1548 0:0058 S11 ¼ ; S21 ¼ ; 0:1057 0:3225 0:0058 0:2916 0:0300 0:0127 0:0282 0:0191 T11 ¼ ; T12 ¼ ; 0:0127 0:0535 0:0191 0:0259 0:0249 0:0008 0:0114 0:0081 ; T22 ¼ ; T21 ¼ 0:0008 0:0261 0:0081 0:0207 0:0205 0:0039 0:0846 0:0062 W11 ¼ ; W21 ¼ ; 0:0039 0:0364 0:0062 0:1528 0:0080 0:0050 0:0075 0:0055 ; X12 ¼ ; X11 ¼ 0:0050 0:0143 0:0055 0:0139 0:0208 0:0082 0:0161 0:0116 X21 ¼ ; X22 ¼ : 0:0082 0:0318 0:0116 0:0293 Hence, system (12) is asymptotically stable for any h 2 Rþ . The delay-independent criteria in [3,7] cannot be satisfied. The delay-dependent stability criteria of [2,3,6] cannot be applied for a sufficiently large time delay h 2 Rþ .
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
735
4. Conclusions In this paper, by making of Lyapunov stability theorem and LMI approach, some generalizations on the stability criteria have been proposed to guarantee asymptotic stability for a class of neutral systems with multiple time-delays. It has been shown by mathematical proof that new sufficient conditions are proved to be less conservative than these results appeared in the current lit is an erature. Furthermore, the suitable choice for the matrices Ei and Fi , i 2 m, open research topic that is not presented in this paper, and we will consider this problem by other useful techniques; such as Genetic Algorithms [24–26]. References [1] S. Boyd, L.El. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA, 1994. [2] J.D. Chen, C.H. Lien, K.K. Fan, J.S. Cheng, Delay-dependent stability criterion for neutral time-delays systems, Electron. Lett. 36 (2000) 1897–1898. [3] J.D. Chen, C.H. Lien, K.K. Fan, J.H. Chou, Criteria for asymptotic stability of a class of neutral systems via a LMI approach, IEE Proc. Control Theory Appl. 148 (2001) 442–447. [4] J.D. Chen, C.H. Lien, J.H. Chou, Flexible stability criteria of a class of neutral systems with multiple time delays via LMI approach, J. Chin. Inst. Eng. 25 (2002) 341–348. [5] L. Dugard, E.I. Verriest, Stability and Control of Time-Delay Systems, Springer-Verlag, London, 1997. [6] K.K. Fan, C.H. Lien, J.G. Hsieh, Asymptotic stability for a class of neutral systems with multiple time delays, J. Optim. Theory Appl. 114 (2002) 705–716. [7] E. Fridman, New Lyapunov–Krasovskii functionals for stability of linear retarded and neutral type systems, Syst. Control Lett. 43 (2001) 309–319. [8] K. Gu, Q.L. Han, A.C.J. Luo, S.I. Niculescu, Discretized Lyapunov functional for systems with distributed delay and piecewise constant coefficients, Int. J. Control 74 (2001) 737–744. [9] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, SpringerVerlag, New Jersy, 1993. [10] G.Di. Hui, G.Da. Hu, Simple criteria for stability of neutral systems with multiple delays, Int. J. Syst. Sci. 28 (1997) 1325–1328. [11] D. Ivanescu, J.M. Dion, L. Dugard, S.I. Niculescu, Dynamical compensation for time-delay systems: an LMI approach, Int. J. Robust Nonlinear Control 10 (2000) 611–628. [12] J.H. Kim, Delay and its-derivative dependent robust stability of time-delayed linear systems with uncertainty, IEEE Trans. Automat. Control 46 (2001) 789–792. [13] V.B. Kolmanovskii, A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999. [14] V.B. Kolmanovskii, J.P. Richard, Stability of some linear systems with delays, IEEE Trans. Automat. Control 44 (1999) 984–989. [15] X. Li, C.E. De Souza, Delay-dependent robust stability and stabilization of uncertain linear delay systems: a linear matrix inequality approach, IEEE Trans. Automat. Control 42 (1997) 1144–1148. [16] C.H. Lien, K.W. Yu, J.G. Hsieh, Stability conditions for a class of neutral systems with multiple time delays, J. Math. Anal. Appl. 245 (2000) 20–27. [17] B. Ni, Q.L. Han, On stability for a class of neutral delay-differential systems, in: Proceedings of the American Control Conference, Arlington, VA, 2001, pp. 25–27.
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[18] S.I. Niculescu, Further remarks on delay-dependent stability of linear neutral systems, in: Proceedings of MTNS 2000, Perpignan, France, 2000. [19] S.I. Niculescu, T.A. Neto, J.M. Dion, L. Dugard, Delay-dependent stability of linear systems with delayed state: An LMI approach, in: Proceedings of 34th Conference Decision and Control, New Orleans, LA, 2 December. 1995, pp. 1495–1496. [20] J.H. Park, S. Won, Asymptotic stability of neutral systems with multiple delays, J. Optim. Theory Appl. 103 (1999) 183–200. [21] J.H. Park, S. Won, Stability analysis for neutral delay-differential systems, J. Franklin Inst. 337 (2000) 1–9. [22] P. Park, A delay-dependent stability criterion for systems with uncertain time-invariant delays, IEEE Trans. Automat. Control 44 (1999) 876–877. [23] J.J. Yan, Robust stability analysis of uncertain time delay systems with delay-dependence, Electron. Lett. 37 (2001) 135–137. [24] L.J. Eshelman, J.D. Schaffer, in: L.D. Whitley (Ed.), Real-Coded Genetic Algorithms and Interval Schemata, Foundation of Genetic Algorithms-2, Morgan Kaufmann Publishers, Los Altas, CA, 1993. [25] M. Gen, R.W. Cheng, Genetic Algorithms and Engineering Design, John Wiley and Sons, New York, 1997. [26] Z. Michalewicz, Genetic Algorithm + Data Structures ¼ Evolution Programs, Springer-Verlag, New York, 1994.
New stability criteria for a class of neutral systems with discrete and distributed time-delays: an LMI approach Jenq-Der Chen
*
Institute of Engineering Science and Technology, National Kaohsiung First University of Science and Technology, Kaohsiung 824, Taiwan, ROC
Abstract In this paper, a problem of the asymptotic stability for a class of neutral systems with multiple discrete and distributed time-delays is considered. Lyapunov stability theory is applied to guarantee the stability for the systems. New discrete-delay-independent and discrete-delay-dependent stability conditions are derived in terms of the spectral radius and linear matrix inequality. By mathematical analysis, the stability criteria are proved to be less conservative than the ones reported in the current literatures. A numerical example is given to illustrate the availability of the proposed results. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Neutral systems; Discrete and distributed time-delays; Lyapunov stability theory; Linear matrix inequality
1. Introduction In the recent years, stability analysis of time-delay systems has received considerable attention and has been one of the most interesting topics in the control systems. This is due to theoretical interests as well as a powerful tool for practical system analysis and design, since delay phenomenon is often encountered in various mechanics, physics, biology, medicine, economy, and engineering systems, such as AIDS epidemic, aircraft stabilization, chemical * Address: Department of Electronic Engineering, Yung-Ta Institute of Technology & Commerce, 316 Chunsan Road, Lin-Lo, Ping-Tung 909, Taiwan, ROC. E-mail address: [email protected] (J.-D. Chen).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00302-3
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J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
Nomenclature C0 set of all continuous functions from ½H ; 0 to Rn Rþ set of all non-negative real numbers Rn n-dimensional real space Rm n set of all real m by n matrices AT (respectively xT ) transpose of matrix A (respectively, vector x) kxk Euclidean norm of vector x kAk spectral norm of matrix A P > 0 P is a positive definite symmetric matrix P < 0 P is a negative definite symmetric matrix qðAÞ spectral radius of real matrix A jAj ½jaij j, with A ¼ ½aij m f1; 2; . . . ; mg I unit matrix engineering systems, control of epidemics, distributed networks, infeed grinding and cutting model, manual control, microwave oscillator, models of lasers, neural network, nuclear reactor, population dynamic model, rolling mill, ship stabilization, and systems with lossless transmission lines [2–23]. Moreover, time-delay is frequently a source of instability and a source of generation of oscillation in many systems; for example, the trivial solution of ða 1Þ ½€xðtÞ þ x_ ðtÞ þ xðtÞ ¼ 0, with a > 1, is stable, but that of system €xðtÞ þ x_ ðtÞ þ xðtÞ ¼ a ½€xðt hÞ þ x_ ðt hÞ þ xðt hÞ is unstable for any h > 0 [13]. Consider the linear system " # m X x_ ðtÞ ¼ A þ Ei xðtÞ; t P 0;
ð1Þ
i¼1
The necessary and sufficient where xðtÞ 2 Rn , A, Ei 2 Rn n , i 2 m. Pm condition for asymptotic stability of system (1) is that the matrix A þ i¼1 Ei is Hurwitz. Recently, many reports are concentrated on systems with discrete delay [4,8– 11,14–23]. By increasing in the equation number of summands and simultaneously decreasing the differences between neighbouring argument values, one naturally arrives at equations with distributed (or continuous) and mixed (both distributed and discrete) delay arguments [2,3,6,7,13]. Now, we consider the neutral systems with discrete and distributed time-delays: Z t m X x_ ðtÞ ¼ AxðtÞ þ Bi xðt hi Þ þ Ci x_ ðt hi Þ þ Di xðsÞ ds ; t P 0; i¼1
tsi
ð2aÞ
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
xðtÞ ¼ /ðtÞ;
t 2 ½H ; 0;
721
ð2bÞ
where x 2 Rn , xt is the state at time t defined by xt ðsÞ :¼ xðt þ sÞ 8s 2 ½H ; 0, are H ¼ maxi2m fhi ; si g P 0, with kxt ks :¼ supH 6 r 6 0 kxðt þ rÞk, hi and si , i 2 m, both non-negative constants, which represent discrete delays and distributed are known, and delays, respectively. The matrices A, Bi , Ci , Di 2 Rn n , i 2 m, the initial vector / 2 C0 . play an important role In view of [2–6,13], the distributed delays si , i 2 m, about the stability of system (2), so our main results will also depend on those Depending on whether the stability criterion itself distributed delays si , i 2 m. contains the discrete delay argument as a parameter, stability criteria for neutral systems can be classified into two categories, namely discrete-delayindependent criteria [4,7,8,16,20,21] and discrete-delay-dependent criteria [2– 4,6,16,17,20,21]. Generally speaking, the latter ones are less conservative than the former ones, but the former ones are also important when the effect of time delay is small. In this paper, the discrete-delay-independent and discrete-delaydependent criteria will be proposed to guarantee the asymptotic stability for neutral systems with multiple time-delays. Many approaches have been used to searching sufficient conditions for the stability problem of time-delay systems. The stability problem for time-delay systems with distributed delay is considered using discretized Lyapunov functional [8]. A robust stability of time-delay systems was investigated by checking the Hamiltonian matrix and solving an algebraic Riccati equation [23]. Appropriate model transformations of original time-delay systems and Lyapunov theory are useful for the analysis of stability of systems [5,11–16,20–23]. Many sharp results are used the linear matrix inequality (LMI) approach to solve the stability problem of time-delay systems [5,9,14,16,20–22]. Furthermore, a model transformation and Lyapunov theory with LMI approach are used in this paper, less conservative criteria are proposed to guarantee the asymptotic stability for the neutral systems with discrete and distributed time-delays. A numerical example is given to illustrate the validity of the proposed results.
2. Problem formulation and main results By some model transformations, system (2) can be written as: " # Z t Z t m X d xðtÞ þ xðsÞ ds Ci xðt hi Þ þ Fi ðh t þ si ÞxðhÞ dh Ei dt thi tsi i¼1 Z t m X b ¼ AxðtÞ þ ðBi Ei Þxðt hi Þ þ ðDi Fi Þ xðhÞ dh ; t P 0; i¼1
tsi
ð3Þ
722
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
are some chosen matrices such that the matrix where Ei P , Fi 2 Rn n , i 2 m, m b A ¼ A þ i¼1 ðEi þ si Fi Þ is Hurwitz in view of (1). By this transformation, it will cause our obtained results are less conservative than other recent literatures. Remark 1. The system (2a) is a general representation for the description of neutral system discrete and distributed time-delays, for example the neutral system Z t x_ ðtÞ ¼ AxðtÞ þ Bxðt hÞ þ C x_ ðt gÞ þ D xðsÞ ds; ts
can be converted into the form of system (2a) with B1 ¼ B;
C1 ¼ D1 ¼ 0;
D3 ¼ D;
h1 ¼ h;
B2 ¼ D2 ¼ 0;
h2 ¼ g;
C2 ¼ C;
B3 ¼ C3 ¼ 0;
s3 ¼ s:
Lemma 1 (Chen et al., 2001 [3], Chen et al., 2002 [4]). For any matrices Ei , Ci , Fi 2 Rn n , if ! m X s2i q hi jEi j þ jCi j þ jFi j < 1; 2 i¼1 then the operator D : C0 ! Rn with Z t Z m X Dðxt Þ ¼ xðtÞ þ Ei xðsÞds Ci xðt hi Þ þ Fi i¼1
thi
t
ðh t þ si ÞxðhÞdh tsi
is stable. Lemma 2. For any vectors x, y 2 Rn and R > 0, we have 2xT y 6 xT Rx þ y T R1 y:
QðyÞ T SðyÞ 1 T QðyÞ SðyÞRðyÞ SðyÞ < 0;
Lemma 3 (Boyd et al., 1994 [1]). The LMI RðyÞ < 0;
SðyÞ < 0 is equivalent to RðyÞ
where QðyÞ ¼ QðyÞT , RðyÞ ¼ RðyÞT , and SðyÞ depend affinely on y. Now we present the main result for asymptotic stability of system (2). b is Theorem 1. System (2) is asymptotically stable provided that the matrix A Hurwitz and satisfying the condition
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
q
m X
hi jEi j þ jCi j þ
i¼1
s2i jFi j 2
723
! ð4aÞ
<1
and there exist some positive definite symmetric matrices P , Mi , Ni , Qi , Ri , Sij , Tij , such that the following LMI condition holds: Uij , Vi , Wij , Xij , and Yij , i, j 2 m, 3 2 U11 U12 U13 U14 U15 U16 U17 U18 6 UT12 U22 0 0 0 0 0 0 7 7 6 T 6U 0 U 0 0 0 0 0 7 33 7 6 13 6 UT 0 0 U44 0 0 0 0 7 7 6 14 U¼6 T 7 < 0; 6 U15 0 0 0 U55 0 0 0 7 7 6 T 6 U16 0 0 0 0 U66 0 0 7 7 6 T 4U 0 0 0 0 0 U77 0 5 17
UT18
0
0
0
0
0
0
U88 ð4bÞ
where bT P þ P A bþ U11 ¼ A
m X i¼1
þ
m X j¼1
U12 ¼
" hi Mi þ Ni þ
s2i Qi þ Ri þ si Vi 2
s2j s2j hj Sij þ Tij þ Uij þ si hj Wij þ si Xij þ si Yij 2 2
!# ;
h pffiffiffiffiffi pffiffiffiffiffi T b PEm ATPC b T PE1 hm A bT b h1 A 1 A PCm T b T PF1 sm A b T PFm ðB1 E1 Þ P ðBm Em ÞT P s1 A i pffiffiffiffi pffiffiffiffiffi T T s1 ðD1 F1 Þ P sm ðDm Fm Þ P ;
U22 ¼ diag ðM1 ; . . . ; Mm ; N1 ; . . . ; Nm ; 2 Q1 ; . . . ; 2 Qm ; R1 ; . . . ; Rm ; V1 ; . . . ; Vm Þ; pffiffiffiffiffi pffiffiffiffiffi T T h1 ðBi Ei Þ PE1 hm ðBi Ei Þ PEm ; U13 ¼ ½X1 Xm ; Xi ¼ b 1; . . . ; X b m Þ; X b i ¼ diag ðSi1 ; . . . ; Sim Þ; U33 ¼ diag ð X U14 ¼ ½R1 Rm ; Ri ¼ ½ðBi Ei ÞT PC1 ðBi Ei ÞT PCm ; b 1; . . . ; R b m Þ; R b i ¼ diag ðTi1 ; . . . ; Tim Þ; U44 ¼ diag ð R T
T
U15 ¼ ½N1 Nm ; Ni ¼ ½s1 ðBi Ei Þ PF1 sm ðBi Ei Þ PFm ; b 1; . . . ; N b m Þ; N b i ¼ diag ð2 Ui1 ; . . . ; 2 Uim Þ; U55 ¼ diag ð N U16 ¼ ½C1 Cm ; pffiffiffiffi pffiffiffiffiffi pffiffiffiffi pffiffiffiffiffi si h1 ðDi Fi ÞT PE1 si hm ðDi Fi ÞT PEm ; Ci ¼ b 1; . . . ; C b m Þ; C b i ¼ diag ðWi1 ; . . . ; Wim Þ; U66 ¼ diag ð C
724
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
U17 ¼ ½K1 Km ;
pffiffiffiffi pffiffiffiffi T T Ki ¼ ½ si ðDi Fi Þ PC1 si ðDi Fi Þ PCm ;
b 1; . . . ; K b m Þ; K b i ¼ diag ðXi1 ; . . . ; Xim Þ; U77 ¼ diag ð K pffiffiffiffi pffiffiffiffi T T U18 ¼ ½W1 Wm ; Wi ¼ ½ si s1 ðDi Fi Þ PF1 si sm ðDi Fi Þ PFm ; b 1; . . . ; W b m Þ; W b i ¼ diag ð2 Yi1 ; . . . ; 2 Yim Þ: U88 ¼ diag ð W Proof. By the schur complement of [1], the condition (4b) is equivalent to pðP ; Mi ; Ni ; Qi ; Ri ; Sij ; Tij ; Uij ; Vi ; Wij ; Xij ; Yij Þ b TP þ P A bþ ¼A
m h X
b T PEi M 1 ET P A b þ Mi Þ hi ð A i i
i
i¼1
þ
þ
m m 2 X X si b T 1 T b b T PCi N 1 C T P A b þ Ni Þ þ ð A ðA PF Q F P A þ Q Þ i i i i i i 2 i¼1 i¼1 m X
½ðBi Ei ÞT PR1 i P ðBi Ei Þ þ Ri
i¼1
þ
m X m X i¼1
þ
j¼1
m X m X i¼1
T
½hj ððBi Ei Þ PEj Sij1 EjT P ðBi Ei Þ þ Sij Þ ½ðBi Ei ÞT PCj Tij1 CjT P ðBi Ei Þ þ Tij
j¼1
# " m X m X s2j T 1 T þ ððBi Ei Þ PFj Uij Fj P ðBi Ei Þ þ Uij Þ 2 i¼1 j¼1 þ
m X T ½si ððDi Fi Þ PVi 1 P ðDi Fi Þ þ Vi Þ i¼1
þ
m X m X i¼1
þ
m X m X i¼1
þ
T
½si ððDi Fi Þ PCj Xij1 CjT P ðDi Fi Þ þ Xij Þ
j¼1
m X m X i¼1
T
½si hj ððDi Fi Þ PEj Wij1 EjT P ðDi Fi Þ þ Wij Þ
j¼1
j¼1
"
# s2j T 1 T si ððDi Fi Þ PFj Yij Fj P ðDi Fi Þ þ Yij Þ < 0: 2 ð5Þ
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
725
The functional given by V ðxt Þ ¼ V1 ðxt Þ þ V2 ðxt Þ þ V3 ðxt Þ þ V4 ðxt Þ þ V5 ðxt Þ;
ð6aÞ
where V1 ðxt Þ ¼ GT ðxt Þ P Gðxt Þ; m X
Gðxt Þ ¼ xðtÞ þ
Z
t
xðsÞds Ci xðt hi Þ þ Fi
Ei thi
i¼1
Z
ðh t þ si ÞxðhÞdh ;
t
tsi
ð6bÞ V2 ðxt Þ ¼
m Z X i¼1
(
t
xT ðsÞ ðs t þ hi Þ Mi þ Ni þ ðBi Ei Þ
T
thi
" R1 i
P
þ
m X
hj
Ej Sij1 EjT
þ
Cj Tij1 CjT
j¼1
s2j þ Fj Uij1 FjT 2
!#
)
P ðBi Ei Þ xðsÞ ds;
V3 ðxt Þ ¼
m Z X i¼1
(
t T
x ðhÞ
tsi
"
P Vi
1
þ
ð6cÞ
1 2 T ðh t þ si Þ Qi þ ðDi Fi Þ 2
m X
hj
Ej Wij1 EjT
j¼1
þ
Cj Xij1 CjT
s2j þ Fj Yij1 FjT 2
!#
)
P ðDi Fi Þ xðhÞ dh;
V4 ðxt Þ ¼
m X m Z X i¼1
j¼1
t
h xT ðsÞ ðs t þ hj Þ ðSij þ si Wij Þ:
thj
i þ ðTij þ si Xij Þ xðsÞ ds;
V5 ðxt Þ ¼
ð6dÞ
m X m Z t 1X 2 ðh t þ sj Þ xT ðhÞðUij þ si Yij ÞxðhÞ dh; 2 i¼1 j¼1 tsj
ð6eÞ
ð6fÞ
is a legitimate Lyapunov functional candidate [9]. The time derivative of Vi ðxt Þ, i¼ 5, along the trajectories of system (3) is given by
726
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
b T P þ P AÞxðtÞ b b TP V_1 ðxt Þ ¼ xT ðtÞð A þ 2xT ðtÞ A
m X
Z
xðsÞ ds thi
i¼1
b TP 2xT ðtÞ A b TP þ 2x ðtÞ A T
m X i¼1 m X
Ci xðt hi Þ Z
m X
t
ðh t þ si ÞxðhÞ dh
Fi tsi
i¼1
þ2
xT ðt hi ÞðBi Ei ÞT PxðtÞ
i¼1
þ2 2
m X m X i¼1 j¼1 m X m X i¼1
þ2 þ2 þ2 2
i¼1 j¼1 m Z t X
þ2
x ðt hi ÞðBi Ei Þ PEj
j¼1
j¼1
xðsÞ ds
T
xT ðt hi ÞðBi Ei Þ PCj xðt hj Þ T
xT ðt hi ÞðBi Ei Þ PFj
Z
t
ðh t þ sj ÞxðhÞ dh tsj
T
xT ðsÞ dsðDi Fi Þ PxðtÞ
Z
m X m Z X i¼1
t
thj
tsi i¼1 m X m Z t X i¼1 j¼1 m X m X
Z
T
T
j¼1
m X m X
i¼1
t
Ei
T
T
x ðhÞ dhðDi Fi Þ PEj
Z
t
xðsÞ ds
tsi
thj
t T
xT ðhÞ dhðDi Fi Þ PCj xT ðt hj Þ tsi t T
xT ðhÞ dhðDi Fi Þ PFj
Z
tsi
t
ðh t þ sj ÞxðhÞ dh:
tsj
By using the Lemma 2, it is ease to see b T P þP AÞxðtÞ b V_1 ðxt Þ 6 xT ðtÞð A " Z m X T T 1 T b b þ hi x ðtÞ A PEi M E P AxðtÞ þ i
i¼1
þ
#
t T
x ðsÞMi xðsÞds
i
thi
m X T b T PCi N 1 C T P AxðtÞþx b ½xT ðtÞ A ðt hi ÞNi xðt hi Þ i i i¼1
" # Z t m X s2i T b T 1 T b T x ðtÞ A PFi Qi Fi P AxðtÞ þ ðht þsi Þx ðhÞQi xðhÞdh þ 2 tsi i¼1 þ
m X T T ½xT ðt hi ÞðBi Ei Þ PR1 i P ðBi Ei Þxðt hi Þþx ðtÞRi xðtÞ i¼1
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
þ
m X m X i¼1
þ þ
Z
hj xT ðt hi ÞðBi Ei ÞT PEj Sij1 EjT P ðBi Ei Þxðt hi Þ #
j¼1 t
xT ðsÞSij xðsÞ ds
thj m X m X i¼1
727
"
T
½xT ðt hi ÞðBi Ei Þ PCj Tij1 CjT P ðBi Ei Þxðt hi Þ
j¼1
þ xT ðt "hj ÞTij xðt hj Þ m X m X s2j T x ðt hi ÞðBi Ei ÞT PFj Uij1 FjT P ðBi Ei Þxðt hi Þ þ 2 i¼1 j¼1 # Z t T ðh t þ sj Þ x ðhÞUij xðhÞ dh þ tsj m Z t X T þ xT ðhÞðDi Fi Þ PVi 1 P ðDi Fi ÞxðhÞ dh þ si xT ðtÞVi xðtÞ tsi i¼1 " Z t m X m X T þ hj xT ðhÞðDi Fi Þ PEj Wij1 EjT P ðDi Fi ÞxðhÞ dh ts i i¼1 j¼1 # Z t
xT ðsÞWij xðsÞ ds
þ si
thj
þ
"Z m X m X i¼1
j¼1
t T
xT ðhÞðDi Fi Þ PCj Xij1 CjT P ðDi Fi ÞxðhÞ dh tsi #
þ si xT ðt hj ÞXij xðt hj Þ " Z t m X m X s2j T þ xT ðhÞðDi Fi Þ PFj Yij1 FjT P ðDi Fi ÞxðhÞ dh 2 ts i i¼1 j¼1 # Z t T ðh t þ sj Þ x ðhÞYij xðhÞ dh ; þ si tsj
V_2 ðxt Þ ¼
m X
(
" T
x ðtÞ hi Mi þ Ni þ ðBi Ei Þ P R1 i þ T
m X
hj Ej Sij1 EjT
i¼1
) j¼1 !# 2 m Z t X s j xT ðsÞMi xðsÞ ds þ Cj Tij1 CjT þ Fj Uij1 FjT P ðBi Ei Þ xðtÞ 2 thi i¼1 ( " m m X X T T x ðt hi Þ Ni þ ðBi Ei Þ P R1 hj Ej Sij1 EjT i þ i¼1
s2j þ Cj Tij1 CjT þ Fj Uij1 FjT 2
!#
j¼1
)
P ðBi Ei Þ xðt hi Þ;
728
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736 m X
V_3 ðxt Þ ¼
i¼1
" s2i T Qi þ ðDi Fi Þ P Vi 1 x ðtÞ 2
þ
(
T
m X
hj
Ej Wij1 EjT
þ
Cj Xij1 CjT
j¼1
m Z X
!#
) P ðDi Fi Þ xðtÞ
t
xT ðhÞðh t þ si Þ Qi xðhÞ dh tsi
i¼1
s2j þ Fj Yij1 FjT 2
m X
(
" T
x ðt si Þ ðDi Fi Þ P Vi 1 T
i¼1
þ
m X
hj
Ej Wij1 EjT
þ
Cj Xij1 CjT
j¼1
s2j þ Fj Yij1 FjT 2
!#
)
P ðDi Fi Þ xðt si Þ;
V_4 ðxt Þ ¼
m X m X i¼1
xT ðtÞ½hj ðSij þ si Wij Þ þ ðTij þ si Xij ÞxðtÞ
j¼1
"Z m X m X i¼1
j¼1
t
xT ðsÞðSij þ si Wij ÞxðsÞ ds thj
# xT ðt hj ÞðTij þ si Xij Þxðt hj Þ ; " m X m X s2j T x ðtÞðUij þ si Yij ÞxðtÞ V_5 ðxt Þ ¼ 2 i¼1 j¼1
Z
#
t T
ðh t þ sj Þ x ðhÞðUij þ si Yij ÞxðhÞ dh :
tsj
Hence the derivative of V ðxt Þ is given by V_ ðxt Þ ¼ V_1 ðxt Þ þ V_2 ðxt Þ þ V_3 ðxt Þ þ V_4 ðxt Þ þ V_5 ðxt Þ 6 xT ðtÞ pðP ; Mi ; Ni ; Qi ; Ri ; Sij ; Tij ; Uij ; Vi ; Wij ; Xij ; Yij Þ xðtÞ: ð7Þ P s2i m In view of Lemma 1, the condition q <1 i¼1 hi jEi j þ jCi j þ 2 jFi j guarantees that the operator
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
Gðxt Þ ¼ xðtÞ þ þ Fi
m X i¼1 Z t
Z
729
t
xðsÞ ds Ci xðt hi Þ
Ei thi
ðh t þ si ÞxðhÞ dh ;
tsi
is stable. Thus, by Theorem 9.8.1 of [9] with (5)–(7), we conclude that systems (2) and (3) are both asymptotically stable. the entries of U in (4b) Remark 2. Notice that for given si , hi , Ei , and Fi , i 2 m, Hence, the are affine in P , Mi , Ni , Qi , Ri , Sij , Tij , Uij , Vi , Wij , Xij , and Yij , i, j 2 m. asymptotic stability problem for the neutral systems with discrete and distributed time-delays can be converted into a strictly feasible LMI problem [11]. it Remark 3. Notice that forR all chosen matrices R t Ei ¼ 0 (or Fi ¼ 0), i 2 m, t do represents that the term Ei thi xðsÞ ds (or Fi tsi ðhi t þ si ÞxðhÞ dh), i 2 m not convert to left-hand side of system (3). By the proof of Theorem 1, the LMI condition (4b) are reduced by deleting their corresponding elements. in Theorem 1, we may obtain the first Simply setting Ei ¼ 0, Fi ¼ 0, i 2 m, discrete-delay-independent criterion for asymptotic stability of system (2). Corollary 1. System (2) is asymptotically stable for any delays hi 2 Rþ provided Pm that A is Hurwitz, the condition q i¼1 jCi j < 1 is satisfied, and there exist some such that positive definite symmetric matrices P , Ni , Ri , Tij , Vi , and Xij , i, j 2 m, the following LMI condition holds: 2 3 b 12 b 13 b 14 b 11 U U U U T 6U b 22 0 0 7 6 b 12 U 7 ð8Þ 6 bT 7 < 0; b 33 4 U 13 0 U 0 5 b 44 bT 0 0 U U 14
where b 11 ¼ AT P þ PA þ U
m X
" N i þ Ri þ s i V i þ
i¼1
m X
# ðTij þ si Xij Þ ;
j¼1
pffiffiffiffiffi b 12 ¼ AT PC1 AT PCm BT P BT P pffiffiffiffi U s1 DT1 P sm DTm P ; 1 m b 22 ¼ diag ðN1 ; . . . ; Nm ; R1 ; . . . ; Rm ; V1 ; . . . ; Vm Þ; U b 13 ¼ ½R1 Rm ; Ri ¼ ½BT PC1 BT PCm ; U i
i
b 33 ¼ diag ð R b 1; . . . ; R b m Þ; R b i ¼ diag ðTi1 ; . . . ; Tim Þ; U p pffiffiffiffi ffi b 14 ¼ ½K1 Km ; Ki ¼ ½ ffiffiffi U si DTi PC1 si DTi PCm ; b 1; . . . ; K b m Þ; K b i ¼ diag ðXi1 ; . . . ; Xim Þ: b 44 ¼ diag ð K U
730
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
Remark 4. Corollary 1 coincides with the Theorem 1 of [3]. Theorem 1 of [3] could be seen as a special case of this result. in Theorem 1, we may obtain the Simply setting Ei ¼ 0, Fi ¼ Di , i 2 m, second discrete-delay-independent criterion for asymptotic stability of system (2). Corollary 2. System (2) is asymptotically stable for any delays hi 2 Rþ provided P P s2i m <1 that A ¼ A þ mi¼1 si Di is Hurwitz, the condition q i¼1 jCi j þ 2 jDi j is satisfied, and there exist some positive definite symmetric matrices P , Ni , Qi , Ri , such that the following LMI condition holds: Tij , and Uij , i; j 2 m, 2 3 U11 U12 U13 U14 6 T 7 0 0 7 6 U12 U22 6 T 7 < 0; ð9Þ 6U 0 U33 0 7 4 13 5 T U14 0 0 U44 where T
U11 ¼ A P þ P A þ
m X i¼1
"
h U12 ¼ AT PC1 AT PCm
m X s2j s2 N i þ i Q i þ Ri þ Tij þ Uij 2 2 j¼1 T
T
s1 A PD1 sm A PDm
!# ;
i BT1 P BTm P ;
U22 ¼ diag ðN1 ; . . . ; Nm ; 2 Q1 ; . . . ; 2 Qm ; R1 ; . . . ; Rm Þ; U13 ¼ ½R1 Rm ; Ri ¼ ½BTi PC1 BTi PCm ; b 1; . . . ; R b m Þ; R b i ¼ diag ðTi1 ; . . . ; Tim Þ; U33 ¼ diag ð R U14 ¼ ½N1 Nm ; Ni ¼ ½s1 BTi PD1 sm BTi PDm ; b 1; . . . ; N b m Þ; N b i ¼ diag ð2 Ui1 ; . . . ; 2 Uim Þ: U44 ¼ diag ð N in Theorem 1, we may obtain the first Simply setting Ei ¼ Bi , Fi ¼ 0, i 2 m, discrete-delay-dependent criterion for asymptotic stability of system (2). P e ¼ A þ m Bi Corollary 3. System (2) is asymptotically stable provided that A i¼1 Pm is Hurwitz, the condition q i¼1 ðhi jBi j þ jCi jÞ < 1 is satisfied, and there exist such some positive definite symmetric matrices P , Mi , Ni , Vi , Wij , and Xij , i; j 2 m, that the following LMI condition holds: 2 3 e 12 e 13 e 14 e 11 U U U U 6 eT 7 e 22 6 U 12 U 0 0 7 6 7 < 0; ð10Þ 6U T e 33 0 U 0 7 4 e 13 5 eT e 44 U 0 0 U 14
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
731
where e TP þ P A eþ e 11 ¼ A U
m X
"
# m X hi Mi þ Ni þ si Vi þ ðsi hj Wij þ si Xij Þ ;
i¼1
e 12 U
j¼1
pffiffiffiffiffi T pffiffiffiffiffi T e PB1 hm A e PBm ¼ h1 A pffiffiffiffi T pffiffiffiffiffi T s1 D1 P sm Dm P ;
e T PCm e T PC1 A A
e 22 ¼ diag ðM1 ; . . . ; Mm ; N1 ; . . . ; Nm ; V1 ; . . . ; Vm Þ; U h ffi pffiffiffiffiffi i pffiffiffiffi pffiffiffiffiffi e 13 ¼ ½C1 Cm ; Ci ¼ pffiffiffi si h1 DTi PB1 si hm DTi PBm ; U e 33 ¼ diag ð C b 1; . . . ; C b m Þ; C b i ¼ diag ðWi1 ; . . . ; Wim Þ; U p pffiffiffiffi ffi e 14 ¼ ½K1 Km ; Ki ¼ ½ ffiffiffi si DTi PC1 si DTi PCm ; U b 1; . . . ; K b m Þ; K b i ¼ diag ðXi1 ; . . . ; Xim Þ: e 44 ¼ diag ð K U Remark 5. By using the Lemma 3 and si ¼ hi , (10) is equivalent to eþ e TP þ P A A
m X
e T PBi M 1 BT P A e þ Mi Þ ½hi ð A i i
i¼1
þ
m X
e T PCi N 1 C T P A e þ Ni Þ þ ðA i i
i¼1
þ þ
½hi ðDTi PVi 1 PDi þ Vi Þ
i¼1
m X m X i¼1 j¼1 m X m X i¼1
m X
½hi hj
ðDTi PBj Wij1 BTj PDi
þ Wij Þ
½hi ðDTi PCj Xij1 CjT PDi þ Xij Þ < 0:
j¼1
By setting some matrices as Vi ¼ ðkPDi k þ eÞ I;
Wij ¼ ðkBTj PDi k þ eÞ I;
Xij ¼ ðkCjT PDi k þ eÞ I;
where e > 0, we have ( m X T e TP þ P A eþ e T PBi M 1 BT P A e þ Mi Þ x ðtÞ A ½hi ð A i
i
i¼1
þ
m m X X e T PCi N 1 C T P A e þ Ni Þ þ ðA ½hi ðDTi PVi 1 PDi þ Vi Þ i i i¼1
þ
i¼1
þ
i¼1
m X m X
m X m X i¼1
½hi hj ðDTi PBj Wij1 BTj PDi þ Wij Þ
j¼1
j¼1
) ½hi
ðDTi PCj Xij1 CjT PDi
þ Xij Þ xðtÞ
732
( J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736 m X e TP þ P A eþ e T PBi M 1 BT P A e þ Mi Þ 6 xT ðtÞ A ½hi ð A i
i
i¼1 2 m X e T PCi N 1 C T P A e þ Ni Þ þ hi kPDi k þ e þ kPDi k ðA þ i i kPDi k þ e i¼1 ! 2 T m X kBj PDi k hi hj kBTj PDi k þ e þ T I þ kBj PDi k þ e j¼1 " ! #) 2 T m X kC PD k i j hi kCjT PDi k þ e þ T I xðtÞ þ kCj PDi k þ e j¼1 ( m X e TP þ P A eþ e T PBi M 1 BT P A e þ Mi Þ 6 xT ðtÞ A ½hi ð A i
! I
i
i¼1
þ
m m X X e T PCi N 1 C T P A e þ Ni Þ þ ðA hi ð2 kPDi k þ eÞ I i i i¼1
þ
i¼1
)
i¼1
m X m X
hi ½hj ð2
kBTj PDi k
þ eÞ þ ð2
kCjT PDi k
þ eÞ I xðtÞ:
j¼1
Note that the parameter e > 0 could be chosen sufficiently small. Corollary 3 is less conservative than the result in [6]. in Theorem 1, we may obtain the Simply setting Ei ¼ Bi , Fi ¼ Di , i 2 m, second discrete-delay-dependent criterion for asymptotic stability of system (2). Corollary 4. System (2) is asymptotically provided that A¼ P stable Aþ Pm s2i m <1 i¼1 ðBi þ si Di Þ is Hurwitz, the condition q i¼1 hi jBi j þ jCi j þ 2 jDi j is satisfied, and there exist some positive definite symmetric matrices P , Mi , Ni , such that the following LMI condition holds: and Q"i , i 2 m, # 12 U11 U ð11Þ T U 22 < 0; U 12 where
m X s2 11 ¼ AT P þ P A þ hi Mi þ Ni þ i Qi ; U 2 i¼1 h pffiffiffiffiffi p ffiffiffiffiffi T T 12 ¼ U h1 A PB1 hm A PBm AT PC1 AT PCm i s1 AT PD1 sm AT PDm ; 22 ¼ diag ðM1 ; . . . ; Mm ; N1 ; . . . ; Nm ; 2 Q1 ; . . . ; 2 Qm Þ: U
Remark 6. Corollary 4 coincides with the Theorem 2 of [3]. Theorem 2 of [3] could be seen as a special case of this result. By Lemma 3 and m ¼ 1, (11) is equivalent to
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
733
AT P þ P A þ h ðAT PBM 1 BT P A þ MÞ þ ðAT PCN 1 C T P A þ N Þ þ
s2 T ðA PDQ1 DT P A þ QÞ < 0; 2
where A ¼ A þ B þ s D. Choose þ eÞ I; M ¼ ðkBT P Ak
þ eÞ I; N ¼ ðkC T P Ak
þ eÞ I; Q ¼ ðkDT P Ak
then, we have T x ðtÞ AT P þ P A þ h ðAT PBM 1 BT P A þ MÞ þ ðAT PCN 1 C T P A þ N Þ s2 þ ðAT PDQ1 DT P A þ QÞ xðtÞ " 2 ! T 2 kB P Ak þ h kBT P Ak þeþ 6 2lðP AÞ þe kBT P Ak ! T 2 þ e þ kC P Ak þ kC T P Ak þe kC T P Ak !# 2 s2 kDT P Ak T þ kD P Ak þ e þ kxðtÞk2 T 2 kD P Ak þ e s2 T T T 6 2 lðP AÞ þ h kB P Ak þ kC P Ak þ kD P Ak 2 e s2 2 þ h þ þ 1 kxðtÞk : 2 2 Note that the parameter e > 0 could be chosen sufficiently small. Corollary 4 is less conservative than the result in [2].
3. Illustrative example Consider the following neutral system: 2 0 1 0:2 x_ ðtÞ ¼ xðtÞ þ xðt 0:776Þ 0 2 0 1 0:1 0 0 0:1 þ xðt hÞ þ x_ ðt 0:776Þ 0 0:1 0:1 0 Z t 0:05 0 0:01 0 þ x_ ðt hÞ þ xðsÞ ds 0 0:05 0 0:01 t0:2 Z t 0:05 0 xðsÞ ds; þ 0 0:05 t0:1
ð12Þ
734
J.-D. Chen / Appl. Math. Comput. 150 (2004) 719–736
where h 2 Rþ is a finite constant. In view of (3) and (12), we have h1 ¼ 0:776, h2 ¼ h, s1 ¼ 0:2, and s2 ¼ 0:1. By Theorem 1 with 0:85 0 E1 ¼ 0 0:85 and E2 ¼ F 1 ¼ F 2 ¼
0 0 ; 0 0
we have q½h1 jE1 j þ jC1 j þ jC2 j < 1; 1:8162 0:0708 4:3979 0:1714 P¼ ; M1 ¼ ; 0:0708 3:4423 0:1714 8:3376 0:7077 0:0175 0:3030 0:2186 N2 ¼ ; N1 ¼ 0:0175 0:7136 0:2186 0:5743 0:4097 0:1244 0:1820 0:0068 R1 ¼ ; R2 ¼ ; 0:1244 0:3795 0:0068 0:3432 0:0217 0:0027 0:0960 0:0052 ; V2 ¼ ; V1 ¼ 0:0027 0:0389 0:0052 0:1750 0:3484 0:1057 0:1548 0:0058 S11 ¼ ; S21 ¼ ; 0:1057 0:3225 0:0058 0:2916 0:0300 0:0127 0:0282 0:0191 T11 ¼ ; T12 ¼ ; 0:0127 0:0535 0:0191 0:0259 0:0249 0:0008 0:0114 0:0081 ; T22 ¼ ; T21 ¼ 0:0008 0:0261 0:0081 0:0207 0:0205 0:0039 0:0846 0:0062 W11 ¼ ; W21 ¼ ; 0:0039 0:0364 0:0062 0:1528 0:0080 0:0050 0:0075 0:0055 ; X12 ¼ ; X11 ¼ 0:0050 0:0143 0:0055 0:0139 0:0208 0:0082 0:0161 0:0116 X21 ¼ ; X22 ¼ : 0:0082 0:0318 0:0116 0:0293 Hence, system (12) is asymptotically stable for any h 2 Rþ . The delay-independent criteria in [3,7] cannot be satisfied. The delay-dependent stability criteria of [2,3,6] cannot be applied for a sufficiently large time delay h 2 Rþ .
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735
4. Conclusions In this paper, by making of Lyapunov stability theorem and LMI approach, some generalizations on the stability criteria have been proposed to guarantee asymptotic stability for a class of neutral systems with multiple time-delays. It has been shown by mathematical proof that new sufficient conditions are proved to be less conservative than these results appeared in the current lit is an erature. Furthermore, the suitable choice for the matrices Ei and Fi , i 2 m, open research topic that is not presented in this paper, and we will consider this problem by other useful techniques; such as Genetic Algorithms [24–26]. References [1] S. Boyd, L.El. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA, 1994. [2] J.D. Chen, C.H. Lien, K.K. Fan, J.S. Cheng, Delay-dependent stability criterion for neutral time-delays systems, Electron. Lett. 36 (2000) 1897–1898. [3] J.D. Chen, C.H. Lien, K.K. Fan, J.H. Chou, Criteria for asymptotic stability of a class of neutral systems via a LMI approach, IEE Proc. Control Theory Appl. 148 (2001) 442–447. [4] J.D. Chen, C.H. Lien, J.H. Chou, Flexible stability criteria of a class of neutral systems with multiple time delays via LMI approach, J. Chin. Inst. Eng. 25 (2002) 341–348. [5] L. Dugard, E.I. Verriest, Stability and Control of Time-Delay Systems, Springer-Verlag, London, 1997. [6] K.K. Fan, C.H. Lien, J.G. Hsieh, Asymptotic stability for a class of neutral systems with multiple time delays, J. Optim. Theory Appl. 114 (2002) 705–716. [7] E. Fridman, New Lyapunov–Krasovskii functionals for stability of linear retarded and neutral type systems, Syst. Control Lett. 43 (2001) 309–319. [8] K. Gu, Q.L. Han, A.C.J. Luo, S.I. Niculescu, Discretized Lyapunov functional for systems with distributed delay and piecewise constant coefficients, Int. J. Control 74 (2001) 737–744. [9] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, SpringerVerlag, New Jersy, 1993. [10] G.Di. Hui, G.Da. Hu, Simple criteria for stability of neutral systems with multiple delays, Int. J. Syst. Sci. 28 (1997) 1325–1328. [11] D. Ivanescu, J.M. Dion, L. Dugard, S.I. Niculescu, Dynamical compensation for time-delay systems: an LMI approach, Int. J. Robust Nonlinear Control 10 (2000) 611–628. [12] J.H. Kim, Delay and its-derivative dependent robust stability of time-delayed linear systems with uncertainty, IEEE Trans. Automat. Control 46 (2001) 789–792. [13] V.B. Kolmanovskii, A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999. [14] V.B. Kolmanovskii, J.P. Richard, Stability of some linear systems with delays, IEEE Trans. Automat. Control 44 (1999) 984–989. [15] X. Li, C.E. De Souza, Delay-dependent robust stability and stabilization of uncertain linear delay systems: a linear matrix inequality approach, IEEE Trans. Automat. Control 42 (1997) 1144–1148. [16] C.H. Lien, K.W. Yu, J.G. Hsieh, Stability conditions for a class of neutral systems with multiple time delays, J. Math. Anal. Appl. 245 (2000) 20–27. [17] B. Ni, Q.L. Han, On stability for a class of neutral delay-differential systems, in: Proceedings of the American Control Conference, Arlington, VA, 2001, pp. 25–27.
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