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Global robust asymptotical stability of multi-delayed interval neural networks: an LMI approach
Physics Letters A 328 (2004) 452–462 www.elsevier.com/locate/pla
Global robust asymptotical stability of multi-delayed interval neural networks: an LMI approach Chuandong Li a,∗ , Xiaofeng Liao a , Rong Zhang b a Department of Computer Science and Engineering, Chongqing University, Chongqing 400030, PR China b College of Business Administration, Chongqing University, Chongqing 400030, PR China
Received 14 March 2004; received in revised form 29 May 2004; accepted 16 June 2004
Communicated by A.P. Fordy
Abstract Based on the Lyapunov–Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) technique, some delay-dependent criteria for interval neural networks (IDNN) with multiple time-varying delays are derived to guarantee global robust asymptotic stability. The main results are generalizations of some recent results reported in the literature. Numerical example is also given to show the effectiveness of our results. 2004 Elsevier B.V. All rights reserved. Keywords: Robust stability; Interval delayed neural networks (IDNN); Lyapunov functional; Linear matrix inequality (LMI); Multiple time-varying delays
1. Introduction In recent years, several neural network models with single or multiple delays have been extensively studied [1–13], particularly regarding their stability analysis. However, it is well known that the stability of a well-designed system may often be destroyed by its unavoidable uncertainty due to the existence of modeling error, external disturbance and parameter fluctuation during the implementation. So it is essential to introduce the robust technique to design a system with such uncertainty. If the uncertainty of a system is only due to the deviations and perturbations of its parameters, and if these deviations and perturbations are all bounded, then the system is called an interval system. Recently, several global and robust stability criteria for interval neural networks with constant or time-varying delays have been proposed [16–19,21,22]. In this Letter, we further extend the results of [16–19] to a * Corresponding author.
E-mail addresses: [email protected] (C. Li), [email protected] (X. Liao). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.06.053
C. Li et al. / Physics Letters A 328 (2004) 452–462
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significantly larger class of systems. Specifically, we consider the following interval neural networks with multiple time-varying delays: r u(t) ˙ = −Du(t) + Ag u(t) + Bk g u t − τk (t) + I,
(1)
k=1
where u(t) = [u1 (t), . . . , un (t)]T is the state vector of the neural network, D = diag(d1 , . . . , dn ) is a positive (k) diagonal matrix, A = (aij )n×n and Bk = (bij )n×n are the connection weight and delayed connection weight matrices, respectively. g(u) = [g1 (u1 ), . . . , gn (u)]T denote the neuron activation functions with g(0) = 0, and I = [I1 , . . . , In ]T is a constant external input vector. 0 < τk (t) τ , k = 1, 2, . . . , r, represent axonal signal transmission delays. Suppose that the parameters of the system (1) are uncertain but bounded, and satisfy 0 < d i di d i ,
a ij aij a ij ,
(k)
(k)
(k)
b ij bij bij ,
i, j = 1, 2, . . . , n.
(2)
For notational convenience, let D = diag( d i )n×n , D = diag(d i )n×n , A = ( a ij )n×n , A = (a ij )n×n , (k) (k) B k = bij n×n , B k = b ij n×n , N[ A, A] = A = (aij ) ∈ R n×n | A ≺ A ≺ A, i.e., a ij aij a ij , i, j = 1, 2, . . . , n , (k) (k) N[ B k , B k ] = Bk = bij(k) ∈ R n×n | B k ≺ Bk ≺ B k , i.e., b(k) ij bij bij , i, j = 1, 2, . . . , n , N[ D, D] = D = diag(di ) ∈ R n×n | D ≺ D ≺ D, i.e., d i di d i , i = 1, 2, . . . , n .
(3)
Then, for the interval system (1), we have D ∈ N[ D, D],
A ∈ N[ A, A] and Bk ∈ N[ B k , B k ].
(4)
The global and robust stability criteria for system (1) obtained in [16–19] are based on the following assumptions for activation functions, respectively: (H1) Each activation function gi is continuous, differentiable, monotonically increasing and bounded [16]. (H2) Each activation function gi is bounded and there exists positive constant σi > 0 such that 0 (gi (x) − gi (y))/ (x − y) σi , ∀x, y ∈ R [17–19]. However, throughout this Letter, we always assume that the activation functions satisfy the following assumption: (H) Each activation function gi is bounded and there exists positive constant σi > 0 such that gi (x) − gi (y) σi |x − y|, ∀x, y ∈ R. Obviously, condition (H) is weaker than both (H1) and (H2). Also, we use P T to denote the transpose of a square matrix P . And we use P > 0 to denote a symmetrical positive definite matrix P . It is well known that bounded activation functions always guarantee the existence of an equilibrium point for system (1). For notational convenience, we will always shift an intended equilibrium point u∗ = [u∗1 , u∗2 , . . . , u∗n ]T ∈ R n of system (1) to the origin by letting x(t) = u(t) − u∗ , which yields the following system: r Bk f x t − τk (t) , x(t) ˙ = −Dx(t) + Af x(t) + k=1
(5)
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where x(t) = [x1 (t), . . . , xn (t)]T is the state vector of the transformed system, f (x) = [f1 (x1 ), . . . , fn (xn )]T denotes the activation function vector with fi (xi ) = gi (xi + u∗i ) − gi (u∗i ), i = 1, 2, . . . , n. From the assumptions (H), we can find that the function vector, f , possesses the following properties: for any x ∈ R, |fi (x)| σi |x|, and fi (x) is bounded with fi (0) = 0. Obviously, the equilibrium point u∗ of system (1) is globally robustly stable if and only if the origin of system (5) is globally robustly stable.
2. Problem transformation and preliminaries In this section, we shift the interval system (5) to another form in order to derive our main results. For this purpose, let 1 1 1 B0(k) = ( B k + B k ), D0 = ( D + D), A0 = ( A + A), 2 2 2 1 1 (k) HA = (A − A ) = (αij )n×n , HB = (B k − B k ) = (βij )n×n , 2 2
(6)
(k)
where D0 , A0 and B0 are called reference matrices of interval matrices D, A and Bk , respectively. Correspondingly, the system r u(t) ˙ = −D0 u(t) + A0 g u(t) + B0(k) g u t − τk (t) + I
(7)
k=1
is called the reference system of system (1). (k) Note that each element of matrices HA and HB is nonnegative. So we can define √ √ √ √ EA = [ α11 e1 , . . . , α1n e1 , . . . , αn1 en , . . . , αnn en ]n×n2 , (k) (k) (k) (k) (k) EB = β11 e1 , . . . , β1n e1 , . . . , βn1 en , . . . , βnn en n×n2 , √ √ √ √ FA = [ α11 e1 , . . . , α1n en , . . . , αn1 e1 , . . . , αnn en ]Tn2 ×n , (k) (k) (k) (k) T FB(k) = β11 e1 , . . . , β1n en , . . . , βn1 e1 , . . . , βnn en n2 ×n ,
(8)
where ei denotes the ith column-vector of the n × n identity matrix. Obviously, we have
n
n n n (k) (k) (k)T (k) T EB EB = diag α1j , . . . , αnj , β1j , . . . , βnj , EA EA = diag
FAT FA
= diag
j =1
j =1
n
n
j =1
αj 1 , . . . ,
αj n ,
(k)T (k) FB FB
= diag
j =1
j =1 n j =1
j =1 (k) βj 1 , . . . ,
n
(k) βj n
.
(9)
j =1
Furthermore, let 2 2 Σ ∗ = Σ ∈ R n ×n | Σ = diag(ε11 , . . . , ε1n , . . . , εn1 , . . . , εnn ), |εij | 1, i, j = 1, 2, . . . , n .
(10)
The following two lemmas are essential to derive our main results. Lemma 1. Let M[ A, A] = {A = A0 + EA ΣA FA | ΣA ∈ Σ ∗ }, M[ B k , B k ] = Bk = B0(k) + EB(k) ΣB(k) FB(k) | ΣB(k) ∈ Σ ∗ . Then, M[ A, A] = N[ A, A] and M[ B k , B k ] = N[ B k , B k ].
(11)
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Lemma 2. For any B = (bij ) ∈ R n×n and any continuous function f : R n → R n defined by (5), if the condition (H) is satisfied, then the following inequality holds: f T (x)B T Bf (x) n
n n
σj2 bij2 xj2 = nx T (t)Σ σ ΣB ΣBT Σ σ x(t),
i=1 j =1 2
where Σ σ = diag(σi )n×n , ΣB = [b11e1 , . . . , bn1 en , b12e1 , . . . , bn2 en , . . . , bnn en ] ∈ R n×n and ei (i = 1, 2, . . . , n) denote the ith column vector of the n × n identity matrix E. From Lemma 1, the system (5) is equivalent to the following system: r (k) x(t) ˙ = −Dx(t) + (A0 + EA ΣA FA )f x(t) + B0 + EB(k) ΣB(k) FB(k) f x t − τk (t) ,
(12)
k=1
where ΣA , ΣB(k) ∈ Σ ∗ . Obviously, system (1) is globally and robustly stable if and only if system (12) is globally and robustly stable. So, we only investigate system (12) in the sequel.
3. Global robust asymptotical stability of interval neural networks with multiple time-varying delays In this section, two criteria are obtained for the robust stability of IDNN with multiple time-varying delays via Lyapunov stability theorem for functional differential equations [20] and linear matrix inequality (LMI) technique [14,15]. Theorem 1. Suppose that τk satisfy 0 τk (t) ηk < 1, k = 1, 2, . . . , r. If condition (H) is satisfied and moreover there exist positive definite matrix P , positive definite diagonal matrices Q, Rk and scalars sk > 0 (k = 1, 2, . . . , r) such that Ω1 = Ω1∗ − P A0 Q−1 AT0 P −
1 1 (k) (k)T T P EA EA P− P B0 R −1 B P s0 1 − ηk k 0 r
k=1
−
r k=1
1 (k) (k)T P EB EB P > 0, sk (1 − ηk )
(13)
i.e., Ω∗ 1 P A0 ˜ P EA P B (1) 0 . .. P B (r) 0 P E˜ B(1) .. . P E˜ (r) B
AT0 P
TP E˜ A
Q 0
0 s0 E
0 0
··· ···
0 .. .
0 .. .
(1 − η1 )R1 .. .
··· .. .
0
0
0
· · · (1 − ηr )Rr
0 .. . 0
0 .. . 0
0 .. . 0
··· .. . ···
(1)T
B0
P
···
(1) E˜ B P
···
(r) E˜ B P
0 0
0 0
··· ···
0 0
0 .. .
0 .. .
··· .. .
0 .. .
0
···
0
(r)T
B0
.. . 0
P
s1 (1 − η1 )E .. . 0
··· 0 .. .. . . · · · sr (1 − ηr )E
> 0,
(14)
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C. Li et al. / Physics Letters A 328 (2004) 452–462
where E denotes the n × n identity matrix, Σ σ = diag(σi )n×n and r (k)T (k) Σσ , Ω1∗ = DP + P D − Σ σ Q + s0 FAT FA + Rk + sk FB FB
n n α1j , . . . , αnj , E˜ A = diag j =1
k=1
n n (k) (k) (k) β ,..., β E˜ = diag . B
nj
1j
j =1
j =1
j =1
Then, the equilibrium of system (1) is unique. Moreover, the unique equilibrium is globally robustly asymptotically stable. A proof of this theorem is given in Appendix A. If we let r = 1 then the system (1) will reduce to the interval neural system with single time-varying delay, and consequently, Theorem 1 reduces to the following. Corollary 1. Suppose that r = 1, τ (t) η < 1. If condition (H) is satisfied and moreover there exist positive definite matrices P , positive diagonal matrices Q, R and constant scalars s0 , s > 0 such that ∗ Ω11 = Ω11 − P A0 Q−1 AT0 P −
i.e.,
∗ Ω11
AT0 P
E˜ A P
1 1 1 T P B0 R −1 B0T P − P EB EBT P > 0, P EA EA P− s0 1−η s(1 − η)
B0T P
E˜ B P
P A0 Q 0 0 0 ˜ 0 s0 E 0 0 P EA > 0, P B0 0 0 (1 − η)R 0 ˜ 0 0 0 s(1 − η)E P EB where E denotes the n × n identity matrix, Σ σ = diag(σi )n×n and ∗ = DP + P D − Σ σ Q + s0 FAT FA + R + sFBT FB Σ σ , Ω11
n n n n α1j , . . . , αnj , β1j , . . . , βnj . E˜ B = diag E˜ A = diag j =1
j =1
j =1
j =1
Then, the equilibrium of system (1) with r = 1 is unique. Moreover, the unique equilibrium is globally robustly asymptotically stable. It is worth noting that the matrices Q and Rk (k = 1, 2, . . . , r) in the theorem above should be diagonal. We can see from the following theorem that these limits can been removed by using some inequality technique, although the same Lyapunov–Krasovskii functional is adopted. Theorem 2. Suppose that 0 τk (t) ηk < 1, k = 1, 2, . . . , r. If condition (H) is satisfied and moreover there exist positive definite matrices P , Q and Rk and the constant scalars ε > 0, sk > 0, k = 0, 1, 2, . . . , r, such that (i)
Ω2 = Ω2∗ − P A0 Q−1 AT0 P −
1 1 T P EA EA P− P B0(k) R −1 B (k)T P s0 1 − ηk k 0 r
k=1
−
r k=1
1 P EB(k) EB(k)T P > 0, sk (1 − ηk )
(15)
C. Li et al. / Physics Letters A 328 (2004) 452–462
i.e.,
Ω∗ 2 P A0 ˜ P EA P B (1) 0 . .. P B (r) 0 P E˜ B(1) .. . (r) P E˜ B
AT0 P
TP E˜ A
B0(1)T P
···
B0(r)T P
E˜ B(1)P
···
E˜ B(r) P
Q
0
0
···
0
0
···
0
0
s0 E
0
···
0
0
···
0
0 .. . 0
0 .. . 0
(1 − η1 )R1 .. . 0
0 .. . 0
··· .. . ···
0 .. . 0
0 .. .
0 .. .
0 .. .
··· .. .
s1 (1 − η1 )E .. .
··· .. .
0 .. .
0
0
0
···
(ii)
457
T + Ω3 = εE − nΣ σ ΣQ ΣQ
r
··· 0 .. .. . . · · · (1 − ηr )Rr .. . 0
> 0;
(16)
· · · sr (1 − ηr )E
0
ΣR(k) ΣR(k)T Σ σ > 0,
k=1
where E denotes the n × n identity matrix, Σ σ = diag(σi )n×n and r (k)T (k) ∗ σ T Ω2 = DP + P D − Σ s0 FA FA + sk FB FB Σ σ − εE, k=1
n n α1j , . . . , αnj , E˜ A = diag j =1
Q = (qij )n×n ,
j =1
R k = γij(k) n×n
E˜ B(k)
n (k) n (k) , = diag β ,..., β nj
1j
j =1
j =1
with QT Q = Q, R Tk R k = Qk , ΣQ = [q11e1 , . . . , qn1 en , q12 e1 , . . . , qn2 en , . . . , qnn en ]n×n2 , (k) (k) (k) (k) (k) ΣR(k) = γ11 e1 , . . . , γn1 en , γ12 e1 , . . . , γn2 en , . . . , γnn en n×n2 ,
ei (i = 1, 2, . . . , n)
denote the ith column vector of the n × n identity matrix E. Then, the equilibrium of system (1) is unique. Moreover, the unique equilibrium is globally robustly asymptotically stable. A proof of this theorem is given in Appendix B.
4. Example In this section, one example is given to show the effectiveness of our results and to compare our results with those reported in Refs. [18,19]. Throughout this section, the LMI is solved by the LMI-Toolbox in M ATLAB, and the delayed differential equations are calculated numerically via the 4th-order Lounge–Kutta approach with the time step 0.001. Example 1. Consider the system (1) with a time-varying delay, i.e., τ (t) = 0.1 arctan(t), and 1 0 2 1 2.01 1.01 1.5 0.1 , A= , A= , B= , D=D= 0 1 0 0.1 0.01 0.2 0 0.1
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C. Li et al. / Physics Letters A 328 (2004) 452–462
Fig. 1. The convergence dynamics of the system in Example 1.
1.55 0.16 B= , 0.05 0.15 and g1 (x) = g2 (x) = 0.5[abs(x + 1) − abs(x − 1)]. In this case, since max{ a 11 , a 11 } = 2.01 > 1, all of the conditions in [18,19] cannot be satisfied. On the other hand, there exists at least one feasible solution, i.e., 0.16189 −0.23833 P= , Q = R = diag(0.0289, 0.0289), s0 = s = 1, −0.23833 8.25852 of the LMI in Corollary 1. Hence, the system is globally robustly stable. In particular, for the corresponding reference system: 1 0 2.005 1.005 1.5025 0.13 , A0 = , B0 = , D0 = 0 1 0.005 0.15 0.025 0.125 u(θ ) = [3.2, −1.2]T
for any θ ∈ [−1, 0],
we can calculate the unique equilibrium point u∗ = [3.3725, −1.245]T , and its convergence behavior is shown in Fig. 1.
5. Conclusions Some criteria for the robust asymptotical stability of the interval multi-delayed neural networks with timevarying delays have been derived via the approach combining the Lyapunov–Krasovskii functional and LMI technique. The problem transformation presented in this Letter allows these criteria less conservative, and moreover, LMI technique makes the verification easier. In addition, as a corollary, one new sufficient condition for the asymptotical stability of the interval neural networks with a single delay has been proposed. Finally, an example has been provided to show the effectiveness of our results. The approach developed in this Letter is generally applicable to the robust stability issue of interval systems described by the ordinary/functional differential equations.
Acknowledgements The authors thank editor Dr. A.P. Fordy and the referees for their valuable suggestions. The work described in this Letter was partially supported by the National Natural Science Foundation of China (Grant Nos. 60271019, 70371030), the Doctorate Foundation of the Ministry of Education of China (Grant No. 20020611007), the Applied Basic Research Grants Committee of Science and Technology of Chongqing (Grant No. 7370).
C. Li et al. / Physics Letters A 328 (2004) 452–462
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Appendix A. A proof of Theorem 1 Obviously, it is enough to prove that the origin is the unique equilibrium of system (12) and it is robustly asymptotically stable. To prove the uniqueness of the equilibrium, we assume that the solution x ∗ = (0, 0, . . . , 0)T is also its equilibrium, i.e., −Dx ∗ + (A0 + EA ΣA FA )f (x ∗ ) + (A0 + EB ΣB FB )f (x ∗ ) = 0,
(A.1)
which implies f (x ∗ ) = 0. Multiplying both sides of (A.1) by 2(x ∗ )T P , we obtain −2(x ∗)T P Dx ∗ + 2(x ∗ )T P A0 f (x ∗ ) + 2(x ∗ )T P EA ΣA FA f (x ∗ )
r
r (k) (k) (k) (k) ∗ T ∗ ∗ T + 2(x ) P B0 f (x ) + 2(x ) P EB ΣB FB f (x ∗ ) = 0. k=1
(A.2)
k=1
Noting that 2x T P A0 f (x) x T P A0 Q−1 AT0 P x + f T (x)Qf (x) x T P A0 Q−1 AT0 P + Σ σ QΣ σ x, 1 R −1 B (k)T P x + f T (x)(1 − ηk )Rk f (x) 2x T P B0(k) f (x) x T P B0(k) 1 − ηk k 0 1 x T P B0(k) Rk−1 B0(k)T P + Σ σ Rk Σ σ x, 1 − ηk
(A.3)
1 T T x P EA ΣA ΣAT EA P x + s0 f T (x)FAT FA f (x) s0 1 T x T P EA EA P x + s0 f T (x)FAT FA f (x) s0 1 T P EA EA P + s0 Σ σ FAT FA Σ σ x, xT (A.4) s0 1 2x T P EB(k) ΣB(k) FB(k) f (x) x T P EB(k) ΣB(k) ΣB(k)T EB(k)T P x + sk (1 − ηk )f T (x)FB(k)T FB(k) f (x) sk (1 − ηk ) 1 (k) (k)T T σ (k)T (k) σ P EB EB P + sk Σ FB FB Σ x. x (A.5) sk (1 − ηk )
2x T P EA ΣA FA f (x)
Introducing (A.3)–(A.5) to (A.2), we can derive
∗ T
−(x )
DP + P D − Σ
σ
Q + s0 FAT FA
+
r
(k)T (k) Rk + sk FB FB
Σ σ − P A0 Q−1 AT0 P
k=1 r
1 1 (k)T T R −1 B P − P EA EA P (1 − ηk ) k 0 s0 k=1 r 1 (k) (k)T − P EB EB P (x ∗ ) 0, sk (1 − ηk )
−
(k)
P B0
(A.6)
k=1
which implies Ω1 0. It leads to a contradiction with Eq. (13). Therefore, there exists unique equilibrium of system (12).
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C. Li et al. / Physics Letters A 328 (2004) 452–462
To prove the robust stability of the origin of system (12), we construct the following Lyapunov functional:
V x(t) = x (t)P x(t) + T
t
r
(k)T (k) f T x(θ ) Rk + sk FB FB f x(θ ) dθ,
(A.7)
k=1 t −τ (t ) k
where P T = P > 0, sk > 0, and Rk is positive diagonal matrix. By (A.3)–(A.5), we can calculate and estimate the Dini’s derivative of V (x(t)) along the solution of system (12) as follows: V˙ x(t) = −x T (t)[DP + P D]x(t) + 2x T (t)P A0 f x(t) + 2x T (t)P EA ΣA FA f x(t) +2
r
r (k) (k) (k) (k) x T (t)P B0 f x(t − τk ) + 2 x T (t)P EB ΣB FB f x(t − τk )
k=1
+
r
f
k=1
T
r x(t) Rk f x(t) + f T x(t) sk FB(k)T FB(k) f x(t)
k=1
−
k=1
r
1 − τk (t) f T x(t − τk ) Rk + sk FB(k)T FB(k) f x(t − τk )
k=1
−x (t) DP + P D − Σ T
σ
Q + s0 FAT FA
r (k)T (k) Rk + sk FB FB Σ σ − P A0 Q−1 AT0 P + k=1
r 1 1 1 (k) (k)T (k) (k)T T − P EA EA P EB EB P x(t) P− P B0 R −1 B P− s0 1 − ηk k 0 sk (1 − ηk ) k=1 k=1 r 1 1 (k) (k)T T T = −x (t) Ω1∗ − P A0 Q−1 AT0 P − P EA EA P− P B0 R −1 B P s0 1 − ηk k 0 k=1 r 1 (k) (k)T − P EB EB P x(t) sk (1 − ηk ) r
k=1
= −x (t)Ω1 x(t). T
T =E ˜ A E˜ A and E (k) E (k)T = E˜ (k) E˜ (k) . Then, by Schur comHence, V˙ (x(t)) < 0 when Ω1 > 0. Noting that EA EA B B B B plement lemma [12,13], Ω1 > 0 if and only if LMI (14) holds, which completes the proof of the theorem. 2
Appendix B. A proof of Theorem 2 By Lemma 2, we have T σ f T x(t) Qf x(t) = f T x(t) QT Qf x(t) nx T (t)Σ σ ΣQ ΣQ Σ x(t), (k) f T x(t) Rk f x(t) = f T x(t) R Tk R k f x(t) nx T (t)Σ σ ΣR ΣR(k)T Σ σ x(t).
(B.1)
To show the uniqueness of the equilibrium of system (12), repeating (A.1), (A.2), (A.4), (A.5) and applying (B.1) to (A.3), we can rewrite (A.6) as
r (k)T (k) −(x ∗ )T DP + P D − Σ σ s0 FAT FA + sk FB FB Σ σ − P A0 Q−1 AT0 P k=1
C. Li et al. / Physics Letters A 328 (2004) 452–462
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r 1 1 1 T P EB(k) EB(k)T P − εE Rk−1 B0(k)T P − P EA EA P− 1 − ηk s0 sk (1 − ηk ) k=1 k=1
r T (x ∗ ) = −(x ∗ )T Ω2 (x ∗ ) − (x ∗ )T Ω3 (x ∗ ) 0, + ΣR(k) ΣR(k)T + εE − nΣ σ ΣQ ΣQ
−
r
P B0(k)
(B.2)
k=1
which implies Ω2 0 or Ω3 0. It leads to a contradiction with condition (i) or (ii) in this theorem. Therefore, there exists unique equilibrium of system (12). The Lyapunov–Krasovskii functional (A.7) is also used here. Moreover, arguing similarly with the proof of Theorem 1, we have 1 T P V˙ x(t) −x T (t) DP + P D − P A0 Q−1 AT0 P − s0 Σ σ FAT FA Σ σ − P EA EA s0 − −
r k=1 r
1 1 P EB(k) EB(k)T P Rk−1 B0(k)T P − 1 − ηk sk (1 − ηk ) r
P B0(k)
k=1
(k)T
sk Σ σ FB
k=1
(k)
FB Σ σ − εE + εE − n
r
(k)
(k)T
Σ σ ΣR ΣR
Σσ
k=1
T σ − nΣ σ ΣQ ΣQ Σ x(t)
= −x (t) DP + P D − Σ T
σ
s0 FAT FA
+
r
(k)T (k) sk FB FB
Σ σ − εE − P A0 Q−1 AT0 P
k=1
r 1 1 1 (k) (k) (k)T −1 (k)T T P EB EB P x(t) P B0 R B P− − P EA EA P − s0 1 − ηk k 0 sk (1 − ηk ) k=1 k=1
r (k) (k)T T − x T (t) εE − nΣ σ Σ σ x(t) ΣR ΣR − ΣQ ΣQ r
k=1
= −x T (t)Ω2 x(t) − x T (t)Ω3 x(t). Hence, V˙ (x(t)) < 0 when Ω2 > 0 and Ω3 > 0, which completes the proof of the theorem. 2
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
S. Arik, Phys. Lett. A 311 (2003) 504. J. Cao, Phys. Lett. A 307 (2003) 136. Q. Zhang, Phys. Lett. A 318 (2003) 182. S. Arik, IEEE Trans. Circuits Systems I 47 (2000) 1089. T. Roska, C.W. Wu, M. Balsi, L.O. Chua, IEEE Trans. Circuits Systems 39 (1992) 487. H. Rao, R.M. Phaneendra, Neural Networks 12 (1999) 455. X. Liao, J. Yu, Int. J. Circuit Theory Appl. 26 (1998) 219. X. Liao, K.W. Wong, J. Yu, Int. J. Bifur. Chaos 11 (7) (2001) 1835. X. Liao, J. Yu, G. Chen, Int. J. Circuit Theory Appl. 30 (2002) 519. X. Liao, K.W. Wong, Z. Wu, G. Chen, IEEE Trans. Circuits Systems 49 (11) (2001) 1355. X. Liao, Z. Wu, J. Yu, J. Comput. Appl. Math. 143 (2002) 29. X. Liao, G. Chen, E.N. Sanchez, IEEE Trans. Circuits Systems 49 (7) (2002) 1033. X. Liao, G. Chen, E.N. Sanchez, Neural Networks 15 (2002) 855.
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[14] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA, 1994. [15] Y. Nesterov, A. Nemirovsky, Interior Point Polynomial Methods in Convex Programming, SIAM, Philadelphia, PA, 1994. [16] X. Liao, J. Yu, IEEE Trans. Neural Networks 9 (1998) 1042. [17] X. Liao, K.-W. Wong, Z. Wu, G. Chen, IEEE Trans. Circuits Systems I 48 (2001) 1355. [18] X. Liao, J. Wang, Int. J. Neural Systems 13 (2) (2003). [19] S. Arik, IEEE Trans. Circuits Systems I 50 (2003) 156. [20] J.K. Hale, S.M.V. Lunel, Introduction to the Theory of Functional Differential Equations, Springer, New York, 1991. [21] J. Cao, Q. Li, J. Math. Anal. Appl. 252 (2000) 50. [22] J. Cao, Phys. Lett. A 270 (2000) 157.
Global robust asymptotical stability of multi-delayed interval neural networks: an LMI approach Chuandong Li a,∗ , Xiaofeng Liao a , Rong Zhang b a Department of Computer Science and Engineering, Chongqing University, Chongqing 400030, PR China b College of Business Administration, Chongqing University, Chongqing 400030, PR China
Received 14 March 2004; received in revised form 29 May 2004; accepted 16 June 2004
Communicated by A.P. Fordy
Abstract Based on the Lyapunov–Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) technique, some delay-dependent criteria for interval neural networks (IDNN) with multiple time-varying delays are derived to guarantee global robust asymptotic stability. The main results are generalizations of some recent results reported in the literature. Numerical example is also given to show the effectiveness of our results. 2004 Elsevier B.V. All rights reserved. Keywords: Robust stability; Interval delayed neural networks (IDNN); Lyapunov functional; Linear matrix inequality (LMI); Multiple time-varying delays
1. Introduction In recent years, several neural network models with single or multiple delays have been extensively studied [1–13], particularly regarding their stability analysis. However, it is well known that the stability of a well-designed system may often be destroyed by its unavoidable uncertainty due to the existence of modeling error, external disturbance and parameter fluctuation during the implementation. So it is essential to introduce the robust technique to design a system with such uncertainty. If the uncertainty of a system is only due to the deviations and perturbations of its parameters, and if these deviations and perturbations are all bounded, then the system is called an interval system. Recently, several global and robust stability criteria for interval neural networks with constant or time-varying delays have been proposed [16–19,21,22]. In this Letter, we further extend the results of [16–19] to a * Corresponding author.
E-mail addresses: [email protected] (C. Li), [email protected] (X. Liao). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.06.053
C. Li et al. / Physics Letters A 328 (2004) 452–462
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significantly larger class of systems. Specifically, we consider the following interval neural networks with multiple time-varying delays: r u(t) ˙ = −Du(t) + Ag u(t) + Bk g u t − τk (t) + I,
(1)
k=1
where u(t) = [u1 (t), . . . , un (t)]T is the state vector of the neural network, D = diag(d1 , . . . , dn ) is a positive (k) diagonal matrix, A = (aij )n×n and Bk = (bij )n×n are the connection weight and delayed connection weight matrices, respectively. g(u) = [g1 (u1 ), . . . , gn (u)]T denote the neuron activation functions with g(0) = 0, and I = [I1 , . . . , In ]T is a constant external input vector. 0 < τk (t) τ , k = 1, 2, . . . , r, represent axonal signal transmission delays. Suppose that the parameters of the system (1) are uncertain but bounded, and satisfy 0 < d i di d i ,
a ij aij a ij ,
(k)
(k)
(k)
b ij bij bij ,
i, j = 1, 2, . . . , n.
(2)
For notational convenience, let D = diag( d i )n×n , D = diag(d i )n×n , A = ( a ij )n×n , A = (a ij )n×n , (k) (k) B k = bij n×n , B k = b ij n×n , N[ A, A] = A = (aij ) ∈ R n×n | A ≺ A ≺ A, i.e., a ij aij a ij , i, j = 1, 2, . . . , n , (k) (k) N[ B k , B k ] = Bk = bij(k) ∈ R n×n | B k ≺ Bk ≺ B k , i.e., b(k) ij bij bij , i, j = 1, 2, . . . , n , N[ D, D] = D = diag(di ) ∈ R n×n | D ≺ D ≺ D, i.e., d i di d i , i = 1, 2, . . . , n .
(3)
Then, for the interval system (1), we have D ∈ N[ D, D],
A ∈ N[ A, A] and Bk ∈ N[ B k , B k ].
(4)
The global and robust stability criteria for system (1) obtained in [16–19] are based on the following assumptions for activation functions, respectively: (H1) Each activation function gi is continuous, differentiable, monotonically increasing and bounded [16]. (H2) Each activation function gi is bounded and there exists positive constant σi > 0 such that 0 (gi (x) − gi (y))/ (x − y) σi , ∀x, y ∈ R [17–19]. However, throughout this Letter, we always assume that the activation functions satisfy the following assumption: (H) Each activation function gi is bounded and there exists positive constant σi > 0 such that gi (x) − gi (y) σi |x − y|, ∀x, y ∈ R. Obviously, condition (H) is weaker than both (H1) and (H2). Also, we use P T to denote the transpose of a square matrix P . And we use P > 0 to denote a symmetrical positive definite matrix P . It is well known that bounded activation functions always guarantee the existence of an equilibrium point for system (1). For notational convenience, we will always shift an intended equilibrium point u∗ = [u∗1 , u∗2 , . . . , u∗n ]T ∈ R n of system (1) to the origin by letting x(t) = u(t) − u∗ , which yields the following system: r Bk f x t − τk (t) , x(t) ˙ = −Dx(t) + Af x(t) + k=1
(5)
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where x(t) = [x1 (t), . . . , xn (t)]T is the state vector of the transformed system, f (x) = [f1 (x1 ), . . . , fn (xn )]T denotes the activation function vector with fi (xi ) = gi (xi + u∗i ) − gi (u∗i ), i = 1, 2, . . . , n. From the assumptions (H), we can find that the function vector, f , possesses the following properties: for any x ∈ R, |fi (x)| σi |x|, and fi (x) is bounded with fi (0) = 0. Obviously, the equilibrium point u∗ of system (1) is globally robustly stable if and only if the origin of system (5) is globally robustly stable.
2. Problem transformation and preliminaries In this section, we shift the interval system (5) to another form in order to derive our main results. For this purpose, let 1 1 1 B0(k) = ( B k + B k ), D0 = ( D + D), A0 = ( A + A), 2 2 2 1 1 (k) HA = (A − A ) = (αij )n×n , HB = (B k − B k ) = (βij )n×n , 2 2
(6)
(k)
where D0 , A0 and B0 are called reference matrices of interval matrices D, A and Bk , respectively. Correspondingly, the system r u(t) ˙ = −D0 u(t) + A0 g u(t) + B0(k) g u t − τk (t) + I
(7)
k=1
is called the reference system of system (1). (k) Note that each element of matrices HA and HB is nonnegative. So we can define √ √ √ √ EA = [ α11 e1 , . . . , α1n e1 , . . . , αn1 en , . . . , αnn en ]n×n2 , (k) (k) (k) (k) (k) EB = β11 e1 , . . . , β1n e1 , . . . , βn1 en , . . . , βnn en n×n2 , √ √ √ √ FA = [ α11 e1 , . . . , α1n en , . . . , αn1 e1 , . . . , αnn en ]Tn2 ×n , (k) (k) (k) (k) T FB(k) = β11 e1 , . . . , β1n en , . . . , βn1 e1 , . . . , βnn en n2 ×n ,
(8)
where ei denotes the ith column-vector of the n × n identity matrix. Obviously, we have
n
n n n (k) (k) (k)T (k) T EB EB = diag α1j , . . . , αnj , β1j , . . . , βnj , EA EA = diag
FAT FA
= diag
j =1
j =1
n
n
j =1
αj 1 , . . . ,
αj n ,
(k)T (k) FB FB
= diag
j =1
j =1 n j =1
j =1 (k) βj 1 , . . . ,
n
(k) βj n
.
(9)
j =1
Furthermore, let 2 2 Σ ∗ = Σ ∈ R n ×n | Σ = diag(ε11 , . . . , ε1n , . . . , εn1 , . . . , εnn ), |εij | 1, i, j = 1, 2, . . . , n .
(10)
The following two lemmas are essential to derive our main results. Lemma 1. Let M[ A, A] = {A = A0 + EA ΣA FA | ΣA ∈ Σ ∗ }, M[ B k , B k ] = Bk = B0(k) + EB(k) ΣB(k) FB(k) | ΣB(k) ∈ Σ ∗ . Then, M[ A, A] = N[ A, A] and M[ B k , B k ] = N[ B k , B k ].
(11)
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Lemma 2. For any B = (bij ) ∈ R n×n and any continuous function f : R n → R n defined by (5), if the condition (H) is satisfied, then the following inequality holds: f T (x)B T Bf (x) n
n n
σj2 bij2 xj2 = nx T (t)Σ σ ΣB ΣBT Σ σ x(t),
i=1 j =1 2
where Σ σ = diag(σi )n×n , ΣB = [b11e1 , . . . , bn1 en , b12e1 , . . . , bn2 en , . . . , bnn en ] ∈ R n×n and ei (i = 1, 2, . . . , n) denote the ith column vector of the n × n identity matrix E. From Lemma 1, the system (5) is equivalent to the following system: r (k) x(t) ˙ = −Dx(t) + (A0 + EA ΣA FA )f x(t) + B0 + EB(k) ΣB(k) FB(k) f x t − τk (t) ,
(12)
k=1
where ΣA , ΣB(k) ∈ Σ ∗ . Obviously, system (1) is globally and robustly stable if and only if system (12) is globally and robustly stable. So, we only investigate system (12) in the sequel.
3. Global robust asymptotical stability of interval neural networks with multiple time-varying delays In this section, two criteria are obtained for the robust stability of IDNN with multiple time-varying delays via Lyapunov stability theorem for functional differential equations [20] and linear matrix inequality (LMI) technique [14,15]. Theorem 1. Suppose that τk satisfy 0 τk (t) ηk < 1, k = 1, 2, . . . , r. If condition (H) is satisfied and moreover there exist positive definite matrix P , positive definite diagonal matrices Q, Rk and scalars sk > 0 (k = 1, 2, . . . , r) such that Ω1 = Ω1∗ − P A0 Q−1 AT0 P −
1 1 (k) (k)T T P EA EA P− P B0 R −1 B P s0 1 − ηk k 0 r
k=1
−
r k=1
1 (k) (k)T P EB EB P > 0, sk (1 − ηk )
(13)
i.e., Ω∗ 1 P A0 ˜ P EA P B (1) 0 . .. P B (r) 0 P E˜ B(1) .. . P E˜ (r) B
AT0 P
TP E˜ A
Q 0
0 s0 E
0 0
··· ···
0 .. .
0 .. .
(1 − η1 )R1 .. .
··· .. .
0
0
0
· · · (1 − ηr )Rr
0 .. . 0
0 .. . 0
0 .. . 0
··· .. . ···
(1)T
B0
P
···
(1) E˜ B P
···
(r) E˜ B P
0 0
0 0
··· ···
0 0
0 .. .
0 .. .
··· .. .
0 .. .
0
···
0
(r)T
B0
.. . 0
P
s1 (1 − η1 )E .. . 0
··· 0 .. .. . . · · · sr (1 − ηr )E
> 0,
(14)
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C. Li et al. / Physics Letters A 328 (2004) 452–462
where E denotes the n × n identity matrix, Σ σ = diag(σi )n×n and r (k)T (k) Σσ , Ω1∗ = DP + P D − Σ σ Q + s0 FAT FA + Rk + sk FB FB
n n α1j , . . . , αnj , E˜ A = diag j =1
k=1
n n (k) (k) (k) β ,..., β E˜ = diag . B
nj
1j
j =1
j =1
j =1
Then, the equilibrium of system (1) is unique. Moreover, the unique equilibrium is globally robustly asymptotically stable. A proof of this theorem is given in Appendix A. If we let r = 1 then the system (1) will reduce to the interval neural system with single time-varying delay, and consequently, Theorem 1 reduces to the following. Corollary 1. Suppose that r = 1, τ (t) η < 1. If condition (H) is satisfied and moreover there exist positive definite matrices P , positive diagonal matrices Q, R and constant scalars s0 , s > 0 such that ∗ Ω11 = Ω11 − P A0 Q−1 AT0 P −
i.e.,
∗ Ω11
AT0 P
E˜ A P
1 1 1 T P B0 R −1 B0T P − P EB EBT P > 0, P EA EA P− s0 1−η s(1 − η)
B0T P
E˜ B P
P A0 Q 0 0 0 ˜ 0 s0 E 0 0 P EA > 0, P B0 0 0 (1 − η)R 0 ˜ 0 0 0 s(1 − η)E P EB where E denotes the n × n identity matrix, Σ σ = diag(σi )n×n and ∗ = DP + P D − Σ σ Q + s0 FAT FA + R + sFBT FB Σ σ , Ω11
n n n n α1j , . . . , αnj , β1j , . . . , βnj . E˜ B = diag E˜ A = diag j =1
j =1
j =1
j =1
Then, the equilibrium of system (1) with r = 1 is unique. Moreover, the unique equilibrium is globally robustly asymptotically stable. It is worth noting that the matrices Q and Rk (k = 1, 2, . . . , r) in the theorem above should be diagonal. We can see from the following theorem that these limits can been removed by using some inequality technique, although the same Lyapunov–Krasovskii functional is adopted. Theorem 2. Suppose that 0 τk (t) ηk < 1, k = 1, 2, . . . , r. If condition (H) is satisfied and moreover there exist positive definite matrices P , Q and Rk and the constant scalars ε > 0, sk > 0, k = 0, 1, 2, . . . , r, such that (i)
Ω2 = Ω2∗ − P A0 Q−1 AT0 P −
1 1 T P EA EA P− P B0(k) R −1 B (k)T P s0 1 − ηk k 0 r
k=1
−
r k=1
1 P EB(k) EB(k)T P > 0, sk (1 − ηk )
(15)
C. Li et al. / Physics Letters A 328 (2004) 452–462
i.e.,
Ω∗ 2 P A0 ˜ P EA P B (1) 0 . .. P B (r) 0 P E˜ B(1) .. . (r) P E˜ B
AT0 P
TP E˜ A
B0(1)T P
···
B0(r)T P
E˜ B(1)P
···
E˜ B(r) P
Q
0
0
···
0
0
···
0
0
s0 E
0
···
0
0
···
0
0 .. . 0
0 .. . 0
(1 − η1 )R1 .. . 0
0 .. . 0
··· .. . ···
0 .. . 0
0 .. .
0 .. .
0 .. .
··· .. .
s1 (1 − η1 )E .. .
··· .. .
0 .. .
0
0
0
···
(ii)
457
T + Ω3 = εE − nΣ σ ΣQ ΣQ
r
··· 0 .. .. . . · · · (1 − ηr )Rr .. . 0
> 0;
(16)
· · · sr (1 − ηr )E
0
ΣR(k) ΣR(k)T Σ σ > 0,
k=1
where E denotes the n × n identity matrix, Σ σ = diag(σi )n×n and r (k)T (k) ∗ σ T Ω2 = DP + P D − Σ s0 FA FA + sk FB FB Σ σ − εE, k=1
n n α1j , . . . , αnj , E˜ A = diag j =1
Q = (qij )n×n ,
j =1
R k = γij(k) n×n
E˜ B(k)
n (k) n (k) , = diag β ,..., β nj
1j
j =1
j =1
with QT Q = Q, R Tk R k = Qk , ΣQ = [q11e1 , . . . , qn1 en , q12 e1 , . . . , qn2 en , . . . , qnn en ]n×n2 , (k) (k) (k) (k) (k) ΣR(k) = γ11 e1 , . . . , γn1 en , γ12 e1 , . . . , γn2 en , . . . , γnn en n×n2 ,
ei (i = 1, 2, . . . , n)
denote the ith column vector of the n × n identity matrix E. Then, the equilibrium of system (1) is unique. Moreover, the unique equilibrium is globally robustly asymptotically stable. A proof of this theorem is given in Appendix B.
4. Example In this section, one example is given to show the effectiveness of our results and to compare our results with those reported in Refs. [18,19]. Throughout this section, the LMI is solved by the LMI-Toolbox in M ATLAB, and the delayed differential equations are calculated numerically via the 4th-order Lounge–Kutta approach with the time step 0.001. Example 1. Consider the system (1) with a time-varying delay, i.e., τ (t) = 0.1 arctan(t), and 1 0 2 1 2.01 1.01 1.5 0.1 , A= , A= , B= , D=D= 0 1 0 0.1 0.01 0.2 0 0.1
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C. Li et al. / Physics Letters A 328 (2004) 452–462
Fig. 1. The convergence dynamics of the system in Example 1.
1.55 0.16 B= , 0.05 0.15 and g1 (x) = g2 (x) = 0.5[abs(x + 1) − abs(x − 1)]. In this case, since max{ a 11 , a 11 } = 2.01 > 1, all of the conditions in [18,19] cannot be satisfied. On the other hand, there exists at least one feasible solution, i.e., 0.16189 −0.23833 P= , Q = R = diag(0.0289, 0.0289), s0 = s = 1, −0.23833 8.25852 of the LMI in Corollary 1. Hence, the system is globally robustly stable. In particular, for the corresponding reference system: 1 0 2.005 1.005 1.5025 0.13 , A0 = , B0 = , D0 = 0 1 0.005 0.15 0.025 0.125 u(θ ) = [3.2, −1.2]T
for any θ ∈ [−1, 0],
we can calculate the unique equilibrium point u∗ = [3.3725, −1.245]T , and its convergence behavior is shown in Fig. 1.
5. Conclusions Some criteria for the robust asymptotical stability of the interval multi-delayed neural networks with timevarying delays have been derived via the approach combining the Lyapunov–Krasovskii functional and LMI technique. The problem transformation presented in this Letter allows these criteria less conservative, and moreover, LMI technique makes the verification easier. In addition, as a corollary, one new sufficient condition for the asymptotical stability of the interval neural networks with a single delay has been proposed. Finally, an example has been provided to show the effectiveness of our results. The approach developed in this Letter is generally applicable to the robust stability issue of interval systems described by the ordinary/functional differential equations.
Acknowledgements The authors thank editor Dr. A.P. Fordy and the referees for their valuable suggestions. The work described in this Letter was partially supported by the National Natural Science Foundation of China (Grant Nos. 60271019, 70371030), the Doctorate Foundation of the Ministry of Education of China (Grant No. 20020611007), the Applied Basic Research Grants Committee of Science and Technology of Chongqing (Grant No. 7370).
C. Li et al. / Physics Letters A 328 (2004) 452–462
459
Appendix A. A proof of Theorem 1 Obviously, it is enough to prove that the origin is the unique equilibrium of system (12) and it is robustly asymptotically stable. To prove the uniqueness of the equilibrium, we assume that the solution x ∗ = (0, 0, . . . , 0)T is also its equilibrium, i.e., −Dx ∗ + (A0 + EA ΣA FA )f (x ∗ ) + (A0 + EB ΣB FB )f (x ∗ ) = 0,
(A.1)
which implies f (x ∗ ) = 0. Multiplying both sides of (A.1) by 2(x ∗ )T P , we obtain −2(x ∗)T P Dx ∗ + 2(x ∗ )T P A0 f (x ∗ ) + 2(x ∗ )T P EA ΣA FA f (x ∗ )
r
r (k) (k) (k) (k) ∗ T ∗ ∗ T + 2(x ) P B0 f (x ) + 2(x ) P EB ΣB FB f (x ∗ ) = 0. k=1
(A.2)
k=1
Noting that 2x T P A0 f (x) x T P A0 Q−1 AT0 P x + f T (x)Qf (x) x T P A0 Q−1 AT0 P + Σ σ QΣ σ x, 1 R −1 B (k)T P x + f T (x)(1 − ηk )Rk f (x) 2x T P B0(k) f (x) x T P B0(k) 1 − ηk k 0 1 x T P B0(k) Rk−1 B0(k)T P + Σ σ Rk Σ σ x, 1 − ηk
(A.3)
1 T T x P EA ΣA ΣAT EA P x + s0 f T (x)FAT FA f (x) s0 1 T x T P EA EA P x + s0 f T (x)FAT FA f (x) s0 1 T P EA EA P + s0 Σ σ FAT FA Σ σ x, xT (A.4) s0 1 2x T P EB(k) ΣB(k) FB(k) f (x) x T P EB(k) ΣB(k) ΣB(k)T EB(k)T P x + sk (1 − ηk )f T (x)FB(k)T FB(k) f (x) sk (1 − ηk ) 1 (k) (k)T T σ (k)T (k) σ P EB EB P + sk Σ FB FB Σ x. x (A.5) sk (1 − ηk )
2x T P EA ΣA FA f (x)
Introducing (A.3)–(A.5) to (A.2), we can derive
∗ T
−(x )
DP + P D − Σ
σ
Q + s0 FAT FA
+
r
(k)T (k) Rk + sk FB FB
Σ σ − P A0 Q−1 AT0 P
k=1 r
1 1 (k)T T R −1 B P − P EA EA P (1 − ηk ) k 0 s0 k=1 r 1 (k) (k)T − P EB EB P (x ∗ ) 0, sk (1 − ηk )
−
(k)
P B0
(A.6)
k=1
which implies Ω1 0. It leads to a contradiction with Eq. (13). Therefore, there exists unique equilibrium of system (12).
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C. Li et al. / Physics Letters A 328 (2004) 452–462
To prove the robust stability of the origin of system (12), we construct the following Lyapunov functional:
V x(t) = x (t)P x(t) + T
t
r
(k)T (k) f T x(θ ) Rk + sk FB FB f x(θ ) dθ,
(A.7)
k=1 t −τ (t ) k
where P T = P > 0, sk > 0, and Rk is positive diagonal matrix. By (A.3)–(A.5), we can calculate and estimate the Dini’s derivative of V (x(t)) along the solution of system (12) as follows: V˙ x(t) = −x T (t)[DP + P D]x(t) + 2x T (t)P A0 f x(t) + 2x T (t)P EA ΣA FA f x(t) +2
r
r (k) (k) (k) (k) x T (t)P B0 f x(t − τk ) + 2 x T (t)P EB ΣB FB f x(t − τk )
k=1
+
r
f
k=1
T
r x(t) Rk f x(t) + f T x(t) sk FB(k)T FB(k) f x(t)
k=1
−
k=1
r
1 − τk (t) f T x(t − τk ) Rk + sk FB(k)T FB(k) f x(t − τk )
k=1
−x (t) DP + P D − Σ T
σ
Q + s0 FAT FA
r (k)T (k) Rk + sk FB FB Σ σ − P A0 Q−1 AT0 P + k=1
r 1 1 1 (k) (k)T (k) (k)T T − P EA EA P EB EB P x(t) P− P B0 R −1 B P− s0 1 − ηk k 0 sk (1 − ηk ) k=1 k=1 r 1 1 (k) (k)T T T = −x (t) Ω1∗ − P A0 Q−1 AT0 P − P EA EA P− P B0 R −1 B P s0 1 − ηk k 0 k=1 r 1 (k) (k)T − P EB EB P x(t) sk (1 − ηk ) r
k=1
= −x (t)Ω1 x(t). T
T =E ˜ A E˜ A and E (k) E (k)T = E˜ (k) E˜ (k) . Then, by Schur comHence, V˙ (x(t)) < 0 when Ω1 > 0. Noting that EA EA B B B B plement lemma [12,13], Ω1 > 0 if and only if LMI (14) holds, which completes the proof of the theorem. 2
Appendix B. A proof of Theorem 2 By Lemma 2, we have T σ f T x(t) Qf x(t) = f T x(t) QT Qf x(t) nx T (t)Σ σ ΣQ ΣQ Σ x(t), (k) f T x(t) Rk f x(t) = f T x(t) R Tk R k f x(t) nx T (t)Σ σ ΣR ΣR(k)T Σ σ x(t).
(B.1)
To show the uniqueness of the equilibrium of system (12), repeating (A.1), (A.2), (A.4), (A.5) and applying (B.1) to (A.3), we can rewrite (A.6) as
r (k)T (k) −(x ∗ )T DP + P D − Σ σ s0 FAT FA + sk FB FB Σ σ − P A0 Q−1 AT0 P k=1
C. Li et al. / Physics Letters A 328 (2004) 452–462
461
r 1 1 1 T P EB(k) EB(k)T P − εE Rk−1 B0(k)T P − P EA EA P− 1 − ηk s0 sk (1 − ηk ) k=1 k=1
r T (x ∗ ) = −(x ∗ )T Ω2 (x ∗ ) − (x ∗ )T Ω3 (x ∗ ) 0, + ΣR(k) ΣR(k)T + εE − nΣ σ ΣQ ΣQ
−
r
P B0(k)
(B.2)
k=1
which implies Ω2 0 or Ω3 0. It leads to a contradiction with condition (i) or (ii) in this theorem. Therefore, there exists unique equilibrium of system (12). The Lyapunov–Krasovskii functional (A.7) is also used here. Moreover, arguing similarly with the proof of Theorem 1, we have 1 T P V˙ x(t) −x T (t) DP + P D − P A0 Q−1 AT0 P − s0 Σ σ FAT FA Σ σ − P EA EA s0 − −
r k=1 r
1 1 P EB(k) EB(k)T P Rk−1 B0(k)T P − 1 − ηk sk (1 − ηk ) r
P B0(k)
k=1
(k)T
sk Σ σ FB
k=1
(k)
FB Σ σ − εE + εE − n
r
(k)
(k)T
Σ σ ΣR ΣR
Σσ
k=1
T σ − nΣ σ ΣQ ΣQ Σ x(t)
= −x (t) DP + P D − Σ T
σ
s0 FAT FA
+
r
(k)T (k) sk FB FB
Σ σ − εE − P A0 Q−1 AT0 P
k=1
r 1 1 1 (k) (k) (k)T −1 (k)T T P EB EB P x(t) P B0 R B P− − P EA EA P − s0 1 − ηk k 0 sk (1 − ηk ) k=1 k=1
r (k) (k)T T − x T (t) εE − nΣ σ Σ σ x(t) ΣR ΣR − ΣQ ΣQ r
k=1
= −x T (t)Ω2 x(t) − x T (t)Ω3 x(t). Hence, V˙ (x(t)) < 0 when Ω2 > 0 and Ω3 > 0, which completes the proof of the theorem. 2
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