LMI-based criterion for global asymptotic stability

Transcription

1 170 Int. J. Systems, Control and Communications, Vol. 1, No. 2, 2008 LMI-based criterion for global asymptotic stability of BAM neural networks with time delays Ju H. Park Robust Control and Nonlinear Dynamics Laboratory, Department of Electrical Engineering, Yeungnam University, Dae-Dong, Kyongsan , Republic of Korea Corresponding author O.M. Kwon School of Electrical and Computer Engineering, 12 Gaeshin-Dong, Heungduk-Gu, Chungbuk National University, Cheongju, Republic of Korea Abstract: This paper presents a stability criterion for global asymptotic stability of the equilibrium point for Bidirectional Associative Memory (BAM) neural networks with fixed time delays. An approach combining the Lyapunov-Krasovskii functional with Linear Matrix Inequality (LMI) is taken to investigate the stability of the system. A delay-dependent LMI criterion is derived. Finally, a numerical example is given to illustrate the results. Keywords: BAM neural networks; LMI; asymptotic stability; delay. Reference to this paper should be made as follows: Park, J.H. and Kwon, O.M. (2008) LMI-based criterion for global asymptotic stability of BAM neural networks with time delays, Int. J. Systems, Control and Communications, Vol. 1, No. 2, pp Biographical notes: Ju H. Park received the PhD Degree in Electrical and Electronics Engineering from Pohang University of Science and Technology (POSTECH), Republic of KOREA in He was a Associate Researcher in ERC-ARC, POSTECH from 1997 to Since 2000, he has been with Yeungnam University as Professor. He is serving as Associate Editor for Applied Mathematics and Computation, Journal of The Franklin Institute, and International Journal of Control, Automation, and Systems. Also, he joined 10 international journals as a member of editorial team. His current research interests are in the area of chaos systems, neural networks, signal processing, and communication networks. O.M. Kwon received the PhD Degree in Electrical and Electronics Engineering from Pohang University of Science and Technology (POSTECH), Republic of KOREA in He was a Senior Researcher in Samsung Heavy Industry Co. Ltd. from 2004 to Since 2006, Copyright 2008 Inderscience Enterprises Ltd.

2 LMI-based criterion for global asymptotic stability 171 he has been with Chungbuk National University as Assistant Professor. He is the Managing Editor for Journal of Mathematical Control Science and Applications. His research interests include time-delay systems, neural networks and secure communications. 1 Introduction Bidirectional Associative Memory (BAM) neural networks are a class of important neural network with the ability to store a collection of pattern pairs via unsupervised learning, which have applications in pattern recognition, artificial intelligence, and automatic control (Kosko, 1987, 1988). Thus, the BAM neural networks has been one of the most interesting research topics and has attracted the attention of many researchers (Gopalsamy and He, 1994; Cao and Wang, 2000; Guo et al., 2003). As is well known, both in biological and man-made neural networks, the delays arise because of the processing of information. More specifically, the delays occur in the communication and response of neurons owing to the finite switching speed of amplifiers in the electronic implementation of analog neural networks. The delay is a source of instability and oscillatory response of the networks. Therefore, the study of the stability problem of BAM with delays has raised considerable interest (Liao and Yu, 1998; Zhang and Yang, 2001; Zhao, 2002; Cao, 2003; Chen et al., 2003; Park, 2006; Huang et al., 2005). In this paper, we investigate the problem of stability analysis for BAM neural networks with fixed time delays. Attention is focused on the derivation of a novel criterion which guarantees the global asymptotic stability of the equilibrium point of the BAM neural network. The criterion developed in this paper is expressed by several LMIs, which can be solved numerically very efficiently by various convex optimisation algorithms (Boyd et al., 1994) and no tuning of parameters are involved. The rest of this paper is organised as follows: in Section 2, the problem to be investigated is stated; in Section 3, a new stability criterion for asymptotic stability of BAM with delays will be established; in Section 4, some conclusions are drawn. In this paper, we use X T,X 1 to denote respectively the transpose of a vector (or matrix) and the inverse of a square matrix. R n denotes the n dimensional Euclidean space, and R n m is the set of all n m real matrices. I denotes the identity matrix with appropriate dimensions. denotes the elements below the main diagonal of a symmetric block matrix. diag{ } denotes the diagonal matrix. For symmetric matrices X and Y, the notation X>Y (respectively, X Y ) means that the matrix X Y is positive definite, (respectively, nonnegative). 2 Problem statement Consider the following BAM neural networks with constant delays: u i (t) = a i u i (t)+ v j (t) = b j v j (t)+ m w ji g j (v j (t τ)) + I i, j=1 i =1, 2,...,n, m v ij g i (u i (t σ)) + J j, j =1, 2,...,m, (1) i=1

3 172 J.H. Park and O.M. Kwon in which u =(u 1,u 2,...,u n ) T R n and v =(v 1,v 2,...,v m ) T R m are the activations of the ith neurons and the jth neurons, respectively, w ji and v ij are the connection weights at the time t, I i and J j denote the external inputs, τ>0 and σ>0 are positive constants which correspond to the finite speed of axonal signal transmission, and a i > 0,b j > 0. In this paper, it is assumed that the activate functions g i possess the following properties: (A1) (A2) g i is bounded on R,i=1, 2,...,max{m, n}. There exist real numbers M i > 0 such that g i (x) g i (y) M i x y for any x, y R,i=1, 2,...,max{m, n}. It is clear that under the assumption (A1) and (A2), system (3) has at least one equilibrium. Assume that u =(u 1,u 2,...,u n) T and v =(v1,v 2,...,v m) T are the equilibrium point of the system, then we will shift the equilibrium points to the origin by the transformation x i (t) =u i (t) u i, y j(t) =v j (t) vj, f i(x i (t)) = g i (u i (t)) g i (u i ), and f j(y j (t)) = g j (v j (t)) g j (vj ). Then, the transformed system is as follows: ẋ i (t) = a i x i (t)+ ẏ j (t) = b j y j (t)+ m w ji f j (y j (t τ)), j=1 m v ij f i (x i (t σ)), i=1 i =1, 2,...,n, j =1, 2,...,m, x i (s) =φ i (s), y j (s) =ψ j (s), s [ max{τ,σ}, 0], i =1, 2,...,n, j =1, 2,...,m, (2) where the activate functions f i satisfy the following properties: (H1) (H2) (H3) f i is bounded on R,i=1, 2,...,max{m, n}. There exist real numbers M i > 0 such that f i (x) f i (y) M i x y for any x, y R,i=1, 2,...,max{m, n}. f i (0) = 0, i=1, 2,...,max = {m, n}. For convenience, we can rewrite (2) in the form: ẋ(t) = Ax(t)+W T f(y(t τ)), ẏ(t) = By(t)+V T f(x(t σ)), (3) where x(t)=(x 1 (t),x 2 (t),...,x n (t)) T,y(t)=(y 1 (t),y 2 (t),...,y m (t)) T,A= diag(a 1, a 2,...,a n ),B = diag(b 1,b 2,...,b m ), W =(w ij ) m n, V =(v ij ) n m,f =(f 1,f 2,..., f m ) T, f =(f 1,f 2,...,f n ) T,M = diag{m 1,M 2,...,M m }, and M = diag{m 1,M 2,..., M n }. 3 Main result In this section, we present a stability criterion for global asymptotic stability of system (3).

4 LMI-based criterion for global asymptotic stability 173 The following fact and lemmas will be used for deriving main result. Fact 1: (Schur complement) Given constant symmetric matrices Σ 1, Σ 2, Σ 3 where Σ 1 =Σ T 1 and 0 < Σ 2 =Σ T 2, then Σ 1 +Σ T 3 Σ 1 2 Σ 3 < 0 if and only if [ ] [ ] Σ1 Σ T 3 Σ2 Σ 3 < 0, or < 0. (4) Σ 3 Σ 2 Σ T 3 Σ 1 Lemma 2 (Moon et al., 2001): Assume that a R na, b R n b, and N R na n b are defined, then for any matrices X R na na, Y R na n b and Z R n b n b, the following holds 2a T Nb [ a b ] T [ X Y N Z ][ ] a, b [ ] X Y 0. (5) Z Lemma 3 (Cao et al., 1998): For any z,y R n m, a positive scalar ɛ, and any positive definite matrix X R n n the following inequality holds. 2z T y ɛz T X 1 z + ɛ 1 y T Xy (6) Now, we have the following theorem. Theorem 4: For given τ and σ, the equilibrium point of system (3) is globally asymptotically stable if there exist positive definite matrices P, Q, S 1, S 2, X 1, X 2, Z 1, Z 2, P i,q i, (i =1, 2), positive diagonal matrices D =diag{d 1,...,d n }, E =diag{e 1,...,e m }, positive scalars ɛ 1,ɛ 2 and R 1 0,R 2 0, satisfying the following LMIs: Π 1 0 R1 T Π DAM DW T 0 Π V Π < 0, P 2 0 ɛ 2 I Π 5 0 R T 2 Π EBM EV T 0 Π W < 0, (7) Π Q 2 0 ɛ 1 I ( ) X1 R 1 > 0, Z 1 ( ) X2 R 2 > 0, Z 2

5 174 J.H. Park and O.M. Kwon where Π 1 = PA AP + MS 1 M + σa T Z 1 A, Π 2 = S 1 + σx 1 R 1 M 1 M 1 R1 T + Q 1 + Q 2 + τvz 2 V T, Π 3 =[PW T σa T Z 1 ], Π 4 =diag{ P 1, ɛ 1 I}, Π 5 = QB BQ + MS 2 M + τb T Z 2 B, Π 6 = S 2 + τx 2 R 2 M 1 M 1 R2 T + P 1 + P 2 + σwz 1 W T, Π 7 =[QV T τb T Z 2 ], Π 8 =diag{ Q 1, ɛ 2 I}. Proof: Consider the following Lyapunov-Krasovskii functional where V = V 1 + V 2 + V 3 + V 4 + V 5 + V 6 + V 7 + V 8 (8) V 1 = x T (t)px(t), V 3 = i=1 0 V 2 = y T (t)qy(t), x T (s)ms 1 Mx(s)ds, V 4 = 0 y T (s)ms 2 My(s)ds, n xi(t) n yi(t) V 5 =2 d i f i (s)ds, V 6 =2 e i f i (s)ds, V 7 = σ t+θ ẋ T (s)z 1 ẋ(s)ds dθ, V 8 = i=1 0 0 τ t+θ The time derivative of V i along the trajectory of system (3) is V 1 =2x T (t)p ( Ax(t)+W T f(y(t τ))) V 2 =2y T (t)q( By(t)+V T f(x(t σ))), V 3 = x T (t)ms 1 Mx(t) x T (t σ)ms 1 Mx(t σ), V 4 = y T (t)ms 2 My(t) y T (t τ)ms 2 My(t τ), ẏ T (s)z 2 ẏ(s)ds dθ. V 5 =2 f T (x(t))dẋ(t) =2 f T (x(t))d( Ax(t)+W T f(y(t τ))), V 6 =2f T (y(t))eẏ(t) =2f T (y(t))e( By(t)+V T f(x(t σ))), V 7 = σẋ T (t)z 1 ẋ(t) ẋ T (s)z 1 ẋ(s)ds, = σ( Ax(t)+W T f(y(t τ))) T Z 1 ( Ax(t)+W T f(y(t τ))) ẋ T (s)z 1 ẋ(s)ds,

6 LMI-based criterion for global asymptotic stability 175 V 8 = τẏ T (t)z 2 ẏ(t) ẏ T (s)z 2 ẏ(s)ds, = τ( By(t)+V T f(x(t σ))) T Z 2 ( By(t)+V T f(x(t σ))) ẏ T (s)z 2 ẏ(s)ds. (9) By the Leibniz-Newton formula, the following equations satisfy ( 2 f T (x(t σ)) x(t) x(t σ) ( 2f T (y(t τ)) y(t) y(t τ) ) ẋ(s)ds =0, Applying Lemma 2 to two terms in equation (10) gives that 2 ) ẏ(s)ds =0. (10) f T (x(t σ))ẋ(s)ds σ f T (x(t σ))x 1 f(x(t σ)) +2 f T (x(t σ))(r 1 I)(x(t) x(t σ)) + ẋ T (s)z 1 ẋ(s)ds, 2 f T (y(t τ))ẏ(s)ds τf T (y(t τ))x 2 f(y(t τ)) +2f T (y(t τ))(r 2 I)(y(t) y(t τ)) + ẏ T (s)z 2 ẏ(s)ds, (11) where [ ] X1 R 1 0, Z 1 [ ] X2 R 2 0. (12) Z 2 Here note that x T (t σ)ms 1 Mx(t σ) f T (x(t σ))s 1 f(x(t σ)), y T (t τ)ms 2 My(t τ) f T (y(t τ))s 2 f(y(t τ)), 2 f(x(t))dax(t) 2 f(x(t))dam 1 f(x(t)), 2f(y(t))EBy(t) 2f(y(t))EBM 1 f(y(t)), 2 f(x(t σ))r 1 x(t σ) 2 f(x(t σ))r 1 M 1 f(x(t σ)), 2f(y(t τ))r 2 y(t τ) 2f(y(t τ))r 2 M 1 f(y(t τ)) (13) and using Lemma 3 we can obtain the following relations: 2x T (t)pw T f(y(t τ)) f T (y(t τ))p 1 f(y(t τ)) + x T (t)pw T P 1 1 WPx(t), 2y T (t)qv T f(x(t σ)) f T (x(t σ))q 1 f(x(t σ)) + y T (t)qv T Q 1 1 VQy(t),

7 176 J.H. Park and O.M. Kwon 2 f T (x(t))dw T f(y(t τ)) f T (x(t))dw T P2 1 WD f(x(t))) + f T (y(t τ))p 2 f(y(t τ)), 2f T (y(t))ev T f(x(t σ)) f T (y(t))ev T Q 1 2 VEf(y(t)) + f T (x(t σ))q 2 f(x(t σ)), 2σx T (t)a T Z 1 W T f(y(t τ)) ɛ 1 1 σ2 x T (t)a T Z 1 Z 1 Ax(t) + ɛ 1 f T (y(t τ))ww T f(y(t τ)), 2τy T (t)b T Z 2 V T f(x(t σ)) ɛ 1 2 τ 2 y T (t)b T Z 2 Z 2 By(t) + ɛ 2 f T (x(t σ))vv T f(x(t σ)). (14) Substituting equations (11), (13) and (14) into equation (9) gives that N 1 0 R1 T N 4 0 R2 T V z1 T (t) N 2 0 z 1 (t)+z2 T (t) N 5 0 z 2 (t) N 3 N 6 where z T 1 (t)ω 1 z 1 (t)+z T 2 (t)ω 2 z 2 (t) (15) z 1 (t) = [ x T (t) f T (x(t)) f T (x(t σ)) ] T, z 2 (t) = [ y T (t)f T (y(t))f T (y(t τ))] T, N 1 = PA AP + MS 1 M + PW T P1 1 WP + σa T Z 1 A + ɛ 1 1 σ2 A T Z 1 Z 1 A, N 2 = 2DAM 1 + DW T P2 1 WD, N 3 = S 1 + σx 1 R 1 M 1 M 1 R1 T + Q 1 + Q 2 + τvz 2 V T + ɛ 2 VV T, N 4 = QB BQ + MS 2 M + QV T Q 1 1 VQ+ τbt Z 2 B + ɛ 1 2 τ 2 B T Z 2 Z 2 B, N 5 = 2EBM 1 + EV T Q 1 2 VE, N 6 = S 2 + τx 2 R 2 M 1 M 1 R2 T + P 1 + P 2 + σwz 1 W T + ɛ 1 WW T. From equation (15), if Ω 1 < 0, and Ω 2 < 0, then there exist positive scalars δ 1 and δ 2 such that V δ 1 x(t) 2 δ 2 y(t) 2. Then, using Fact 1, the inequalities Ω 1 < 0 and Ω 2 < 0 are equivalent to (5) and (6) respectively. This completes our proof. Remark 5: The criterion given in Theorem 4 is delay-dependent. It is well known that the delay-dependent criteria are generally less conservative than delay-independent criteria when the delay is small. The LMI solutions of Theorem 4 can be obtained by solving the eigenvalue problem with respect to solution variables, which is a convex optimisation problem. In this paper, we utilise Matlab s Robust Control Toolbox (Gahinet et al., 1995) which implements interior-point algorithm. This algorithm is significantly faster than classical convex optimisation algorithms (Boyd et al., 1994).

8 LMI-based criterion for global asymptotic stability 177 Remark 6: By iteratively solving the LMIs given in Theorem 4 with respect to τ and σ, one can find the maximum upper bounds of time delay τ and σ for guaranteeing asymptotic stability of system (3). Example 7: Consider the following BAM neural networks ẋ i (t) = a i x i (t)+ ẏ j (t) = b j y j (t)+ 2 w ji f j (y j (t σ), j=1 2 v ij f i (x i (t σ), i =1, 2, j =1, 2. (16) i=1 To take f i (x) = 1 2 ( x i +1 x i 1 ) and f j (y) = 1 2 ( y j +1 y j 1 ), then we have M i = M j =1for all i and j, i.e., M = M = I. Let [ ] 1 0 A =, B = 0 1 [ W = [ ] 1 0, 0 1 ], V = [ ] By applying Theorem 4 to the system (16), one can see that the LMIs given in Theorem 4 are feasible for any σ. This implies that the system (16) is asymptotically stable for any delay σ>0. 4 Concluding remarks The global asymptotic stability of BAM neural networks with fixed time delay has been investigated. By using the Lyapunov theory and matrix inequality framework, a new delay-dependent stability criterion expressed by four LMIs is presented. To show the effectiveness of the proposed criterion, a numerical example is given. References Boyd, B., Ghaoui, L.E., Feron, E. and Balakrishnan, V. (1994) Linear Matrix Inequalities in Systems and Control Theory, SIAM, Philadelphia. Cao, Y.Y., Sun, Y.X. and Lam, J. (1998) Delay-dependent robust H control for uncertain systems with timevarying delays, IEEE Proceedings-Control Theory and Applications, Vol. 145, pp Cao, J. and Wang, L. (2000) Periodic oscillatory solution of bidirectional associative memory networks, Phy. Rev. E, Vol. 61, pp Cao, J. (2003) Global asymptotic stability of delayed bi-directional associative memory neural networks, Applied Mathematics and Computation, Vol. 142, pp Chen, A., Huang, L. and Cao, J. (2003) Existence and stability of almost periodic solution for BAM neural networks with delays, Applied Mathematics and Computation, Vol. 137, pp

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