State Estimation for Discrete-time Markovian Jumping Neural Networks with Mixed Mode-Dependent Delays

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1 SUBMITTED 1 State Estimation for Discrete-time Markovian Jumping Neural Networks with Mied Mode-Dependent Delays Yurong Liu, Zidong Wang and Xiaohui Liu Abstract In this paper, we investigate the state estimation problem for a new class of discrete-time neural networks with Markovian jumping parameters as well as mode-dependent mied time-delays. The parameters of the discrete-time neural networks are subject to the switching from one mode to another at different times according to a Markov chain, and the mied time-delays consist of both discrete and distributed delays that are dependent on the Markovian jumping mode. New techniques are developed to deal with the mied time-delays in the discrete-time setting, and a novel Lyapunov-Krasovskii functional is put forward to reflect the mode-dependent time-delays. Sufficient conditions are established in terms of linear matri inequalities (LMIs) that guarantee the eistence of the state estimators. We show that both the eistence conditions and the eplicit epression of the desired estimator can be characterized in terms of the solution to an LMI. A numerical eample is eploited to show the usefulness of the derived LMI-based conditions. Keywords Discrete-time neural networks; Mied time delays; Markovian jumping parameters; State estimator; Linear matri inequality. I. Introduction It is now well known that many biological and artificial neural networks contain inherent time delays in signal transmission, which may cause oscillation and instability (see e.g. 1 3). In recent years, a great number of papers have been published on various neural networks with time delays. The eistence of equilibrium point, global asymptotic stability, global eponential stability, and the eistence of periodic solutions have been intensively investigated, see e.g For the dynamical behavior analysis of delayed neural networks, different types of time delays, such as constant delays, time-varying delays, and distributed delays, have been taken into account by using a variety of techniques that include linear matri inequality (LMI) approach, Lyapunov functional method, M-matri theory, topological degree theory, and techniques of inequality analysis. For eample, in 10, 13, the global asymptotic stability analysis problem has been dealt with for a class of neural networks with discrete and distributed time-delays by using an effective LMI approach. In many applications, since the neuron states are not often fully available in the network outputs, the neuron state estimation problem becomes important. Thus, the state estimation problem for neural networks has recently drawn particular research interests, see e.g. 3, For eample, in 14, an adaptive state estimator has been described by using techniques of optimization theory, the calculus of variations and gradient descent dynamics. In 3, the neuron state estimation problem has been addressed for recurrent neural networks with time-varying delays, and an effective LMI approach has been developed to verify the stability of the estimation This work was supported in part by the Biotechnology and Biological Sciences Research Council (BBSRC) of the UK under Grants BB/C50664/1 and 100/EGM17735, the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grants GR/S7658/01 and EP/C54586/1, an International Joint Project sponsored by the Royal Society of the UK, the Natural Science Foundation of Jiangsu Province of China under Grant BK007075, the National Natural Science Foundation of China under Grant , and the Aleander von Humboldt Foundation of Germany. Y. Liu is with the Department of Mathematics, Yangzhou University, Yangzhou 500, P. R. China. Z. Wang and X. Liu are with the Department of Information Systems and Computing, Brunel University, Ubridge, Middlese, UB8 3PH, United Kingdom. addresses: liuyurong@gmail.com (Y. Liu), Zidong.Wang@brunel.ac.uk (Z. Wang). Corresponding author.

2 SUBMITTED error dynamics. Very recently, in 17, the robust state estimation problem has been studied for a class of uncertain neural networks with time-varying delay, and a sufficient condition has been presented to guarantee the eistence of the desired state estimator for the uncertain delayed neural networks. In practice, neural networks may be subject to network mode switching, which is governed by a Markovian chain 18. In 19, the eponential stability has been studied for delayed recurrent neural networks with Markovian jumping parameters. In 0, the eponential synchronization problem has been investigated for a class of continuous-time comple networks with Markovian jump and mied delays. On the other hand, discrete-time neural networks could be more suitable to model digitally transmitted signals in a dynamical way. However, to the best of the authors knowledge, the state estimation problem for discrete-time Markovian jumping neural networks with or without mode-dependent mied time-delays has not been adequately addressed in the literature yet, and the purpose of this paper is therefore to shorten such a gap. In this paper, we deal with the state estimation problem for a new class of discrete-time neural networks with Markovian jumping parameters and mode-dependent mied time-delays. By utilizing the Lyapunov functional method and some new techniques, we derive several delay-dependent sufficient conditions under which the estimation error dynamics is asymptotically stable. The criteria obtained in this paper are in the form of LMIs whose solution can be easily calculated by using the standard numerical software. Note that Markovian jumping parameters are introduced for discrete-time neural networks, and both the discrete and distributed delays are mode-dependent. A numerical eample is presented to illustrate the usefulness of our results. Notations: Throughout this paper, R n and R n m denote, respectively, the n dimensional Euclidean space and the set of all n m real matrices. The superscript T denotes the transpose and the notation X Y (respectively, X > Y ) where X and Y are symmetric matrices, means that X Y is positive semi-definite (respectively, positive definite); I is the identity matri with compatible dimension. refers to the Euclidean vector norm. If A is a symmetric matri, λ min ( ) and λ ma ( ) denote the minimum and the maimum eigenvalue, respectively. In symmetric block matrices, we use an asterisk to represent a term that is induced by symmetry and diag{ stands for a block-diagonal matri. E and E y will, respectively, mean the epectation of and the epectation of conditional on y. Matrices, if their dimensions are not eplicitly stated, are assumed to be compatible for algebraic operations. II. Problem formulation Let r(k) (k 0) be a Markov chain taking values in a finite state space S = {1,,..., N with probability transition matri Π = (π ij ) N N given by { P r(k + 1) = j r(k) = i = π ij, i,j S where π ij 0(i,j S) is the transition rate from i to j and N j=1 π ij = 1, i S. Consider a discrete-time n-neuron neural network with N modes described by the following dynamical equation: (k+1) = D(r(k))(k)+A(r(k))F((k))+B(r(k))G((k 1,r(k) ))+C(r(k)),r(k) H((k v))+j(r(k)), (1) where (k) = ( 1 (k), (k),..., n (k)) T is the neural state vector; the constant matrices D(r(k)) = diag{d 1 (r(k)), d (r(k)),...,d n (r(k)) describe the rate with which the each neuron will reset its potential to the resting state in isolation when disconnected from the networks and eternal inputs; A(r(k)) = a ij (r(k)) n n, B(r(k)) = b ij (r(k)) n n and C(r(k)) = c ij (r(k)) n n are, respectively, the connection weight matri, the discretely delayed connection weight matri and the distributively delayed connection weight matri; 1,r(k) denotes discrete time delay while,r(k) characterizes distributed time delay, and both kinds of time delays are dependent on the system mode r(k). In (1), F((k)) = (f 1 ( 1 (k)),f ( (k)),...,f n ( n (k))) T, G((k 1,r(k) )) = (g 1 ( 1 (k

3 SUBMITTED 3 1,r(k) )), g ( (k 1,r(k) )),..., g n ( n (k 1,r(k) ))) T and H((k)) = (h 1 ( 1 (k)),h ( (k)),...,h n ( n (k))) T are the nonlinear activation functions; and J(r(k))) is constant vector representing the eternal bias on the neurons. Remark 1: The discrete-time recurrent neural network (1) can be considered as a discrete analogue of the continuous-time recurrent neural network of the form: d dt = D(r(t))(t) + A(r(t))F((t)) + B(r(t))G((t 1,r(t))) + C(r(t)) t t,r(t) H((s))ds + J(r(t)), () where r(t) corresponds to a Markov Chain. Note that the continuous-time neural networks without Markov chain have been etensively investigated; see, e.g. 13 and the references therein. However, to the best of authors knowledge, discrete-time mode-dependent neural networks with or without mied time delays, have received very little attention. Throughout this paper, we make the following assumption. Assumption 1 ( 13) For the activation functions F( ),G( ) and H( ), there eist constants λ i,λ+ i,σ i,σ+ i, υ i and υ + i such that, for any s 1,s R,s 1 s,1 i n λ i f i(s 1 ) f i (s ) λ + i s 1 s, σ i g i(s 1 ) g i (s ) σ i + s 1 s, υ i h i(s 1 ) h i (s ) υ i + s 1 s. (3) Remark : As pointed out in 13, the constants λ i, λ+ i, σ i, σ+ i, υ i and υ i + in Assumption 1 are allowed to be positive, negative or zero. Hence, the resulting activation functions could be non-monotonic, and are more general than the usual sigmoid functions and the recently commonly used Lipschitz conditions. Such a description is very precise/tight in quantifying the lower and upper bounds of the activation functions, hence very helpful for using LMI-based approach to reduce the possible conservatism. For neural networks, whether biological or artificial, it is usually difficult to get a complete access to their states and, in many applications, it consequently becomes necessary to estimate the states of neural networks based only on the output from the networks. In this paper, we assume that the output from the neural network (1) is of the form: y(k) = E(r(k))(k) + S(k,(k)) (4) where y(k) = (y 1 (k),y (k),...,y m (k)) T denotes the measurement output of the neural network, E(i) R m n (i S) is a constant matri, and S(k,(k)) R m is a nonlinear disturbance dependent on the neuron state satisfying the following Lipschitz condition: S(k,) S(k,y) W( y), k R,,y R n (5) with W being a known constant matri of appropriate dimension. Remark 3: Usually, the relationship m < n holds, which means that only partial information (or a combination of the information) on the system states can be accessed via output measurements. In order to track the state of system (1), we construct a full-order state estimator for (1): ˆ(k + 1) = D(r(k))ˆ(k) + A(r(k))F(ˆ(k)) + B(r(k))G(ˆ(k 1,r(k) )) + C(r(k)),r(k) H(ˆ(k v)) + J(r(k)) K(r(k))y(k) E(r(k))ˆ(k) S(k, ˆ(k)), (6) where ˆ(k) is the state estimate, and K(i) R n m (i S) is the estimate gain matri to be designed. Let ǫ(k) = (ǫ 1 (k),ǫ (k),...,ǫ n (k)) T = ˆ(k) (k) be the state estimation error. Definition 1: System (6) is said to be an asymptotic state estimator of neural network (1) if the estimation error satisfies lim k + E ǫ(k) = 0. (7) This paper aims to design the state estimator of (1). By constructing new Lyapunov Krasovskii functional, we will derive the sufficient conditions under which system (6) becomes an asymptotic state estimator of neural network (1), and the resulting gain matrices K(i) (i S) will also be given eplicitly.

4 SUBMITTED 4 III. Main results and proofs The following lemmas are needed in deriving our results. Lemma 1 ( 13) Let X, Y be any n-dimensional real vectors and P be a n n positive semi-definite matri. Then, the following matri inequality holds: X T PY X T PX + Y T PY. Lemma ( 1) Let M R n n be a positive semi-definite matri, i R n be a vector and a i 0 (i = 1,,...) be scalars. If the series concerned are convergent, then the following inequality holds: ( + ) T ( + ) ( + ) + a i i M a i i a i a i T i M i (8) i=1 i=1 Lemma 3: Suppose that B = diag{β 1,β,...,β n 0 is a diagonal matri. Let = ( 1,,..., n ) T R n, and H() = ( 1 ( 1 ), ( ),..., n ( n )) T be a continuous nonlinear function satisfying i=1 i=1 l i i(s) s l + i, s 0, s R, i = 1,,...,n (9) with l i and l + i being constant scalars. Then H() T BL1 BL BL B H() 0 or T BL 1 T BL H() + H T ()BH() 0 where L 1 = diag { l 1 + l 1,l+ l,...,l+ n ln and L = diag { l + 1 +l 1, l+ +l,..., l+ n +ln Proof: Notice that (9) is equivalent to or H() T ( i ( i ) l + i i)( i ( i ) l i i) 0, l + i l i e ie T i l+ i +l i e i e T i l+ i +l i e i e T i e i e T i H() 0, i = 1,...,n, where e k denotes the unit column vector having 1 element on its kth row and zeros elsewhere. Since β i 0, it follows readily that namely n β i i=1 H() T H() l + i l i e ie T i l+ i +l i e i e T i l+ i +l i e i e T i e i e T i T BL1 BL BL B H() H() 0. 0, (10) This completes the proof of this lemma. Lemma 4 ( 19Schur Complement) Given constant matrices Ω 1,Ω,Ω 3 where Ω 1 = Ω T 1 and Ω > 0, then Ω 1 + Ω T 3 Ω 1 Ω 3 < 0 if only if Ω1 Ω T 3 Ω 3 Ω < 0.

5 SUBMITTED 5 In what follows, for presentation convenience, we denote 1 = ma{ 1,i i S, = ma{,i i S, = ma{ 1,, 1 = min{ 1,i i S, = min{,i i S, π = min{π ii i S, Λ 1 = diag { λ + 1 λ 1,λ+ λ,...,λ+ n λ n, Λ = diag{ λ λ 1 Σ 1 = diag { σ + 1 σ 1,σ+ σ,...,σ+ n σ n, Σ = diag{ σ σ 1 Υ 1 = diag { υ + 1 υ 1,υ+ υ,...,υ+ n υ n, Υ = diag{ υ υ 1, λ+ + λ, σ+ + σ, υ+ + υ,..., λ+ n + λ n,..., σ+ n + σ n,..., υ+ n + υ n Theorem 1: Suppose that Assumption 1 and condition (5) hold and Let K(i)(i S) be known constant matrices. Then system (6) becomes an asymptotic state estimator of neural network (1) if there eist a set of scalar constants ϑ i > 0(i S), a set of matrices P i > 0(i S), two matrices Q > 0 and R > 0, and three sets of diagonal matrices Ω i > 0,Θ i > 0 and i > 0(i S) such that the following LMIs hold: Φ i = where Ξ i Ω i Λ Θ i Σ 0 i Υ 0 0 DT (i)p i Ω i Λ Ω i A T (i)p i Θ i Σ 0 κ 0 Q Θ i Q B T (i)p i i Υ κ i R i ,i R 0 C T (i)p i ϑ i I P i P i D(i) P i A(i) 0 P i B(i) 0 P i C(i) P i P i P i =,,. < 0, (i S) (11) N π ij P j, D(i) = D(i) + K(i)E(i), (1) j=1 Ξ i = P i Ω i Λ 1 Θ i Σ 1 i Υ 1 + ϑ i W T W, (13) κ 0 = (1 π)( 1 1 ) + 1, (14) Proof: Denote κ i =,i + (1 π ii )( ) + 1 (1 π)( )( + 1). (15) F(ǫ(k)) = f 1 (ǫ 1 (k)), f (ǫ 1 (k)),..., f n (ǫ n (k)) T = F(ˆ(k)) F((k)), (16) G(ǫ(k)) = g 1 (ǫ 1 (k)), g (ǫ 1 (k)),..., g n (ǫ n (k)) T = G(ˆ(k)) G((k)), (17) H(ǫ(k)) = h 1 (ǫ 1 (k)), h (ǫ 1 (k)),..., h n (ǫ n (k)) T = H(ˆ(k)) H((k)), (18) S(k,ǫ(k)) = S(k, ˆ(k)) S(k,(k)). (19) It should be pointed out here that F(ǫ(k)), G(ǫ(k)), H(ǫ(k)) and S(k,ǫ(k)) are all dependent on (k) or ˆ(k), as well as ǫ(k). However, in order to avoid cumbersome notations, we shall use simpler symbols. For instance, we use F(ǫ(k)) instead of F(ǫ(k),(k)) and so on. With the above notations and from Eqs. (1) and (6), we obtain the dynamics of estimation error governed by ǫ(k + 1) = D(r(k))ǫ(k) + A(r(k)) F(ǫ(k)) + B(r(k)) G(ǫ(k 1,r(k) )) + C(r(k)),r(k) H(ǫ(k v)) + K(r(k)) S(k,ǫ(k)). (0)

6 SUBMITTED 6 For notation convenience, in the sequel, we denote ǫ k = ǫ T (k),ǫ T (k 1),...,ǫ T (k ) T, ξ(k, i) = ǫ T (k) F T (ǫ(k)) GT (ǫ(k)) GT (ǫ(k 1,i )) HT (ǫ(k)) Z(i) = D(i) A(i) 0 B(i) 0 C(i) I.,i H T (ǫ(k v)) ST (k,ǫ(k)) T, Now, in order to ensure that (6) is the state estimator, we just need to show that the estimation error system (0) is asymptotically stable in the mean square, i.e., lim k E ǫ(k) = 0. To this end, we construct the following Lyapunov-Krasovskii functional V ( k,k,r(k)) by where V (ǫ k,k,r(k)) = V 1 (ǫ k,k,r(k)) + V (ǫ k,k,r(k)) + V 3 (ǫ k,k,r(k)) + V 4 (ǫ k,k,r(k)) + V 5 (ǫ k,k,r(k)) (1) V 1 (ǫ k,k,r(k)) = ǫ T (k)p r(k) ǫ(k), () V (ǫ k,k,r(k)) = V 3 (ǫ k,k,r(k)) = V 4 (ǫ k,k,r(k)) = V 5 (ǫ k,k,r(k)) = with Q = (1 π)q and R = (1 π)r. For i S, we have v=k 1,r(k) GT (ǫ(v))q G(ǫ(v)), (3) 1 1 ι= 1 v=k ι,r(k) ι=1 v=k ι s 1 s= +1 ι=1 v=k ι G T (ǫ(v)) Q G(ǫ(v)), (4) H T (ǫ(v))r H(ǫ(v)), (5) EV 1 (ǫ k+1,k + 1,r(k + 1)) ǫ k,r(k) = i V 1 (ǫ k,k,i) H T (ǫ(v)) R H(ǫ(v)) (6) = ξ T (k,i)z T (i)p i Z(i)ξ(k,i) ǫ T (k)p i ǫ(k). (7) EV (ǫ k+1,k + 1,r(k + 1)) ǫ k,r(k) = i V (ǫ k,k,i) N k = π ij G T (ǫ(v))q G(ǫ(v)) GT (ǫ(v))q G(ǫ(v)) j=1 v=k 1,j +1 v=k 1,i k k = π ii π ij v=k 1,i +1 v=k 1,i G T (ǫ(v))q G(ǫ(v)) + j i v=k 1,j +1 G T (ǫ(k))q G(ǫ(k)) G T (ǫ(k 1,i ))Q G(ǫ(k 1,i )) + (1 π) k 1 v=k 1 +1 v=k 1,i G T (ǫ(k))q G(ǫ(k)) G T (ǫ(v))q G(ǫ(v)), (8) = EV 3 (ǫ k+1,k + 1,r(k + 1)) ǫ k,r(k) = i V 3 (ǫ k,k,i) ( GT (ǫ(k)) Q G(ǫ(k)) G T (ǫ(k ι)) Q G(ǫ(k ι))) 1 1 ι= 1 = (1 π)( 1 1 ) G T (ǫ(k))q G(ǫ(k)) (1 π) k 1 v=k 1 +1 G T (ǫ(v))q G(ǫ(v)), (9)

7 SUBMITTED 7 and = EV 4 (ǫ k+1,k + 1,r(k + 1)) ǫ k,r(k) = i V 4 (ǫ k,k,i) N,j k π ij H T (ǫ(v))r H(ǫ(v)),i H T (ǫ(v))r H(ǫ(v)) j=1 ι=1 v=k ι+1 ι=1 v=k ι,i = π HT ii (ǫ(k))r H(ǫ(k)) H T (ǫ(k ι))r H(ǫ(k ι)) ι=1 +,j k,i π ij H T (ǫ(v))r H(ǫ(v)) j i ι=1 v=k ι+1 ι=1 v=k ι,i = π HT ii (ǫ(k))r H(ǫ(k)) H T (ǫ(k ι))r H(ǫ(k ι)) ι=1 +,i k,i π ij H T (ǫ(v))r H(ǫ(v)) j i ι=1 v=k ι+1 ι=1 v=k ι +,j k,i k π ij H T (ǫ(v))r H(ǫ(v)) j i ι=1 v=k ι+1 ι=1 v=k ι+1,i ( HT (ǫ(k))r H(ǫ(k)) H T (ǫ(k ι))r H(ǫ(k ι))) ι=1 + π ij j i k ι= +1 v=k ι+1 H T (ǫ(v))r H(ǫ(v)),i = HT,i (ǫ(k))r H(ǫ(k)) H T ((k ι))r H((k ι)) + (1 π ii ) k ι= +1 v=k ι+1 ι=1 H T (ǫ(v))r H(ǫ(v)) (,i + (1 π ii )( )) H T (ǫ(k))r H(ǫ(k)) H T (ǫ(k v))r H(ǫ(k v)) + (1 π) ι= +1 v=k ι+1,i H T (ǫ(v))r H(ǫ(v)), (30) = EV 5 (ǫ k+1,k + 1,r(k + 1)) ǫ k,r(k) = i V 5 (ǫ k,k,i) s 1 HT (ǫ(k)) R H(ǫ(k)) H T (ǫ(k ι)) R H(ǫ(k ι)) s= +1 ι=1 = (1 π) 1 ( )( + 1) H T (ǫ(k))r H(ǫ(k)) ι= +1 v=k ι+1 H T (ǫ(v))r H(ǫ(v)). (31)

8 SUBMITTED 8 From (7)-(31), it follows readily that EV (ǫ k+1,k + 1,r(k + 1)) ǫ k,r(k) = i V (ǫ k,k,i) ξ T (k,i)z T (i)p i Z(i)ξ(k,i) ǫ T (k)p i ǫ(k) + κ GT 0 (ǫ(k))q G(ǫ(k)) G T (ǫ(k 1,i ))Q G(ǫ(k 1,i )) + κ HT i (ǫ(k))r H(ǫ(k)) H T (ǫ(k v))r H(ǫ(k v)), (3),i where κ 0 and κ i are defined in (14) and (15), respectively. In terms of Lemma, one knows,i H T (ǫ(k v))r H(ǫ(k v)) 1,i,i TR,i H(ǫ(k v)) H(ǫ(k v)). (33) From Assumption 1 and (16)-(19), it is not difficult to see that for any s 1,s R,s 1 s,1 i n λ i f i (s 1 ) f i (s ) λ + i s 1 s, σ i g i(s 1 ) g i (s ) σ i + s 1 s, υ i h i (s 1 ) h i (s ) υ i + s 1 s and f i (0) = g i (0) = h i (0) = 0. Consequently, it follows that for any s R,s 0,1 i n λ i f i (s) s λ + i, σ i g i(s) s σ + i, υ i h i (s) s υ + i. (34) Also, from Lemma 3, it implies that Moreover, from (5) and (19), it follows that and S(t,0) = 0, which implies that ǫ T (k)ω i Λ 1 ǫ(k) ǫ T (k)ω i Λ F(ǫ(k)) + F T (ǫ(k))ω i F(ǫ(k)) 0, (35) ǫ T (k)θ i Σ 1 ǫ(k) ǫ T (k)θ i Σ G(ǫ(k)) + GT (ǫ(k))θ i G(ǫ(k)) 0, (36) ǫ T (k) i Υ 1 ǫ(k) ǫ T (k) i Υ H(ǫ(k)) + HT (ǫ(k)) i H(ǫ(k)) 0. (37) Combination of (33)-(38) with (3) results in S(k,) S(k,y) W( y),,y R n, ϑ i ST (k,ǫ(k)) S(k,ǫ(k)) ϑ i ǫ T (k)w T Wǫ(k) 0. (38) EV (ǫ k+1,k + 1,r(k + 1)) ǫ k,r(k) = i V (ǫ k,k,i) ξ T (k,i)z T (i)p i Z(i)ξ(k,i) ǫ T (k)p i ǫ(k) + κ GT 0 (ǫ(k))q G(ǫ(k)) G T (ǫ(k 1,i ))Q G(ǫ(k 1,i )) + κ HT i (ǫ(k))r H(ǫ(k)) 1,i TR,i ( H(ǫ(k v)) H(ǫ(k v)) ǫ T (k)ω i Λ 1 ǫ(k),i ) ( ǫ T (k)ω i Λ F(ǫ(k)) + F T (ǫ(k))ω i F(ǫ(k)) ǫ T (k)θ i Σ 1 ǫ(k) ǫ T (k)θ i Σ G(ǫ(k)) + G ) ) T (ǫ(k))θ i G(ǫ(k)) (ǫ T (k) i Υ 1 ǫ(k) ǫ T (k) i Υ H(ǫ(k)) + HT (ǫ(k)) i H(ǫ(k)) ( ϑ ST i (k,ǫ(k)) S(k,ǫ(k)) ) ϑ i ǫ T (k)w T Wǫ(k) = ξ T (k,i) Ψ i ξ(k,i), (39)

9 SUBMITTED 9 where Ψ i = Ψ i + Z T (i)p i Z(i) with Ξ i Ω i Λ Θ i Σ 0 i Υ 0 0 Ω i Λ Ω i Θ i Σ 0 κ 0 Q Θ i Ψ i = Q i Υ κ i R i ,i R ϑ i I and Ξ i defined in (13). By applying Lemma 4 to (11), we can deduce that Ψ { i < 0,(i S). Let α 0 = ma λma ( Ψ i ), then α 0 < 0 i S and it follows readily from (39) that which implies EV (ǫ k+1,k + 1,r(k + 1)) ǫ k,r(k) = i V (ǫ k,k,i) α 0 ǫ(k), EV (ǫ k+1,k + 1,r(k + 1)) EV (ǫ k,k,r(k)) α 0 ǫ(k). (40) For an arbitrary positive number s, it can be inferred from (40) that which results in s EV (ǫ s+1,s + 1,r(s + 1)) EV (ǫ 0,0,r(0)) α 0 E ǫ(k), s E ǫ(k) 1 EV (ǫ 0,0,r(0)). α 0 k=0 It can now be concluded that the series + E ǫ(k) is convergent, and therefore k=0 lim k + E ǫ(k) = 0. This completes the proof of the theorem. Now we are in a position to deal with the design problem of estimator. The following result is derived easily from Theorem 1, hence its proof is omitted here. Theorem : Let Assumption 1 and condition (5) hold. Then system (6) becomes an asymptotic state estimator of neural network (1) if there eist a set of scalar constants ϑ i > 0(i S), a set of matrices P i > 0(i S), a set of matrices X i (i S), two matrices Q > 0 and R > 0, and three sets of diagonal matrices Ω i > 0,Θ i > 0 and i > 0(i S) such that the following LMIs hold: Φ i = Ξ i Ω i Λ Θ i Σ 0 i Υ 0 0 Y T i Ω i Λ Ω i A T (i)p i Θ i Σ 0 κ 0 Q Θ i Q B T (i)p i i Υ κ i R i ,i R 0 C T (i)p i ϑ i I P i Y i P i A(i) 0 P i B(i) 0 P i C(i) P i P i k=0 < 0, (i S) (41) where Y i = P i D(i) + X i E(i), and P i,ξ i,κ 0 and κ i are defined as in Theorem 1. Furthermore, the estimate gain matri can be taken as K(i) = P 1 i X i (i S).

10 SUBMITTED 10 Remark 4: In Theorem 1 and Theorem, sufficient conditions are provided for the system (6) to be an asymptotic state estimator. Such conditions are epressed in the form of LMIs, which could be easily checked by utilizing the recently developed interior-point methods available in Matlab toolbo, and no turning of parameters will be needed. It should be mentioned that, in the past decade, LMIs have gained much attention for their computational tractability and usefulness in many areas because the so-called interior point method has been proven to be numerically very efficient for solving the LMIs. IV. A numerical eamples In this section, we present a simple eample to demonstrate the approach addressed. Eample 1: Consider a three-neuron neural network (1) with the following parameters: 1,1 = 7, 1, = 8,,1 = 3,, = 5, D(1) = diag{1.,0.7,0.8, D() = diag{1.3,0.8,0.6, A(1) = , A() = , B(1) = , B() = E(1) = E() = Take the activation functions as follows:, C(1) =, Π = , C() = f 1 (s) = g 1 (s) = h 1 (s) = tanh(0.6s), f (s) = g (s) = h (s) = tanh(0.s), f 3 (s) = g 3 (s) = h 3 (s) = 0.4tanh(s). It is easy to see that Λ 1 = Σ 1 = Υ 1 = 0, Λ = Σ = Υ = diag{ 0.3,0.1,0.. In addition, letting ( S(k,) = 0.sin ) ( ) T , , it is also easy to verify that W =. With the above parameters, by using Matlab LMI Toolbo, we solve the LMIs (41) and obtain the feasible solution as follows: P 1 = , P = , Q = X 1 = , R =, X = Ω 1 = diag{4.0414, , , Ω = diag{8.83,7.5886,1.45 Θ 1 = diag{4.9918,11.061,1.956, Θ = diag{4.9115,10.89, , 1 = diag{9.3179, ,1.4407, = diag{0.5649, ,.4857, ϑ 1 = , ϑ = From Theorem, (6) becomes an asymptotic state estimator of neural network (1) with the given parameters,,,,

11 SUBMITTED 11 and the estimate gain matrices can be taken as K(1) = , K() = V. Conclusions In this paper, we have studied the state estimation problem for a class of discrete-time neural networks with Markovian Parameters and mode-dependent mied time-delays. An asymptotic state estimator is designed to estimate the neuron states, through available output measurements, such that the dynamics of the estimation error is globally stable in the mean square. By using new Lyapunov-Krasovskii functional, we have established an LMI approach to derive the sufficient conditions guaranteeing the eistence of the asymptotic state estimators. The eplicit epression of the desired estimator has been parameterized by means of the solution to an LMI. A simulation eample has been used to illustrate the usefulness of the derived LMI-based conditions. It should be pointed out that the main results presented in this paper can be etended to stability analysis, filter design and control applications for other discrete-time delayed systems (e.g. genetic regulatory networks), see 7 and the references therein. References 1 S. Arik, IEEE Transactions on Circuits Systems -I 47(000)1089. M. P. Joy, Neural Networks 13(000) Z. Wang, D. Ho and X. Liu, IEEE Trans. Neural Networks 16(1)(005)79. 4 J. Cao, D.-S. Huang and Y. Qu, Chaos, Solitons & Fractals 3(1)(005)1. 5 J. Cao and D. Ho, Chaos, Solitons & Fractals 4(5)(005) H. Huang and J. Cao, Applied Mathematics and Computation 14(1)(003) V. Singh, Chaos, Solitons & Fractals 9()(006) Q. Song, Z. Zhao and Y. Li, Physics Letters A 335(-3)(005)13. 9 Q. Song and Z. Zhao, Chaos, Solitons & Fractals 5()(005) Z. Wang, Y. Liu and X. Liu, Physics Letters A 345(4-6)(005) H. Zhao, Neural Networks 17(004)47. 1 H. Zhao, Physics Letters A 336(005) Y. Liu, Z. Wang and X. Liu, Neural Networks 19(5)(006) F. M. Salam and J. Zhang, Adaptive neural observer with forward co-state propagation. In: Proc. International Joint Conference on Neural Networks (IJCNN 01), Washington, U.S.A., Vol.1, 001, pp R. Habtom and L. Litz, Estimation of unmeasured inputs using recurrent neural networks and the etended Kalman filter. In: Proc. International Conference on Neural Networks, Houston, U.S.A., Vol. 4, 1997, pp V. Elanayar and Y. Shin, IEEE Trans. Neural Networks 5(4)(1994) H. Huang, G. Feng and J. Cao, IEEE Trans. Neural Networks 19(8)(008) P. Tino, M. Cernansky and L. Benuskova, IEEE Trans. Neural Networks 15(1)(004)6. 19 Z. Wang, Y. Liu, L. Yu and X. Liu, Phys. Lett. A 356(4-5)(006) Y. Liu, Z. Wang and X. Liu, Physics Letters A 37()(008) Y. Liu, Z. Wang, J. Liang and X. Liu, IEEE Transactions on Systems, Man, and Cybernetics - Part B 38(5)(008)1314. Z. Wang, H. Gao, J. Cao and X. Liu, IEEE Trans. NanoBioscience 7()(008) Z. Wang, Y. Liu and X. Liu, Automatica 44(5)(008) Z. Wang, F. Yang, D. W. C. Ho, S. Swift, A. Tucker and X. Liu, IEEE Trans. NanoBioscience 7(1)(008)44. 5 H. Gao and T. Chen, IEEE Trans. Automatic Control 5()(007)38. 6 H. Gao and C. Wang, Asian Journal of Control 7()(005)81. 7 H. Gao and C. Wang, IEEE Trans. Signal Processing 5(6)(004)1631..