Delay-dependent robust exponential state estimation of Markovian jumping fuzzy Hopfield neural networks with mixed random time-varying delays

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Abstract

This paper investigates delay-dependent robust exponential state estimation of Markovian jumping fuzzy neural networks with mixed random time-varying delay. In this paper, the Takagi–Sugeno (T–S) fuzzy model representation is extended to the robust exponential state estimation of Markovian jumping Hopfield neural networks with mixed random time-varying delays. Moreover probabilistic delay satisfies a certain probability-distribution. By introducing a stochastic variable with a Bernoulli distribution, the neural networks with random time delays is transformed into one with deterministic delays and stochastic parameters. The main purpose is to estimate the neuron states, through available output measurements such that for all admissible time delays, the dynamics of the estimation error is globally exponentially stable in the mean square. Based on the Lyapunov–Krasovskii functional and stochastic analysis approach, several delay-dependent robust state estimators for such T–S fuzzy Markovian jumping Hopfield neural networks can be achieved by solving a linear matrix inequality (LMI), which can be easily facilitated by using some standard numerical packages. The unknown gain matrix is determined by solving a delay-dependent LMI. Finally some numerical examples are provided to demonstrate the effectiveness of the proposed method.

Introduction

Studying artificial neural networks (NNs) has been the central focus of intensive research activities during the last decades since these artificial networks have found wide applications in areas like associative memory, pattern classification, reconstruction of moving images, signal processing, solving optimization problems, etc., see [1]. Different models of neural networks such as Hopfield-type neural networks, cellular neural networks, Cohen-Grossberg neural networks, and bidirectional associative memory neural networks have been extensively investigated in the literature, see [2], [3], [4], [5] and the references cited therein. The dynamic behavior of NNs is known to strongly influence their wide applications in the literature. The problem of stability analysis of NNs has attracted the attention of numerous investigators. Within the electronic implementations of NNs, the finite switching speed of amplifiers and active devices as well as the inherent transmission time of neurons will incur time delays in the interaction among the neurons. Therefore stability studies of delayed neural networks (DNNs) have also received considerable investigations [6], [7], [8], [9], [10]. In the design of NNs, the issue of global exponential stability is of prime concern since it guarantees the DNNs to converge fast enough in order to attain fast and satisfactory response. Accordingly, the problem of global exponential stability analysis for DNNs has been studied by many investigators in the past years. In the case with differentiable time-varying delays, sufficient conditions were reported in [11], [12] for global exponential stability. When time-varying delays may not be differentiable, global exponential stability results can be found in Zhang et al. [13]. However, in a real system, time delay often exists in a random form, that is, some values of the time delay are very large but the probability of the delay taking such large values is very small and it may lead to a more conservative result only if the information of variation range of the time delay is considered. In addition, its probabilistic characteristic such as the Bernoulli distribution and the Poisson distribution, can also be obtained by statistical methods. Therefore, it is necessary and realizable to investigate the probability-distribution delay. Recently, the stability of discrete case of stochastic NNs with probability-distribution delay are investigated in [14], [15] and the stability of continuous case of stochastic NNs with probability-distribution delay are investigated in [16]. In addition, the problem of NNs with probability-distribution delay is investigated in [17].

The well known Takagi–Sugeno (T–S) fuzzy model [18] is recognized as an appealing and efficient tool in approximating a complex nonlinear systems. Takagi and Sugeno proposed an effective way to transform a nonlinear dynamic system to a set of linear sub-models via some fuzzy models by defining a linear input/output relationship as its consequence of individual plant rule. In T–S fuzzy models, local dynamics in different state space regions is represented by linear models. The overall fuzzy model of the system is obtained by fuzzy “mixing” of these linear models. Then the analysis of the nonlinear system is based on these linear models. Moreover, in [19], the standard T–S fuzzy model was extended to one with time delays, and some stability conditions were presented in terms of linear matrix inequalities (LMIs). Originally, Tanaka and his colleagues have provided a sufficient condition for the quadratic stability of the T–S fuzzy systems in the sense of Lyapunov in a series of papers [20], [21] by considering a Lyapunov function of the sub-fuzzy systems of the T–S fuzzy systems. Recently, problems of stability analysis for fuzzy NNs with time-varying delays have been discussed in [22], [23], [24], [25], [26].

In practice, sometimes a neural network has finite state representations (also called modes, patterns, or clusters) and modes may switch (or jump) from one to another at different times [27], [28], [29]. Recently, it has been revealed in [30] that, switching (or jumping) between different NNs modes can be governed by a Markovian chain. Specifically, the class of NNs with Markovian jump parameters has two components in the state vector. The first one which carries continuously is referred to be the continuous state of the NNs and the second one which varies discretely is referred to be the mode of the NNs. For a specific mode, the dynamics of the NNs is continuous, but the parameter jumps among different modes may be seen as discrete events, see for example [31].

On the other hand the neuron states in relatively large scale NNs are not often completely available in the network outputs. Thus, in many applications [32], [33], one often needs to estimate the neuron states through available measurements and then utilizes the estimated neuron states to achieve certain design objectives such as state feedback control. For example, in [34], a recurrent neural network was applied to model an unknown system and the neuron states of the designed neural network were then utilized by the control law. Therefore, from the point of view of control, the state estimation problem for NNs is of significance for many applications. From these investigations, it is concluded that delay effect might be the source of instability, hidden oscillations, divergence, chaos or other poor performance behavior. In practical situations, since the neuron states are not often fully available in the network outputs in many applications. This emphasizes the importance of the neuron state estimation problem, for which some partial results are available [35], [36], [37], [38], [39], [40], [41]. In [42], [43], the authors studied state estimation for Markovian jumping recurrent NNs with interval time-varying delays by constructing new Lyapunov–Krasovkii functionals and LMIs. Recently, Ahn [44] studied new delay-dependent state estimation of T–S fuzzy NNs with time-varying delays. To the best of the authors knowledge, delay-dependent exponential state estimation of Markovian jumping fuzzy NNs with random time-varying delay have not been studied in the literature and it is very important in both theories and applications.

Motivated by the above discussions, delay-dependent robust exponential state estimation of Markovian jumping fuzzy NNs with mixed random time-varying delay are considered in this paper. The information of delay-probability-distribution is introduced into the NNs model and a new method is proposed to remove the constraint on the upper bound of the delay derivative. By constructing a Lyapunov–Krasovskii functional, employing some analysis techniques, sufficient conditions are derived for the considered NNs in terms of LMIs, which can be easily calculated by Matlab LMI control Toolbox. The unknown gain matrix is determined by solving a delay-dependent LMI. Numerical examples are given to illustrate the effectiveness of the proposed method.

Notations: Throughout this paper, Rn and Rn×n denote the n-dimensional Euclidean space and the set of all n × n real matrices respectively. The superscript T denotes the transposition and the notation X  Y (similarly, X > Y), where X and Y are symmetric matrices, means that XY is positive semi-definite (similarly, positive definite). ∥ · ∥ is the Euclidean norm in Rn. Moreover, let (Ω,F,P) be a complete probability space with a filtration {Ft}t0 satisfying the usual conditions. That is the filtration contains all P-null sets and is right continuous. The notation * always denotes the symmetric block in one symmetric matrix. Sometimes, the arguments of a function or a matrix will be omitted in the analysis when no confusion can arise.

Section snippets

Problem description and preliminaries

Consider the following Hopfield neural networks with time-varying delay described byx˙(t)=-Ax(t)+W0g(x(t))+W1g(x(t-τ(t)))+W2t-σ(t)tg(x(s))ds+J,here x(·)=[x1(·),x2(·),,xn(·)]TRn is neuron state vector, x(t) = ξ(t), t  0; ξ(t) is the initial condition. A = diag{a1,  , an} is a diagonal matrix with ai > 0, i = 1,  , n, W0, W1 and W2 represent the connection weighting matrices, respectively, g(x(·))=[g1(x1(·)),,gn(xn(·))]TRn denotes the neuron activation function, and J=[J1,,Jn]TRn is a constant input

Main results

In this section, we derive a new delay-dependent criterion for global exponential stability of the system (14) using the Lyapunov functional method combining with LMI approach.

Theorem 3.1

Given scalars τ1 > 0, τ2 > 0, μ1, μ2, 0 < α0 < 1 and satisfying α0μ1 < 1, if there exist definite matrices Pi=PiT>0,Q1=Q1T>0,Q2=Q2T>0,Q3=Q3T>0,Q4=Q4T>0,R1=R1T>0,R2=R2T>0,S=ST>0, such that the following LMI holds for any i = 1, 2,  , N and l = 1, 2,  , rΘil=Σilτ1ϒ1il(τ2-τ1)ϒ1ilβτ1ϒ2ilβ(τ2-τ1)ϒ2il-2Pi+R1000-2Pi+R200-2Pi+R10-2Pi+R2<0,

Robust exponential state estimation criterion

In this section, we will derive the delay-dependent global robust exponential state estimation criterion for the following Markovian jumping fuzzy NNs with time-varying delayse˙(t)=-((A¯il+ΔA¯il(t))+K¯jiC¯il)e(t)+(W¯0il+ΔW¯0il(t))ϕ(t)+α0(W¯1il+ΔW¯1il(t))ϕ(t-τ1(t))+(1-α0)(W¯1il+ΔW¯1il(t))ϕ(t-τ2(t))+(α(t)-α0)((W¯1il+ΔW¯1il(t))ϕ(t-τ1(t))-(W¯1il+ΔW¯1il(t))ϕ(t-τ2(t)))+(W¯2il+ΔW¯2il(t))t-σ(t)tϕ(s)ds-K¯jiD¯ilψ(t),where the parametric uncertainties are assumed to be of the form[ΔA¯il(t)ΔW¯0il(t)ΔW¯1il(

Numerical examples

Example 1

Consider the error system (14) with parameters defined as:A11=3005,A12=4006,A21=5005,A22=7009,W011=0.6-0.70.50.4,W111=0.30.40.2-0.5,W211=0.70.60.80.5,W012=-0.4-0.51.20.9,W112=-0.60.3-1.3-0.8,W212=0.40.11.50.6,W021=-0.45-0.451.251.0,W121=-0.650.35-1.35-0.5,W221=0.450.151.550.65,W022=-0.8-0.60.2-0.7,W122=0.20.1-0.6-0.5,W222=0.60.50.50.7,C11=C12=D11=D12=D21=D22=0.1I,C21=0.2I,C22=0.6I,J1=J2=2cos(t)+0.03t22sin(t)-0.03t2,Γ=-887-7,W=0.5000.5,F=1.0001.0.The activation function g(x(t))=14|x(t)+1|-|x(t)-1

Conclusion

In this paper, new sufficient conditions guaranteeing the robust exponential stability (in the mean square sense) for delay-dependent robust exponential state estimation of Markovian jumping fuzzy NNs with mixed random time-varying delay have been proposed. Stability condition for the Markovian jumping fuzzy NNs have been obtained in the form of LMIs. Probability distribution of the time-varying delays is introduced into the stability criteria and this new method removes the constraint that the

Acknowledgement

The authors sincerely thank the Associate Editor and anonymous reviewer for their constructive comments and fruitful suggestions to improve the quality of the manuscript. The work of the first and second author was supported by UGC-SAP (DRS-II) Grant No. F.510/2/DRS/2009(SAP-I). The work of the third author was supported by CSIR-SRF under Grant No: 09/715(0013)/2009-EMR-I.

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