Non-fragile exponential state estimation for continuous-time fuzzy stochastic neural networks with time-varying delays 1

Abstract

This paper investigates non-fragile exponential state estimation problems for continuous-time fuzzy stochastic neural networks with time-varying delays. The Takagi-Sugeno (T-S) fuzzy model representation is extended to the exponential state estimator design for fuzzy stochastic neural networks with time-varying delays. The neuron activation function and the nonlinear measurement equation are assumed to be satisfy sector-bounded conditions and standard Lipschitz conditions. For these two conditions, delay-dependent sufficient conditions are presented to guarantee the existence of the desired state estimators for fuzzy stochastic neural networks. Finally, two numerical examples are given to demonstrate that the proposed approaches are effective and that the sector-bounded conditions are weaker than the standard Lipschitz conditions.

Keywords

Non-fragile state estimation Takagi-Sugeno stochastic neural networks exponential stability

1 Introduction

Takagi-Sugeno (T-S) fuzzy models are often used to represent complex nonlinear systems by means of fuzzy sets and fuzzy reasoning applied to a set of linear sub-models [24]. The dynamics of nonlinear system are captured by a set of fuzzy rules which characterize local correlation in the state space. During the last decade, the stability analysis and controller synthesis problem for systems in T-S fuzzy model have been studied extensively and numerous methods have been proposed, see [1 –6] and the references therein.

In order to model a system realistically, a degree of randomness must be incorporated into the model. So the stability problems for stochastic neural networks have been studied [10 , 15]. Recently, the Takagi-Sugeno fuzzy model approach has been used to describe neural networks with time delay. And the problem of stability analysis for T-S fuzzy neural networks has been extensively reported [16, 19] and the references therein. Especially, stability results for T-S fuzzy stochastic neural networks are also established [14].

Recently, the state estimation and filtering problem for neural networks has been dealt with in [7 , 27]. Then based on T-S fuzzy model, the state estimation and filtering problem for Takagi-Sugeno fuzzy delayed Hopfield neural networks also have been developed. For example, a delay-dependent state estimator for T-S fuzzy delayed Hopfield neural networks is presented [1, 4]. Based on Lyapunov-Krasovskii stability approach, a new H_∞ state estimator for Takagi-Sugeno fuzzy delayed Hopfield neural networks is also considered [2]. And a new passive and exponential filter for Takagi-Sugeno fuzzy Hopfield neural networks, with time delay and external disturbance is proposed [3]. The delay-dependent robust asymptotic state estimation of fuzzy neural networks with mixed interval time-varying delay is investigated [5]. Meanwhile, the state estimation for fuzzy neural networks with Markovian jumping parameters has been addressed [6, 17]. The problem of state estimation for Markovian jumping fuzzy cellular neural networks using sampled-data with mode-dependent probabilistic time-varying delays is investigated, which the information of the delayed states can be taken into full consideration [23]. The problem of state estimation for T-S fuzzy neural network with discrete and distributed time-varying interval delays has been studied [25].

On the other hand, all the above filter or state estimation problems for neural networks are based on an implicit assumption that estimator can be implemented exactly [1 , 27]. However, the designed observer should be able to tolerate some uncertainty in its coefficients due to the fact that the uncertainty is not avoided, which is caused by many reasons, for example, the imprecision inherent in analog systems and the need for additional tuning of parameters in the final controller implementation [21]. So the non-fragile state observer for neural networks needs to considered. The non-fragile synchronization of neural networks with time-varying delay and randomly occurring controller gain fluctuation is considered [11]. Recently, the non-fragile observer based design for neural networks with mixed time-varying delays and Markovian jumping parameters is investigated [26]. Meanwhile, the mixed H_∞ and passivity based state estimation for a class of discrete-time fuzzy neural networks with the estimator gain change is considered [29]. However, it is should be pointed that the stochastic disturbance is not taken into account when dealing with the state estimation problem for fuzzy neural networks [11 , 29].

To the best of our knowledge, the non-fragile state estimation problems for Takagi-Sugeno fuzzy stochastic neural networks has not been investigated in the existing literature. Motivated by this consideration, we consider a class of Takagi-Sugeno fuzzy stochastic neural networks with time-varying delays in this paper. Both sector-bounded conditions and standard Lipschitz conditions are considered for the neuron activation function and the nonlinear measurement equation. By employing a suitable new Lyapunov-Krasovskii functional and the new integral inequality in the stochastic setting [28], the non-fragile state estimators can be achieved in terms of LMIs. The designed estimators ensure the mean-square exponential stability of the resulting error system. The desired the non-fragile estimator can be constructed through a convex optimization problem that can be efficiently solved by using standard numerical algorithms [8]. Finally, two numerical examples are given to show the effectiveness of the proposed method.

Notations: $ℝ^{n}$ and $ℝ^{m \times n}$ denote the n-dimensional Euclidean space and the set of real m × n matrix, respectively. For a real symmetric matrix X, X > 0 (X ≥ 0) means that X is positive definite (positive semi-definite). The superscript T denotes the transpose of a matrix or a vector. The symbol ∗ in a matrix stands for the transposed elements in the symmetric positions. λ_min (·) means the minimal eigenvalue of a matrix. E {·} is the expectation operator. L₂ [0, ∞) is the space of square-integrable vector functions over [0, ∞). | · | refers to the Euclidean norm, and || · || stands for the usual L₂ [0, ∞) norm.

2 Problem description

The Hopfield neural network is described as follows:

$\begin{matrix} {\dot{u}}_{i} (t) = - a_{i} u_{i} (t) + \sum_{j = 1}^{n} w_{ij}^{1} f_{j} (u_{j} (t - τ (t))) \\ + \sum_{j = 1}^{n} w_{ij}^{0} f_{j} (u_{j} (t)) + I_{i}, i = 1, 2, \dots, n, \end{matrix}$ (1) or equivalently in the vector form

$\begin{matrix} \dot{u} (t) = - Au (t) + W_{1} f (u (t - τ (t))) \\ + W_{0} f (u (t)) + I \end{matrix}$ (2) where $u (t) = [u_{1} (t), u_{2} (t), \dots, u_{n} (t)]^{T} \in ℝ^{n}$ denotes the state vector associated with n neurons. The matrix A = diag {a₁, a₂, …, a_n} is a diagonal matrix with positive entries a_i > 0, i = 1, 2, ⋯ , n. And $W_{0} = (w_{ij}^{0})_{n \times n}$ and $W_{1} = (w_{ij}^{1})_{n \times n}$ are the connection weights matrices representing the weighting coefficients of the neurons. f (u (t)) =[f₁ (u₁ (t)), $f_{2} (u_{2} (t)), \dots, f_{n} (u_{n} (t))]^{T} \in ℝ^{n}$ denotes the neuron activation function with f (0) =0. I = [I₁, I₂, …, I_n] ^T is a constant external input vector. The delay τ (t) satisfies 0 ⩽ τ (t) ⩽ τ and the delay derivative μ satisfies $| \dot{τ} (t) | ⩽ μ < 1$ .

Let $u^{*} = [u_{1}^{*}, u_{2}^{*}, \dots, u_{n}^{*}]^{T}$ is an equilibrium point of the system (1), and we choose the coordinate transformation x (t) = u (t) - u^∗, then system (2) is changed into the following system:

$\begin{matrix} \dot{x} (t) = - Ax (t) + W_{1} σ (x (t - τ (t)) + W_{0} σ (x (t)) \end{matrix}$ (3) where $x (t) = [x_{1} (t), x_{2} (t), \dots, x_{n} (t)]^{T} \in ℝ^{n}$ is the state vector of the transformed system, σ (x (t)) = [σ₁ (x₁ (t)) , σ₂ (x₂ (t)) , ⋯ , σ_n (x_n (t))] ^T, and σ_i (x (t)) = $f_{i} (x_{i} (t) + x_{i}^{*}) - f_{i} (x_{i}^{*})$ with σ_i (x (0)) = 0, i = 1, 2, ⋯ , n.

(H₁) The neuron activation function σ (·) is bounded and satisfies the following Lipschitz condition:

$| σ (x) - σ (y) | \leq | L (x - y) |, \forall x \neq y \in ℝ$ (4) where $L = diag {L_{1}, L_{2}, \dots, L_{n}} \in ℝ^{n}$ is a known constant diagonal matrix.

The well-known Takagi-Sugeno (T-S) fuzzy model [24] is a popular and convenient tool in functional approximations. Recently, the T-S fuzzy models have been extended to the stochastic neural networks [14]. In this paper, we will consider the following stochastic fuzzy Hopfield neural network with time-varying delays, which is represented by a T-S fuzzy model composed of a set of fuzzy implications and each implications is expressed as a linear system model. The ith rule of this T-S fuzzy model is given as follows:

Fuzzy rule i: IF θ₁ (t) is $η_{1}^{i}$ and … and θ_p (t) is $η_{p}^{i}$ THEN $\begin{matrix} dx (t) = [- A_{i} x (t) + W_{0 i} σ (x (t)) \\ + W_{1 i} σ (x (t - τ (t)))] dt \\ + [C_{i} x (t) + D_{i} x (t - τ (t)) \\ + W_{2 i} σ (x (t)) + W_{3 i} σ (x (t - τ (t)))] dw (t) \end{matrix}$ (5) $y (t) = E_{i} x (t) + φ (t, x (t))$ (6) where r is the number of IF-THEN rules, $η_{1}^{i}, \dots, η_{p}^{i}$ is the fuzzy set and θ (t) = [θ₁ (t) , θ₂ (t) , …, θ_p (t)] ^T is the premise variable vector. w (t) is a zero-mean real scalar Wiener process on a probability space (Ω,ℱ,𝒫) relative to an increasing family (ℱ_t) _t>0 of σ-algebras ℱ_t ⊂ℱ generated by w (t). The stochastic process {w (t)} is independent, which is assumed to satisfy E {dw (t)} =0, E {dw² (t)} = dt. The matrices A_i, W_0i, W_1i, C_i, D_i, W_2i, W_3i and E_i (i = 1, 2, …, r) are known constant matrices. y (t) is the measurement equation of neural network (6). And φ (t, x (t)) is the neuron-dependent nonlinear disturbance on the network outputs.

(H₂) The nonlinear disturbance φ (t, x (t)) is bounded and satisfies the following Lipschitz condition:

$| φ (x) - φ (y) | \leq | G (x - y) |, \forall x \neq y \in ℝ$ (7) where $G = diag {G_{1}, G_{2}, \dots, G_{n}} \in ℝ^{n}$ is a known constant diagonal matrix.

For comparison, in this paper, the following assumptions are also given.

(H₃) The neuron activation function σ (·) is assumed to satisfy the following sector-bounded condition:

$\begin{matrix} [σ (x) - σ (y) - Σ_{1} (x - y)]^{T} \\ \times [σ (x) - σ (y) - Σ_{2} (x - y)] \leq 0, \forall x, y \in ℝ \end{matrix}$ (8) where $Σ_{1}, Σ_{2} \in ℝ^{n}$ are known constant matrices.

(H₄) The nonlinear disturbance φ (t, x (t)) is assumed to satisfy the following sector-bounded condition:

$\begin{matrix} [φ (x) - φ (y) - φ_{1} (x - y)]^{T} \\ \times [φ (x) - φ (y) - φ_{2} (x - y)] \leq 0, \forall x, y \in ℝ \end{matrix}$ (9) where $φ_{1}, φ_{2} \in ℝ^{n}$ are known constant matrices.

Remark 1. In this paper, the above assumptions (H₁) and (H₂) are called Lipschitz conditions. Recently, [9 , 27] present the sector-bounded neuron activation function (H₃) and (H₄), which cover the general Lipschitz nonlinearities [1–6 , 25].

Based on (H₁, H₂) and (H₃, H₄), in the next section, we will give state estimators design method for these two conditions. It is assumed that the premise variables do not depend on the variables w (t) explicitly. The defuzzified output of system (5)-(6) is inferred as follows: $\begin{matrix} dx (t) = \sum_{i = 1}^{r} h_{i} (θ (t)) {[- A_{i} x (t) + W_{0 i} σ (x (t)) \\ + W_{1 i} σ (x (t - τ (t)))] dt \\ + [C_{i} x (t) + D_{i} x (t - τ (t)) + W_{2 i} σ (x (t)) \\ + W_{3 i} σ (x (t - τ (t)))] dw (t)} \end{matrix}$ (10) $y (t) = \sum_{i = 1}^{r} h_{i} (θ (t)) {E_{i} x (t) + φ (t, x (t))}$ (11) where $h_{i} (θ (t)) = \frac{ν_{i} (θ (t))}{\sum_{j = 1}^{r} h_{j} (θ (t))}$ , $ν_{i} (θ (t)) = \prod_{j = 1}^{p} η_{j}^{i}$ (θ_j (t)) in which $η_{j}^{i} (θ_{j} (t))$ is the grade of membership of θ_j (t) in $η_{j}^{i}$ . According to the theory of fuzzy sets, we have $ν_{i} (θ (t)) ⩾ 0, (i = 1, 2, \dots, r), \sum_{i = 1}^{r} ν_{i} (θ (t))$ >0 for all t. Therefore, it implied that $h_{i} (θ (t)) ⩾ 0, (i = 1, 2, \dots, r), \sum_{i = 1}^{r} h_{i} (θ (t)) = 1$ for all t. In order to estimate the neuron state of (10)-(11), we construct the following non-fragile state estimator:

Plant rule i: IF θ₁ (t) is $η_{1}^{i}$ and … and θ_p (t) is $η_{p}^{i}$ THEN

$\begin{matrix} d \hat{x} (t) = [- A_{i} \hat{x} (t) + W_{0 i} σ (\hat{x} (t)) \\ + W_{1 i} σ (\hat{x} (t - τ (t)))] dt \\ + (K_{i} + Δ K_{i}) [y (t) - E_{i} \hat{x} (t) - φ (t, \hat{x} (t))] dt \\ + [C_{i} \hat{x} (t) + D_{i} \hat{x} (t - τ (t)) + W_{2 i} σ (\hat{x} (t)) \\ + W_{3 i} σ (\hat{x} (t - τ (t)))] dw (t) \end{matrix}$ (12) where K_i is the gain matrix of the state estimator to be designed. And ΔK_i represents estimator gain variations which is assumed to be:

$Δ K_{i} = M_{i} F (t) N_{i}$ (13) where M_i and N_i are known real constant matrices with appropriate dimensions, F_i (t) is an unknown matrix satisfying $F_{i}^{T} (t) F_{i} (t) ⩽ I$ .

Using a standard fuzzy inference method, the defuzzified output system (12) is inferred as follows:

$\begin{matrix} d \hat{x} (t) = \sum_{i = 1}^{r} h_{i} (θ (t)) {[- A_{i} \hat{x} (t) \\ + W_{0 i} σ (\hat{x} (t)) + W_{1 i} σ (\hat{x} (t - τ (t)))] dt \\ + (K_{i} + Δ K_{i}) [y (t) - E_{i} \hat{x} (t) - φ (t, \hat{x} (t))] dt \\ + [C_{i} \hat{x} (t) + D_{i} \hat{x} (t - τ (t)) + W_{2 i} σ (\hat{x} (t)) \\ + W_{3 i} σ (\hat{x} (t - τ (t)))] dw (t)} \end{matrix}$ (14)

Let the estimation error be $e (t) = x (t) - \hat{x} (t)$ , then we can obtain the following estimation error system

$\begin{matrix} de (t) = \sum_{i = 1}^{r} h_{i} (θ (t)) {[- A_{i} e (t) \\ - (K_{i} + Δ K_{i}) E_{i} e (t) + W_{0 i} f_{1} (t) \\ + W_{1 i} f_{1} (t - τ (t)) - (K_{i} + Δ K_{i}) f_{2} (t)] dt \\ + [C_{i} e (t) + D_{i} e (t - τ (t)) \\ + W_{2 i} f_{1} (t) + W_{3 i} f_{1} (t - τ (t))] dw (t)} \end{matrix}$ (15) where

$\begin{matrix} f_{1} (t) = σ (x (t)) - σ (\hat{x} (t)) \\ f_{2} (t) = φ (t, x (t)) - φ (t, \hat{x} (t)) \end{matrix}$ (16)

Remark 2. In general, the T-S fuzzy logic is used for linearizing the nonlinear terms. And based on the properties of linear system theory, the nonlinear systems could be studied. Due to the existence of the nonlinear functions f₁ (t) , f₂ (t), the systems (15) described with the fuzzy rules is still nonlinear system. In this paper, similar to the nonlinear neuron activation functions in [1–6 , 25], after liberalization manipulation, the nonlinear functions of fuzzy neural networks could be dealt with accordingly.

The following definition and lemma will be used in establishing our main results.

Definition 1. [14] The trivial solution of the stochastic fuzzy Hopfield neural networks (10) is said to exponentially stable in the mean square sense, if there exist positive scalars α and β, such that

$E {∥ x (t, ψ) ∥^{2}} ⩽ α e^{- β t} E {sup_{- τ \leq s \leq 0} ∥ ψ (s) ∥^{2}}$

Lemma 1.[8] For given matricesM, NandF (t) withF^T (t) F (t) ≤ Iand scalarɛ > 0, the following inequality holds:

$MF (t) N + (MF (t) N)^{T} \leq ɛ {MM}^{T} + ɛ^{- 1} N^{T} N$

3 Main results

In this section, two non-fragile delay-dependent state estimators for fuzzy stochastic neural networks with time-varying delays (10)-(11) are designed. Firstly, based on (H₃, H₄), the following theorem presents a sufficient condition to ensure the existence of the desired estimator in terms of a LMI.

Theorem 1.Consider (H₃, H₄). Given scalarτ > 0 andμ ⩾ 0, the estimation error system (15) is exponentially mean-square stable, if there exist matricesP > 0,Q > 0, R > 0, Z > 0, T_i, scalarsλ_1i, λ_2i, λ_3iandɛ_i, (i = 1, 2, …, r) such that the following LMI holds:

$[\begin{matrix} Γ_{1 i} & Γ_{2 i} & Γ_{3 i} & Γ_{4 i} \\ * & - P & 0 & 0 \\ * & * & - 2 P + Z & - τ {PM}_{i} \\ * & * & * & - ɛ_{i} I \end{matrix}] < 0$ (17) where

$\begin{matrix} Γ_{1 i} = [\begin{matrix} Γ_{1 i}^{11} & 0 & Z & Γ_{1 i}^{14} & {PW}_{1 i} & Γ_{1 i}^{16} \\ * & Γ_{1 i}^{22} & 0 & 0 & - λ_{2 i} F_{2} & 0 \\ * & * & Γ_{1 i}^{33} & 0 & 0 & 0 \\ * & * & * & - λ_{1 i} I & 0 & 0 \\ * & * & * & * & - λ_{2 i} I & 0 \\ * & * & * & * & * & Γ_{1 i}^{66} \end{matrix}] \\ Γ_{1 i}^{11} = - {PA}_{i} - A_{i}^{T} P - T_{i} E_{i} - E_{i}^{T} T_{i}^{T} + Q + R \\ - Z - λ_{1 i} F_{1} - λ_{3 i} J_{1} + ɛ_{i} E_{i}^{T} N_{i}^{T} N_{i} E_{i} \\ Γ_{1 i}^{14} = {PW}_{0 i} - λ_{1 i} F_{2} \\ Γ_{1 i}^{16} = - T_{i} - λ_{3 i} J_{2} + ɛ_{i} E_{i}^{T} N_{i}^{T} N_{i} \\ Γ_{1 i}^{22} = - (1 - μ) Q - λ_{2 i} F_{1}, Γ_{1 i}^{33} = - R - Z \\ Γ_{1 i}^{66} = - λ_{3 i} I + ɛ_{i} N_{i}^{T} N_{i} \\ Γ_{2 i} = [{PC}_{i} {PD}_{i} 0 {PW}_{2 i} {PW}_{3 i} 0]^{T} \\ Γ_{3 i} = τ [- ({PA}_{i} + T_{i} E_{i}) 0 0 {PW}_{0 i} {PW}_{1 i} - T_{i}]^{T} \\ Γ_{4 i} = [- {PM}_{i} 0 0 0 0 0]^{T} \end{matrix}$

And the state estimator gain matrix K_i can be designed as K_i = P^-1T_i, i = 1, 2, …, r.

Proof. For simplicity, let $\begin{matrix} φ (t) = \sum_{i = 1}^{r} h_{i} (θ (t)) [- A_{i} e (t) \\ - (K_{i} + Δ K_{i}) E_{i} e (t) + W_{0 i} f_{1} (t) \\ + W_{1 i} f_{1} (t - τ (t)) - (K_{i} + Δ K_{i}) f_{2} (t)] \end{matrix}$ (18) $\begin{matrix} g (t) = \sum_{i = 1}^{r} h_{i} (θ (t)) [C_{i} e (t) + D_{i} e (t - τ (t)) \\ + W_{2 i} f_{1} (t) + W_{3 i} f_{1} (t - τ (t))] \end{matrix}$ (19)

Then the Equation. (15) can be rewritten as

$de (t) = φ (t) dt + g (t) dw (t)$ (20)

Choose the Lyapunov functional candidate as

$\begin{matrix} V_{1} (e (t), t) = e^{T} (t) Pe (t) + \int_{t - τ}^{t} e^{T} (s) Re (s) ds \\ + \int_{t - τ (t)}^{t} e^{T} (s) Qe (s) ds \\ + τ \int_{- τ}^{0} \int_{t + θ}^{t} φ^{T} (s) Z φ (s) dsd θ \end{matrix}$ (21)

By Itô’s differential rule, the stochastic differential dV₁ (e (t) , t) along the trajectories of system (14) is given by

$\begin{matrix} {dV}_{1} (e (t), t) = ℒ V_{1} (e (t), t) dt \\ + 2 e^{T} (t) Pg (t) dw (t) \end{matrix}$ (22) where

$ℒ V_{1} (e (t), t) ⩽ 2 \sum_{i = 1}^{r} h_{i} (θ (t)) e^{T} (t) P [- A_{i} e (t)$

$\begin{matrix} - (K_{i} + Δ K_{i}) E_{i} e (t) + W_{0 i} f_{1} (t) \\ + W_{1 i} f_{1} (t - τ (t)) - (K_{i} + Δ K_{i}) f_{2} (t)] \\ - (1 - μ) e^{T} (t - τ (t)) Qe (t - τ (t)) \\ + e^{T} (t) Qe (t) + g^{T} (t) Pg (t) + e^{T} (t) Re (t) \\ - e^{T} (t - τ) Re (t - τ) + τ^{2} φ^{T} (t) Z φ (t) \\ - τ \int_{t - τ}^{t} φ^{T} (s) Z φ (s) ds \end{matrix}$ (23)

By Lemma 1 in [28], we have

$\begin{matrix} - τ \int_{t - τ}^{t} φ^{T} (s) Z φ (s) ds \\ ⩽ {[\begin{matrix} e (t) \\ e (t - τ) \end{matrix}]}^{T} [\begin{matrix} - Z & Z \\ Z & - Z \end{matrix}] [\begin{matrix} e (t) \\ e (t - τ) \end{matrix}] \\ + 2 {[\begin{matrix} e (t) \\ e (t - τ) \end{matrix}]}^{T} [\begin{matrix} Z \\ - Z \end{matrix}] \int_{t - τ}^{t} g (s) dw (s) \end{matrix}$ (24)

From (8), (9) and (16), we have

$\begin{matrix} [f_{1} (t) - Σ_{1} e (t)]^{T} [f_{1} (t) - Σ_{2} e (t)] ⩽ 0 \\ [f_{1} (t - τ (t)) - Σ_{1} e (t - τ (t))]^{T} \\ \times [f_{1} (t - τ (t)) - Σ_{2} e (t - τ (t))] ⩽ 0 \\ [f_{2} (t) - φ_{1} e (t)]^{T} [f_{2} (t) - φ_{2} e (t)] ⩽ 0 \end{matrix}$

It is clear that for any scalars λ_1i > 0, λ_2i > 0 and λ_3i > 0, we have $- λ_{1 i} {[\begin{matrix} e (t) \\ f_{1} (t) \end{matrix}]}^{T} [\begin{matrix} F_{1} & F_{2} \\ * & I \end{matrix}] [\begin{matrix} e (t) \\ f_{1} (t) \end{matrix}] ⩾ 0$ (25) $- λ_{2 i} {[\begin{matrix} e (t - τ (t)) \\ f_{1} (t - τ (t)) \end{matrix}]}^{T} [\begin{matrix} F_{1} & F_{2} \\ * & I \end{matrix}] [\begin{matrix} e (t - τ (t)) \\ f_{1} (t - τ (t)) \end{matrix}] ⩾ 0$ (26) $- λ_{3 i} {[\begin{matrix} e (t) \\ f_{2} (t) \end{matrix}]}^{T} [\begin{matrix} J_{1} & J_{2} \\ * & I \end{matrix}] [\begin{matrix} e (t) \\ f_{2} (t) \end{matrix}] ⩾ 0$ (27) where

$\begin{matrix} F_{1} = \frac{Σ_{1}^{T} Σ_{2} + Σ_{2}^{T} Σ_{1}}{2}, F_{2} = - \frac{Σ_{1}^{T} + Σ_{2}^{T}}{2} \\ J_{1} = \frac{φ_{1}^{T} φ_{2} + φ_{2}^{T} φ_{1}}{2}, J_{2} = - \frac{φ_{1}^{T} + φ_{2}^{T}}{2} \end{matrix}$

Then, by adding the terms on the left sides of (25)-(27) to the right side of (23), and considering (24), we can obtain

$ℒ V_{1} (e (t), t) ⩽ \sum_{i = 1}^{r} h_{i} (θ (t)) {ξ^{T} (t) Π_{i} ξ (t) + ν (t)}$ (28) where $\begin{matrix} Π_{i} = Π_{1 i} + Π_{2 i} P^{- 1} (Π_{2 i})^{T} + Π_{3 i} Z^{- 1} (Π_{3 i})^{T} \\ Π_{1 i} = [\begin{matrix} Π_{1 i}^{11} & 0 & Z & Π_{1 i}^{14} & {PW}_{1 i} & Π_{1 i}^{16} \\ * & Π_{1 i}^{22} & 0 & 0 & - λ_{2 i} F_{2} & 0 \\ * & * & Π_{1 i}^{33} & 0 & 0 & 0 \\ * & * & * & - λ_{1 i} I & 0 & 0 \\ * & * & * & * & - λ_{2 i} I & 0 \\ * & * & * & * & * & - λ_{3 i} I \end{matrix}] \\ Π_{1 i}^{11} = - P (A_{i} + (K_{i} + Δ K_{i}) E_{i}) + Q + R \\ - Z - (A_{i} + (K_{i} + Δ K_{i}) E_{i})^{T} P \\ - λ_{1 i} F_{1} - λ_{3 i} J_{1} \\ Π_{1 i}^{14} = {PW}_{0 i} - λ_{1 i} F_{2} \\ Π_{1 i}^{16} = - P (K_{i} + Δ K_{i}) - λ_{3 i} J_{2} \\ Π_{1 i}^{22} = - (1 - μ) Q - λ_{2 i} F_{1}, Π_{1 i}^{33} = - R - Z \\ Π_{2 i} = [{PC}_{i} {PD}_{i} 0 {PW}_{2 i} {PW}_{3 i} 0]^{T} \\ Π_{3 i} = [- τ Z (A_{i} + (K_{i} + Δ K_{i}) E_{i}) 0 0 \end{matrix}$ $\begin{matrix} τ {ZW}_{0 i} τ {ZW}_{1 i} - τ Z (K_{i} + Δ K_{i})]^{T} \\ ξ (t) = [e^{T} (t) e^{T} (t - τ (t)) e^{T} (t - τ) \\ f_{1}^{T} (t) f_{1}^{T} (t - τ (t)) f_{2}^{T} (t)]^{T} \\ ν (t) = 2 {[\begin{matrix} e (t) \\ e (t - τ) \end{matrix}]}^{T} [\begin{matrix} Z \\ - Z \end{matrix}] \int_{t - τ}^{t} g (s) dw (s) \end{matrix}$

In order to show that the estimation error system (14) is exponentially mean-square stable, we just need the condition Π_i < 0, by Schur complement, which is equivalent to the following LMI:

$[\begin{matrix} Π_{1 i} & Π_{2 i} & Π_{3 i} \\ * & - P & 0 \\ * & * & - Z \end{matrix}] < 0$ (29)

Then the above LMI could be written as:

$\begin{matrix} [\begin{matrix} {\tilde{Π}}_{1 i} & Π_{2 i} & {\tilde{Π}}_{3 i} \\ * & - P & 0 \\ * & * & - Z \end{matrix}] + Π_{4 i}^{T} F_{i} (t) Π_{5 i} \\ + Π_{5 i}^{T} F_{i}^{T} (t) Π_{4 i} < 0 \end{matrix}$ (30) where

$\begin{matrix} {\tilde{Π}}_{1 i} = [\begin{matrix} {\tilde{Π}}_{1 i}^{11} & 0 & Z & Π_{1 i}^{14} & {PW}_{1 i} & {\tilde{Π}}_{1 i}^{16} \\ * & Π_{1 i}^{22} & 0 & 0 & - λ_{2 i} F_{2} & 0 \\ * & * & Π_{1 i}^{33} & 0 & 0 & 0 \\ * & * & * & - λ_{1 i} I & 0 & 0 \\ * & * & * & * & - λ_{2 i} I & 0 \\ * & * & * & * & * & - λ_{3 i} I \end{matrix}] \\ {\tilde{Π}}_{1 i}^{11} = - P (A_{i} + K_{i} E_{i}) - (A_{i} + K_{i} E_{i})^{T} P \\ + Q + R - Z - λ_{1 i} F_{1} - λ_{3 i} J_{1} \\ {\tilde{Π}}_{1 i}^{16} = - {PK}_{i} - λ_{3 i} J_{2} \\ {\tilde{Π}}_{3 i} = [- τ Z (A_{i} + K_{i} E_{i}) 0 0 τ {ZW}_{0 i} \\ τ {ZW}_{1 i} - τ {ZK}_{i}]^{T} \\ Π_{4 i} = [- M_{i}^{T} P 0 0 0 0 0 0 - τ M_{i}^{T} Z] \\ Π_{5 i} = [N_{i} E_{i} 0 0 0 0 N_{i} 0 0] \end{matrix}$

By Lemma 1, there exist ɛ_i > 0 such that the following inequality is satisfied

$[\begin{matrix} {\hat{Π}}_{1 i} & Π_{2 i} & {\tilde{Π}}_{3 i} & {\tilde{Π}}_{4 i} \\ * & - P & 0 & 0 \\ * & * & - Z & - τ {ZM}_{i} \\ * & * & * & - ɛ_{i} I \end{matrix}] < 0$ (31) where

$\begin{matrix} {\hat{Π}}_{1 i} = [\begin{matrix} {\hat{Π}}_{1 i}^{11} & 0 & Z & Π_{1 i}^{14} & {PW}_{1 i} & {\hat{Π}}_{1 i}^{16} \\ * & Π_{1 i}^{22} & 0 & 0 & - λ_{2 i} F_{2} & 0 \\ * & * & Π_{1 i}^{33} & 0 & 0 & 0 \\ * & * & * & - λ_{1 i} I & 0 & 0 \\ * & * & * & * & - λ_{2 i} I & 0 \\ * & * & * & * & * & {\hat{Π}}_{1 i}^{66} \end{matrix}] \\ {\hat{Π}}_{1 i}^{11} = - P (A_{i} + K_{i} E_{i}) - (A_{i} + K_{i} E_{i})^{T} P + Q \\ + R - Z - λ_{1 i} F_{1} - λ_{3 i} J_{1} + ɛ_{i} E_{i}^{T} N_{i}^{T} N_{i} E_{i} \\ {\hat{Π}}_{1 i}^{16} = - {PK}_{i} - λ_{3 i} J_{2} + ɛ_{i} E_{i}^{T} N_{i}^{T} N_{i} \\ {\hat{Π}}_{1 i}^{66} = - λ_{3 i} I + ɛ_{i} N_{i}^{T} N_{i} \\ {\tilde{Π}}_{4 i} = [- M_{i}^{T} P 0 0 0 0 0] \end{matrix}$

For any symmetric matrices P > 0 and Z > 0, we have PZ^-1P - 2P + Z = (P - Z) ^TZ^-1 (P - Z) ⩾0.

Then we get

$- {PZ}^{- 1} P ⩽ - 2 P + Z$ (32)

Pre- and postmultiplying the aforementioned inequality (31) by diag {I, I, I, I, I, I, I, PZ^-1, I} and its transport matrix, respectively. By setting T_i = PK_i, it can be easily seen that the resulting LMI is equivalent to LMI (17). Taking expectation for both sides of (28), we obtain

$\begin{matrix} E {{dV}_{1} (e (t), t)} ⩽ E {ℒ V_{1} (e (t), t)} \\ < - a (| | x (t) | |^{2} + | | x (t - τ (t)) | |^{2}) < 0 \end{matrix}$ (33) where a = min {λ_min (- Π_i)} >0, then using a similar method, which is used in [14], we can prove that the estimation error system (15) is exponentially stable. This completes the proof.

Next, based on (H₁, H₂), the following theorem presents a sufficient condition to ensure the existence of the desired estimator in terms of a LMI.

Theorem 2.Consider (H₁, H₂). Given scalarτ > 0, μ ⩾ 0, the estimation error system (15) is exponentially mean-square stable, if there exist matricesP > 0,Q > 0, R > 0, Z > 0, T_i, scalarsλ_1i, λ_2i, λ_3iandɛ_i, (i = 1, 2, …, r), such that the following LMI holds:

$[\begin{matrix} {\hat{Γ}}_{1 i} & Γ_{2 i} & Γ_{3 i} & Γ_{4 i} \\ * & - P & 0 & 0 \\ * & * & - 2 P + Z & - τ {PM}_{i} \\ * & * & * & - ɛ_{i} I \end{matrix}] < 0$ (34) where

$\begin{matrix} {\hat{Γ}}_{1 i} = [\begin{matrix} {\hat{Γ}}_{1 i}^{11} & 0 & Z & {PW}_{0 i} & {PW}_{1 i} & {\hat{Γ}}_{1 i}^{16} \\ * & {\hat{Γ}}_{1 i}^{22} & 0 & 0 & 0 & 0 \\ * & * & {\hat{Γ}}_{1 i}^{33} & 0 & 0 & 0 \\ * & * & * & - λ_{1 i} I & 0 & 0 \\ * & * & * & * & - λ_{2 i} I & 0 \\ * & * & * & * & * & {\hat{Γ}}_{1 i}^{66} \end{matrix}] \\ {\hat{Γ}}_{1 i}^{11} = - {PA}_{i} - A_{i}^{T} P - T_{i} E_{i} - E_{i}^{T} T_{i}^{T} + Q + R \\ - Z + λ_{1 i} L^{T} L + λ_{3 i} G^{T} G + ɛ_{i} E_{i}^{T} N_{i}^{T} N_{i} E_{i} \\ {\hat{Γ}}_{1 i}^{16} = - T_{i} + ɛ_{i} E_{i}^{T} N_{i}^{T} N_{i} \\ {\hat{Γ}}_{1 i}^{22} = - (1 - μ) Q + λ_{2 i} L^{T} L \\ {\hat{Γ}}_{1 i}^{33} = - R - Z, {\hat{Γ}}_{1 i}^{66} = - λ_{3 i} I + ɛ_{i} N_{i}^{T} N_{i} \end{matrix}$ where Γ_2i, Γ_3i and Γ_4i are defined in Theorem 1. Moreover, the state estimator gain matrix K_i can be designed as K_i = P^-1T_i, i = 1, 2, …, r.

Proof. Choose the same Lyapunov-Krasovskii functional candidate V₁ (e (t) , t) as in (21), and from (4), (7) and (16), it is clear that for any scalars λ_1i > 0, λ_2i > 0 and λ_3i > 0, we have $- λ_{1 i} [f_{1}^{T} (t) f_{1} (t) - e^{T} (t) L^{T} Le (t)] ⩾ 0$ (35) $\begin{matrix} - λ_{2 i} [f_{1}^{T} (t - τ (t)) f_{1} (t - τ (t)) \\ - e^{T} (t - τ (t)) L^{T} Le (t - τ (t))] ⩾ 0 \end{matrix}$ (36) $- λ_{3 i} [f_{2}^{T} (t) f_{2} (t) - e^{T} (t) G^{T} Ge (t)] ⩾ 0$ (37)

In the proof of Theorem 1, replacing (25)-(27) with (35)-(37), then follows directly from the similar way of proof of Theorem 1, hence it is omitted. This completes the proof.

Remark 3. In [7 , 27], the state estimation problem for stochastic neural networks have been investigated. However, the above stochastic neural networks [7 , 27] are all discrete-time neural networks. For continuous-time stochastic neural networks, the state estimation problem is not studied.

If there are no stochastic perturbations and φ (t, x (t)) = B_ix (t - τ (t)), the neural networks (4)-(5) can be reduced as follows:

Fuzzy rule i: IF θ₁ (t) is $η_{1}^{i}$ and … and θ_p (t) is $η_{p}^{i}$ THEN $\begin{matrix} x (t) = - A_{i} x (t) + W_{0 i} σ (x (t)) \\ + W_{1 i} σ (x (t - τ (t))) \end{matrix}$ (38) $y (t) = E_{i} x (t) + B_{i} x (t - τ (t))$ (39)

Then by using the fuzzy inference method, the non-fragile state estimator is designed as follows:

$\begin{matrix} d \hat{x} (t) = \sum_{i = 1}^{r} h_{i} (θ (t)) {- A_{i} \hat{x} (t) + W_{0 i} σ (\hat{x} (t)) \\ + W_{1 i} σ (\hat{x} (t - τ (t))) + (K_{i} + Δ K_{i}) \\ \times [y (t) - E_{i} \hat{x} (t) - B_{i} x (t - τ (t))]} \end{matrix}$ (40)

Corollary 1.Consider (H₃, H₄). Given scalarτ > 0 andμ ⩾ 0, the system (38)-(39) with estimator (40) are exponentially mean-square stable, if there exist matricesP > 0, Q > 0, Z > 0, T_i, scalarsλ_1i, λ_2i and ɛ_i, (i = 1, 2, …, r) such that the following LMI holds: $\begin{matrix} [\begin{matrix} ϒ_{1} & ϒ_{2} & ϒ_{3} \\ * & - 2 P + Z & - τ {PM}_{i} \\ * & * & - ɛ_{i} I \end{matrix}] < 0 \end{matrix}$ where $\begin{matrix} ϒ_{1} = [\begin{matrix} ϒ_{1}^{11} & ϒ_{1}^{12} & Z & ϒ_{1}^{14} & {PW}_{1 i} \\ * & ϒ_{1}^{22} & 0 & 0 & - λ_{2 i} F_{2} \\ * & * & - Z & 0 & 0 \\ * & * & * & - λ_{1 i} I & 0 \\ * & * & * & * & - λ_{2 i} I \end{matrix}] \\ ϒ_{1}^{11} = - {PA}_{i} - A_{i}^{T} P - T_{i} E_{i} - E_{i}^{T} T_{i}^{T} + Q \\ - Z - λ_{1 i} F_{1} + ɛ_{i} E_{i}^{T} N_{i}^{T} N_{i} E_{i} \\ ϒ_{1}^{12} = - T_{i} B_{i} + ɛ_{i} E_{i}^{T} N_{i}^{T} N_{i} B_{i} \\ ϒ_{1}^{14} = {PW}_{0 i} - λ_{1 i} F_{2} \\ ϒ_{1}^{22} = - (1 - μ) Q - λ_{2 i} F_{1} + ɛ_{i} B_{i}^{T} N_{i}^{T} N_{i} B_{i} \\ ϒ_{2 i} = [- τ ({PA}_{i} + T_{i} E_{i}) - τ T_{i} B_{i} 0 \\ τ {PW}_{0 i} τ {PW}_{1 i}]^{T} \\ ϒ_{3 i} = [- {PM}_{i} 0 0 0 0]^{T} \end{matrix}$ The state estimator gain matrix K_i can be designed as K_i = P^-1T_i.

Remark 4. The state estimation for fuzzy delayed neural networks has considered in [1]. It should be noted that delayed neural networks studied in [1] did not take time-varying delay into account. Meanwhile, the author only design a fuzzy-rule-independent state estimator gain matrix K. However, the fuzzy-rule-dependent state estimator gain matrices K_i is considered in this paper, which is less conservativeness. Moreover, in this paper the non-fragile state estimator is designed. For comparison with the Theorem 1 in [1], the following state estimator is designed for systems (38)-(39): $\begin{matrix} d \hat{x} (t) = \sum_{i = 1}^{r} h_{i} (θ (t)) {- A_{i} \hat{x} (t) + W_{0 i} σ (\hat{x} (t)) \\ + W_{1 i} σ (\hat{x} (t - τ (t))) \\ + K_{i} [y (t) - E_{i} \hat{x} (t) - B_{i} x (t - τ (t))]} \end{matrix}$

Corollary 2.Consider (H₃, H₄). Given scalarτ > 0, μ ⩾ 0, the systems (38)-(39) with estimator (42) are exponentially mean-square stable, if there exist matricesP > 0, Q > 0, Z > 0, T_i, scalarsλ_1iandλ_2i, (i = 1, 2, …, r) such that the following LMI holds: $\begin{matrix} [\begin{matrix} {\hat{ϒ}}_{1} & ϒ_{2} \\ * & - 2 P + Z \end{matrix}] < 0 \end{matrix}$ where

$\begin{matrix} {\hat{ϒ}}_{1} = [\begin{matrix} {\hat{ϒ}}_{1}^{11} & - T_{i} B_{i} & Z & ϒ_{1}^{14} & {PW}_{1 i} \\ * & {\hat{ϒ}}_{1}^{22} & 0 & 0 & - λ_{2 i} F_{2} \\ * & * & - Z & 0 & 0 \\ * & * & * & - λ_{1 i} I & 0 \\ * & * & * & * & - λ_{2 i} I \end{matrix}] \\ {\hat{ϒ}}_{1}^{11} = - {PA}_{i} - A_{i}^{T} P - T_{i} E_{i} - E_{i}^{T} T_{i}^{T} \\ + Q - Z - λ_{1 i} F_{1} \\ {\hat{ϒ}}_{1}^{22} = - (1 - μ) Q - λ_{2 i} F_{1} \end{matrix}$

The state estimator gain matrix K_i can be designed as K_i = P^-1T_i.

Remark 5. The state estimators can be achieved in terms of LMIs and the designed estimators ensure the mean-square exponential stability of the estimation error system. These methods are also could be extend to the concerned Takagi-Sugeno fuzzy switched neural networks, see reference [4].

4 Numerical examples

In this section, two examples are given to demonstrate the effectiveness of the proposed state estimators.

Example 1. Consider the fuzzy stochastic neural networks (38)-(39) with the following parameters:

$\begin{matrix} A_{1} = [\begin{matrix} 5 & 0 \\ 0 & 6 \end{matrix}], W_{01} = [\begin{matrix} 0.2 & - 1 \\ 0.1 & 0.2 \end{matrix}] \\ E_{1} = [\begin{matrix} - 0.2 & 0.2 \end{matrix}], B_{1} = [\begin{matrix} 0.3 & 0.6 \end{matrix}] \\ A_{2} = [\begin{matrix} 4 & 0 \\ 0 & 5 \end{matrix}], W_{02} = [\begin{matrix} 0.5 & - 0.5 \\ 0.4 & 0.3 \end{matrix}] \\ E_{2} = [\begin{matrix} 0.2 & - 0.1 \end{matrix}], B_{2} = [\begin{matrix} - 0.2 & 0.4 \end{matrix}] \\ W_{11} = [\begin{matrix} 0.1 & 0.2 \\ 0.2 & 0.7 \end{matrix}], W_{12} = [\begin{matrix} 0.4 & 0.3 \\ 0.1 & 0.6 \end{matrix}] \end{matrix}$

The activation functions σ (x (t)) is chosen as $[\frac{1}{1 + e^{- x_{1} (t)}}$ $\frac{1}{1 + e^{- x_{2} (t)}}]^{T}$ , and the fuzzy membership functions are taken as h₁ (θ (t)) = sin² (3x₁) and h₂ (θ (t)) = cos² (3x₁).

According to assumptions H₃ and H₄, we get Σ₁ = 0, Σ₂ = 0.25I, then we have F₁ = 0, F₂ = -0.125I. By Corollary 1, when μ = 0, we can obtain the maximum allowable delay τ = 0.8067, and the non-fragile estimator gain matrices are given as follows:

$K_{1} = [\begin{matrix} 0.0537 \\ 0.1305 \end{matrix}], K_{2} = [\begin{matrix} 0.0042 \\ 0.2348 \end{matrix}]$

Specially, when W₀₁ = W₀₂ = 0, the system became the one in [1]. The Theorem 1 in [1] holds only for constant time delay. However, by Corollary 2, we conclude that our conditions are delay-dependent and hold for any μ. For different μ, the comparison results are given in Table 1, in which “–” means that the result is not applicable to the corresponding case.

Meanwhile, when μ = 0, the maximum allowable delay is τ = 0.8085, and the fuzzy-rule-dependent state estimator gain matrices are given as follows:

$K_{1} = [\begin{matrix} 0.0417 \\ 0.1448 \end{matrix}], K_{2} = [\begin{matrix} 0.0290 \\ 0.2215 \end{matrix}]$

It is should be noted that the fuzzy-rule-independent state estimator gain matrix K is obtained by Theorem 1 in [1]. Therefore, our method is less conservativeness in some degree than that in [1].

With the initial conditions x (0) = [-3 1] ^T and $\hat{x} (0) = [2 - 2]^{T}$ , respectively, the response of the state dynamics for fuzzy stochastic neural networks(38)-(39) which converges to zero exponentially in the mean square is shown in Fig. 1. Figure 2 depicts the estimation error $e (t) = x (t) - \hat{x} (t)$ .

Example 2. Consider the fuzzy stochastic neural networks (5)-(6) as follows:

Fuzzy rule 1: IFθ₁ (t) is $η_{1}^{1}$ and … and θ_p (t) is $η_{p}^{1}$ THEN

$\begin{matrix} dx (t) = [- A_{1} x (t) + W_{11} σ (x (t - τ (t)) \\ + W_{01} σ (x (t))] dt + [C_{1} x (t) + D_{1} x (t - τ (t)) \\ + W_{21} σ (x (t)) + W_{31} σ (x (t - τ (t))] dw (t) \\ y (t) = E_{1} x (t) + φ (t, x (t)) \end{matrix}$

Fuzzy rule 2: IFθ₁ (t) is $η_{2}^{1}$ and … and θ_p (t) is $η_{p}^{2}$ THEN

$\begin{matrix} dx (t) = [- A_{2} x (t) + W_{12} σ (x (t - τ (t)) \\ + W_{02} σ (x (t))] dt + [C_{2} x (t) + D_{2} x (t - τ (t)) \\ + W_{22} σ (x (t)) + W_{32} σ (x (t - τ (t))] dw (t) \\ y (t) = E_{2} x (t) + φ (t, x (t)) \end{matrix}$ with the following parameters $\begin{matrix} A_{1} = [\begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix}], W_{01} = [\begin{matrix} - 0.5 & 0.1 \\ - 0.2 & 0.3 \end{matrix}] \\ W_{11} = [\begin{matrix} - 0.2 & 0.4 \\ 0.5 & - 0.1 \end{matrix}], C_{1} = [\begin{matrix} 0.5 & - 0.2 \\ 0.3 & 0 \end{matrix}] \\ D_{1} = [\begin{matrix} 0.1 & - 0.5 \\ - 0.5 & 0 \end{matrix}], W_{21} = [\begin{matrix} 0.2 & 0 \\ 0 & 0.2 \end{matrix}] \\ W_{31} = [\begin{matrix} 0.2 & 0 \\ 0 & 0.2 \end{matrix}], E_{1} = [\begin{matrix} - 0.3 & 0.3 \\ 0.1 & 0.4 \end{matrix}] \\ A_{2} = [\begin{matrix} 4 & 0 \\ 0 & 5 \end{matrix}], W_{02} = [\begin{matrix} 0.4 & - 0.7 \\ 0.1 & 0 \end{matrix}] \\ W_{12} = [\begin{matrix} - 0.2 & 0.6 \\ 0.5 & - 0.1 \end{matrix}], C_{2} = [\begin{matrix} 0.5 & 1 \\ 0.2 & 0.1 \end{matrix}] \\ D_{2} = [\begin{matrix} 0.2 & 0.2 \\ - 0.3 & 0.2 \end{matrix}], W_{22} = [\begin{matrix} 0.3 & 0 \\ 0 & 0.3 \end{matrix}] \\ W_{32} = [\begin{matrix} 0.1 & 0 \\ 0 & 0.1 \end{matrix}], E_{2} = [\begin{matrix} 0.3 & - 0.4 \\ - 0.3 & 0.1 \end{matrix}] \end{matrix}$ The activation functions σ (x) = (|x + 1| - |x - 1|)/2, and the neuron-dependent nonlinear disturbance φ (t, x (t)) = tanh (t). The fuzzy membership functions are taken as h₁ (θ (t)) = sin² (x₂ + 0.5) and h₂ (θ (t)) = cos² (x₂ + 0.5). Meanwhile, M₁ = M₂ = 0.2I, N₁ = N₂ = 0.1I, F₁ (t) = F₂ (t) = sin (t).

Firstly, we consider (H₃, H₄) and get Σ₁ = 0, Σ₂ = I, φ₁ = 0, φ₂ = I, then we have F₁ = 0, F₂ = -0.5I, J₁ = 0, J₂ = -0.5I. By Theorem 1, when μ = 0.5, we can obtain the maximum allowable delay τ = 0.7758, and the non-fragile estimator gain matrices K₁, K₂ as follows:

$\begin{matrix} K_{1} = [\begin{matrix} 0.0089 & 0.0018 \\ - 0.0021 & 0.0031 \end{matrix}] \\ K_{2} = [\begin{matrix} - 0.0178 & 0.0015 \\ - 0.0084 & - 0.0188 \end{matrix}] \end{matrix}$

With the initial conditions x (0) = [ 3 - 1] ^T and $\hat{x} (0) = [- 1 2]^{T}$ , respectively, the response of the state dynamics for fuzzy stochastic neural networks (5)-(6) which converges to zero exponentially in the mean square is shown in Fig. 3. Figure 4 depicts the estimation error $e (t) = x (t) - \hat{x} (t)$ .

Secondly, we consider (H₁, H₂) and get L = I, G = I. By Theorem 2, when μ = 0.5, we can obtain the maximum allowable delay τ = 0.6149, and the non-fragile estimator gain matrices K₁, K₂ as follows:

$\begin{matrix} K_{1} = [\begin{matrix} - 0.0092 & 0.0006 \\ 0.0205 & 0.0139 \end{matrix}] \\ K_{2} = [\begin{matrix} - 0.0035 & - 0.0005 \\ - 0.0057 & 0.0003 \end{matrix}] \end{matrix}$

With the initial conditions x (0) = [ 3 - 1] ^T and $\hat{x} (0) = [- 1 2]^{T}$ , respectively, the response of the state dynamics for fuzzy stochastic neural networks (5)-(6) which converges to zero exponentially in the mean square is shown in Fig. 5. Figure 6 depicts the estimation error $e (t) = x (t) - \hat{x} (t)$ .

Remark 6. Notice that when μ = 0.5, the maximum allowable delay τ is 0.7758 for (H₃, H₄), the same maximum allowable delay for the (H₁, H₂) is 0.6149, which has illustrated that the sector-bounded conditions (H₃, H₄) are weaker than Lipschitz conditions (H₁, H₂).

To further compare the (H₃, H₄) and (H₁, H₂) in this paper, the maximum allowable time delay τ for different μ by Theorem 1 and Theorem 2 are listed in Table 2, in which “–” means that the result is not applicable to the corresponding case. The maximum time delay τ obtained by Theorem 1 is greater than those given in Theorem 2. On the contrary, Theorem 2 is not applicable to this example for μ = 0.8. It also can been seen that sector-bounded conditions are weaker than Lipschitz conditions.

5 Conclusion

The non-fragile state estimator is firstly designed for fuzzy stochastic neural networks with time-varying delays in this paper. The neuron activation function and the nonlinear measurement equation are assumed to be satisfy sector-bounded conditions and standard Lipschitz conditions. Based on the new Lyapunov-Krasovskii functional and the new integral inequality in the stochastic setting, delay-dependent sufficient conditions are presented to guarantee the existence of the desired non-fragile state estimators for fuzzy stochastic neural networks. Both the sector-bounded conditions and Lipschitz conditions are given to study the non-fragile state estimation for fuzzy stochastic neural networks, respectively. The state estimators for both sector-bounded conditions and standard Lipschitz conditions have been designed in terms of LMIs, which could be solved easily and efficiently. Two numerical examples have been provided to demonstrate the effectiveness of the proposed result. In future, our research would be the extension of the present results to more general cases: the fuzzy stochastic neural networks with Markovian jumping parameters and the fuzzy switched stochastic neural networks.

References

Ahn

, Delay-dependent state estimation for T-S fuzzy delayed Hopfield neural networks, Nonlinear Dynamics61 (2010), 483–489.

Ahn

, H_∞ state estimation for Takagi-Sugeno fuzzy delayed Hopfield neural networks, International Journal of Intelligent Systems4 (2011), 855–862.

Ahn

, Passive and exponential filter design for fuzzy neural networks, Information Sciences238 (2013), 126–137.

Ahn

, Receding horizon disturbance attenuation for Takagi-Sugeno fuzzy switched dynamic neural network, Information Sciences280 (2014), 53–63.

Balasubramaniam

, Vembarasan

and Rakkiyappan

, Delay-dependent robust asymptotic state estimation of Takagi-Sugeno fuzzy Hopfield neural networks with mixed interval time-varying delays, Expert Systems with Applications39 (2010), 472–481.

Balasubramaniam

, Vembarasan

and Rakkiyappan

, Delay-dependent robust exponential state estimation of Markovian jumping fuzzy Hopfield neural networks with mixed random time-varying delays, Communications in Nonlinear Science and Numerical Simulation16 (2011), 2109–2129.

Bao

and Cao

, Delay-distribution-dependent state estimation for discrete-time stochastic neural networks with random delay, Neural Networks24 (2011), 19–28.

Boyd

, EI Ghaoui

, Feron

and Balakrishnan

, Linear matrix inequalities in system and control theory, SIAM, Philadelphia, 1994.

Chen

and Zheng

, Stochastic state estimation for neural networks with distributed delays and Markovian jump, Neural Networks25 (2012), 14–20.

10.

Deng

, Hua

, Liu

, Peng

and Fei

, Robust delaydependent exponential stability for uncertain stochastic neural networks with mixed delays, Neurocomputing74 (2011), 1503–1509.

11.

Fang

and Park

, Non-fragile synchronization of neural networks with time-varying delay and randomly occurring controller gain fluctuation, Applied Mathematics and Computation219 (2013), 8009–8017.

12.

Hua

, Liu

, Deng

and Fei

, New results on robust exponential stability of uncertain stochastic neural networks with mixed time-varying delays, Neural Processing Letters32 (2010), 219–233.

13.

Hua

, Tan

and Chen

, Delay-dependent H1 and generalized H2 filtering for stochastic neural networks with timevarying delay and noise disturbance, Neural Computing and Applications25(3-4) (2013), 613–624.

14.

, Chen

, Lin

and Zhou

, Mean square exponential stability of stochastic fuzzy Hopfield neural networks with discrete and distributed time-varying delays, Neurocomputing72 (2009), 2017–2023.

15.

, Chen

, Zhou

and Fang

, Robust exponential stability for uncertain stochastic neural networks with discrete and distributed time-varying delays, Physics Letters A372 (2008), 3385–3394.

16.

, Chen

, Zhou

and Qian

, Robust stability for uncertain delayed fuzzy Hopfield neural networks with Markovian jumping parameters, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics39 (2009), 94–102.

17.

and Rakkiyappan

, Robust asymptotic state estimation of Takagi-Sugeno fuzzy Markovian jumping Hopfield neural networks with mixed interval time-varying delays, Mathematical Methods in The Applied Sciences34 (2011), 2197–2207.

18.

Liang

, Wang

and Liu

, State estimation for coupled uncertain stochastic networks with missing measurements and time-varying delays: The discrete-time case, IEEE Transactions on Neural Networks20 (2009), 781–793.

19.

Lou

and Cui

, Robust asymptotic stability of uncertain fuzzy BAM neural networks with time-varying delays, Fuzzy Sets and Systems158 (2007), 2746–2756.

20.

Lou

and Cui

, Design of state estimator for uncertain neural networks via the integral-inequality method, Nonlinear Dynamics53 (2008), 223–235.

21.

Keel

and Bhattacharyya

, Robust, fragile, or optimal?IEEE Transactions on Automatic Control42 (1997), 1098–1105.

22.

Rakkiyappan

, Chandrasekar

, Lakshmanan

and Park

, State estimation of memristor-based recurrent neural networks with time-varying delays based on passivity theory, Complexity19 (2014), 32–43.

23.

Rakkiyappan

, Sakthivel

, Park

and Kwon

, Sampleddata state estimation for Markovian jumping fuzzy cellular neural networks with mode-dependent probabilistic time-varying delays, Applied Mathematics and Computation221 (2013), 741–769.

24.

Takagi

and Sugeno

, Fuzzy identification of systems and its application to modeling and control, IEEE Transactions on Systems, Man, and Cybernetics15 (1985), 116–132.

25.

Tseng

, Tsai

and Lu

, Design of delay-dependent exponential estimator for T-S fuzzy neural networks with mixed timevarying interval delays using hybrid Taguchi-Genetic algorithm, Neural Processing Letters36 (2012), 49–67.

26.

Vembarasan

, Balasubramaniam

and Chan

, Non-fragile state observer design for neural networks with Markovian jumping parameters and time-delays, Nonlinear Analysis: Hybrid Systems14 (2014), 61–73.

27.

Wang

, Ding

, Zhang

and Hao

, Robust state estimation for discrete-time stochastic genetic regulatory networks with probabilistic measurement delays, Neurocomputing111 (2013), 1–12.

28.

, Wang

and Shi

, A delay decomposition approach to L2-L1 filter design for stochastic systems with time-varying delay, Automatica47 (2011), 1482–1488.

29.

Zhang

, Cai

and Wang

, Mixed H_∞ and passivity based state estimation for fuzzy neural networks with Markovian-type estimator gain change, Neurocomputing139 (2014), 321–327.