State Estimation for Neural Networks with Leakage Delay and Time-Varying Delays

Abstract

The state estimation problem is investigated for neural networks with leakage delay and time-varying delay as well as for general activation functions. By constructing appropriate Lyapunov-Krasovskii functionals and employing matrix inequality techniques, a delay-dependent linear matrix inequalities (LMIs) condition is developed to estimate the neuron state with some observed output measurements such that the error-state system is globally asymptotically stable. An example is given to show the effectiveness of the proposed criterion.

1. Introduction

In the past few years, neural networks have been extensively studied and successfully applied in many areas such as combinatorial optimization, signal processing, associative memory, affine invariant matching, and pattern recognition [1]. In such applications, the stability analysis is a necessary step for the practical design of neural networks [2]. In hardware implementation, time delays occur due to finite switching speed of the amplifiers and communication time. The existence of time delay may lead to some complex dynamic behaviors such as oscillation, divergence, chaos, instability, or other poor performance of the neural networks [3]. Therefore, the issue of stability analysis of neural networks with time delay attracts many researchers, and a number of remarkable results have been built up in the open literature; for example, see [2–5] and references therein.

When a neural network is designed to handle complex nonlinear problems, a great number of neurons with tremendous connections are often required. In such relatively large-scale neural networks, it may be very difficult and expensive (or even impossible) to obtain the complete information of the neuron states. On the other hand, in many practical applications, one needs to know the neuron states and then use them to achieve certain objectives. For instance, a recurrent neural network was presented in [6] to model an unknown nonlinear system, and the neuron states were utilized to implement a control law. Therefore, it is of great importance to study the state estimation problem of neural networks.

Recently, some results related to the state estimation problem for neural networks have been reported; for example, see [7–38] and references therein. In [7], authors initially studied the state estimation problem of delayed neural networks, where a delay-independent condition was obtained in terms of a linear matrix inequality (LMI). In [8], authors proposed a free-weighting matrix approach to discuss the state estimation problem for neural networks with time-varying delay. By using the Newton-Leibniz formula, some slack variables were introduced to derive a less conservative condition. In [11], attention was focused on the design of a state estimator to estimate the neuron states by using the delay-fractioning technique to reduce the possible conservatism. The authors in [13] first investigated the robust state estimator problem of delayed neural networks with parameter uncertainties. Delay-dependent conditions were presented to guarantee the global asymptotical stability of the error system. In [16], a further result on design problem of state estimator for a class of neural networks of neutral type was presented. A delay-dependent LMI criterion for existence of the estimator was derived. In [20], the state estimation problem for discrete neural networks with partially unknown transition probabilities and time-varying delays was discussed. By utilizing a novel Lyapunov functional integrating both lower and upper time-delay bounds and some new techniques, some delay-range-dependent sufficient conditions under which the estimation error dynamics were stochastically stable are established. In [22], authors investigated the state estimation problem for neural networks with discrete time-varying delay and distributed time-varying delay; a delay-interval-dependent condition is developed to estimate the neuron state with some observed output measurements such that the error-state system was globally asymptotically stable. In [25], leakage delay in the leakage term was used to destabilize the neuron states. By constructing the Lyapunov-Krasovskii functional which contains a triple-integral term, an improved delay-dependent stability criterion was derived in terms of LMIs. In [27], the state estimation problem for a class of discrete-time stochastic neural networks with random delays was considered. By employing a Lyapunov-Krasovskii functional, sufficient delay distribution- dependent conditions were established in terms of LMIs that guarantee the existence of the state estimator. In [33], authors discussed the state estimation problem for Takagi-Sugeno (T-S) fuzzy Hopfield neural networks via strict output passivation of the error system. In [36–38], authors investigated the distributed state estimation problem for sensor networks and presented several new sufficient conditions to guarantee the convergence of the estimation error systems. To the best of the author’s knowledge, there are no results on the problem of state estimation for neural networks with leakage delay and time-varying delays. As pointed out in [39], neural networks with leakage delay are a class of important neural networks; time delay in the leakage term also has great impact on the dynamics of neural networks since time delay in the stabilizing negative feedback term has a tendency to destabilize a system. Therefore, it is necessary to investigate the state estimation problem for neural networks with leakage delay [25].

Motivated by the previous discussions, the objective of this paper is to study the state estimation for neural networks with leakage delay and time-varying delays by employing new Lyapunov-Krasovskii functionals and using matrix inequality techniques. The obtained sufficient condition does not require the differentiability of time-varying delays and is expressed in terms of linear matrix inequalities, which can be checked numerically using the effective LMI toolbox in Matlab. An example is given to show the effectiveness of the proposed criterion.

Notations. The notations are quite standard. 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. ‖⋅‖ refers to the Euclidean vector norm. 𝐴^𝑇 represents the transpose of matrix 𝐴, and the asterisk “∗” in a matrix is used to represent the term which is induced by symmetry. 𝐼 is the identity matrix with compatible dimension. 𝑋>𝑌 means that 𝑋 and 𝑌 are symmetric matrices and that 𝑋−𝑌 is positive definite. Matrices, if not explicitly specified, are assumed to have compatible dimensions.

2. Model Description and Preliminaries

Consider the following neural networks with leakage delay and time-varying delays: 𝑥̇(𝑡)=−𝐶𝑥(𝑡−𝛿)+𝐴𝑓(𝑥(𝑡))+𝐵𝑓(𝑥(𝑡−𝜏(𝑡)))+𝐽(𝑡) (1) for 𝑡≥0, where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈ℝ^𝑛 is the state vector of the network at time 𝑡, and 𝑛 corresponds to the number of neurons; 𝐶∈ℝ^𝑛×𝑛 is a positive diagonal matrix, and 𝐴∈ℝ^𝑛×𝑛 and 𝐵∈ℝ^𝑛×𝑛 are the connection weight matrix and the delayed connection weight matrix, respectively; 𝑓(𝑥(𝑡))=(𝑓₁(𝑥₁(𝑡)),𝑓₂(𝑥₂(𝑡)),…,𝑓_𝑛(𝑥_𝑛(𝑡)))^𝑇∈ℝ^𝑛 denotes the neuron activation at time 𝑡; 𝐽(𝑡)∈ℝ^𝑛 is an external input vector; 𝛿 and 𝜏(𝑡) denote the leakage delay and time-varying delay, respectively.

Throughout this paper, we make the following assumptions.

(H1) (see [40]) For any 𝑗∈{1,2,…,𝑛} there exist constants 𝐹⁻_𝑗 and 𝐹⁺_𝑗 such that 𝐹⁻_𝑗≤𝑓_𝑗𝛼₁−𝑓_𝑗𝛼₂𝛼₁−𝛼₂≤𝐹⁺_𝑗 (2) for all 𝛼₁≠𝛼₂.

(H2) (see [25]) The leakage delay 𝛿 and time-varying delays 𝜏(𝑡) satisfy the following conditions: 0≤𝛿,0≤𝜏(𝑡)≤𝜏, (3) where 𝛿 and 𝜏 are constants.

As is well known, information about the neuron states is often incomplete from the network measurements (outputs) in applications, and the network measurements are subject to nonlinear disturbances. Similar to [7], we assume that the network measurements satisfy 𝑦(𝑡)=𝐷𝑥(𝑡)+𝑔(𝑡,𝑥(𝑡)), (4) where 𝑦(𝑡)∈ℝ^𝑚 is the measurement output, and 𝐷∈ℝ^𝑚×𝑛 is a known constant matrix with appropriate dimension. 𝑔∶ℝ×ℝ^𝑛→ℝ^𝑚 is the neuron-dependent nonlinear disturbances on the network outputs and satisfies the following condition.

(H3) (see [22]) For any 𝑗∈{1,2,…,𝑚} there exist constants 𝐺⁻_𝑗 and 𝐺⁺_𝑗 such that 𝐺⁻_𝑗≤𝑔_𝑗𝑡,𝛼₁(𝑡)−𝑔_𝑗𝑡,𝛼₂(𝑡)𝛼₁(𝑡)−𝛼₂(𝑡)≤𝐺⁺_𝑗 (5) for 𝑡≥0 and all 𝛼₁(𝑡)≠𝛼₂(𝑡).

Remark 1. As pointed out in [40], the constants 𝐹⁻_𝑗 and 𝐹⁺_𝑗 in assumption (H1) of this paper are allowed to be positive, negative, or zero. Hence, assumption (H1), first proposed by Liu et al. in [40], is weaker than the Lipschitz condition.

For the delay neural networks (1), we construct the full-order state estimation as follows: 𝑢̇(𝑡)()(𝑢(𝑡(𝑢((𝑡(𝑡)𝑦(𝑡)(𝑡)𝑣(𝑡)(𝑡)((𝑡,=−𝐶𝑢𝑡−𝛿+𝐴𝑓))+𝐵𝑓𝑡−𝜏)))+𝐽+𝐾−𝑣=𝐷𝑢+𝑔𝑡,𝑢)) (6)(7) where 𝑢(𝑡)∈ℝ^𝑛 is an estimation of the state 𝑥(𝑡) of (1), and 𝐾∈ℝ^𝑛×𝑚, to be determined, is a gain matrix of the state estimator.

Let the error state be 𝑒(𝑡)=𝑥(𝑡)−𝑢(𝑡) then it follows from (1), (4), (6), and (7) that 𝑒(̇𝑡)(𝑡)()(𝑒(𝑡(𝑒((𝑡((𝑡,=−𝐾𝐷𝑒−𝐶𝑒𝑡−𝛿+𝐴ℎ))+𝐵ℎ𝑡−𝜏)))−𝐾𝑟𝑡,𝑒)) (8) where ℎ(𝑒(𝑡))=𝑓(𝑥(𝑡))−𝑓(𝑢(𝑡)), 𝑟(𝑡,𝑒(𝑡))=𝑔(𝑡,𝑥(𝑡))−𝑔(𝑡,𝑢(𝑡)).

The problem to be addressed in this study is to find out the gain matrix 𝐾 such that the system (8) is globally asymptotically stable.

To prove our results, the following lemmas that can be found in [22] are necessary.

Lemma 2 (see [22]). For any constant matrix 𝑊∈ℝ^𝑚×𝑚, 𝑊>0, scalar 0<ℎ(𝑡)<ℎ, and vector function 𝜔(⋅)∶[0,ℎ]→ℝ^𝑚 such that the integrations concerned are well defined, then ₀^ℎ(𝑡)𝜔(𝑠)𝑑𝑠^𝑇𝑊₀^ℎ(𝑡)𝜔(𝑠)(𝑡)𝑑𝑠≤ℎ₀^ℎ(𝑡)𝜔^𝑇(𝑠)𝑊𝜔(𝑠)𝑑𝑠. (9)

Lemma 3 (see [22]). Given constant matrices 𝑃, 𝑄, and 𝑅, where 𝑃^𝑇=𝑃, 𝑄^𝑇=𝑄, then 𝑅𝑃𝑅_𝑇−𝑄<0 (10) is equivalent to the following conditions: 𝑄>0,𝑃+𝑅𝑄⁻¹𝑅^𝑇<0. (11)

3. Main Results

For presentation convenience, in the following, we denote 𝐹₁=diag𝐹⁻₁,𝐹⁻₂,…,𝐹⁻_𝑛,𝐹₂=diag𝐹⁺₁,𝐹⁺₂,…,𝐹⁺_𝑛,𝐹₃=diag𝐹⁻₁𝐹⁺₁,𝐹⁻₂𝐹⁺₂,…,𝐹⁻_𝑛𝐹⁺_𝑛,𝐹₄𝐹=diag⁻¹+𝐹⁺₁2,𝐹⁻₂+𝐹⁺₂2𝐹,…,^−𝑛+𝐹⁺_𝑛2,𝐺₁=diag𝐺⁻₁𝐺⁺₁,𝐺⁻₂𝐺⁺₂,…,𝐺⁻_𝑛𝐺⁺_𝑛,𝐺₂𝐺=diag⁻¹+𝐺⁺₁2,𝐺⁻₂+𝐺⁺₂2𝐺,…,^−𝑛+𝐺⁺_𝑛2. (12)

Theorem 4. Assume that the assumptions (H1)–(H3) hold. If there exist four symmetric positive definite matrices 𝑃_𝑖(𝑖=1,2,3,4), five positive diagonal matrices 𝑊₁, 𝑊₂, 𝑅₁, 𝑅₂, and 𝑅₃, and seven matrices 𝑄₁, 𝑄₂, 𝑄₃, 𝑋₁₁, 𝑋₁₂, 𝑋₂₂, and 𝑍, such that the following LMIs hold: 𝑋𝑋=¹¹𝑋₁₂∗𝑋₂₂Ω>0,Ω=¹¹Ω₁₂𝑄₃0Ω₁₅𝑋₂₂Ω₁₇𝑄₁𝐵Ω₁₉𝑄₃0∗Ω₂₂00−𝑄₁𝐶𝑋₁₂Ω₂₇𝑄₁𝐵−𝑍00∗∗Ω₃₃𝑄₂00𝐹₄𝑅₂000𝑄₂∗∗∗−𝑃₃0000000∗∗∗∗−𝑃₁−𝑋₂₂00000∗∗∗∗∗−𝑃₂00000∗∗∗∗∗∗−𝑅₁0000∗∗∗∗∗∗∗−𝑅₂000∗∗∗∗∗∗∗∗−𝑅₃100∗∗∗∗∗∗∗∗∗−𝜏𝑃₄01∗∗∗∗∗∗∗∗∗0−𝜏𝑃₄<0, (13)(14)where Ω₁₁=𝑋₁₂+𝑋^𝑇₁₂+𝑃₁+𝛿²𝑃₂+𝑃₃−𝑍𝐷−𝐷^𝑇𝑍^𝑇−𝑄₃−𝑄^𝑇₃−𝐹₃𝑅₁−𝐺₁𝑅₃, =𝑋₁₁−𝐹₁𝑊₁+𝐹₂𝑊₂−𝑄₁−𝐷^𝑇𝑍^𝑇, Ω₁₅=−𝑋₁₂−𝑄₁𝐶, Ω₁₇=𝑄₁𝐴+𝐹₄𝑅₁, Ω₁₉=−𝑍+𝐺₂𝑅₃, Ω₂₂=−𝑄₁−𝑄^𝑇₁+𝜏𝑃₄, Ω₂₇=𝑊₁−𝑊₂+𝑄₁𝐴, Ω₃₃=−𝑄₂−𝑄^𝑇₂−𝐹₃𝑅₂, then error-state system (8) of the delayed neural network described by (1) and (6) is globally asymptotically stable, and the estimator gain matrix 𝐾 can be designed as 𝐾=𝑄₁⁻¹𝑍.

Proof. From assumption (H1), we know that ^𝑒𝑖₀^(𝑡)ℎ_𝑖(𝑠)−𝐹⁻_𝑖𝑠𝑑𝑠≥0,_{𝑒𝑖0(𝑡)}𝐹⁺_𝑖𝑠−ℎ_𝑖(𝑠)𝑑𝑠≥0,𝑖=1,2,…,𝑛. (15)(16)
Let 𝑊₁=diag(𝑤₁₁,𝑤₁₂,…,𝑤_1𝑛), 𝑊₂=diag(𝑤₂₁,𝑤₂₂,…,𝑤_2𝑛), and consider the following Lyapunov-Krasovskii functional as 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡)+𝑉₄(𝑡), (17) where 𝑉₁(𝑡)=𝑒(𝑡)_{𝑡𝑡−𝛿}𝑒(𝑠)𝑑𝑠^𝑇𝑋₁₁𝑋₁₂∗𝑋₂₂𝑒(𝑡)_{𝑡𝑡−𝛿}𝑒(𝑠),𝑉𝑑𝑠₂(𝑡)=2^𝑛_𝑖=1𝑤_1𝑖^𝑒𝑖₀^(𝑡)ℎ_𝑖(𝑠)−𝐹⁻_𝑖𝑠𝑑𝑠+2^𝑛_𝑖=1𝑤_2𝑖^𝑒𝑖₀^(𝑡)𝐹⁺_𝑖𝑠−ℎ_𝑖(𝑠𝑉)𝑑𝑠,₃(𝑡)=^𝑡_𝑡−𝛿𝑒^𝑇(𝑠)𝑃₁𝑒(𝑠)𝑑𝑠+𝛿⁰_−𝛿^𝑡_𝑡+𝜉𝑒^𝑇(𝑠)𝑃₂𝑒(𝑠)𝑉𝑑𝑠𝑑𝜉,₄(𝑡)=^𝑡_𝑡−𝜏𝑒^𝑇(𝑠)𝑃₃𝑒(𝑠)𝑑𝑠+⁰_−𝜏^𝑡_𝑡+𝜉𝑒^𝑇̇(𝑠)𝑃₄𝑒(̇𝑠)𝑑𝑠𝑑𝜉. (18)
Calculating the time derivative of 𝑉_𝑖(𝑡) (𝑖=1,2,3,4), we obtain 𝑉̇₁(𝑡)=2𝑒(𝑡)_{𝑡𝑡−𝛿}𝑒(𝑠)𝑑𝑠^𝑇𝑋₁₁𝑋₁₂∗𝑋₂₂𝑒(̇𝑡)𝑒(𝑡)−𝑒(𝑡−𝛿)=𝑒^𝑇(𝑡)𝑋₁₂+𝑋^𝑇₁₂𝑒(𝑡)+2𝑒^𝑇(𝑡)𝑋₁₁𝑒(̇𝑡)−2𝑒^𝑇(𝑡)𝑋₁₂𝑒(𝑡−𝛿)+2𝑒^𝑇(𝑡)𝑋₂₂^𝑡_𝑡−𝛿𝑒(𝑠)𝑒̇𝑑𝑠+2^𝑇(𝑡)𝑋₁₂^𝑡_𝑡−𝛿𝑒(𝑠)𝑑𝑠−2𝑒^𝑇(𝑡−𝛿)𝑋₂₂^𝑡_𝑡−𝛿𝑒(𝑠)𝑉̇𝑑𝑠,₂(𝑡)=2𝑒^𝑇̇(𝑡)𝑊₁ℎ(𝑒(𝑡))−𝐹₁𝑒(𝑡)+2𝑒^𝑇̇(𝑡)𝑊₂𝐹₂𝑒(𝑡)−ℎ(𝑒(𝑡))=2𝑒^𝑇(𝑡)−𝐹₁𝑊₁+𝐹₂𝑊₂𝑒(̇𝑡)+2𝑒^𝑇̇(𝑡)𝑊₁−𝑊₂ℎ(𝑒(𝑡,𝑉̇))₃(𝑡)=𝑒^𝑇(𝑡)𝑃₁+𝛿²𝑃₂𝑒(𝑡)−𝑒^𝑇(𝑡−𝛿)𝑃₁𝑒(𝑡−𝛿)−𝛿^𝑡_𝑡−𝛿𝑒^𝑇(𝑠)𝑃₂𝑒(𝑠)𝑑𝑠≤𝑒^𝑇(𝑡)𝑃₁+𝛿²𝑃₂𝑒(𝑡)−𝑒^𝑇(𝑡−𝛿)𝑃₁𝑒()−𝑡−𝛿_{𝑡𝑡−𝛿}𝑒(𝑠)𝑑𝑠^𝑇𝑃₂^𝑡_𝑡−𝛿𝑒(𝑠),𝑉̇𝑑𝑠₄(𝑡)=𝑒^𝑇(𝑡)𝑃₃𝑒(𝑡)−𝑒^𝑇(𝑡−𝜏)𝑃₃𝑒()𝑒̇𝑡−𝜏+𝜏^𝑇(𝑡)𝑃₄𝑒(̇𝑡)−^𝑡_𝑡−𝜏𝑒^𝑇̇(𝑠)𝑃₄𝑒(̇𝑠)𝑑𝑠. (19)(20)(21)(22)
In deriving inequality (21), we have made use of Lemma 2. It follows from inequalities (19)–(22) that 𝑉̇(𝑡)≤𝑒^𝑇(𝑡)𝑋₁₂+𝑋^𝑇₁₂+𝑃₁+𝛿²𝑃₂+𝑃₃𝑒(𝑡)+2𝑒^𝑇(𝑡)𝑋₁₁−𝐹₁𝑊₁+𝐹₂𝑊₂𝑒(̇𝑡)−2𝑒^𝑇(𝑡)𝑋₁₂𝑒(𝑡−𝛿)+2𝑒^𝑇(𝑡)𝑋₂₂^𝑡_𝑡−𝛿𝑒(𝑠)𝑒̇𝑑𝑠+𝜏^𝑇(𝑡)𝑃₄𝑒(̇𝑡)+2𝑒^𝑇̇(𝑡)𝑋₁₂^𝑡_𝑡−𝛿𝑒(𝑠)𝑒̇𝑑𝑠+2^𝑇(𝑡)𝑊₁−𝑊₂ℎ(𝑒(𝑡))−𝑒^𝑇(𝑡−𝛿)𝑃₁𝑒(𝑡−𝛿)−2𝑒^𝑇(𝑡−𝛿)𝑋₂₂^𝑡_𝑡−𝛿𝑒(𝑠)𝑑𝑠−𝑒^𝑇(𝑡−𝜏)𝑃₃𝑒()−𝑡−𝜏_{𝑡𝑡−𝛿}𝑒(𝑠)𝑑𝑠^𝑇𝑃₂^𝑡_𝑡−𝛿𝑒(𝑠)𝑑𝑠−^𝑡_𝑡−𝜏𝑒^𝑇̇(𝑠)𝑃₄𝑒(̇𝑠)𝑑𝑠. (23)
From model (8), we have 0=2𝑒^𝑇(𝑡)+𝑒^𝑇̇(𝑡)𝑄₁×[−𝑒(̇𝑡)(𝑡)()(𝑒(𝑡(𝑒((𝑡((𝑡−𝐾𝐷𝑒−𝐶𝑒𝑡−𝛿+𝐴ℎ))+𝐵ℎ𝑡−𝜏)))−𝐾𝑟𝑡,𝑒))]=−𝑒^𝑇(𝑡)𝑄₁𝐾𝐷+𝐷^𝑇𝐾^𝑇𝑄^𝑇₁𝑒(𝑡)−2𝑒^𝑇(𝑡)𝑄₁𝑒(̇𝑡)−2𝑒^𝑇(𝑡)𝑄₁𝐶𝑒(𝑡−𝛿)+2𝑒^𝑇(𝑡)𝑄₁𝐴ℎ(𝑒(𝑡))+2𝑒^𝑇(𝑡)𝑄₁×(𝑒((𝑡𝐵ℎ𝑡−𝜏)))−2𝑒^𝑇(𝑡)𝑄₁((𝑡−𝑒̇𝐾𝑟𝑡,𝑒))_𝑇(𝑡)𝑄₁+𝑄^𝑇₁𝑒(̇𝑡)−2𝑒^𝑇(𝑡)𝐷^𝑇𝐾^𝑇𝑄^𝑇₁𝑒(̇𝑡)−2𝑒^𝑇̇(𝑡)𝑄₁𝐶𝑒(𝑡−𝛿)+2𝑒^𝑇̇(𝑡)𝑄₁(𝑒(𝑡𝑒̇𝐴ℎ))+2^𝑇(𝑡)𝑄₁(𝑒((𝑡𝑒̇𝐵ℎ𝑡−𝜏)))−2^𝑇(𝑡)𝑄₁𝐾𝑟(𝑡,𝑒(𝑡). (24)
By Newton-Leibniz formulation and assumption (H2), we have 0=−2𝑒^𝑇(𝑡−𝜏(𝑡))𝑄₂×𝑒(𝑡−𝜏(𝑡))−𝑒(𝑡−𝜏)−^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝑒(̇𝑠)𝑑𝑠≤−2𝑒^𝑇(𝑡−𝜏(𝑡))𝑄₂𝑒(𝑡−𝜏(𝑡))+2𝑒^𝑇(𝑡−𝜏(𝑡))𝑄₂𝑒(𝑡−𝜏)+𝜏𝑒^𝑇(𝑡−𝜏(𝑡))𝑄₂𝑃₄⁻¹𝑄^𝑇₂𝑒(𝑡−𝜏(𝑡))+^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝑒^𝑇̇(𝑠)𝑃₄𝑒(̇𝑠)𝑑𝑠,0=−2𝑒^𝑇(𝑡)𝑄₃×𝑒(𝑡)−𝑒(𝑡−𝜏(𝑡))−^𝑡_{(𝑡)𝑡−𝜏}𝑒(̇𝑠)𝑑𝑠≤−2𝑒^𝑇(𝑡)𝑄₃𝑒(𝑡)+2𝑒^𝑇(𝑡)𝑄₃𝑒(𝑡−𝜏(𝑡))+𝜏𝑒^𝑇(𝑡)𝑄₃𝑃₄⁻¹𝑄^𝑇₃𝑒(𝑡)+^𝑡_{(𝑡)𝑡−𝜏}𝑒^𝑇̇(𝑠)𝑃₄𝑒(̇𝑠)𝑑𝑠. (25)
In addition, for positive diagonal matrices 𝑅_𝑖>0 (𝑖=1,2,3), we can get from assumption (H1) and assumption (H3) that [15] 𝑒(𝑡)ℎ(𝑒(𝑡))^𝑇𝐹₃𝑅₁−𝐹₄𝑅₁−𝐹₄𝑅₁𝑅₁𝑒(𝑡)ℎ(𝑒(𝑡𝑒((𝑡ℎ(𝑒((𝑡))≤0,𝑡−𝜏))𝑡−𝜏)))^𝑇𝐹₃𝑅₂−𝐹₄𝑅₂−𝐹₄𝑅₂𝑅₂×𝑒((𝑡ℎ(𝑒((𝑡𝑒(𝑡)𝑟((𝑡𝑡−𝜏))𝑡−𝜏)))≤0,𝑡,𝑒))^𝑇𝐺₁𝑅₃−𝐺₂𝑅₃−𝐺₂𝑅₃𝑅₃𝑒(𝑡)𝑟(𝑡,𝑒(𝑡))≤0. (26) It follows from (23)–(26) that 𝑉̇(𝑡)≤𝑒^𝑇(𝑡)𝑋₁₂+𝑋^𝑇₁₂+𝑃₁+𝛿²𝑃₂+𝑃₃−𝑄₁𝐾𝐷−𝐷^𝑇𝐾^𝑇𝑄^𝑇₁−𝑄₃−𝑄^𝑇₃+𝜏𝑄₃𝑃₄⁻¹𝑄^𝑇₃−𝐹₃𝑅₁−𝐺₁𝑅₃𝑒(𝑡)+2𝑒^𝑇(𝑡)𝑋₁₁−𝐹₁𝑊₁+𝐹₂𝑊₂−𝑄₁−𝐷^𝑇𝐾^𝑇𝑄^𝑇₁𝑒(̇𝑡)+2𝑒^𝑇(𝑡)𝑄₃𝑒(𝑡−𝜏(𝑡))−2𝑒^𝑇(𝑡)𝑋₁₂+𝑄₁𝐶𝑒(𝑡−𝛿)+2𝑒^𝑇(𝑡)𝑋₂₂^𝑡_𝑡−𝛿𝑒(𝑠)𝑑𝑠+2𝑒^𝑇(𝑡)𝑄₁𝐴+𝐹₄𝑅₁ℎ(𝑒(𝑡))+2𝑒^𝑇(𝑡)𝑄₁𝐵ℎ(𝑒(𝑡−𝜏(𝑡)))+2𝑒^𝑇(𝑡)−𝑄₁𝐾+𝐺₂𝑅₃𝑟((𝑡+𝑒̇𝑡,𝑒))_𝑇(𝑡)−𝑄₁−𝑄^𝑇₁+𝜏𝑃₄𝑒(̇𝑡)−2𝑒^𝑇̇(𝑡)𝑄₁()𝑒̇𝐶𝑒𝑡−𝛿+2^𝑇(𝑡)𝑋₁₂^𝑡_𝑡−𝛿𝑒(𝑠)𝑒̇𝑑𝑠+2^𝑇(𝑡)𝑊₁−𝑊₂+𝑄₁𝐴ℎ(𝑒(𝑡𝑒̇))+2^𝑇(𝑡)𝑄₁(𝑒((𝑡𝑒̇𝐵ℎ𝑡−𝜏)))−2^𝑇(𝑡)𝑄₁𝐾𝑟(𝑡,𝑒(𝑡)+𝑒^𝑇((𝑡×𝑡−𝜏))−𝑄₂−𝑄^𝑇₂+𝜏𝑄₂𝑃₄⁻¹𝑄^𝑇₂−𝐹₃𝑅₂𝑒(𝑡−𝜏(𝑡))+2𝑒^𝑇(𝑡−𝜏(𝑡))𝑄₂𝑒(𝑡−𝜏)+2𝑒^𝑇(𝑡−𝜏(𝑡))𝐹₄𝑅₂ℎ(𝑒(𝑡−𝜏(𝑡)))−𝑒^𝑇(𝑡−𝜏)𝑃₃𝑒(𝑡−𝜏)−𝑒^𝑇(𝑡−𝛿)𝑃₁𝑒×(𝑡−𝛿)−2𝑒^𝑇(𝑡−𝛿)𝑋₂₂^𝑡_𝑡−𝛿𝑒(𝑠)−𝑑𝑠_{𝑡𝑡−𝛿}𝑒(𝑠)𝑑𝑠^𝑇𝑃₂^𝑡_𝑡−𝛿𝑒(𝑠)𝑑𝑠−ℎ^𝑇(𝑒(𝑡))𝑅₁ℎ(𝑒(𝑡))−ℎ^𝑇(𝑒(𝑡−𝜏(𝑡)))𝑅₂ℎ(𝑒(𝑡−𝜏(𝑡)))−𝑟^𝑇(𝑡,𝑒(𝑡))𝑅₃𝑟(𝑡,𝑒(𝑡))=𝜉^𝑇(𝑡)Π𝜉(𝑡), (27) where 𝜉(𝑡)=(𝑒^𝑇(𝑡), 𝑒^𝑇̇(𝑡), 𝑒^𝑇(𝑡−𝜏(𝑡)), 𝑒^𝑇(𝑡−𝜏), 𝑒^𝑇(𝑡−𝛿), ∫^𝑡_𝑡−𝛿𝑒^𝑇(𝑠)𝑑𝑠, ℎ^𝑇(𝑒(𝑡)), ℎ^𝑇(𝑒(𝑡−𝜏(𝑡))), 𝑟^𝑇(𝑡,𝑒(𝑡)))^𝑇,ΠΠ=¹¹Π₁₂𝑄₃0−𝑋₁₂−𝑄₁𝐶𝑋₂₂𝑄₁𝐴+𝐹₄𝑅₁𝑄₁𝐵−𝑄₁𝐾+𝐺₂𝑅₃∗Π₂₂00−𝑄₁𝐶𝑋₁₂𝑊₁−𝑊₂+𝑄₁𝐴𝑄₁𝐵−𝑄₁𝐾∗∗Π₃₃𝑄₂00𝐹₄𝑅₂00∗∗∗−𝑃₃00000∗∗∗∗−𝑃₁−𝑋₂₂000∗∗∗∗∗−𝑃₂000∗∗∗∗∗∗−𝑅₁00∗∗∗∗∗∗∗−𝑅₂0∗∗∗∗∗∗∗∗−𝑅₃ (28)with Π₁₁=𝑋₁₂+𝑋^𝑇₁₂+𝑃₁+𝛿²𝑃₂+𝑃₃−𝑄₁𝐾𝐷−𝐷^𝑇𝐾^𝑇𝑄^𝑇₁−𝑄₃−𝑄^𝑇₃+𝜏𝑄₃𝑃₄⁻¹𝑄^𝑇₃−𝐹₃𝑅₁−𝐺₁𝑅₃, Π₁₂=𝑋₁₁−𝐹₁𝑊₁+𝐹₂𝑊₂−𝑄₁−𝐷^𝑇𝐾^𝑇𝑄^𝑇₁, Π₂₂=−𝑄₁−𝑄^𝑇₁+𝜏𝑃₄, Π₃₃=−𝑄₂−𝑄^𝑇₂+𝜏𝑄₂𝑃₄⁻¹𝑄^𝑇₂−𝐹₃𝑅₂.
By using Lemma 3, and noting 𝐾=𝑄₁⁻¹𝑍, it is easy to verify the equivalence of Π<0 and Ω<0. Thus, one can derive from (14) and (27) that 𝑉̇(𝑡)≤0, (29) which implies that the error-state system (8) of the delayed neural networks described by (1) and (6) is globally asymptotically stable. The proof is completed.

4. Numerical Example

To verify the effectiveness of the theoretical result of this paper, consider the following example.

Example 1. Consider a two-neuron neural network (1), where ,(𝑡)=()(),𝑓𝐶=1.3000.9,𝐴=0.5−0.20.70.5𝐵=0.6−0.1−1.2−0.8,𝐽1.6cos21𝑡−1.3sin1.1𝑡₁(𝑥)=𝑓₂(𝑥)(𝑥),(𝑡)|(.=tanh𝛿=0.1,𝜏=0.27sin7𝑡)| (30)
Figure 1 shows that the considered neural network has a chaotic attractor, where the initial condition is 𝑥₁(𝑡)=0.5cos(0.5𝑡), 𝑥₂(𝑡)=−0.2sin(18𝑡), and 𝑡∈[−0.27,0].
It can be verified that assumptions (H1) and (H2) are satisfied, and 𝐹₁=0, 𝐹₂=𝐼, 𝐹₃=0, 𝐹₄=diag{0.5,0.5}, 𝜏=0.27.
Choose network measurement (4), where ,𝑔((𝑡(𝑡)𝐷=0.7−0.20.10.5𝑡,𝑥))=0.1cos𝑥+0.3. (31)
It is obvious that assumption (H3) is satisfied with 𝐺₁=−0.01𝐼 and 𝐺₂=0. By the Matlab LMI Control Toolbox, we find a solution to the LMIs in (13) and (14) as follows: 𝑃₁=0.50420.00680.00680.1895,𝑃₂=,𝑃2.74640.08960.08962.0468₃=0.3288−0.0027−0.00270.1564,𝑃₄=,𝑊0.19070.01380.01380.1027₁=0.1184000.0779,𝑊₂=,𝑅0.2313000.1183₁=0.9466000.4360,𝑅₂=,𝑅0.4704000.2050₃=2.9274003.4376,𝑄₁=,𝑄0.21680.01340.01440.0812₂=0.20250.00320.00320.1054,𝑄₃=,𝑋−0.01340.0159−0.01620.0155₁₁=0.47530.01230.01230.1554,𝑋₁₂=,𝑋−0.5667−0.0378−0.0362−0.2141₂₂=.0.83760.05080.05080.4503,𝑍=0.58580.1452−0.02490.4239 (32)
Subsequently, we can obtain from 𝐾=𝑄₁⁻¹𝑍 that 𝐾=2.75110.3508−0.79495.1560. (33)
Therefore, we know from Theorem 4 that error-state system (8) of the delayed neural network described by (1) and (6) is globally asymptotically stable. The simulation results are shown in Figures 2 and 3, which demonstrate the effectiveness of the developed approach for the design of the state estimator for neural networks with leakage delay and time-varying delay.

Figure 1 

Phase trajectory of model (1).

Figure 2 

The responses of the true state 𝑥1(𝑡) (solid line) and its estimation 𝑢1(𝑡) (dash line).

Figure 3 

The responses of the true state 𝑥2(𝑡) (solid line) and its estimation 𝑢2(𝑡) (dash line).

5. Conclusions

In this paper, the state estimation problem has been investigated for neural networks with leakage delay and time-varying delay as well as general activation functions. By employing Lyapunov functional method and the matrix inequality techniques, a delay-dependent LMIs condition has been established to estimate the neuron state with some observed output measurements such that the error-state system is globally asymptotically stable. An example has been provided to show the effectiveness of the proposed criterion.

Acknowledgments

The authors would like to thank the reviewers and the editor for their valuable suggestions and comments which have led to a much improved paper. This work was supported by the National Natural Science Foundation of China under Grants 61273021, 60974132, 11172247, and 51208538 and in part by the Natural Science Foundation Project of CQ cstc2013jjB40008.