New Improved Exponential Stability Criteria for Discrete-Time Neural Networks with Time-Varying Delay

Abstract

The robust stability of uncertain discrete-time recurrent neural networks with time-varying delay is investigated. By decomposing some connection weight matrices, new Lyapunov-Krasovskii functionals are constructed, and serial new improved stability criteria are derived. These criteria are formulated in the forms of linear matrix inequalities (LMIs). Compared with some previous results, the new results are less conservative. Three numerical examples are provided to demonstrate the less conservatism and effectiveness of the proposed method.

1. Introductionn

In recent years, recurrent neural networks (see [1–7]), such as Hopfield neural networks, cellular neural networks, and other networks have been widely investigated and successfully applied in all kinds of science areas such as pattern recognition, image processing, and fixed-point computation. However, because of the finite switching speed of neurons and amplifiers, time delay is unavoidable in nature and technology. It can make important effects on the stability of dynamic systems. Thus, the studies on stability are of great significance. There has been a growing research interest on the stability analysis problems for delayed neural networks, and many excellent papers and monographs have been available. On the other hand, during the design of neural network and its hardware implementation, the convergence of a neural network may often be destroyed by its unavoidable uncertainty due to the existence of modeling error, the deviation of vital data, and so on. Therefore, the studies on robust convergence of delayed neural network have been a hot research direction. Up to now, many sufficient conditions, either delay-dependent or delay-independent, have been proposed to guarantee the global robust asymptotic or exponential stability for different class of delayed neural networks (see [8–13]).

It is worth pointing out that most neural networks have been assumed to be in continuous time, but few in discrete time. In practice, the discrete-time neural networks are more applicable to problems that are inherently temporal in nature or related to biological realities. And they can ideally keep the dynamic characteristics, functional similarity, and even the physical or biological reality of the continuous-time networks under mild restriction. Thus, the stability analysis problems for discrete-time neural networks have received more and more interest, and some stability criteria have been proposed in literature (see [14–25]). In [14], Liu et al. researched a class of discrete-time RNNs with time-varying delay, and proposed a delay-dependent condition guaranteeing the global exponential stability. By using a similar technique to that in [21], the result obtained in [14] has been improved by Song and Wang in [15]. The results in [15] are further improved by Zhang et al. in [16] by introducing some useful terms. In [17], Yu et al. proposed a new less conservative result than that obtained in [16] via constructing a new augment Lyapunov-Krasovskii functional.

In this paper, the connection weight matrix 𝐶 is decomposed, and some new Lyapunov-Krasovskii functionals are constructed. Combined with linear matrix inequality (LMI) technique, serial new improved stability criteria are derived. Numerical examples show that these new criteria are less conservative than those obtained in [14–17].

Notation 1. The notations are used in our paper except where otherwise specified. ‖⋅‖ denotes a vector or a matrix norm; ℝ,ℝ^𝑛 are real and n-dimension real number sets, respectively; ℕ⁺ is nonnegative integer set. 𝐼 is identity matrix; ∗ represents the elements below the main diagonal of a symmetric block matrix; Real matrix 𝑃>0<(0) denotes that 𝑃 is a positive definite (negative definite) matrix; ℕ[𝑎,𝑏]={𝑎,𝑎+1,…,𝑏}; 𝜆_min(𝜆_max) denotes the minimum and maximum eigenvalue of a real matrix.

2. Preliminaries

Consider a discrete-time recurrent neural network with time-varying delays [17] described by Σ∶𝑥(𝑘+1)=𝐶(𝑘)𝑥(𝑘)+𝐴(𝑘)𝑓(𝑥(𝑘))+𝐵(𝑘)𝑓(𝑥(𝑘−𝜏(𝑘)))+𝐽,𝑘=1,2,…, (2.1) where 𝑥(𝑘)=[𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘)]^𝑇∈ℝ^𝑛 denotes the neural state vector; 𝑓(𝑥(𝑘))=[𝑓₁(𝑥₁(𝑘)),𝑓₂(𝑥₂(𝑘)),…,𝑓_𝑛(𝑥_𝑛(𝑘))]^𝑇, 𝑓(𝑥(𝑘−𝜏(𝑘)))=[𝑓₁(𝑥₁(𝑘−𝜏(𝑘))),𝑓₂(𝑥₂(𝑘−𝜏(𝑘))),…,𝑓_𝑛(𝑥_𝑛(𝑘−𝜏(𝑘)))]^𝑇 are the neuron activation functions; 𝐽=[𝐽₁,𝐽₂,…,𝐽_𝑛]^𝑇 is the external input vector; Positive integer 𝜏(𝑘) represents the transmission delay that satisfies 0<𝜏_𝑚≤𝜏(𝑘)≤𝜏_𝑀, where 𝜏_𝑚,𝜏_𝑀 are known positive integers representing the lower and upper bounds of the delay. 𝐶(𝑘)=𝐶+Δ𝐶(𝑘), 𝐴(𝑘)=𝐴+Δ𝐴(𝑘), 𝐵(𝑘)=𝐵+Δ𝐵(𝑘). 𝐶=diag(𝑐₁,𝑐₂,…,𝑐_𝑛) with |𝑐_𝑖|<1 describes the rate with which the 𝑖th neuron will reset its potential to the resting state in isolation when disconnected from the networks and external inputs; 𝐶,𝐴,𝐵∈ℝ^𝑛×𝑛 represent the weighting matrices; Δ𝐶(𝑘),Δ𝐴(𝑘),Δ𝐵(𝑘) denote the time-varying structured uncertainties which are of the following form: [Δ𝐶(𝑘),Δ𝐴(𝑘),Δ𝐵(𝑘)]=𝐾𝐹(𝑘)[𝐸_𝑐,𝐸_𝑎,𝐸_𝑏], (2.2) where 𝐾,𝐸_𝑐,𝐸_𝑎,𝐸_𝑏 are known real constant matrices with appropriate dimensions, 𝐹(𝑘) is unknown time-varying matrix function satisfying 𝐹^𝑇(𝑘)𝐹(𝑘)≤𝐼,forall𝑘∈ℕ⁺.

The nominal Σ₀ of Σ can be defined as Σ₀∶𝑥(𝑘+1)=𝐶𝑥(𝑘)+𝐴𝑓(𝑥(𝑘))+𝐵𝑓(𝑥(𝑘−𝜏(𝑘)))+𝐽,𝑘=1,2,…, (2.3)

To obtain our main results, we need to introduce the following assumption, definition and lemmas.

Assumption 1. For any 𝑥,𝑦∈ℝ, 𝑥≠𝑦, 𝜎⁻_𝑖≤𝑓_𝑖(𝑥)−𝑓_𝑖(𝑦)𝑥−𝑦≤𝜎⁺_𝑖,𝑖=1,2,…,𝑛, (2.4) where 𝜎⁻_𝑖,𝜎⁺_𝑖 are known constant scalars.
As pointed out in [16] under Assumption 1, system (2.3) has equilibrium points. Assume that 𝑥^∗=[𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛]^𝑇 is an equilibrium point of (2.3) and let 𝑦_𝑖(𝑘)=𝑥_𝑖(𝑘)−𝑥^∗_𝑖, 𝑔_𝑖(𝑦_𝑖(𝑘))=𝑓_𝑖(𝑦_𝑖(𝑘)+𝑥^∗_𝑖)−𝑓_𝑖(𝑥^∗_𝑖). Then, system (2.3), can be transformed into the following form: 𝑦(𝑘+1)=𝐶𝑦(𝑘)+𝐴𝑔(𝑦(𝑘))+𝐵𝑔(𝑦(𝑘−𝜏(𝑘))),𝑘=1,2,…, (2.5) where 𝑦(𝑘)=[𝑦₁(𝑘),𝑦₂(𝑘),…,𝑦_𝑛(𝑘)]^𝑇, 𝑔(𝑦(𝑘))=[𝑔₁(𝑦₁(𝑘)),𝑔₂(𝑦₂(𝑘)),…,𝑔_𝑛(𝑦_𝑛(𝑘))]^𝑇, 𝑔(𝑦(𝑘−𝜏(𝑘)))=[𝑔₁(𝑦₁(𝑘−𝜏(𝑘))),𝑔₂(𝑦₂(𝑘−𝜏(𝑘))),…,𝑔_𝑛(𝑦_𝑛(𝑘−𝜏(𝑘)))]^𝑇. From Assumption 1, for any 𝑥,𝑦∈ℝ, 𝑥≠𝑦, functions 𝑔_𝑖(⋅) satisfy 𝜎⁻_𝑖≤(𝑔_𝑖(𝑥)−𝑔_𝑖(𝑦))/(𝑥−𝑦)≤𝜎⁺_𝑖,𝑖=1,2,…,𝑛, and 𝑔_𝑖(0)=0.

Remark 2.1. Assumption 1 is widely used for dealing with the stability problem for neural networks. As pointed out in [13, 14, 16, 17, 26, 27], constants 𝜎⁻_𝑖,𝜎⁺_𝑖(𝑖=1,2,…,𝑛) can be positive, negative, and zero. Thus, this assumption is less restrictive than traditional Lipschitz condition.

Definition 2.2. The delayed discrete-time recurrent neural network in (2.5) is said to be globally exponentially stable if there exist two positive scalars  𝛼>0 and 0<𝛽<1 such that ‖𝑦(𝑘)‖≤𝛼⋅𝛽^𝑘sup_{𝑠∈ℕ[−𝜏𝑀,0]}‖𝑦(𝑠)‖,∀𝑘≥1. (2.6)

Lemma 2.3 (Tchebychev Inequality [28]). For any given vectors 𝑣_𝑖∈ℝ^𝑛,𝑖=1,2,…,𝑛, the following inequality holds: [^𝑛_𝑖=1𝑣_𝑖]^𝑇[^𝑛_𝑖=1𝑣_𝑖]≤𝑛^𝑛_𝑖=1𝑣^𝑇_𝑖𝑣_𝑖. (2.7)Lemma 2.4 (see [29]). For given matrices 𝑄=𝑄^𝑇,𝐻,𝐸 and 𝑅=𝑅^𝑇>0 of appropriate dimensions, then 𝑄+𝐻𝐹𝐸+𝐸^𝑇𝐹^𝑇𝐻^𝑇<0, (2.8) for all 𝐹 satisfying 𝐹^𝑇𝐹≤𝑅, if and only if there is an 𝜀>0, such that 𝑄+𝜀⁻¹𝐻𝐻^𝑇+𝜀𝐸^𝑇𝑅𝐸<0. (2.9)Lemma 2.5 (see [16]). If Assumption 1 holds, then for any positive-definite diagonal matrix 𝐷=diag(𝑑₁,𝑑₂,…,𝑑_𝑛)>0, the following inequality holds: 𝑔(𝑦(𝑘))^𝑇𝐷𝑔(𝑦(𝑘))−𝑦^𝑇(𝑘)𝐷(₁+₂)𝑔(𝑦(𝑘))+𝑦^𝑇(𝑘)₁𝐷₂𝑦(𝑘)≤0,𝑘∈ℕ⁺, (2.10) where ∏₁=diag(𝜎⁻₁,𝜎⁻₂,…,𝜎⁻_𝑛), ∏₂=diag(𝜎⁺₁,𝜎⁺₂,…,𝜎⁺_𝑛).Lemma 2.6 (see [30]). Given constant symmetric matrices Σ₁,Σ₂,Σ₃ where Σ^𝑇₁=Σ₁ and 0<Σ₂=Σ^𝑇₂, then Σ₁+Σ^𝑇₃Σ₂⁻¹Σ₃<0 if and only if Σ₁Σ^𝑇₃Σ₃−Σ₂<0,or,−Σ₂Σ₃Σ^𝑇₃Σ₁<0. (2.11)Lemma 2.7 (see [13]). Let 𝑁 and 𝐸 be real constant matrices with appropriate dimensions, matrix 𝐹(𝑘) satisfying 𝐹^𝑇(𝑘)𝐹(𝑘)≤𝐼, then, for any 𝜖>0, 𝐸𝐹(𝑘)𝑁+𝑁^𝑇𝐹^𝑇(𝑘)𝐸^𝑇≤𝜖⁻¹𝐸𝐸^𝑇+𝜖𝑁^𝑇𝑁,𝑘∈ℕ⁺.

3. Main Results

To obtain our main results, we decompose the connection weight matrix 𝐶 as follows: 𝐶=𝐶₁+𝐶₂. (3.1) Then, we can get the following stability results.

Theorem 3.1. For any given positive scalars 0<𝜏_𝑚<𝜏_𝑀, then, under Assumption 1, system (2.5) without uncertainty is globally exponentially stable for any time-varying delay 𝜏(𝑘) satisfying 𝜏_𝑚≤𝜏(𝑘)≤𝜏_𝑀, if there exist positive-definite matrices 𝑃₁,𝑄₁,𝑄₂,𝑄₃, positive-definite diagonal matrices 𝐷₁,𝐷₂,𝑄₄,𝑄₅,𝑄₆, and arbitrary matrices 𝑃_𝑖,𝐻_𝑖,𝑖=2,3,…,21 with appropriate dimensions, such that the following LMI holds: ΞΞ≜¹¹Ξ₁₂Ξ₁₃Ξ₁₄Ξ₁₅Ξ₁₆Ξ₁₇Ξ₁₈Ξ₁₉Ξ_1,10∗Ξ₂₂Ξ₂₃Ξ₂₄Ξ₂₅Ξ₂₆Ξ₂₇Ξ₂₈Ξ₂₉Ξ_2,10∗∗Ξ₃₃Ξ₃₄Ξ₃₅Ξ₃₆Ξ₃₇Ξ₃₈Ξ₃₉Ξ_3,10∗∗∗Ξ₄₄Ξ₄₅Ξ₄₆Ξ₄₇Ξ₄₈Ξ₄₉Ξ_4,10∗∗∗∗Ξ₅₅Ξ₅₆Ξ₅₇Ξ₅₈Ξ₅₉Ξ_5,10∗∗∗∗∗Ξ₆₆Ξ₆₇Ξ₆₈Ξ₆₉Ξ_6,10∗∗∗∗∗∗Ξ₇₇Ξ₇₈Ξ₇₉Ξ_7,10∗∗∗∗∗∗∗Ξ₈₈Ξ₈₉Ξ_8,10∗∗∗∗∗∗∗∗Ξ₉₉Ξ_9,10∗∗∗∗∗∗∗∗∗Ξ_10,10<0, (3.2) where Ξ₁₁=2𝐶^𝑇₁𝑃₁𝐶₁−𝐶^𝑇₁𝑃₁₂𝐶₂−𝐶^𝑇₂𝑃^𝑇₁₂𝐶₁−2𝑃₁+𝐻₁₂𝐶₂+𝐶^𝑇₂𝐻^𝑇₁₂+𝑄₂+𝑄₃+(𝜏_𝑀−𝜏_𝑚+1)𝑄₁+(1+𝜏_𝑚)𝑄₄+(1+𝜏_𝑀)𝑄₅+(𝜏_𝑀−𝜏_𝑚)𝑄₆−2₁𝐷₁₂Ξ₁₂,=𝐶^𝑇₂(𝐻₁₃−𝑃₁₃)^𝑇,Ξ₁₃=𝐶^𝑇₁(𝑃₁+𝑃₁₂)−𝐻₁₂+𝐶^𝑇₂(𝐻₁₄−𝑃₁₄−𝑃₁₂)^𝑇+𝐶^𝑇₁𝑃^𝑇₁Ξ₁₄,=𝐶^𝑇₁𝑃₂−𝐻₂+𝐶^𝑇₂(𝐻₁₅−𝑃₁₅)^𝑇,Ξ₁₅=−𝐶^𝑇₁𝑃₂+𝐻₂+𝐶^𝑇₂(𝐻₁₆−𝑃₁₆)^𝑇Ξ₁₆,=−𝐶^𝑇₁𝑃₁₂𝐴+𝐻₁₂𝐴+𝐶^𝑇₂(𝐻₁₇−𝑃₁₇)^𝑇+𝐷₁(₁+₂Ξ),₁₇=−𝐶^𝑇₁𝑃₁₂𝐵+𝐻₁₂𝐵+𝐶^𝑇₂(𝐻₁₈−𝑃₁₈)^𝑇,Ξ₁₈=−𝐶^𝑇₁𝑃₂+𝐻₂+𝐶^𝑇₂(𝐻₁₉−𝑃₁₉)^𝑇Ξ₁₉,=𝐶^𝑇₁𝑃₂−𝐻₂+𝐶^𝑇₂(𝐻₂₀−𝑃₂₀)^𝑇,Ξ_1,10=𝐶^𝑇₁𝑃₂−𝐻₂+𝐶^𝑇₂(𝐻₂₁−𝑃₂₁)^𝑇Ξ₂₂,=−𝑄₁−2₁𝐷₂₂,Ξ₂₃=𝑃₁₃−𝐻₁₃,Ξ₂₄=𝑃₃−𝐻₃Ξ₂₅,=−𝑃₃+𝐻₃,Ξ₂₆=−𝑃₁₃𝐴+𝐻₁₃Ξ𝐴,₂₇=−𝑃₁₃𝐵+𝐻₁₃𝐵+𝐷₂(₁+₂),Ξ₂₈=−𝑃₃+𝐻₃Ξ₂₉,=𝑃₃−𝐻₃,Ξ_2,10=𝑃₃−𝐻₃Ξ₃₃,=𝑃₁₂+𝑃₁₄−𝐻₁₄+𝑃^𝑇₁₂+𝑃^𝑇₁₄−𝐻^𝑇₁₄+2𝑃₁,Ξ₃₄=𝑃₂+𝑃₄−𝐻₄+𝑃^𝑇₁₅−𝐻^𝑇₁₅Ξ₃₅=𝐻₄−𝑃₂−𝑃₄+𝑃^𝑇₁₆−𝐻^𝑇₁₆,Ξ₃₆=(𝐻₁₄−𝑃₁₂−𝑃₁₄)𝐴+𝑃^𝑇₁₇−𝐻^𝑇₁₇Ξ₃₇,=(𝐻₁₄−𝑃₁₂−𝑃₁₄)𝐵+𝑃^𝑇₁₈−𝐻^𝑇₁₈,Ξ₃₈=𝐻₄−𝑃₂−𝑃₄+𝑃^𝑇₁₉−𝐻^𝑇₁₉Ξ₃₉,=𝑃₄+𝑃₂−𝐻₄+𝑃^𝑇₂₀−𝐻^𝑇₂₀,Ξ_3,10=𝑃₄+𝑃₂−𝐻₄+𝑃^𝑇₂₁−𝐻^𝑇₂₁Ξ₄₄,=𝑃₅−𝐻₅+𝑃^𝑇₅−𝐻^𝑇₅−𝑄₃,Ξ₄₅=𝐻₅−𝑃₅+𝑃^𝑇₆−𝐻^𝑇₆Ξ₄₆,=𝐻₁₅𝐴−𝑃₁₅𝐴+𝑃^𝑇₇−𝐻^𝑇₇,Ξ₄₇=𝐻₁₅𝐵−𝑃₁₅𝐵+𝑃^𝑇₈−𝐻^𝑇₈Ξ₄₈,=𝐻₅−𝑃₅+𝑃^𝑇₉−𝐻^𝑇₉,Ξ₄₉=𝑃₅−𝐻₅+𝑃^𝑇₁₀−𝐻^𝑇₁₀Ξ₄₉,=𝑃₅−𝐻₅+𝑃^𝑇₁₁−𝐻^𝑇₁₁Ξ₅₅,=𝐻₆−𝑃₆+𝐻^𝑇₆−𝑃^𝑇₆−𝑄₂,Ξ₅₆=𝐻₁₆𝐴−𝑃₁₆𝐴+𝐻^𝑇₇−𝑃^𝑇₇Ξ₅₇,=𝐻₁₆𝐵−𝑃₁₆𝐵+𝐻^𝑇₈−𝑃^𝑇₈,Ξ₅₈=𝐻₆−𝑃₆+𝐻^𝑇₉−𝑃^𝑇₉Ξ₅₉,=𝑃₆−𝐻₆+𝐻^𝑇₁₀−𝑃^𝑇₁₀,Ξ_5,10=𝑃₆−𝐻₆+𝐻^𝑇₁₁−𝑃^𝑇₁₁Ξ₆₆,=𝐻₁₇𝐴−𝑃₁₇𝐴+𝐴^𝑇𝐻^𝑇₁₇−𝐴^𝑇𝑃^𝑇₁₇−𝐷₁−𝐷^𝑇₁Ξ₆₇,=𝐻₁₇𝐵−𝑃₁₇𝐵+𝐴^𝑇𝐻^𝑇₁₈−𝐴^𝑇𝑃^𝑇₁₈Ξ₆₈,=𝐻₇−𝑃₇+𝐴^𝑇𝐻^𝑇₁₉−𝐴^𝑇𝑃^𝑇₁₉,Ξ₆₉=𝑃₇−𝐻₇+𝐴^𝑇𝐻^𝑇₂₀−𝐴^𝑇𝑃^𝑇₂₀Ξ_6,10,=𝑃₇−𝐻₇+𝐴^𝑇𝐻^𝑇₂₁−𝐴^𝑇𝑃^𝑇₂₁Ξ₇₇,=𝐻₁₈𝐵−𝑃₁₈𝐵+𝐵^𝑇𝐻^𝑇₁₈−𝐵^𝑇𝑃^𝑇₁₈−𝐷₂−𝐷^𝑇₂Ξ₇₈,=𝐻₈−𝑃₈+𝐵^𝑇𝐻^𝑇₁₉−𝐵^𝑇𝑃^𝑇₁₉,Ξ₇₉=𝑃₈−𝐻₈+𝐵^𝑇𝐻^𝑇₂₀−𝐵^𝑇𝑃^𝑇₂₀Ξ_7,10,=𝑃₈−𝐻₈+𝐵^𝑇𝐻^𝑇₂₁−𝐵^𝑇𝑃^𝑇₂₁Ξ₈₈,=𝐻₉−𝑃₉+𝐻^𝑇₉−𝑃^𝑇₉−(1+𝜏_𝑀)⁻¹𝑄₅,Ξ₈₉=𝑃₉−𝐻₉+𝐻^𝑇₁₀−𝑃^𝑇₁₀Ξ_8,10,=𝑃₉−𝐻₉+𝐻^𝑇₁₁−𝑃^𝑇₁₁Ξ_9,9,=𝑃₁₀−𝐻₁₀+𝑃^𝑇₁₀−𝐻^𝑇₁₀−(1+𝜏_𝑚)⁻¹𝑄₄,Ξ_9,10=𝑃₁₀−𝐻₁₀+𝐻^𝑇₁₁−𝑃^𝑇₁₁Ξ_10,10,=𝑃₁₁−𝐻₁₁+𝑃^𝑇₁₁−𝐻^𝑇₁₁−(𝜏_𝑀−𝜏_𝑚)⁻¹𝑄₆. (3.3)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32)

Proof. Construct a new augmented Lyapunov-Krasovskii functional candidate as follows: 𝑉(𝑘)=𝑉₁(𝑘)+𝑉₂(𝑘)+𝑉₃(𝑘)+𝑉₄(𝑘)+𝑉₅(𝑘)+𝑉₆(𝑘), (3.4) where 𝑉₁(𝑘)=2𝑌^𝑇𝑃(𝑘)¹0⋯000⋯0⋮⋮⋮⋮00⋯0_{10𝑛×10𝑛}𝑌(𝑘), (3.5)𝑌^𝑇(𝑘)=[𝑦^𝑇(𝑘),𝑦^𝑇(𝑘−𝜏(𝑘)),𝜂^𝑇(𝑘),𝑦^𝑇(𝑘−𝜏_𝑀),𝑦^𝑇(𝑘−𝜏_𝑚),𝑔^𝑇(𝑦(𝑘)),𝑔^𝑇(𝑦(𝑘−𝜏_𝑀)),  ∑^𝑘_{𝑖=𝑘−𝜏𝑀}𝑦^𝑇(𝑖),∑^𝑘_{𝑖=𝑘−𝜏𝑚}𝑦^𝑇(𝑖),∑^{𝑘−𝜏𝑚}_{𝑖=𝑘−𝜏𝑀+1}𝑦^𝑇(𝑖)]^𝑇,𝜂(𝑘)=𝑦(𝑘+1)−𝐶₁𝑦(𝑘); 0 is zero matrix with appropriate dimensions: 𝑉₂(𝑘)=^𝑘−1_{𝑖=k−𝜏(𝑘)}𝑦^𝑇(𝑖)𝑄₁𝑉𝑦(𝑖),₃(𝑘)=^𝑘−1_{𝑖=𝑘−𝜏𝑚}𝑦^𝑇(𝑖)𝑄₂𝑦(𝑖)+^𝑘−1_{𝑖=𝑘−𝜏𝑀}𝑦^𝑇(𝑖)𝑄₃𝑉𝑦(𝑖),₄(𝑘)=^𝑘−1_{𝑗=𝑘−𝜏𝑚}^𝑘−1_𝑖=𝑗𝑦^𝑇(𝑖)𝑄₄𝑦(𝑖)+^𝑘−1_{𝑗=𝑘−𝜏𝑀}^𝑘−1_𝑖=𝑗𝑦^𝑇(𝑖)𝑄₅𝑉𝑦(𝑖),₅(𝑘)=^{𝑘−𝜏𝑚}_{𝑗=𝑘−𝜏𝑀+1}^𝑘−1_𝑖=𝑗𝑦^𝑇(𝑖)𝑄₁𝑉𝑦(𝑖),₆(𝑘)=^{𝑘−𝜏𝑚}_{𝑗=𝑘−𝜏𝑀+1}^𝑘−1_𝑖=𝑗𝑦^𝑇(𝑖)𝑄₆𝑦(𝑖). (3.6) Set 𝑌^𝑇(𝑘+1)=[𝑦^𝑇(𝑘+1),𝑦^𝑇(𝑘−𝜏(𝑘)),𝜂^𝑇(𝑘),𝑦^𝑇(𝑘−𝜏_𝑀),𝑦^𝑇(𝑘−𝜏_𝑚),𝑔^𝑇(𝑦(𝑘)),𝑔^𝑇(𝑦(𝑘−𝜏_𝑀)),∑^𝑘_{𝑖=𝑘−𝜏𝑀}𝑦^𝑇(𝑖), ∑^𝑘_{𝑖=𝑘−𝜏𝑚}𝑦^𝑇(𝑖), ∑^{𝑘−𝜏𝑚}_{𝑖=𝑘−𝜏𝑀+1}𝑦^𝑇(𝑖)]^𝑇=[𝑦^𝑇(𝑘)𝐶^𝑇₁+𝜂^𝑇(𝑘),𝜂^𝑇(𝑘),𝑦^𝑇(𝑘−𝜏_𝑀), 𝑦^𝑇(𝑘−𝜏_𝑚), 𝑔^𝑇(𝑦(𝑘)),𝑔^𝑇(𝑦(𝑘−𝜏_𝑀)),∑^𝑘_{𝑖=𝑘−𝜏𝑀}𝑦^𝑇(𝑖), ∑^𝑘_{𝑖=𝑘−𝜏m}𝑦^𝑇(𝑖),∑^{𝑘−𝜏𝑚}_{𝑖=𝑘−𝜏𝑀+1}𝑦^𝑇(𝑖)]^𝑇. Define Δ𝑉(𝑘)=𝑉(𝑘+1)−𝑉(𝑘). Then along the solution of system (2.5) we have Δ𝑉₁(𝑘)=2𝑌^𝑇𝑃(𝑘+1)¹0⋯000⋯0⋮⋮⋮⋮00⋯0𝑌(𝑘+1)−2𝑌^𝑇𝑃(𝑘)¹𝑌0⋯000⋯0⋮⋮⋮⋮00⋯0𝑌(𝑘)=2^𝑇𝑃(𝑘+1)¹0⋯000⋯0⋮⋮⋮⋮00⋯0𝑌(𝑘+1)−2𝑌^𝑇𝑃(𝑘)¹0⋯000⋯0⋮⋮⋮⋮00⋯0𝑌(𝑘)≜2𝐼₁−2𝐼₂,𝐼₁=𝑌^𝑇𝐶(𝑘)^𝑇1000000000𝑃0𝐼00000000𝐼0𝐼0000000000𝐼0000000000𝐼0000000000𝐼0000000000𝐼0000000000𝐼0000000000𝐼0000000000𝐼¹000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000𝑌(𝑘+1)=𝑌^𝑇𝐶(𝑘)^𝑇1000000000𝑃0𝐼00000000𝐼0𝐼0000000000𝐼0000000000𝐼0000000000𝐼0000000000𝐼0000000000𝐼0000000000𝐼0000000000𝐼¹𝑃₂𝑃₁₂0𝑃₃𝑃₁₃0𝑃₄𝑃₁₄0𝑃₅𝑃₁₅0𝑃₆𝑃₁₆0𝑃₇𝑃₁₇0𝑃₈𝑃₁₈0𝑃₉𝑃₁₉0𝑃₁₀𝑃₂₀0𝑃₁₁𝑃₂₁00.𝑦(𝑘+1) (3.7)(3.8) On the other hand, since 𝜂(𝑘)−𝐶₂𝑦(𝑘)−𝐴𝑔(𝑦(𝑘))−𝐵𝑔(𝑦(𝑘−𝜏(𝑘)))−𝐾Υ(𝑘)=0, ∑^𝑘_{𝑖=𝑘−𝜏𝑚}𝑦(𝑖)−∑^𝑘_{𝑖=𝑘−𝜏𝑀}𝑦(𝑖)+∑^{𝑘−𝜏𝑚}_{𝑖=𝑘+1−𝜏𝑀}𝑦(𝑖)−𝑦^𝑇(𝑘−𝜏_𝑚)+𝑦^𝑇(𝑘−𝜏_𝑀)=0, we have 00=𝐶𝑦(𝑘+1)¹𝑦(𝑘)+𝜂(𝑘)_{𝑘𝑖=𝑘−𝜏𝑚}𝑦(𝑖)−^𝑘_{𝑖=𝑘−𝜏𝑀}𝑦(𝑖)+^{𝑘−𝜏𝑚}_{𝑖=𝑘+1−𝜏𝑀}𝑦(𝑖)−𝑦^𝑇(𝑘−𝜏_𝑚)+𝑦^𝑇(𝑘−𝜏_𝑀)𝜂(𝑘)−𝐶₂𝑦(𝑘)−𝐴𝑔(𝑦(𝑘))−𝐵𝑔(𝑦(𝑘−𝜏(𝑘)))−=𝐶𝐾Υ(𝑘)₁0𝐼00000000000𝐼−𝐼00−𝐼𝐼𝐼0−𝐶₂0𝐼00−𝐴−𝐵000−𝐾𝑌(𝑘), (3.38)𝐼₂=𝑌^𝑇𝑃(𝑘)¹𝐻₂𝐻₁₂0𝐻₃𝐻₁₃0𝐻₄𝐻₁₄0𝐻₅𝐻₁₅0𝐻₆𝐻₁₆0𝐻₇𝐻₁₇0𝐻₈𝐻₁₈0𝐻₉𝐻₁₉0𝐻₁₀𝐻₂₀0𝐻₁₁𝐻₂₁0𝐻₂₂𝐻₂₃𝐼0000000000000𝐼−𝐼00−𝐼𝐼𝐼0−𝐶₂0𝐼00−𝐴−𝐵000−𝐾𝑌(𝑘). (3.39) Noting that Υ^𝑇(𝑘)Υ(𝑘)≤[𝑦^𝑇(𝑘),𝑔^𝑇(𝑦(𝑘)),𝑔^𝑇(𝑦(𝑘−𝜏(𝑘)))]^𝑇×𝐸^𝑇_𝑐𝐸_𝑐000𝐸^𝑇_𝑎𝐸_𝑎000𝐸^𝑇_𝑏𝐸_𝑏.y(𝑘)𝑔(𝑦(𝑘))𝑔(𝑦(𝑘−𝜏(𝑘))) (3.40) Combining (3.12)–(3.17), (3.36)–(3.40), similar to the proof of Theorem 3.1, one can easily obtain this result, which completes the proof.

Remark 3.10. Compared with the augmented Lyapunov functional constructed in Theorem 3.1, this new augmented Lyapunov functional include the term Υ(𝑘), which makes the conservatism of the stability criterion be reduced further (details for more, see Example 4.2).

4. Numerical Examples

In this section, three numerical examples will be presented to show the validity of the main results derived above.

Example 4.1. For the convenience of comparison, let us consider a delayed discrete-time recurrent neural network in (2.5) with parameters given by 𝐶=0.8000.9,𝐴=0.001000.005,𝐵=−0.10.01−0.2−0.1. (4.1) The activation functions are given by 𝑔₁(𝑥)=𝑔₂(𝑥)=tanh(𝑥). It is easy to see that the activation functions satisfy Assumption 1 with 𝜎⁻₁=𝜎⁻₂=0, 𝜎⁺₁=𝜎⁺₂=1. For 𝜏_𝑚=2,4,6,8,10,20, references [15–17] gave out the allowable upper bound 𝜏_𝑀 of the time-varying delay, respectively. Decompose matrix 𝐶 as 𝐶=𝐶₁+𝐶₂, where 𝐶₁=0.4000.5,𝐶₂=0.4000.4. (4.2) Table 1 shows that our results are less conservative than these previous results.

Table 1 

Allowable upper bounds 𝜏_𝑀 for given 𝜏_𝑚.

Example 4.2. Consider an uncertain delayed discrete-time recurrent neural network in (2.1) with parameters given by ,𝐸𝐶=0.25000.1,𝐴=0.120.24−0.150.2,𝐵=−0.250.10.020.09,𝐾=0.2000.3_𝑐=0.150.10−0.7,𝐸_𝑎=0.10.3−0.20.05,𝐸_𝑏=00.0.130.06−0.050.15,𝐽= (4.3) The activation functions are given by 𝑓₁(𝑥)=tanh(0.55𝑥)+sin(0.45𝑥), 𝑓₂(𝑥)=tanh(0.65𝑥)+sin(0.45𝑥). It is easy to see that the activation functions satisfy Assumption 1 with 𝜎⁻₁=0.1,𝜎⁻₂=0.2, 𝜎⁺₁=1,𝜎⁺₂=1.1. For 𝜏_𝑚=2,4,6,8,10, references [16, 17] gave out the allowable upper bound 𝜏_𝑀 of the time-varying delay, respectively. Decompose matrix 𝐶 as 𝐶=𝐶₁+𝐶₂, where 𝐶₁=0.2−10.0120.05,𝐶₂=0.051−0.0120.05. (4.4) Set 𝜖=50, by using the MATLAB toolbox, the allowable upper bounds 𝜏_𝑀 for given 𝜏_𝑚 are showed in Table 2. Obviously, our results are less conservative than these previous results.

Table 2 

Allowable upper bounds 𝜏_𝑀 for given 𝜏_𝑚.

Example 4.3. Consider an uncertain delayed discrete-time recurrent neural network in (2.1) with parameters given by ,𝐸𝐶=0.8000.9,𝐴=0.070.100.05,𝐵=−0.10.01−0.2−0.1,𝐾=0.02000.03_𝑐=0.150.10−0.7,𝐸_𝑎=0.10.3−0.20.05,𝐸_𝑏=00.0.130.06−0.050.15,𝐽= (4.5) And the activation functions are the same as given in Example 4.2. Decompose matrix 𝐶 as 𝐶=𝐶₁+𝐶₂, where 𝐶₁=0.4000.5,𝐶₂=0.4000.4. (4.6) Set 𝜖=50, by using the MATLAB toolbox, the allowable upper bounds 𝜏_𝑀 for given 𝜏_𝑚 are showed in Table 3.
The free-weighting matrices are obtained as follows when 𝜏_𝑚=2,𝜏_𝑀=60: 𝑄₁=0.0074−0.0002−0.00020.0007,𝑄₂=𝑄0.0627−0.0191−0.01910.0094₃,=0.0627−0.0191−0.01910.0094,𝐷₁=𝑄0.8703000.2672₄,=0.0079000.0012,𝑄₅𝑄=1.0𝑒−0030.2326000.0303₆,=1.0𝑒−0030.2493000.0320,𝑃₁=𝑃6.3936−0.9854−0.98540.2599₅,=−304.8096−16.0366−131.0952−930.9992,𝐷₂=𝑃0.0612000.0083₆,=1.0𝑒+0031.19310.10970.33650.0269,𝑃₉𝑃=1.0𝑒+003−1.4764−0.1509−0.07911.0626₁₀,=𝑃−350.0434−852.4925−26.8470904.2654₁₁,=1.0𝑒+003−0.5682−0.0257−0.04131.1255,𝑃₁₂𝑃=1.0𝑒+004−0.6548−1.62950.93994.7692₁₃,𝑃=1.0𝑒+0032.2001−3.7370−0.3034−0.3479₁₄,=1.0𝑒+0040.31860.7851−0.4972−2.3726,𝑃₁₇=𝑃273.4789−200.5789−165.3251−31.3854₁₈,𝐻=1.0𝑒+0030.0888−3.84541.60804.3515₅,=−304.4622−16.0335−131.0922−930.6433,𝐻₆𝐻=1.0𝑒+0031.19270.10970.33650.0265₉,𝐻=1.0𝑒+003−1.4768−0.1509−0.07911.0622₁₀,=−349.6840−852.4933−26.8478904.6230,𝐻₁₁𝐻=1.0𝑒+003−0.5678−0.0257−0.04131.1259₁₂,𝐻=1.0𝑒+004−0.2616−0.65220.46982.3832₁₃,=1.0𝑒+0032.2001−3.7370−0.3034−0.3479,𝐻₁₄𝐻=1.0𝑒+004−0.3344−0.84430.44222.4005₁₇,=1.0𝑒+004274.5315−200.9016−167.3579−33.2092,𝐻₁₈𝑃=1.0𝑒+0030.0893−3.84401.59704.3553₂,=𝑃₃=𝑃₄=𝑃₇=𝑃₈=𝑃₁₅=𝑃₁₆=𝑃₁₉=𝑃₂₀=𝑃₂₁=𝐻₂=𝐻₃=𝐻₄=𝐻₇=𝐻₈=𝐻₁₅=𝐻₁₆=𝐻₁₉=𝐻₂₀=𝐻₂₁=0. (4.7)(33)(34)(35)(36)(37)(38)(39)(40)(41)(42)(43)(44)(45)(46)(47)(48)(49)(50)

Table 3 

Allowable upper bounds 𝜏_𝑀 for given 𝜏_𝑚.

5. Conclusion

By decomposing some connection weight matrices , combined with linear matrix inequality (LMI) technique, some new augmented Lyapunov-Krasovskii functionals are constructed, and serial new improved sufficient conditions ensuring exponential stability or robust exponential stability are obtained. Numerical examples show that the new criteria derived in this paper are less conservative than some previous results obtained in the references cited therein.

Acknowlegments

This work was supported by the program for New Century Excellent Talents in University (NCET-06-0811) and the Research Fund for the Doctoral Program of Guizhou College of Finance and Economics (200702).