Delay-Dependent Stability Criteria of Uncertain Periodic Switched Recurrent Neural Networks with Time-Varying Delays

Abstract

This paper deals with the problem of delay-dependent stability criterion of uncertain periodic switched recurrent neural networks with time-varying delays. When uncertain discrete-time recurrent neural network is a periodic system, it is expressed as switched neural network for the finite switching state. Based on the switched quadratic Lyapunov functional approach (SQLF) and free-weighting matrix approach (FWM), some linear matrix inequality criteria are found to guarantee the delay-dependent asymptotical stability of these systems. Two examples illustrate the exactness of the proposed criteria.

1. Introduction

Recurrent neural networks (RNNs) are a very important tool for many application areas such as associative memory, pattern recognition, signal processing, model identification, and combinatorial optimization. With the development of research on RNNs in theory and application, the model is more and more complex. When the continuous-time RNNs are simulated using computer, they should be discretized into discrete-time RNNs [1–3]. Simultaneously, in implementations of artificial neural networks, time-varying delay may occur due to finite switching speeds of the amplifiers and communication time [4, 5]. Therefore, researchers have considered that discrete-time RNNs with time-varying delay are incorporated in the processing and/or transmission parts of the network architectures [6–9]. Parameter uncertainties and nonautonomous phenomena often exist in real systems due to modeling inaccuracies [4]. Particularly when we consider a long-term dynamical behaviors of the system and consider seasonality of the changing environment, the parameters of the system usually will change with time [10–14]. In order to model those systems with neural networks, the uncertain (or switched or jumping) neural networks with time-varying delay appear in many papers [6, 15–24]. So in this paper we consider the stability of the following discrete-time recurrent neural networks with time-varying delay: 𝑢(𝑘+1)=+(𝐴𝑊+Δ𝐴(𝑘))𝑢(𝑘)+𝑊₁+Δ𝑊₁(𝑘)𝑔(𝑢(𝑘))₂+Δ𝑊₂(𝑘)𝑓(𝑢(𝑘−𝑑(𝑘)))+(𝐼+Δ𝐼(𝑘)),(1.1) where 𝑢(𝑘)={𝑢₁(𝑘),𝑢₂(𝑘),…,𝑢_𝑛(𝑘)}∈𝑅^𝑛 is the state vector associated with 𝑛 neurons, 𝐴=diag{𝑎₁,𝑎₂,…,𝑎_𝑛} is a diagonal matrix with positive entries, 𝑊₁ and 𝑊₂ are, respectively, the connection weight matrix and the delayed connection weight matrix, 𝐼 is input vector, 𝑔(𝑢)={𝑔₁(𝑢),𝑔₂(𝑢),…,𝑔_𝑛(𝑢)} and 𝑓(𝑢)={𝑓₁(𝑢),𝑓₂(𝑢),…,𝑓_𝑛(𝑢)} are the neuron activation function vectors, and 𝑑(𝑘) is nonnegative differential time-varying functions which denote the time delays and satisfy0≤𝑑₁≤𝑑(𝑘)≤𝑑₂.(1.2)

In most literatures it is required that parameter uncertainty matrices, such as Δ𝐴(𝑘), Δ𝑊₁(𝑘), Δ𝑊₂(𝑘), and Δ𝐼(𝑘), should be in the formΔ𝐴(𝑘)Δ𝑊₁(𝑘)Δ𝑊₂(𝑘)Δ𝐼(𝑘)=𝐷𝐹(𝑘)𝐸_𝑎𝐸_𝑤1𝐸_𝑤2𝐸_𝑖,(1.3) where 𝐸_𝑎, 𝐸_𝑤1, 𝐸_𝑤2, and 𝐸_𝑖 are given constant matrices of appropriate dimensions and 𝐹(𝑘) is an uncertain matrix such that𝐹^𝑇(𝑘)𝐹(𝑘)≤𝐼.(1.4) In practice, however, Δ𝐴(𝑘), Δ𝑊₁(𝑘), Δ𝑊₂(𝑘), and Δ𝐼(𝑘) are generally difficult to have the decomposition of matrices for 𝐷, 𝐸_𝑎, 𝐸_𝑤1, 𝐸_𝑤2, and 𝐸_𝑖. In addition, periodic oscillation in recurrent neural networks is an interesting dynamic behavior as many biological and cognitive activities require repetition [7, 10, 11, 25]. Simultaneously, periodic oscillations in recurrent neural networks have been found in many applications such as associative memories, pattern recognition, machine learning, and robot motion control [25]. So, if (1.1) is an uncertain periodic neural network in which the period is less than a constant 𝑏, then (1.1) can be expressed as switched neural network for the finite switching state, that is, if (1.1) is a neural network with period 𝑎 (0<𝑎≤𝑏), then Δ𝐴(𝑘)=Δ𝐴(𝑘+𝑎),Δ𝑊₁(𝑘)=Δ𝑊₁(𝑘+𝑎),Δ𝑊₂(𝑘)=Δ𝑊₂(𝑘+𝑎), and Δ𝐼(𝑘)=Δ𝐼(𝑘+𝑎), which is corresponding to a switched neural network set: Ω=⋃^𝑏𝑎=1𝑈_𝑎, where 𝑈_𝑎={𝑆_𝑎𝑖=(𝐴_𝑎𝑖,𝑊_1𝑎𝑖,𝑊_2𝑎𝑖,𝐼_𝑎𝑖)∣0<𝑖≤𝑎}, 𝐴_𝑎𝑖=𝐴+Δ𝐴(𝑘+𝑖), 𝑊_1𝑎𝑖=𝑊₁+Δ𝑊₁(𝑘+𝑖), 𝑊_2𝑎𝑖=𝑊₂+Δ𝑊₂(𝑘+𝑖), and 𝐼_𝑎𝑖=𝐼+Δ𝐼(𝑘+𝑖). Suppose 𝑁 is the number of elements of Ω; then (1.1) is actually modified by𝑢(𝑘+1)=𝐴_𝑟(𝑘)𝑢(𝑘)+𝑊_1𝑟(𝑘)𝑔(𝑢(𝑘))+𝑊_2𝑟(𝑘)𝑓(𝑢(𝑘−𝑑(𝑘)))+𝐼_𝑟(𝑘),(1.5) where 𝑟(𝑘) is a switching rule defined by 𝑟(𝑘)∶𝑁→Ω with Ω={𝑆₁,𝑆₂,…,𝑆_𝑁}. Moreover, 𝑟(𝑘)=𝑗 means the sub-recurrent neural network (sub-RNN) 𝑆_𝑗, which is corresponding to (𝐴_𝑗,𝑊_1𝑗,𝑊_2𝑗,𝐼_𝑗), is active.

The dynamic behaviors of those models are foundations for applications. Under (1.5) most papers discuss the stability of uncertain neural networks with the common Lyapunov function approach [5, 6, 15–23, 25]. To the best of the authors’ knowledge, up to now, there is scarcely any paper that studies the uncertain periodic neural networks using the SQLF. This situation motivates this research.

Motivated by the above discussions, the authors intend to study a problem of the delay-dependent stability criterion of uncertain discrete-time recurrent neural networks with time-varying delays that the uncertain recurrent neural networks have a finite number of sub-RNNs, and the sub-RNNs may change from one to another according to arbitrary switching and restricted switching. The contributions of this paper are the following. (1) Using a switching graph, uncertain periodic recurrent neural networks with time-varying delays are transformed into switched recurrent neural networks; (2) the derivative of the SQLF (3.7) of the literature [8] is improved in (3.11), please see Remark 4.3 and Table 3; (3) based on the switching graph, the delay-dependent stability criteria of switched recurrent neural networks are studied by FWM and SQLF. Then an effective LMI approach is developed to solve the problem.

This paper is organized as follows. In Section 2, we give some basic definitions. We analyze the stability of the system (2.2) with the SQLF and FWM in Section 3. Some examples are given in Section 4. Section 5 offers the conclusions of this paper.

2. Preliminaries

In many electronic circuits, nonmonotonic functions can be more appropriate to describe the neuron activation in designing and implementing an artificial neural network [7]; hence, we have the following assumption.

For any 𝑗∈{1,2,…,𝑛}, there exist constants 𝑔^−𝑗, 𝑔^+𝑗, 𝑓^−𝑗, and 𝑓^+𝑗 such that 𝑔^−𝑗≤𝑔^𝑗𝜃₁−𝑔_𝑗𝜃₂𝜃₁−𝜃₂𝑓≤𝑔^+𝑗,∀𝜃₁,𝜃₂∈𝑅,_−𝑗≤𝑓^𝑗𝜃₁−𝑓_𝑗𝜃₂𝜃₁−𝜃₂≤𝑓^+𝑗,∀𝜃₁,𝜃₂∈𝑅.(2.1)

Under the assumption, the equilibrium points of UDNN (1.1) exist by the fixed point theorem [1]. In the following, let 𝑢^∘={𝑢^∘1,𝑢^∘2,…,𝑢^∘𝑛} be the equilibrium point of (1.1); then 𝑥(⋅)=𝑢(⋅)−𝑢^∘. The systems (1.1) and (1.5) are, respectively, shifted to the following form:𝑥(𝑘+1)𝑥=((𝑘𝐴++1Δ)=𝐴𝐴(𝑘))_𝑟(𝑘)𝑥(𝑘)+𝑊₁+Δ𝑊₁(𝑘)𝑔(𝑥(𝑘))+𝑊₂+Δ𝑊₂(𝑘)𝑓(𝑥(𝑘−𝜏(𝑘))),𝑥(𝑘)+𝑊_1𝑟(𝑘)𝑔(𝑥(𝑘))+𝑊_2𝑟(𝑘)𝑓(𝑥(𝑘−𝑑(𝑘))),𝑟(𝑘)∈Ω.(2.2) For convenience, the switching graph is defined.

Definition 2.1. Let Γ=(Ω,𝐖) be a switching graph, where Ω is the set of sub-RNNs 𝑆_𝑖 and 𝐖 is the set of weighted arcs 𝐰_𝑖𝑗∈{0,1}. 𝐰_𝑖𝑗=1 (or 0) represents the sub-RNN 𝑆_𝑖 switches (or does not switch) to the sub-RNN 𝑆_𝑗.

Remark 2.2. When 𝐰_𝑖𝑗=1, if ∑^𝑁𝑙=1𝐰_𝑗𝑙=0, that is sub-RNN 𝑆_𝑗 cannot switch to any other sub-RNN, we suppose that the uncertain neural networks will always stay in the sub-RNN 𝑆_𝑖 that means 𝐰_𝑖𝑖=1 and 𝐰_𝑖𝑗=0.

Throughout this paper, the superscript 𝑇 stands for the transpose of a matrix, 𝑃>0 means that the matrix 𝑃 is positive definite, and the symmetric terms in a symmetric matrix are denoted by ∗, for example, 𝐺₁𝑀∗𝑁𝑂=𝑂𝑀𝑁𝑂^𝑇,𝐺₂=diag𝑔⁻¹𝑔⁺¹,𝑔⁻²𝑔⁺²,…,𝑔^−𝑛𝑔^+𝑛,=diag−𝑔⁻¹+𝑔⁺¹2,−𝑔⁻²+𝑔⁺²2,…,−𝑔^−𝑛+𝑔^+𝑛𝐹₁2,𝐹₂=diag𝑓⁻¹𝑓⁺¹,𝑓⁻²𝑓⁺²,…,𝑓^−𝑛𝑓^+𝑛,=diag−𝑓⁻¹+𝑓⁺¹2,−𝑓⁻²+𝑓⁺²2,…,−𝑓^−𝑛+𝑓^+𝑛2.(2.3)

3. Asymptotical Stability of Uncertain Periodic Switched Recurrent Neural Networks

Theorem 3.1. Let 𝑑₁ and 𝑑₂ be positive integers such that 0≤𝑑₁≤𝑑₂. Based on a switching graph Γ, the system (2.2) is asymptotical stable if, when 𝐰_𝑖𝑗=1(𝑆_𝑖,𝑆_𝑗∈Ω), there exist the corresponding symmetric matrices 𝑃_𝑖=𝑃^𝑇𝑖>0, 𝑄₁=𝑄^𝑇1>0, 𝑄₂=𝑄^𝑇2>0, 𝑅=𝑅^𝑇>0, 𝑍₁=𝑍^𝑇1>0, 𝑍₂=𝑍^𝑇2>0, 𝑋^𝑖𝑗=(𝑋^𝑖𝑗)^𝑇≥0, 𝑈^𝑖𝑗=(𝑈^𝑖𝑗)^𝑇≥0, 𝐻^𝑖𝑗=diag{ℎ^1𝑖𝑗,ℎ^2𝑖𝑗,…,ℎ^𝑛𝑖𝑗}≥0, 𝑂^𝑖𝑗=diag{𝑜^1𝑖𝑗,𝑜^2𝑖𝑗,…,𝑜^𝑛𝑖𝑗}≥0, and any appropriate dimensional matrices 𝑁^𝑖𝑗, 𝑀^𝑖𝑗, and 𝑇^𝑖𝑗 such that the following LMIs hold: Φ^𝑖𝑗=⎡⎢⎢⎢⎢⎢⎢⎣Φ^𝑖𝑗11Φ^𝑖𝑗12Φ^𝑖𝑗13Φ^𝑖𝑗14Φ^𝑖𝑗15Φ^𝑖𝑗16∗∗Φ∗^𝑖𝑗22ΦΦ^𝑖𝑗23_𝑖𝑗33Φ^𝑖𝑗2400∗∗∗ΦΦ^𝑖𝑗34_𝑖𝑗44000−𝐹₂𝑂^𝑖𝑗∗∗∗∗Φ^𝑖𝑗55Φ^𝑖𝑗56∗∗∗∗∗Φ^𝑖𝑗66Λ⎤⎥⎥⎥⎥⎥⎥⎦<0,(3.1)_1𝑖𝑗=𝑋^𝑖𝑗𝑁^𝑖𝑗Λ∗𝑍₁>0,(3.2)_2𝑖𝑗=𝑈^𝑖𝑗𝑇^𝑖𝑗Λ∗𝑍₂>0,(3.3)_3𝑖𝑗=𝑋^𝑖𝑗+𝑈^𝑖𝑗𝑀^𝑖𝑗∗𝑍₁+𝑍₂>0,(3.4) where Φ^𝑖𝑗11=+𝐴^𝑇𝑖𝑁𝑃_𝑗𝐴_𝑖−𝑃_𝑖+𝑄₁+𝑄₂+𝑑₂−𝑑₁+1𝑅+𝐴_𝑖−𝐸^𝑇𝑑₂𝑍_1𝑗+𝑑₂−𝑑₁𝑍_2𝑗𝐴_𝑖−𝐸_1𝑖𝑗Φ_𝑖𝑗12^𝑇+𝑁^1𝑖𝑗+𝑑₂𝑋^𝑖𝑗11+𝑑₂−𝑑₁𝑈^𝑖𝑗11−𝐺₁𝐻^𝑖𝑗,Φ_𝑖𝑗13=𝑁^2𝑖𝑗^𝑇+𝑇^1𝑖𝑗+𝑑₂𝑋^𝑖𝑗12+𝑑₂−𝑑₁𝑈^𝑖𝑗12,Φ_𝑖𝑗14=𝑁^3𝑖𝑗^𝑇−𝑀^1𝑖𝑗+𝑑₂𝑋^𝑖𝑗13+𝑑₂−𝑑₁𝑈^𝑖𝑗13,Φ_𝑖𝑗15=−𝑁^1𝑖𝑗+𝑁^4𝑖𝑗^𝑇+𝑀^1𝑖𝑗−𝑇^1𝑖𝑗+𝑑₂𝑋^𝑖𝑗14+𝑑₂−𝑑₁𝑈^𝑖𝑗14,Φ_𝑖𝑗16=𝐴^𝑇𝑖𝑃_𝑗𝑊_1𝑖+𝐴_𝑖−𝐸^𝑇𝑑₂𝑍₁+𝑑₂−𝑑₁𝑍₂𝑊_1𝑖−𝐺₂𝐻^𝑖𝑗,Φ_𝑖𝑗22=𝐴^𝑇𝑖𝑃_𝑗𝑊_2𝑖+𝐴_𝑖−𝐸^𝑇𝑑₂𝑍₁+𝑑₂−𝑑₁𝑍₂𝑊_2𝑖,Φ_𝑖𝑗23=−𝑄₁−𝑅+𝑇^2𝑖𝑗^𝑇+𝑇^2𝑖𝑗+𝑑₂𝑋^𝑖𝑗22+𝑑₂−𝑑₁𝑈^𝑖𝑗22,Φ_𝑖𝑗24=−𝑀^2𝑖𝑗+𝑇^3𝑖𝑗^𝑇+𝑑₂𝑋^𝑖𝑗23+𝑑₂−𝑑₁𝑈^𝑖𝑗23,Φ_𝑖𝑗33=−𝑁^2𝑖𝑗+𝑇^4𝑖𝑗^𝑇+𝑀^2𝑖𝑗−𝑇^2𝑖𝑗+𝑑₂𝑋^𝑖𝑗24+𝑑₂−𝑑₁𝑈^𝑖𝑗24,Φ_𝑖𝑗34=−𝑄₂−𝑅−𝑀^3𝑖𝑗−𝑀^3𝑖𝑗^𝑇+𝑑₂𝑋^𝑖𝑗33+𝑑₂−𝑑₁𝑈^𝑖𝑗33,Φ_𝑖𝑗44=−𝑁^3𝑖𝑗−𝑀^4𝑖𝑗^𝑇+𝑀^3𝑖𝑗−𝑇^3𝑖𝑗+𝑑₂𝑋^𝑖𝑗34+𝑑₂−𝑑₁𝑈^𝑖𝑗34,Φ_𝑖𝑗55=−𝑁^4𝑖𝑗−𝑁^4𝑖𝑗^𝑇+𝑀^4𝑖𝑗+𝑀^4𝑖𝑗^𝑇−𝑇^4𝑖𝑗−𝑇^4𝑖𝑗^𝑇+𝑑₂𝑋^𝑖𝑗44+𝑑₂−𝑑₁𝑈^𝑖𝑗44−𝐹₁𝑂^𝑖𝑗,Φ_𝑖𝑗56=𝑊^𝑇1𝑖𝑃_𝑗𝑊_1𝑖+𝑊^𝑇1𝑖𝑑₂𝑍₁+𝑑₂−𝑑₁𝑍₂𝑊_1𝑖−𝐻^𝑖𝑗,Φ_𝑖𝑗66=𝑊^𝑇1𝑖𝑃_𝑗𝑊_2𝑖+𝑊^𝑇1𝑖𝑑₂𝑍₁+𝑑₂−𝑑₁𝑍₂𝑊_2𝑖,=𝑊^𝑇2𝑖𝑃_𝑗𝑊_2𝑖+𝑊^𝑇2𝑖𝑑₂𝑍₁+𝑑₂−𝑑₁𝑍₂𝑊_2𝑖−𝑂^𝑖𝑗.(3.5)

Proof. Suppose that 𝑦(𝑙)=𝑥(𝑙+1)−𝑥(𝑙); then we have 𝑥(𝑘+1)=𝑥(𝑘)+𝑦(𝑘) and 𝑥(𝑘)=𝑥(𝑘−𝑑(𝑘))+∑^{𝑘−1𝑖=𝑘−𝑑(𝑘)}𝑦(𝑙).
We consider the following SQLF: 𝑉_𝑟(𝑘)(𝑘)=𝑉𝑉_1𝑟(𝑘)(𝑘)+𝑉_2𝑟(𝑘)(𝑘)+𝑉_3𝑟(𝑘)(𝑘)+𝑉_4𝑟(𝑘)(𝑘),_1𝑟(𝑘)𝑉(𝑘)=𝑥^𝑇(𝑘)𝑃_𝑟(𝑘)𝑥(𝑘),_2𝑟(𝑘)(𝑘)=^𝑘−1_𝑙=_𝑘−𝑑1𝑥^𝑇(𝑙)𝑄_1𝑟(𝑘)𝑥(𝑙)+^𝑘−1_𝑙=_𝑘−𝑑2𝑥^𝑇(𝑙)𝑄_2𝑟(𝑘)𝑥(𝑙),(3.6)𝑉_3𝑟(𝑘)(𝑘)=^𝑘−𝑑1_{𝜃=𝑘−𝑑2}^𝑘−1_𝑙=𝜃𝑥^𝑇(𝑙)𝑅_𝑟(𝑘)𝑥(𝑙),(3.7)𝑉_4𝑟(𝑘)(𝑘)=⁻¹_{𝜃=−𝑑2}^𝑘−1_{𝑙=𝑘+𝜃}𝑦^𝑇(𝑙)𝑍_1𝑟(𝑘)𝑦(𝑙)+^−𝑑1−1_{𝜃=−𝑑2}^𝑘−1_{𝑙=𝑘+𝜃}𝑦^𝑇(𝑙)𝑍_2𝑟(𝑘)𝑦(𝑙).(3.8)
It is clear that the following equations are true: ^𝑘−¹_{𝑙=𝑘−𝑑2}𝑦^𝑇(𝑙)𝑍_1𝑟(𝑘)𝑦(𝑙)=^{𝑘−𝑑(𝑘)−1}_𝑙=_𝑘−𝑑2𝑦^𝑇(𝑙)𝑍_1𝑟(𝑘)𝑦(𝑙)+^𝑘−1_𝑙=𝑘_{−𝑑(𝑘)}𝑦^𝑇(𝑙)𝑍_1𝑟(𝑘)𝑦(𝑙),_{𝑘−𝑑1−1𝑙=}_𝑘−𝑑2𝑦^𝑇(𝑙)𝑍_2𝑟(𝑘)𝑦(𝑙)=^{𝑘−𝑑(𝑘)−1}_𝑙=_𝑘−𝑑2𝑦^𝑇(𝑙)𝑍_2𝑟(𝑘)𝑦(𝑙)+^{𝑘−𝑑1−1}_𝑙=𝑘_{−𝑑(𝑘)}𝑦^𝑇(𝑙)𝑍_2𝑟(𝑘)𝑦(𝑙).(3.9)
Firstly, we prove that under 𝐰_𝑖𝑗=1 the SQLF is less than 0. Suppose that 𝑟(𝑘)=𝑖 and 𝑟(𝑘+1)=𝑗, that means the sub-RNN 𝑆_𝑖 switches to the sub-RNN 𝑆_𝑗; we obtain Δ𝑉_1𝑖(𝑘)=𝑉_1𝑗(𝑘+1)−𝑉_1𝑖(𝑘)=𝐴_𝑖𝑥(𝑘)+𝑊_1𝑖𝑔(𝑥(𝑘))+𝑊_2𝑖𝑓(𝑥(𝑘−𝑑(𝑘)))^𝑇Δ𝑉_2𝑖×𝑃_𝑗𝐴_𝑖𝑥(𝑘)+𝑊_1𝑖𝑔(𝑥(𝑘))+𝑊_2𝑖𝑓(𝑥(𝑘−𝑑(𝑘)))−𝑥^𝑇(𝑘)𝑃_𝑖𝑥(𝑘),(𝑘)=−𝑉𝑥_2𝑗𝑇(𝑘+1)−𝑉_2𝑖(𝑘)=𝑥^𝑇(𝑘)𝑄_1𝑗+𝑄_2𝑗𝑥(𝑘)−𝑥^𝑇𝑘−𝑑₁𝑄_1𝑖𝑥𝑘−𝑑₁+𝑘−𝑑₂𝑄_2𝑖𝑥𝑘−𝑑₂+^𝑘−1_{𝑙=𝑘+1−𝑑1}𝑥^𝑇(𝑙)𝑄_1𝑗−𝑄_1𝑖𝑥(𝑙)_{𝑘−1𝑙=𝑘}_+1−𝑑2Δ𝑉_3𝑖𝑥^𝑇(𝑙)𝑄_2𝑗−𝑄_2𝑖𝑥(𝑙),(3.10)(𝑘)=−𝑉𝑥_3𝑗𝑇(𝑘+1)−𝑉_3𝑖(𝑘)≤𝑑₂−𝑑₁+1𝑥^𝑇(𝑘)𝑅_𝑗𝑥(𝑘)+𝑘−𝑑₁𝑅_𝑖𝑥𝑘−𝑑₁−𝑥^𝑇𝑘−𝑑₂𝑅_𝑖𝑥𝑘−𝑑₂−𝑥^𝑇(𝑘−𝑑(𝑘))𝑅_𝑖𝑥(𝑘−𝑑(𝑘))_{𝑘−1𝑙=𝑘}_+1−𝑑1𝑥^𝑇(𝑙)𝑅_𝑗−𝑅_𝑖𝑥(𝑙)+^𝑘−𝑑1_{𝜃=𝑘+1−𝑑2}^𝑘−1_𝑙=𝜃Δ𝑉_4𝑖𝑥^𝑇(𝑙)𝑅_𝑗−𝑅_𝑖𝑥(𝑙),(3.11)(𝑘)=−𝑉_4𝑗(𝑘+1)−𝑉_4𝑖(𝑘)=𝑑₂𝑦^𝑇(𝑘)𝑍_1𝑗𝑦(𝑘)+𝑑₂−𝑑₁𝑦^𝑇(𝑘)𝑍_2𝑗𝑦(𝑘)_{𝑘−1𝑙=𝑘}_{−𝑑(𝑘)}𝑦^𝑇(𝑙)𝑍_1𝑖𝑦(𝑙)−^{𝑘−𝑑1−1}_𝑙=𝑘_{−𝑑(𝑘)}−𝑦^𝑇(𝑙)𝑍_2𝑖𝑦(𝑙)_{𝑘−𝑑(𝑘)−1𝑙=}_𝑘−𝑑2𝑦^𝑇(𝑙)𝑍_1𝑖+𝑍_2𝑖𝑦(𝑙)+⁻¹_{𝜃=−𝑑2}^𝑘−1_{𝑙=𝑘+1+𝜃}+𝑦^𝑇(𝑙)𝑍_1𝑗−𝑍_1𝑖𝑦(𝑙)_−𝑑1−1_{𝜃=−𝑑2}^𝑘−1_{𝑙=𝑘+1+𝜃}𝑦^𝑇(𝑙)𝑍_2𝑗−𝑍_2𝑖𝑦(𝑙).(3.12)
In order to strictly guarantee Δ𝑉_2𝑖(𝑘)<0,∑^{𝑘−1𝑙=𝑘+1−𝑑}₁𝑥^𝑇(𝑙)(𝑄_1𝑗−𝑄_1𝑖)𝑥(𝑙) should be less than 0. In the switching graph Γ if there exists sub-RNN 𝑆_𝑖(𝑆_𝑖∈Ω), which satisfied ∑^𝑁𝑗=1𝐰_𝑖𝑗=0, which means the 𝑆_𝑖 cannot switch to any other sub-RNN, the equation 𝑄_1𝑖=𝑄_1𝑗 can be grounded, otherwise the switching sequence must be 𝛽={𝑆_𝛼1→𝑆_𝛼2→,…,→𝑆_𝛼𝑗→⋯,→𝑆_𝛼𝐿→𝑆_𝛼𝐿+1}(𝑆_𝛼𝑗∈Ω), and there exist 𝑙 such that 𝑆_𝛼𝐿+1=𝑆_𝛼𝑙(1≤𝑙<𝐿). Because the affection of the sub-RNNs 𝑆₁,𝑆₂…,𝑆_𝑙−1 and 𝑆_𝑙 on the whole system is before time 𝛼_𝑙, after 𝛼_𝑙 the 𝛽 changes to a periodic sequence 𝛽^={𝑆_𝛼𝑙→⋯,→𝑆_𝛼𝐿→𝑆_𝛼𝐿+1}. Suppose that 𝑖=𝛼_𝑙=𝛼_𝐿+1; then in switching sequence 𝛽^ the following LMIs all hold: 𝑄_1𝛼𝑙−𝑄𝑄_1𝛼𝑙+1≥0𝑄_1𝑖−𝑄_1(𝑙+1)≥0_1𝛼𝑙+1,𝑄−𝑄⋮_1𝛼𝑙+2≥0,_1𝛼𝐿−𝑄_1𝛼𝐿+1≥0𝑄_1(𝐿)−𝑄_1𝑖≥0;(3.13) then the solution of (3.13) is 𝑄_1𝑖=𝑄_1𝛼𝑙=𝑄_1𝛼𝑙+1=⋯=𝑄_1𝛼𝐿. Thus, we suppose that 𝑄₁=𝑄_1𝑖=𝑄_1𝑗.(3.14) Similar to ∑^{𝑘−1𝑙=𝑘+1−𝑑}₁𝑥^𝑇(𝑙)(𝑄_1𝑗−𝑄_1𝑖)𝑥(𝑙), together with ∑^{𝑘−1𝑙=𝑘+1−𝑑}₂𝑥^𝑇(𝑙)(𝑄_2𝑗−𝑄_2𝑖)𝑥(𝑙), ∑^{𝑘−1𝑙=𝑘+1−𝑑}₁𝑥^𝑇(𝑙)(𝑅_𝑗−𝑅_𝑖)𝑥(𝑙), ∑^𝑘−𝑑1_{𝜃=𝑘+1−𝑑2}∑^{𝑘−1𝑙=𝜃}𝑥^𝑇(𝑙)(𝑅_𝑗−𝑅_𝑖)𝑥(𝑙), ∑^{−1𝜃=−𝑑}₂∑^{𝑘−1𝑙=𝑘+1+𝜃}𝑦^𝑇(𝑙)(𝑍_1𝑗−𝑍_1𝑖)𝑦(𝑙) and ∑^{−𝑑1−1𝜃=−𝑑}₂∑^{𝑘−1𝑙=𝑘+1+𝜃}𝑦^𝑇(𝑙)(𝑍_2𝑗−𝑍_2𝑖)𝑦(𝑙), we suppose that 𝑄𝑅₂==𝑄𝑅_2𝑖𝑖=𝑄_2𝑗,𝑍=𝑅_𝑗,(3.15)₁𝑍=𝑍_1𝑖=𝑍_1𝑗,(3.16)₂=𝑍_2𝑖=𝑍_2𝑗.(3.17) On the other hand, for any appropriately dimensioned matrices 𝑁^𝑖𝑗, 𝑀^𝑖𝑗, and 𝑇^𝑖𝑗 the following equations are true: 2𝜉^𝑇(𝑘)𝑁^𝑖𝑗𝑥(𝑘)−𝑥(𝑘−𝑑(𝑘))−^𝑘−1_{𝑙=𝑘−𝑑(𝑘)}2𝜉_𝑇𝑦(𝑙)=0,(3.18)(𝑘)𝑀^𝑖𝑗⎡⎢⎢⎣𝑥(𝑘−𝑑(𝑘))−𝑥𝑘−𝑑₂2𝜉_𝑇−^{𝑘−𝑑(𝑘)−1}_{𝑙=𝑘−𝑑2}𝑦(𝑙)⎤⎥⎥⎦=0,(𝑘)𝑇^𝑖𝑗⎡⎢⎢⎣𝑥𝑘−𝑑₁−𝑥(𝑘−𝑑(𝑘))−^{𝑘−𝑑1−1}_{𝑙=𝑘−𝑑(𝑘)}𝑦(𝑙)⎤⎥⎥⎦=0,(3.19) where 𝜉(𝑘)=[𝑥^𝑇(𝑘),𝑥^𝑇(𝑘−𝑑₁),𝑥^𝑇(𝑘−𝑑₂),𝑥^𝑇(𝑘−𝑑(𝑘))]^𝑇.
In addition, for any semipositive definite matrix 𝑋^𝑖𝑗=(𝑋^𝑖𝑗)^𝑇 and 𝑈^𝑖𝑗=(𝑈^𝑖𝑗)^𝑇, the following equations hold: 𝑑₂𝜉^𝑇(𝑘)𝑋^𝑖𝑗𝜉(𝑘)−^{𝑘−𝑑(𝑘)−1}_𝑙=_𝑘−𝑑2𝜉^𝑇(𝑘)𝑋^𝑖𝑗𝜉(𝑘)−^𝑘−1_𝑙=𝑘_{−𝑑(𝑘)}𝑑𝜉^𝑇(𝑘)𝑋^𝑖𝑗𝜉(𝑘)=0,₂−𝑑₁𝜉^𝑇(𝑘)𝑈^𝑖𝑗𝜉(𝑘)−^{𝑘−𝑑(𝑘)−1}_𝑙=_𝑘−𝑑2𝜉^𝑇(𝑘)𝑈^𝑖𝑗𝜉(𝑘)−^{𝑘−𝑑1−1}_𝑙=𝑘_{−𝑑(𝑘)}𝜉^𝑇(𝑘)𝑈^𝑖𝑗𝜉(𝑘)=0.(3.20) From the assumption, we have 𝑓𝑔_𝑙(𝑥(𝑘))−𝑔^−𝑙𝑥_𝑙(𝑘)×𝑔_𝑙(𝑥(𝑘))−𝑔^+𝑙𝑥_𝑙(𝑘)≤0,𝑙=1,2,…,𝑛,_𝑙𝑥𝑘−𝑑_𝑘−𝑓^−𝑙𝑥_𝑙𝑘−𝑑_𝑘×𝑓_𝑙𝑥𝑘−𝑑_𝑘−𝑓^+𝑙𝑥_𝑙𝑘−𝑑_𝑘≤0,𝑙=1,2,…,𝑛.(3.21)
Similar to the conclusion in [8], for 𝐻^𝑖𝑗=diag{ℎ₁,ℎ₂,…,ℎ_𝑛}≥0,𝑂^𝑖𝑗=diag{𝑜₁,𝑜₂,…,𝑜_𝑛}≥0 the following inequalities are also true: −𝑔𝑥(𝑥(𝑘(𝑘)))^𝑇𝐺¹𝐻^𝑖𝑗𝐺₂𝐻^𝑖𝑗−𝑓𝑥(𝑥(𝑘(𝑘−−𝑑𝑑(𝑘(𝑘)))))_𝑇⋆𝐻^𝑖𝑗𝑔𝑥(𝑥(𝑘(𝑘)))≥0,(3.22)𝐹¹𝑂^𝑖𝑗𝐹₂𝑂^𝑖𝑗⋆𝑂^𝑖𝑗𝑓𝑥(𝑥(𝑘(𝑘−−𝑑𝑑(𝑘(𝑘)))))≥0.(3.23)
Then we add the terms on the right side of (3.16)–(3.23) to yield Δ𝑉_𝑖(𝑘)=≤Δ𝜂_𝑇1𝑉_1𝑖(𝑘)+Δ𝑉_2𝑖(𝑘)+Δ𝑉_3𝑖(𝑘)+Δ𝑉_4𝑖(𝑘)(𝑘)Φ^𝑖𝑗𝜂₁(𝑘)−^𝑘−1_𝑙=𝑘_{−𝑑(𝑘)}−𝜂^𝑇2(𝑘,𝑙)Λ^1𝑖𝑗𝜂₂(𝑘,𝑙)_{𝑘−𝑑1−1𝑙=𝑘}_{−𝑑(𝑘)}𝜂^𝑇2(𝑘,𝑙)Λ^2𝑖𝑗𝜂₂(𝑘,𝑙)−^{𝑘−𝑑(𝑘)−1}_𝑙=_𝑘−𝑑2𝜂^𝑇2(𝑘,𝑙)Λ^3𝑖𝑗𝜂₂(𝑘,𝑙),(3.24) where 𝜂^𝑇1(𝑘)=[𝜉^𝑇(𝑘),𝑔^𝑇(𝑥(𝑘)),𝑓^𝑇(𝑥(𝑘−𝑑(𝑘)))]^𝑇,𝜂^𝑇2(𝑘,𝑙)=[𝜉^𝑇(𝑘),𝑦^𝑇(𝑙)]^𝑇.
And 𝜉(𝑘) is defined in (3.18). Φ^𝑖𝑗, Λ^1𝑖𝑗, Λ^2𝑖𝑗, and Λ^3𝑖𝑗 are defined in (3.1)–(3.4). Therefore, when the corresponding LMIs satisfy Φ^𝑖𝑗<0, Λ^1𝑖𝑗≥0, Λ^2𝑖𝑗≥0, and Λ^3𝑖𝑗≥0,𝑆_𝑖,𝑆_𝑗∈Ω, Δ𝑉_𝑖(𝑘)≤0.
Secondly, based on the switching graph Γ, when 𝐰_𝑖𝑗=1(𝑆_𝑖,𝑆_𝑗∈Ω), all corresponding Δ𝑉_𝑖(𝑘) are less than 0 that means the system (2.2) is asymptotical stable. This completes the proof of Theorem 3.1.

Remark 3.2. Using the method in [8], it is easily to know that the system is the globally exponentially stable.

Remark 3.3. In 𝑉_3𝑖(𝑘) of the literature [8], −∑^{𝑘−𝑑(𝑘)𝑙=𝑘−𝑑}₂𝑥^𝑇(𝑙)𝑅_𝑖𝑥(𝑙)≤−𝑥^𝑇(𝑘−𝑑(𝑘))𝑅_𝑖𝑥(𝑘−𝑑(𝑘)) may lead to considerable conservativeness. Then, it is improved as −∑^{𝑘−𝑑(𝑘)𝑙=𝑘−𝑑}₂𝑥^𝑇(𝑙)𝑅_𝑖𝑥(𝑙)≤−𝑥^𝑇(𝑘−𝑑₂)𝑅_𝑖𝑥(𝑘−𝑑₂)−𝑥^𝑇(𝑘−𝑑(𝑘))𝑅_𝑖𝑥(𝑘−𝑑(𝑘))−𝑥^𝑇(𝑘−𝑑₁)𝑅_𝑖𝑥(𝑘−𝑑₁). Please see Table 3.
Combined with Theorem 3.1, we consider the common Lyapunov function approach; then we have the following.

Corollary 3.4. Let 𝑑₁ and 𝑑₂ be positive integers such that 0≤𝑑₁≤𝑑₂. The system (2.2) is asymptotical stable if there exist symmetric matrices 𝑃=𝑃^𝑇>0, 𝑄₁=𝑄^𝑇1>0, 𝑄₂=𝑄^𝑇2>0, 𝑅=𝑅^𝑇>0, 𝑍₁=𝑍^𝑇1>0, 𝑍₂=𝑍^𝑇2>0, 𝑋=𝑋^𝑇≥0, 𝑈=𝑈^𝑇≥0, 𝐻=diag{ℎ₁,ℎ₂,…,ℎ_𝑛}≥0, 𝑂=diag{𝑜₁,𝑜₂,…,𝑜_𝑛}≥0, and any appropriate dimensional matrices 𝑁, 𝑀, and 𝑇 such that the following LMIs hold: Φ_𝑖=⎡⎢⎢⎢⎢⎣Φ¹¹Φ₁₂Φ₁₃Φ₁₄Φ₁₅Φ₁₆∗∗Φ∗₂₂ΦΦ₂₃₃₃Φ₂₄00∗∗∗ΦΦ₃₄₄₄00∗∗∗∗Φ0₅₅−𝐹₂𝑂Φ₅₆∗∗∗∗∗Φ₆₆⎤⎥⎥⎥⎥⎦<0,𝑆_𝑖Λ∈Ω,₁Λ=𝑋∗𝑍𝑁₁>0,₂Λ=𝑈∗𝑍𝑇₂>0,₃=𝑋+∗𝑈𝑍₁𝑀+𝑍₂>0,(3.25) where Φ₁₁=+𝐴𝑁^𝑇𝑖𝑃_𝑇1𝐴_𝑖−𝑃+𝑄₁+𝑄₂+𝑑₂−𝑑₁+1𝑅+𝐴_𝑖−𝐸^𝑇𝑑₂𝑍₁+𝑑₂−𝑑₁𝑍₂𝐴_𝑖−𝐸Φ+𝑁₁+𝑑₂𝑋₁₁+𝑑₂−𝑑₁𝑈₁₁−𝐺₁𝐻,₁₂Φ₁₃=𝑁^𝑇2+𝑇₁+𝑑₂𝑋₁₂+𝑑₂−𝑑₁𝑈₁₂,Φ₁₄=𝑁^𝑇3−𝑀₁+𝑑₂𝑋₁₃+𝑑₂−𝑑₁𝑈₁₃,Φ₁₅=−𝑁₁+𝑁^𝑇4+𝑀₁−𝑇₁+𝑑₂𝑋₁₄+𝑑₂−𝑑₁𝑈₁₄,Φ=𝐴^𝑇𝑖𝑃𝑊_1𝑖+𝐴_𝑖−𝐸^𝑇𝑑₂𝑍₁+𝑑₂−𝑑₁𝑍₂𝑊_1𝑖−𝐺₂𝐻,₁₆Φ₂₂=𝐴^𝑇𝑖𝑃𝑊_2𝑖+𝐴_𝑖−𝐸^𝑇𝑑₂𝑍₁+𝑑₂−𝑑₁𝑍₂𝑊_2𝑖,Φ₂₃=−𝑄₁−𝑅+𝑇^𝑇2+𝑇₂+𝑑₂𝑋₂₂+𝑑₂−𝑑₁𝑈₂₂,Φ₂₄=−𝑀₂+𝑇^𝑇3+𝑑₂𝑋₂₃+𝑑₂−𝑑₁𝑈₂₃,Φ₃₃=−𝑁₂+𝑇^𝑇4+𝑀₂−𝑇₂+𝑑₂𝑋₂₄+𝑑₂−𝑑₁𝑈₂₄,Φ₃₄=−𝑄₂−𝑅−𝑀₃−𝑀^𝑇3+𝑑₂𝑋₃₃+𝑑₂−𝑑₁𝑈₃₃,Φ₄₄=−𝑁₃−𝑀^𝑇4+𝑀₃−𝑇₃+𝑑₂𝑋₃₄+𝑑₂−𝑑₁𝑈₃₄,Φ=−𝑁₄−𝑁^𝑇4+𝑀₄+𝑀^𝑇4−𝑇₄−𝑇^𝑇4−𝐹₁𝑂+𝑑₂𝑋₄₄+𝑑₂−𝑑₁𝑈₄₄−𝑅,₅₅Φ=𝑊^𝑇1𝑖𝑃𝑊_1𝑖+𝑊^𝑇1𝑖𝑑₂𝑍₁+𝑑₂−𝑑₁𝑍₂𝑊_1𝑖−𝐻,₅₆Φ₆₆=𝑊^𝑇1𝑖𝑃𝑊_2𝑖+𝑊^𝑇1𝑖𝑑₂𝑍₁+𝑑₂−𝑑₁𝑍₂𝑊_2𝑖,=𝑊^𝑇2𝑖𝑃𝑊_2𝑖+𝑊^𝑇2𝑖𝑑₂𝑍₁+𝑑₂−𝑑₁𝑍₂𝑊_2𝑖−𝑂.(3.26)

4. Examples

Example 4.1. Consider the discrete-time recurrent neural network (2.2) with 𝐴𝐴₁=000.7000.2000.5,𝑊₁₁=000..13−−−000...121−000.1.4,𝑊₂₁=−000.2.3−−000..1.012−00.0.01.7₂2,𝐴=000.1000.2000.4,𝑊₁₂=−000..42.3−−−000...122−000.5,𝑊₂₂=−000.5.3−−000..1.023−000.5₃.2,=000.𝑔1000.4000.4,𝑊₁₃=−00.0.0Γ2.41=−−000001..36100−0010₁00.3.1,,𝑊₂₃=−000..208−−000.1..42−−−000.0..125,𝑓(𝑥)=tanh(0.4𝑥),𝑔₂(𝑥)=tanh(0.2𝑥),𝑔₃(𝑥)=tanh(−0.8𝑥),₁(𝑥)=tanh(−0.6𝑥),𝑓₂(𝑥)=tanh(0.4𝑥),𝑓₃(𝑥)=tanh(0.2𝑥).(4.1) Then we have 𝐺₁=𝐹₁=000000000,𝐺₂=000.2000.1−000.4,𝐹₂=000.3−000.2−000.1.(4.2)
Employing the LMIs in Theorem 3.1 yields upper bounds on 𝑑₂ that guarantee the stability of system (1.1) for various lower bounds 𝑑₁, which are listed in Table 1. When 𝑑₁=1 and 𝑑₂=3, it can be seen from Figure 1 that all the state solutions corresponding to the 10 random initial points are convergent asymptotically to the unique equilibrium 𝑥^∗={0,0,0}, and according to Theorem 3.1, LMIs (3.1)–(3.4) are solvable in Matlab 7.0.1.

Table 1 

Allowable upper bound of 𝑑₂ with given 𝑑₁.

Figure 1 

Global convergence of states 𝑥1, 𝑥2, and 𝑥3 in Example 4.1, when 𝑑1=1 and 𝑑2=3.

Example 4.2. Consider the discrete-time recurrent neural network (2.2) with 𝑔𝐴=000.1000.2000.3,𝑊₁=−000..51.4−−−000...121−000.2.3,𝑊₂=−000.2−−000.1..33000...0213₁,(𝑥)=𝑓₁(𝑥)=tanh(0.6𝑥),𝑔₂(𝑥)=𝑓₂(𝑥)=tanh(0.4𝑥),𝑔₃(𝑥)=𝑓₃(𝑥)=tanh(0.2𝑥).(4.3) Then we have 𝐺₁=𝐹₁=000000000,𝐺₂=𝐹₂=000.3000.2000.1.(4.4)
Employing the LMIs in [8] and those in Corollary 3.4 yields upper bounds on 𝑑₂ that guarantee the stability of system for various lower bounds 𝑑₁, which are listed in Table 2. It is clear that the obtained upper bounds of this paper are better than those of [8]. It can be seen from Figure 2 that, when 𝑑₁=1 and 𝑑₂=3, all the state solutions corresponding to the 10 random initial points are convergent asymptotically to the unique equilibrium 𝑥^∗={0,0,0}.

Table 2 

Allowable upper bound of 𝑑₂ with given 𝑑₁.

Table 3 

Allowable upper bound of 𝑑₂ with given 𝑑₁.

Figure 2 

Global convergence of states 𝑥1, 𝑥2, and 𝑥3 in Example 4.2, when 𝑑1=1 and 𝑑2=3.

Remark 4.3. Employing the LMIs in [8] and those in Corollary 3.4 yields upper bounds on 𝑑₂ that guarantee the stability of system (1.1) of the Example  1 of [8] for various lower bounds 𝑑₁, which are listed in Table 3.

5. Conclusions

This paper was dedicated to the delay-dependent stability of uncertain periodic switched recurrent neural networks with time-varying delay. A less conservative LMI-based globally stability criterion is obtained with the switched quadratic Lyapunov functional approach and free-weighting matrix approach for periodic uncertain discrete-time recurrent neural networks with a time-varying delay. One example illustrates the exactness of the proposed criterion. Another example demonstrates that the proposed method is an improvement over the existing one.

Acknowledgments

This work was supported by the Sichuan Science and Technology Department under Grant 2011JY0114. The authors would like to thank the Associate Editor and the anonymous reviewers for their detailed comments and valuable suggestions which greatly contributed to this paper.