Delay-Range-Dependent Global Robust Passivity Analysis of Discrete-Time Uncertain Recurrent Neural Networks with Interval Time-Varying Delay

Abstract

This paper examines a passivity analysis for a class of discrete-time recurrent neural networks (DRNNs) with norm-bounded time-varying parameter uncertainties and interval time-varying delay. The activation functions are assumed to be globally Lipschitz continuous. Based on an appropriate type of Lyapunov functional, sufficient passivity conditions for the DRNNs are derived in terms of a family of linear matrix inequalities (LMIs). Two numerical examples are given to illustrate the effectiveness and applicability.

1. Introduction

Recurrent neural networks have been extensively studied in the past decades. Two popular examples are Hopfield neural networks and cellular neural networks. Increasing attention has been draw to the potential applications of recurrent neural networks in information processing systems such as signal processing, model identification, optimization, pattern recognition, and associative memory. However, these successful applications are greatly dependent on the dynamic behavior of recurrent neural networks (RNNs). On the other hand, time delay is inevitably encountered in RNNs, since the interactions between different neurons are asynchronous. Generally, time delays, both constant and time varying, are often encountered in various engineering, biological, and economic systems due to the finite switching speed of amplifiers in electronic networks, or to the finite signal propagation time in biological networks [1]. The existence of time delay could make delayed RNNs be instable or have poor performance. Therefore, many research interests have been attracted to the stability analysis for delayed RNNs. A lot of results related to this issue have been reported [2–6].

The theory of passivity plays an important role for analyzing the stability of nonlinear system [7] and has received much attention in the literature from the control community since 1970s [8–14]. It is well known that the passivity theory plays an important role in both electrical network and nonlinear control systems, provides a nice tool for analyzing the stability of system [15], and has found applications in diverse areas such as signal processing, chaos control and synchronization, and fuzzy control. The passivity condition for delayed neural networks with or without time-varying parametric uncertainties using LMIs [16] has been proposed in [17]. By constructing proper Lyapunov functionals and using some analytic techniques, sufficient conditions are given to ensure the passivity of the integro-differential neural networks with time-varying delays in [18]. In [19, 20], the authors studied the delay-dependent robust passivity criterion for the delayed cellular neural networks and the delayed recurrent neural networks. It should be pointed out that the aforementioned results are the continuous-time neural networks.

Recently, the stability analysis problems for discrete-time neural networks with time delay have received considerable research interests. For instance in [21], global exponential stability of a class of discrete-time Hopfield neural networks with variable delays is considered. By making use of a difference inequality, a new global exponential stability result is provided. Under different assumptions on the activation functions, a unified linear matrix inequality (LMI) approach has been developed to establish sufficient conditions for the discrete-time recurrent neural networks with interval variable time to be globally exponentially stable in [22]. Delay-dependent results on the global exponential stability problem for discrete-time neural networks with time-varying delays were presented in [23, 24], respectively. However, no delay-range-dependent passivity conditions on discrete-time uncertain recurrent neural networks with interval time-varying delay are available in the literature and remain essentially open. The objective of this paper is to address this unsolved problem.

The purpose of this paper is to deal with the problem of passivity conditions for discrete-time uncertain recurrent neural networks with interval time-varying delay. The interval time-varying delay includes both lower and upper bounds of delay, and the parameter uncertainties are assumed to be time varying but norm bounded which appear in all the matrices in the state equation. It is then established that the resulting passivity condition can be cast in a linear matrix inequality format which can be conveniently solved by using the numerically effective Matlab LMI Toolbox. In particular, when the interval time-delay factor is known, it is emphasized that delay-range-dependent passivity condition yields more general and practical results. Finally, two numerical examples are given to demonstrate the effectiveness.

Throughout this paper, the notation 𝑋≥𝑌(𝑋>𝑌) for symmetric matrices 𝑋 and 𝑌 indicates that the matrix 𝑋−𝑌 is positive and semidefinite (resp., positive definite); 𝑍^𝑇 represents the transpose of matrix 𝑍.

2. Preliminaries

Consider a discrete-time recurrent neural network with interval time-varying delay described by 𝑥(𝑘+1)=(𝐴+Δ𝐴(𝑘))𝑥(𝑘)+(𝑊+Δ𝑊(𝑘))𝑓(𝑥(𝑘))+(𝑊₁+Δ𝑊₁(𝑘))𝑓(𝑥(𝑘−𝜏(𝑘)))+𝑢(𝑘), (2.1) where 𝑥(𝑘)=(𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘))^𝑇 is the state vector, 𝐴=diag(𝑎₁,𝑎₂,…,𝑎_𝑛) with |𝑎_𝑖|<1, 𝑖=1,2,…,𝑛, is the state feedback coefficient matrix, 𝑊^𝑛×𝑛 and 𝑊₁^𝑛×𝑛 are the interconnection matrices representing the weighting coefficients of the neurons, 𝑓(𝑥(𝑘))=[𝑓₁(𝑥₁(𝑘)),…,𝑓_𝑛(𝑥_𝑛(𝑘))]^𝑇∈𝑅^𝑛 is the neuron activation function with 𝑓(0)=0, 𝜏(𝑘) is the time-varying delay of the system satisfying 𝜏₁≤𝜏(𝑘)≤𝜏₂,𝑘∈N, (2.2) where 0≤𝜏₁≤𝜏₂ are known integers. Let 𝑦(𝑘)=𝑓(𝑥(𝑘)) be the output of the neural networks. 𝑢(𝑘) is the input vector. Δ𝐴(𝑘), Δ𝑊(𝑘), and Δ𝑊₁(𝑘) are unknown matrices representing time-varying parameter uncertainties, which are assumed to be of the form Δ𝐴(𝑘)Δ𝑊(𝑘)Δ𝑊₁(𝑘)=𝑀𝐹(𝑘)𝑁₁𝑁₂𝑁₃, (2.3) where 𝑀, 𝑁₁, 𝑁₂, and 𝑁₃ are known real constant matrices, and 𝐹(⋅)∶𝑅→𝑅^𝑘×𝑙 is unknown time-varying matrix function satisfying 𝐹^𝑇(𝑘)𝐹(𝑘)≤𝐼,𝑘∈N. (2.4) The uncertain matrices Δ𝐴(𝑘), Δ𝑊(𝑘), and Δ𝑊₁(𝑘) are said to be admissible if both (2.3) and (2.4) hold.

In order to obtain our main results, the activation functions in (2.1) are assumed to be bounded and satisfy the following assumption.

Assumption 1. The activation functions 𝑔_𝑖(⋅), 𝑖=1,2,…,𝑛, are globally Lipschitz and monotone nondecreasing; that is, there exist constant scalars 𝛼_𝑖 such that for any 𝜍₁,𝜍₂∈𝑅 and 𝜍₁≠𝜍₂, 𝑔0≤^𝑖(𝜍₁)−𝑔_𝑖(𝜍₂)𝜍₁−𝜍₂≤𝛼_𝑖,𝑖=1,2,…,𝑛. (2.5)

Definition 2.1 ([25]). System (2.1) is called passive if there exists a scalar 𝛽≥0 such that 2^𝑘𝑝_𝑘=0𝑦^𝑇(𝑘)𝑢(𝑘)≥−𝛽^𝑘𝑝_𝑘=0𝑢^𝑇(𝑘)𝑢(𝑘) (2.6) for all 𝑘_𝑝≥0 and for all solutions of (2.1) with 𝑥(0)=0.

3. Mathematical Formulation of the Proposed Approach

This section explores the globally robust delay-range-dependent passivity conditions of the discrete-time recurrent uncertain neural network with interval time-varying delay given in (2.1). Specially, an LMI approach is employed to solve the robust delay-range-dependent passivity condition if the system in (2.1) is globally asymptotically stable for all admissible uncertainties Δ𝐴(𝑘), Δ𝑊(𝑘), and Δ𝑊₁(𝑘) satisfying (2.6). The analysis commences by using the LMI approach to develop some results which are essential to introduce the followingLemma 3.1for the development of our main theorem.

Lemma 3.1. Let 𝐴, 𝐷, 𝑆, 𝐹, and 𝑃 be real matrices of appropriate dimensions with 𝑃>0 and 𝐹 satisfying 𝐹^𝑇(𝑘)𝐹(𝑘)≤𝐼. Then the following statements hold.(a)For any 𝜀>0 and vectors 𝑥, 𝑦∈𝑅^𝑛2𝑥^𝑇𝐷𝐹𝑆𝑦≤𝜀⁻¹𝑥^𝑇𝐷𝐷^𝑇𝑥+𝜀𝑦^𝑇𝑆^𝑇𝑆𝑦. (3.1)(b)For vectors 𝑥, 𝑦∈𝑅^𝑛2𝑥^𝑇𝐷𝑆𝑦≤𝑥^𝑇𝐷𝑃𝐷^𝑇𝑥+𝑦^𝑇𝑆^𝑇𝑃⁻¹𝑆𝑦. (3.2)For any matrices 𝐸_𝑖, 𝑆_𝑖, 𝑇_𝑖, and 𝐻_𝑖(𝑖=1,2,…,8) of appropriate dimensions, it follows from null equations that Φ₁=2[𝑥^𝑇(𝑘)𝐸₁+𝑥^𝑇(𝑘−𝜏(𝑘))𝐸₂+𝑥^𝑇(𝑘−𝜏₂)𝐸₃+𝑥^𝑇(𝑘−𝜏₁)𝐸₄+𝑓^𝑇(𝑥(𝑘))𝐸₅+𝑓^𝑇(𝑥(𝑘−𝜏(𝑘)))𝐸₆+𝑥^𝑇(𝑘+1)𝐸₇+𝑢^𝑇(𝑘)𝐸₈]×[𝑥(𝑘)−𝑥(𝑘−𝜏(𝑘))−^𝑘_{𝑗=𝑘−𝜏(𝑘)+1}Φ(𝑥(𝑗)−𝑥(𝑗−1))]=0,₂=2[𝑥^𝑇(𝑘)𝑆₁+𝑥^𝑇(𝑘−𝜏(𝑘))𝑆₂+𝑥^𝑇(𝑘−𝜏₂)𝑆₃+𝑥^𝑇(𝑘−𝜏₁)𝑆₄+𝑓^𝑇(𝑥(𝑘))𝑆₅+𝑓^𝑇(𝑥(𝑘−𝜏(𝑘)))𝑆₆+𝑥^𝑇(𝑘+1)𝑆₇+𝑢^𝑇(𝑘)𝑆₈]×[𝑥(𝑘−𝜏(𝑘))−𝑥(𝑘−𝜏₂)−^{𝑘−𝜏(𝑘)}_{𝑗=𝑘−𝜏2+1}Φ(𝑥(𝑗)−𝑥(𝑗−1))]=0,₃=−2[𝑥^𝑇(𝑘)𝑇₁+𝑥^𝑇(𝑘−𝜏(𝑘))𝑇₂+𝑥^𝑇(𝑘−𝜏₂)𝑇₃+𝑥^𝑇(𝑘−𝜏₁)𝑇₄+𝑓^𝑇(𝑥(𝑘))𝑇₅+𝑓^𝑇(𝑥(𝑘−𝜏(𝑘)))𝑇₆+𝑥^𝑇(𝑘+1)𝑇₇+𝑢^𝑇(𝑘)𝑇₈]×[𝑥(𝑘−𝜏₁)−𝑥(𝑘−𝜏(𝑘))−^𝑘−𝜏1_{𝑗=𝑘−𝜏(𝑘)+1}Φ(𝑥(𝑗)−𝑥(𝑗−1))]=0,₄=−2[𝑥^𝑇(𝑘)𝐻₁+𝑥^𝑇(𝑘−𝜏(𝑘))𝐻₂+𝑥^𝑇(𝑘−𝜏₂)𝐻₃+𝑥^𝑇(𝑘−𝜏₁)𝐻₄+𝑓^𝑇(𝑥(𝑘))𝐻₅+𝑓^𝑇(𝑥(𝑘−𝜏(𝑘)))𝐻₆+𝑥^𝑇(𝑘+1)𝐻₇+𝑢^𝑇(𝑘)𝐻₈]×[𝑥(𝑘+1)−(𝐴+Δ𝐴(𝑘))𝑥(𝑘)−(𝑊+Δ𝑊(𝑘))𝑓(𝑥(𝑘))−(𝑊₁+Δ𝑊₁Φ(𝑘))𝑓(𝑥(𝑘−𝜏(𝑘)))−𝑢(𝑘)]=0,₅=2𝑓^𝑇(𝑥(𝑘))𝑅₁𝑓(𝑥(𝑘))−2𝑓^𝑇(𝑥(𝑘))𝑅₁𝑓(𝑥(𝑘))+2𝑓^𝑇(𝑥(𝑘−𝜏(𝑘)))𝑅₂𝑥(𝑘−𝜏(𝑘))−2𝑓^𝑇(𝑥(𝑘−𝜏(𝑘)))𝑅₂𝑥(𝑘−𝜏(𝑘))=0. (3.3)(3.4)(3.5)(3.6)(3.7)

To study the globally robust delay-range-dependent passivity conditions of the discrete-time uncertain recurrent neural network with interval time-varying delay, the following theorem reveals that such conditions can be expressed in terms of LMIs.

Theorem 3.2. Under Assumption 1, given scalars 0≤𝜏₁<𝜏₂, system (2.1) with interval time-varying delay 𝜏(𝑘) satisfying (2.2) is globally asymptotically robust stability, if there exist matrices 𝑃>0, 𝑄₁>0, 𝑄₂>0, 𝑍₁>0, 𝑍₂>0, diagonal matrices 𝑅₁>0, 𝑅₂>0, and matrices 𝐸_𝑖, 𝑆_𝑖 , 𝑇_𝑖 and 𝐻_𝑖(𝑖=1,2,…,8) of appropriate dimensions, a positive scalar 𝜀>0 and a scalar 𝛽≥0 such that the following LMI holds: Ω𝜏₂𝐸𝜏₂₁𝑆𝜏₂₁𝜏𝑇𝐻𝑀𝜀𝑁₂𝐸^𝑇−𝜏₂Z₁𝜏0000₂₁𝑆^𝑇0−𝜏₂₁(Z₁+Z₂𝜏)000₂₁𝑇^𝑇00−𝜏₂₁Z₂𝑀00_𝑇𝐻^𝑇000−𝜀𝐼0𝜀𝑁^𝑇0000−𝜀𝐼<0, (3.8) where ΩΩ=¹¹Ω₁₂Ω₁₃Ω₁₄Ω₁₅Ω₁₆Ω₁₇Ω₁₈Ω^𝑇₁₂Ω₂₂Ω₂₃Ω₂₄Ω₂₅Ω₂₆Ω₂₇Ω₂₈Ω^𝑇₁₃Ω^𝑇₂₃Ω₃₃Ω₃₄Ω₃₅Ω₃₆Ω₃₇Ω₃₈Ω^𝑇₁₄Ω^𝑇₂₄Ω^𝑇₃₄Ω₄₄Ω₄₅Ω₄₆Ω₄₇Ω₄₈Ω^𝑇₁₅Ω^𝑇₂₅Ω^𝑇₃₅Ω^𝑇₄₅Ω₅₅Ω₅₆Ω₅₇Ω₅₈Ω^𝑇₁₆Ω^𝑇₂₆Ω^𝑇₃₆Ω^𝑇₄₆Ω^𝑇₅₆Ω₆₆Ω₆₇Ω₆₈Ω^𝑇₁₇Ω^𝑇₂₇Ω^𝑇₃₇Ω^𝑇₄₇Ω^𝑇₅₇Ω^𝑇₆₇Ω₇₇Ω₇₈Ω^𝑇₁₈Ω^𝑇₂₈Ω^𝑇₃₈Ω^𝑇₄₈Ω^𝑇₅₈Ω^𝑇₆₈Ω^𝑇₇₈Ω₈₈,Ω₁₁=−𝑃+(𝜏₂₁+1)𝑄₁+𝜏₂𝑍₁+𝜏₂₁𝑍₂+𝐸₁+𝐸^𝑇₁+𝐻₁𝐴+𝐴^𝑇𝐻^𝑇₁+𝑄₂,Ω₁₂=𝐸^𝑇₂−𝐸₁+𝑆₁+𝑇₁+𝐴^𝑇𝐻^𝑇₂,Ω₁₃=𝐸^𝑇₃−𝑆₁+𝐴^𝑇𝐻^𝑇₃,Ω₁₄=𝐸^𝑇₄−𝑇₁+𝐴^𝑇𝐻^𝑇₄,Ω₁₅=𝐸^𝑇₅+𝐻₁𝑊₀+𝐴^𝑇𝐻^𝑇₅+Γ^𝑇𝑅₁,Ω₁₆=𝐸^𝑇₆+𝐻₁𝑊₁+𝐴^𝑇𝐻^𝑇₆,Ω₁₇=𝐸^𝑇₇−𝜏₂𝑍₁−𝜏₂₁𝑍₂−𝐻₁+𝐴^𝑇𝐻^𝑇₇,Ω₁₈=𝐸^𝑇₈+𝐻₁+𝐴^𝑇𝐻^𝑇₈,Ω₂₂=−𝑄₁−𝐸₂−𝐸^𝑇₂+𝑆₂+𝑆^𝑇₂+𝑇₂+𝑇^𝑇₂,Ω₂₃=−𝐸^𝑇₃−𝑆₂+𝑆^𝑇₃+𝑇^𝑇₃,Ω₂₄=−𝐸^𝑇₄−𝑇₂+𝑆^𝑇₄+𝑇₄,Ω₂₅=−𝐸^𝑇₅+𝑆^𝑇₅+𝑇^𝑇₅+𝐻₂𝑊₀,Ω₂₆=−𝐸^𝑇₆+𝑆^𝑇₆+𝑇^𝑇₆+𝐻₂𝑊₁+𝑅^𝑇₂,Ω₂₇=−𝐸^𝑇₇+𝑆^𝑇₇+𝑇^𝑇₇−𝐻₂,Ω₂₈=−𝐸^𝑇₈+𝑆^𝑇₈+𝑇^𝑇₈+𝐻₂,Ω₃₃=−𝑄₂−𝑆₃−𝑆^𝑇₃,Ω₃₄=−𝑆^𝑇₄−𝑇₃,Ω₃₅=−𝑆^𝑇₅+𝐻₃𝑊₀,Ω₃₆=−𝑆^𝑇₆+𝐻₃𝑊₁,Ω₃₇=−𝑆^𝑇₇−𝐻₃,Ω₃₈=−𝑆^𝑇₈+𝐻₃,Ω₄₄=−𝑇₄−𝑇^𝑇₄,Ω₄₅=𝑇^𝑇₅+𝐻₄𝑊₀,Ω₄₆=−𝑇^𝑇₆+𝐻₄𝑊₁,Ω₄₇=−𝑇^𝑇₇−𝐻₄,Ω₄₈=−𝑇^𝑇₈+𝐻₄,Ω₅₅=−𝑅₁−𝑅^𝑇₁+𝐻₅𝑊₀+𝑊^𝑇₀𝐻^𝑇₅,Ω₅₆=𝐻₅𝑊₁+𝑊^𝑇₀𝐻^𝑇₆,Ω₅₇=−𝐻₅+𝑊^𝑇₀𝐻^𝑇₇,Ω₅₈=𝐻₅+𝑊^𝑇₀𝐻^𝑇₈Ω−𝐼,₆₆=−𝑅₂Γ⁻¹−(𝑅₂Γ⁻¹)^𝑇+𝐻₆𝑊₁+𝑊^𝑇₁𝐻^𝑇₆,Ω₆₇=−𝐻₆+𝑊^𝑇₁𝐻^𝑇₇,Ω₆₈=𝐻₆+𝑊^𝑇₁𝐻^𝑇₈,Ω₇₇=−𝐻₇−𝐻^𝑇₇+𝑃+𝜏₂𝑍₁+𝜏₂₁𝑍₂,Ω₇₈=𝐻₇−𝐻₈,Ω₈₈=𝐻₈+𝐻^𝑇₈−𝛽𝐼,𝐸=𝐸^𝑇₁𝐸^𝑇₂𝐸^𝑇₃𝐸^𝑇₄𝐸^𝑇₅𝐸^𝑇₆𝐸^𝑇₇𝐸^𝑇₈^𝑇,𝑆=𝑆^𝑇₁𝑆^𝑇₂𝑆^𝑇₃𝑆^𝑇₄𝑆^𝑇₅𝑆^𝑇₆𝑆^𝑇₇𝑆^𝑇₈^𝑇,𝑇=𝑇^𝑇₁𝑇^𝑇₂𝑇^𝑇₃𝑇^𝑇₄𝑇^𝑇₅𝑇^𝑇₆𝑇^𝑇₇𝑇^𝑇₈^𝑇,𝐻=𝐻^𝑇₁𝐻^𝑇₂𝐻^𝑇₃𝐻^𝑇₄𝐻^𝑇₅𝐻^𝑇₆𝐻^𝑇₇𝐻^𝑇₈^𝑇,𝑁=𝑁^𝑇₁000𝑁^𝑇₂𝑁^𝑇₃00^𝑇, (3.9) in which Γ=𝑑𝑖𝑎𝑔(𝛼₁,𝛼₂,…,𝛼_𝑛), 𝜏₂₁=𝜏₂−𝜏₁. Then system (2.1) satisfying (3.8) with interval time-varying delay is robust delay-range-dependent passivity condition in the sense of Definition 2.1.

Proof. Choose the Lyapunov-Krasovskii functional candidate for the system in (2.1) as 𝑉(𝑘)=𝑉₁(𝑘)+𝑉₂(𝑘)+𝑉₃(𝑘)+𝑉₄(𝑘)=𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+⁻¹_{𝑖=−𝜏2}^𝑘_{𝑗=𝑘+𝑖+1}(𝑥(𝑗)−𝑥(𝑗−1))^𝑇𝑍₁+(𝑥(𝑗)−𝑥(𝑗−1))_{𝑘−1𝑗=𝑘−𝜏(𝑘)}𝑥^𝑇(𝑗)𝑄₁𝑥(𝑗)+^−1−𝜏1_{𝑖=−𝜏2}^𝑘−1_{𝑗=𝑘+𝑖+1}𝑥^𝑇(𝑗)𝑄₁𝑥(𝑗)+^𝑘−1_{𝑗=𝑘−𝜏2}𝑥^𝑇(𝑗)𝑄₂𝑥(𝑗)+^−1−𝜏1_{𝑖=−𝜏2}^𝑘_{𝑗=𝑘+𝑖+1}(𝑥(𝑗)−𝑥(𝑗−1))^𝑇𝑍₂(𝑥(𝑗)−𝑥(𝑗−1)). (3.10) Then, the difference Δ𝑉(𝑘)=𝑉(𝑘+1)−𝑉(𝑘) of 𝑉(𝑘) along the solution of (2.1) gives Δ𝑉₁(𝑘)=𝑥^𝑇(𝑘+1)𝑃𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘),Δ𝑉₂(𝑘)=⁻¹_{𝑖=−𝜏2}^𝑘+1_{𝑗=𝑘+𝑖+2}(𝑥(𝑗)−𝑥(𝑗−1))^𝑇𝑍₁−(𝑥(𝑗)−𝑥(𝑗−1))_{−1𝑖=−𝜏2𝑘𝑗=𝑘+𝑖+1}(𝑥(𝑗)−𝑥(𝑗−1))^𝑇𝑍₁(𝑥(𝑗)−𝑥(𝑗−1))≤𝜏₂(𝑥(𝑘+1)−𝑥(𝑘))^𝑇𝑍₁(𝑥(𝑘+1)−𝑥(𝑘))−^{𝑘−𝜏(𝑘)}_{𝑗=𝑘−𝜏2+1}(𝑥(𝑗)−𝑥(𝑗−1))^𝑇𝑍₁(𝑥(𝑗)−𝑥(𝑗−1))−^𝑘_{𝑗=𝑘−𝜏(𝑘)+1}(𝑥(𝑗)−𝑥(𝑗−1))^𝑇𝑍₁(𝑥(𝑗)−𝑥(𝑗−1)),Δ𝑉₃(𝑘)≤𝑥^𝑇(𝑘){𝑄₂+[(𝜏₂−𝜏₁)+1]𝑄₁}𝑥(𝑘)−𝑥^𝑇(𝑘−𝜏(𝑘))𝑄₁𝑥(𝑘−𝜏(𝑘))−𝑥^𝑇(𝑘−𝜏₂)𝑄₂𝑥(𝑘−𝜏₂),Δ𝑉₄(𝑘)=(𝜏₂−𝜏₁)(𝑥(𝑘+1)−𝑥(𝑘))^𝑇𝑍₂(𝑥(𝑘+1)−𝑥(𝑘))−^{𝑘−𝜏(𝑘)}_{𝑗=𝑘−𝜏2+1}(𝑥(𝑗)−𝑥(𝑗−1))^𝑇𝑍₂(𝑥(𝑗)−𝑥(𝑗−1))−^𝑘−𝜏1_{𝑗=𝑘−𝜏(𝑘)+1}(𝑥(𝑗)−𝑥(𝑗−1))^𝑇𝑍₂(𝑥(𝑗)−𝑥(𝑗−1)). (3.11) Defining the following new variables: 𝜂(𝑘)=[𝑥^𝑇(𝑘)𝑥^𝑇(𝑘−𝜏(𝑘))𝑥^𝑇(𝑘−𝜏₂)𝑥^𝑇(𝑘−𝜏₁)𝑓^𝑇(𝑥(𝑘))𝑓^𝑇(𝑥(𝑘−𝜏(𝑘)))𝑥^𝑇(𝑘+1)𝑢^𝑇(𝑘)]^𝑇,𝐸=𝐸^𝑇₁𝐸^𝑇₂𝐸^𝑇₃𝐸^𝑇₄𝐸^𝑇₅𝐸^𝑇₆𝐸^𝑇₇𝐸^𝑇₈^𝑇,𝑆=𝑆^𝑇₁𝑆^𝑇₂𝑆^𝑇₃𝑆^𝑇₄𝑆^𝑇₅𝑆^𝑇₆𝑆^𝑇₇𝑆^𝑇₈^𝑇,𝑇=𝑇^𝑇₁𝑇^𝑇₂𝑇^𝑇₃𝑇^𝑇₄𝑇^𝑇₅𝑇^𝑇₆𝑇^𝑇₇𝑇^𝑇₈^𝑇,𝐻=𝐻^𝑇₁𝐻^𝑇₂𝐻^𝑇₃𝐻^𝑇₄𝐻^𝑇₅𝐻^𝑇₆𝐻^𝑇₇𝐻^𝑇₈^𝑇,𝑁=𝑁^𝑇₁000𝑁^𝑇₂𝑁^𝑇₃00^𝑇, (3.12) and combining null equations (3.3)–(3.7), it yields Δ𝑉(𝑘)−2𝑦^𝑇(𝑘)𝑢(𝑘)−𝛽𝑢^𝑇(𝑘)𝑢(𝑘)=Δ𝑉₁(𝑘)+Δ𝑉₂(𝑘)+Δ𝑉₃(𝑘)+Δ𝑉₄(𝑘)−2𝑦^𝑇(𝑘)𝑢(𝑘)−𝛽𝑢^𝑇(𝑘)𝑢(𝑘)+Φ₁+Φ₂+Φ₃+Φ₄+Φ₅≤𝑥^𝑇(𝑘+1)𝑃𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+𝜏₂(𝑥(𝑘+1)−𝑥(𝑘))^𝑇𝑍₁(𝑥(𝑘+1)−𝑥(𝑘))−^{𝑘−𝜏(𝑘)}_{𝑗=𝑘−𝜏2+1}(𝑥(𝑗)−𝑥(𝑗−1))^𝑇𝑍₁(𝑥(𝑗)−𝑥(𝑗−1))−^𝑘_{𝑗=𝑘−𝜏(𝑘)+1}(𝑥(𝑗)−𝑥(𝑗−1))^𝑇𝑍₁(𝑥(𝑗)−𝑥(𝑗−1))+𝑥^𝑇(𝑘){𝑄₂+[(𝜏₂−𝜏₁)+1]𝑄₁}𝑥(𝑘)−𝑥^𝑇(𝑘−𝜏(𝑘))𝑄₁𝑥(𝑘−𝜏(𝑘))−𝑥^𝑇(𝑘−𝜏₂)𝑄₂𝑥(𝑘−𝜏₂)+(𝜏₂−𝜏₁)(𝑥(𝑘+1)−𝑥(𝑘))^𝑇𝑍₂−(𝑥(𝑘+1)−𝑥(𝑘))_{𝑘−𝜏(𝑘)𝑗=𝑘−𝜏2+1}(𝑥(𝑗)−𝑥(𝑗−1))^𝑇𝑍₂(𝑥(𝑗)−𝑥(𝑗−1))−^𝑘−𝜏1_{𝑗=𝑘−𝜏(𝑘)+1}(𝑥(𝑗)−𝑥(𝑗−1))^𝑇𝑍₂(𝑥(𝑗)−𝑥(𝑗−1))+2𝜂^𝑇(𝑘)𝐸[𝑥(𝑘)−𝑥(𝑘−𝜏(𝑘))−^𝑘_{𝑗=𝑘−𝜏(𝑘)+1}(𝑥(𝑗)−𝑥(𝑗−1))]+2𝜂^𝑇(𝑘)𝑆[𝑥(𝑘−𝜏(𝑘))−𝑥(𝑘−𝜏₂)−^{𝑘−𝜏(𝑘)}_{𝑗=𝑘−𝜏2+1}(𝑥(𝑗)−𝑥(𝑗−1))]−2𝜂^𝑇(𝑘)𝑇[𝑥(𝑘−𝜏₁)−𝑥(𝑘−𝜏(𝑘))−^𝑘−𝜏1_{𝑗=𝑘−𝜏(𝑘)+1}(𝑥(𝑗)−𝑥(𝑗−1))]−2𝜂^𝑇(𝑘)𝐻[𝑥(𝑘+1)−(𝐴+Δ𝐴(𝑘))𝑥(𝑘)−(𝑊+Δ𝑊(𝑘))𝑓(𝑥(𝑘))−(𝑊₁+Δ𝑊₁(𝑘))𝑓(𝑥(𝑘−𝜏(𝑘)))−𝑢(𝑘)]+2𝑓^𝑇(𝑥(𝑘))𝑅₁𝑓(𝑥(𝑘))−2𝑓^𝑇(𝑥(𝑘))𝑅₁𝑓(𝑥(𝑘))+2𝑓^𝑇(𝑥(𝑘−𝜏(𝑘)))𝑅₂𝑥(𝑘−𝜏(𝑘))−2𝑓^𝑇(𝑥(𝑘−𝜏(𝑘)))𝑅₂𝑥(𝑘−𝜏(𝑘))−2𝑦^𝑇(𝑘)𝑢(𝑘)−𝛽𝑢^𝑇(𝑘)𝑢(𝑘). (3.13)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) Moreover, −2𝜂^𝑇(𝑘)𝐸^𝑘_{𝑗=𝑘−𝜏(𝑘)+1}(𝑥(𝑗)−𝑥(𝑗−1))≤𝜏₂𝜂^𝑇(𝑘)𝐸𝑍₁⁻¹𝐸^𝑇𝜂(𝑘)+^𝑘_{𝑗=𝑘−𝜏(𝑘)+1}(𝑥(𝑗)−𝑥(𝑗−1))^𝑇𝑍₁(𝑥(𝑗)−𝑥(𝑗−1)),−2𝜂^𝑇(𝑘)𝑆^{𝑘−𝜏(𝑘)}_{𝑗=𝑘−𝜏2+1}(𝑥(𝑗)−𝑥(𝑗−1))≤(𝜏₂−𝜏₁)𝜂^𝑇(𝑘)𝑆(𝑍₁+𝑍₂)⁻¹𝑆^𝑇𝜂(𝑘)+^{𝑘−𝜏(𝑘)}_{𝑗=𝑘−𝜏2+1}(𝑥(𝑗)−𝑥(𝑗−1))^𝑇(𝑍₁+𝑍₂)(𝑥(𝑗)−𝑥(𝑗−1)),2𝜂^𝑇(𝑘)𝑇^𝑘−𝜏1_{𝑗=𝑘−𝜏(𝑘)+1}(𝑥(𝑗)−𝑥(𝑗−1))≤(𝜏₂−𝜏₁)𝜂^𝑇(𝑘)𝑇𝑍₂⁻¹𝑇^𝑇𝜂(𝑘)+^𝑘−𝜏1_{𝑗=𝑘−𝜏(𝑘)+1}(𝑥(𝑗)−𝑥(𝑗−1))^𝑇𝑍₂(𝑥(𝑗)−𝑥(𝑗−1)). (3.14)(3.15)(3.16) Using Assumption 1 and noting that 𝑅₁>0 and 𝑅₂>0 are diagonal matrices, one has 2𝑓^𝑇(𝑥(𝑘))𝑅₁𝑓(𝑥(𝑘))≤2𝑓^𝑇(𝑥(𝑘))𝑅₁Γ𝑥(𝑘),−2𝑓^𝑇(𝑥(𝑘−𝜏(𝑘)))𝑅₂𝑥(𝑘−𝜏(𝑘))≤−2𝑓^𝑇(𝑥(𝑘−𝜏(𝑘)))𝑅₂Γ⁻¹𝑓(𝑥(𝑘−𝜏(𝑘))), (3.17)(3.18) where Γ=diag(𝛼₁,𝛼₂,…,𝛼_𝑛).
Following from Lemma 3.1(a) results in 2𝜂^𝑇(𝑘)𝐻(Δ𝐴(𝑘)𝑥(𝑘)+Δ𝑊₀(𝑘)𝑓(𝑥(𝑘))+Δ𝑊₁(𝑘)𝑓(𝑥(𝑘−𝜏(𝑘))))=2𝜂^𝑇(𝑘)𝐻𝑀𝐹(𝑘)𝑁^𝑇𝜂(𝑘)≤𝜀⁻¹𝜂^𝑇(𝑘)𝐻𝑀𝑀^𝑇𝐻^𝑇𝜂(𝑘)+𝜀𝜂^𝑇(𝑘)𝑁𝑁^𝑇𝜂(𝑘). (3.19) Substituting (3.14)–(3.19) into (3.13), it is not difficult to deduce that Δ𝑉(𝑘)−2𝑦^𝑇(𝑘)𝑢(𝑘)−𝛽𝑢^𝑇(𝑘)𝑢(𝑘)≤𝜂^𝑇(𝑘)[Ω+𝜏₂𝐸𝑍₁⁻¹𝐸^𝑇+𝜏₂₁𝑆(𝑍₁+𝑍₂)⁻¹𝑆^𝑇+𝜏₂₁𝑇𝑍₂⁻¹𝑇^𝑇+𝜀⁻¹𝐻𝑀𝑀^𝑇𝐻^𝑇+𝜀𝑁𝑁^𝑇]𝜂(𝑘). (3.20) Using the Schur complement to the (3.20) and in view of LMI (3.8), it follows that Δ𝑉(𝑘)−2𝑦^𝑇(𝑘)𝑢(𝑘)−𝛽𝑢^𝑇(𝑘)𝑢(𝑘)<0. (3.21) It follows from (3.21) that 2^𝑘𝑝_𝑘=0𝑦^𝑇(𝑘)𝑢(𝑘)≥𝑉(𝑥(𝑘_𝑝))−𝑉(𝑥(0))−𝛽^𝑘𝑝_𝑘=0𝑢^𝑇(𝑘)𝑢(𝑘), (3.22) for 𝑥(0)=0, one has 𝑉(𝑥(0))=0, so (2.6) holds, and hence the system is robust delay-range-dependent passivity condition in the sense of Definition 2.1. This completes the proof of Theorem 3.2.

Remark 3.3. Theorem 3.2 provides a sufficient passivity condition for the globally robust stability of the discrete-time uncertain recurrent neural network with interval time-varying delay given in (2.1) and proposes a delay-range-dependent criterion. Even for 𝜏₁=0, the result in Theorem 3.2 may lead to the delay-dependent stability criteria. In fact, if 𝑍₂=𝜀₁𝐼, with 𝜀₁>0, being sufficiently small scalar, 𝑇_𝑖=0, 𝑖=1,2,…,8, 𝐸₄=0, 𝑆₄=0, 𝐻₄=0, Theorem 3.2 yields the following delay-dependent passivity criterion.

Corollary 3.4. Under Assumption 1, given scalars 𝜏₂>0, 𝜏₁=0, system (2.1) with time-varying delay satisfying (2.3) is globally asymptotically robust stability, if there exist matrices 𝑃>0, 𝑄₁>0, 𝑄₂>0, 𝑍₁>0, diagonal matrices 𝑅₁>0, 𝑅₂>0, and matrices 𝐸_𝑖, 𝑆_𝑖, and 𝐻_𝑖(𝑖=1,2,…,8) of appropriate dimensions and 𝐸₄=𝑆₄=𝐻₄=0, a positive scalar 𝜀>0, and a scalar 𝛽≥0 such that the following LMI holds: Ω𝜏₂𝐸𝜏₂𝜏𝑆𝐻𝑀𝜀𝑁₂𝐸^𝑇−𝜏₂Z₁𝜏000₂𝑆^𝑇0−𝜏₂Z₁𝑀00_𝑇𝐻^𝑇00−𝜀𝐼0𝜀𝑁^𝑇000−𝜀𝐼<0, (3.23) where ΩΩ=¹¹Ω₁₂Ω₁₃Ω₁₅Ω₁₆Ω₁₇Ω₁₈Ω^𝑇₁₂Ω₂₂Ω₂₃Ω₂₅Ω₂₆Ω₂₇Ω₂₈Ω^𝑇₁₃Ω^𝑇₂₃Ω₃₃Ω₃₅Ω₃₆Ω₃₇Ω₃₈Ω^𝑇₁₅Ω^𝑇₂₅Ω^𝑇₃₅Ω₅₅Ω₅₆Ω₅₇Ω₅₈Ω^𝑇₁₆Ω^𝑇₂₆Ω^𝑇₃₆Ω^𝑇₅₆Ω₆₆Ω₆₇Ω₆₈Ω^𝑇₁₇Ω^𝑇₂₇Ω^𝑇₃₇Ω^𝑇₅₇Ω^𝑇₆₇Ω₇₇Ω₇₈Ω^𝑇₁₈Ω^𝑇₂₈Ω^𝑇₃₈Ω^𝑇₅₈Ω^𝑇₆₈Ω^𝑇₇₈Ω₈₈,Ω₁₁=−𝑃+(𝜏₂+1)𝑄₁+𝜏₂𝑍₁+𝐸₁+𝐸^𝑇₁+𝐻₁𝐴+𝐴^𝑇𝐻^𝑇₁+𝑄₂,Ω₁₂=𝐸^𝑇₂−𝐸₁+𝑆₁+𝐴^𝑇𝐻^𝑇₂,Ω₁₃=𝐸^𝑇₃−𝑆₁+𝐴^𝑇𝐻^𝑇₃,Ω₁₅=𝐸^𝑇₅+𝐻₁𝑊₀+𝐴^𝑇𝐻^𝑇₅+Γ^𝑇𝑅₁,Ω₁₆=𝐸^𝑇₆+𝐻₁𝑊₁+𝐴^𝑇𝐻^𝑇₆,Ω₁₇=𝐸^𝑇₇−𝜏₂𝑍₁−𝐻₁+𝐴^𝑇𝐻^𝑇₇,Ω₁₈=𝐸^𝑇₈+𝐻₁+𝐴^𝑇𝐻^𝑇₈,Ω₂₂=−𝑄₁−𝐸₂−𝐸^𝑇₂+𝑆₂+𝑆^𝑇₂,Ω₂₃=−𝐸^𝑇₃−𝑆₂+𝑆^𝑇₃,Ω₂₅=−𝐸^𝑇₅+𝑆^𝑇₅+𝐻₂𝑊₀,Ω₂₆=−𝐸^𝑇₆+𝑆^𝑇₆+𝐻₂𝑊₁+𝑅^𝑇₂,Ω₂₇=−𝐸^𝑇₇+𝑆^𝑇₇−𝐻₂,Ω₂₈=−𝐸^𝑇₈+𝑆^𝑇₈+𝐻₂,Ω₃₃=−𝑄₂−𝑆₃−𝑆^𝑇₃,Ω₃₅=−𝑆^𝑇₅+𝐻₃𝑊₀,Ω₃₆=−𝑆^𝑇₆+𝐻₃𝑊₁,Ω₃₇=−𝑆^𝑇₇−𝐻₃,Ω₃₈=−𝑆^𝑇₈+𝐻₃,Ω₅₅=−𝑅₁−𝑅^𝑇₁+𝐻₅𝑊₀+𝑊^𝑇₀𝐻^𝑇₅,Ω₅₆=𝐻₅𝑊₁+𝑊^𝑇₀𝐻^𝑇₆,Ω₅₇=−𝐻₅+𝑊^𝑇₀𝐻^𝑇₇,Ω₅₈=𝐻₅+𝑊^𝑇₀𝐻^𝑇₈Ω−𝐼,₆₆=−𝑅₂Γ⁻¹−(𝑅₂Γ⁻¹)^𝑇+𝐻₆𝑊₁+𝑊^𝑇₁𝐻^𝑇₆,Ω₆₇=−𝐻₆+𝑊^𝑇₁𝐻^𝑇₇,Ω₆₈=𝐻₆+𝑊^𝑇₁𝐻^𝑇₈,Ω₇₇=−𝐻₇−𝐻^𝑇₇+𝑃+𝜏₂𝑍₁,Ω₇₈=𝐻₇−𝐻₈,Ω₈₈=𝐻₈+𝐻^𝑇₈−𝛽𝐼,𝐸=𝐸^𝑇₁𝐸^𝑇₂𝐸^𝑇₃𝐸^𝑇₅𝐸^𝑇₆𝐸^𝑇₇𝐸^𝑇₈^𝑇,𝑆=𝑆^𝑇₁𝑆^𝑇₂𝑆^𝑇₃𝑆^𝑇₅𝑆^𝑇₆𝑆^𝑇₇𝑆^𝑇₈^𝑇,𝐻=𝐻^𝑇₁𝐻^𝑇₂𝐻^𝑇₃𝐻^𝑇₅𝐻^𝑇₆𝐻^𝑇₇𝐻^𝑇₈^𝑇,𝑁=𝑁^𝑇₁00𝑁^𝑇₂𝑁^𝑇₃00^𝑇. (3.24) Therefore, the discrete-time recurrent neural network with time-varying delay in (2.1) (i.e., the lower bounds 𝜏₁=0 and the given upper bounds 𝜏₂) approaches globally robustly delay-dependent passivity condition in the sense of Definition 2.1.

Remark 3.5. In the stochastic context, robust delay-dependent passivity conditions are studied in [26] for discrete-time stochastic neural networks with time-varying delays. In this paper, however, robust delay-range-dependent passivity conditions are studied in the deterministic context. It should be noted that deterministic systems and stochastic systems have different properties and need to be dealt with separately. The results given in Theorem 3.2 provide an LMI approach to the robust delay-range-dependent passivity conditions for deterministic discrete-time recurrent neural networks with interval time-varying delay, which is new and represents a contribution to recurrent neural networks systems.

Two numerical examples are now presented to demonstrate the usefulness of the proposed approach.

4. Examples

Example 4.1. Consider the following discrete-time uncertain recurrent neural network: 𝑥(𝑘+1)=(𝐴+Δ𝐴(𝑘))𝑥(𝑘)+(𝑊+Δ𝑊(𝑘))𝑓(𝑥(𝑘))+(𝑊₁+Δ𝑊₁(𝑘))𝑓(𝑥(𝑘−𝜏(𝑘)))+𝑢(𝑘), (4.1) where 𝐴=0.2000.14,𝑊=0.20.250.250.1,𝑊₁=,−0.20.10.10.3𝑀=0.20.10.10.2,𝑁₁=0.10.20.10.2,𝑁₂=0.30.100.1,𝑁₃=.0.20.10.10.1 (4.2) The activation functions in this example are assumed to satisfy Assumption 1 with 𝛼₁=0.5633, 𝛼₂=0.0478. For interval time-varying delay, the best approached values of 𝛽 by Theorem 3.2, for the given upper bound 𝜏₂=10 and the various lower bounds 𝜏₁, are listed inTable 1by the Matlab LMI Control Toolbox. Therefore, using Theorem 3.2, the discrete-time uncertain recurrent neural network with interval time-varying delay (4.1) satisfies robustly delay-range-dependent passivity conditions in the sense of Definition 2.1 for various 𝛽 levels.

Table 1 

The various lower bounds 𝜏₁ and 𝛽 levels for the given upper bound 𝜏₂=10.

Example 4.2. Consider the discrete-time uncertain recurrent neural network with the following parameters: 𝐴=0.10000.10000.2,𝑊=0.1−0.10.20.1−0.20.3−0.10−0.1,𝑊₁=,−0.10.20.1−0.10.40.20.2−0.10.4𝑀=0.100.10.10.300.20.10,𝑁₁=00.020.02−0.10.10.1−0.10.10.1,𝑁₂=,𝑁0−0.020.020.1−0.100.10−0.1₃=.0.010.020−0.10.30.10.100.2 (4.3) The activation functions in this example are assumed to satisfy Assumption 1 with 𝛼₁=0.034, 𝛼₂=0.429, 𝛼₃=0.508. By the Matlab LMI Control Toolbox, it can be verified that Corollary 3.4 in this paper is feasible solution for all delays 0≤𝜏(𝑘)≤15 (i.e., the lower bound 𝜏₁=0 and the upper bound 𝜏₂=15) as follows: 𝑃=2.3189−0.2657−0.0023−0.26572.2981−0.0574−0.0023−0.05742.3612,𝑄₁=,𝑍0.0565−0.00560.0005−0.00560.0679−0.00520.0005−0.00520.0657₁=0.0199−0.00380.0008−0.00380.01500.00020.00080.00020.0144,𝑍₂=,𝑅0.0215−0.00420.0009−0.00420.01580.00020.00090.00020.0153₁=1.89980001.62980001.8378,𝑅₂=,1.29910000.29800000.2635𝜀=3.4160,𝛽=5.0169. (4.4) Thus, by Corollary 3.4, the discrete-time uncertain recurrent neural network with time-varying delay in (2.1) (i.e., the lower bound 𝜏₁=0 and the given upper bound 𝜏₂=15) attains globally robustly delay-dependent passivity condition in the sense of Definition 2.1.

5. Conclusions

This study has investigated the problem of globally robust passivity conditions for a discrete-time recurrent uncertain neural network with interval time-varying delay. A sufficient condition for the solvability of this problem, which takes into account the range for the time delay, has been established that the passivity conditions can be cast in linear matrix inequalities format. It has been shown that the bound for the time-varying delay in a range which ensures that the discrete-time recurrent uncertain neural network with interval time-varying delay attains globally robust passivity conditions can be obtained by solving a convex optimization problem. Two numerical examples have been presented to demonstrate the effectiveness of the proposed approach.