Robust finite time stabilization analysis for uncertain neural networks with leakage delay and probabilistic time-varying delays

Abstract

This paper investigates the problem of robust finite time stabilization for a uncertain neural networks with leakage delay and probabilistic time-varying delays. By introducing a stochastic variable which satisfies Bernoulli distribution, the information of probabilistic time-varying delay is equivalently transformed into the deterministic time-varying delay with stochastic parameters. The main objective of this paper is to design a memoryless state feedback control such that the resulting proposed system is robustly finite time stable with admissible uncertainties. Based on a suitable Lyapunov–Krasovskii functional, model transformation technique and Wirtinger-based double integral inequality, the general framework is obtained in terms of linear matrix inequalities to determine the finite time stability and to achieve the control design. Finally, three numerical examples are presented to validate the effectiveness and less conservatism of the proposed method.

Introduction

Neural networks have been widely studied in the past few decades for their practical importance and successful application in various areas such as, signal processing [1], optimization solvers [2], speech recognition [3], and target tracking [4]. During the implementation of artificial neural networks, the finite switching speed of amplifiers and the inherent communication time between neurons inevitably introduce time delays which might cause oscillation, divergence, and even instability. Moreover, it should be mentioned that the stability conditions have obtained based on only the deterministic time-delay case or information of variation range of the time delay in the recent available literature (see e.g., [27], [28], [29], [30], [32], [33], [40], [41]). In point of fact, time delay in a stochastic form exists usually in large amount of many industrial and engineering systems, such as chemical, biological and networked control systems [5], [6], [8]. The probabilistic nature of this type of delay can be obtained by statistical methods, for example, binomial and Poisson distributions. Besides, it often exists in many neural networks in which some values of time delay are very large, but the probabilities of this time delay are very small in practice. In this case, only the constant time delay or variation range of time delay is employed to obtain the stability criteria where the results are more conservative. As an example, in [5], the effects of both variation range and distribution probability of time delays are taken into account for the state estimation problem of discrete time stochastic neural networks. Furthermore, the authors in [6], investigated the problem of asymptotic stability analysis for delayed neural networks with probabilistic time-varying delay by using general convex combination method.

Different from above time delays, a new type of time delay called leakage delay and its effect in dynamical neural networks is one of the important research topics in this domain. As correctly pointed out in [7], the time delay in stabilizing negative feedback term has a tendency to destabilize the system. Hence, the leakage term also has great impact on the dynamical behavior of neural networks. It is inspiring that, many authors (see e.g., [8], [9], [11]) considered the problem of stability analysis of neural networks and system involving time delay in the leakage term. For example, in [11], the stability problem of recurrent neural networks with time delay in leakage term under impulsive perturbations has been investigated. The authors in [8] presented the delay-dependent stability criteria for bidirectional associative memory neural networks with leakage delay through the Jensen׳s inequality, Lyapunov–Krosovskii functional and stochastic analysis approach.

Furthermore, the parameters of neural networks may exhibit some deviations because of the existence of modeling errors, external disturbance, and parameter fluctuations, which would cause the parameter uncertainties. Therefore, it is of practical interest to take into account the uncertainties when studying stability of neural networks and it has gained much research attention (refer [12], [13], [14], [15] and references therein). In [12], the authors analyzed robust stability of neural networks with the time delays and uncertainty. The authors discussed the passivity criteria for memristive delayed neural networks with uncertain parameters and leakage delays in [13]. The authors in [15], investigated the passivity for uncertain neural network with the both leakage delay and time-varying delays.

In the practical sense, the concept of finite-time stability is very important to obtain fast or even finite time convergent speed. The finite-time stability is a different stability concept in which the state does not exceed certain bound and larger values are not permitted during specified time interval in systems such as, the vehicle active suspension, Chuas circuit, temperature control and so on [18], [19], [20]. In addition, finite-time stabilization problems concern with the design of a feedback controller, which ensures finite-time stability of the closed-loop system. Moreover, the authors indicated as in [17], a finite-time controller possesses not only fast convergence but also better robustness and disturbance attenuation properties. A lot of interesting results on finite-time stability and stabilization have been obtained in the literature (see e.g., [21], [22], [23], [24], [25], [26]). In [23], based on set-valued analysis and Kakutani׳s fixed point theorem of set-valued maps, the existence of equilibrium point can be guaranteed for memristor-based neural networks. Then, by designing novel discontinuous controller, some sufficient conditions are proposed to stabilize the states of such neural networks in finite time. The finite-time stabilization analysis of delayed neural networks have been investigated via generalized Jensen inequality in [24]. Recently, the problem of finite-time robust stabilization for delayed neural networks with discontinuous activations and parameter uncertainties based on the nonsmooth analysis and control theory is discussed in [21].

On the other hand, in order to reduce the conservatism of stability criteria, various methods are developed such as Jensen׳s inequality [9], reciprocally convex combination technique [38] free-weighting matrices techniques [32], [36], matrix-based quadratic convex approach [30], secondary delay partitioning method [35] and delay decomposition approach [34], [37]. Recently, the Wirtinger-based integral inequality is an essential technique and newly introduced in [28] to reduce conservatism of stability criteria. In [29], the authors extended the Wirtinger based integral inequality in the double integral form to obtain the less conservatism results of stability problem. So it has been utilized for estimating upper bound of time delay for the time derivative of constructed Lyapunov–Krasovskii functional (see e.g., [33], [31], [39], [42]). Inspired by the above, in this manuscript, the authors utilized Wirtinger-based double integral inequality to study the finite-time stability problem. To the best of author׳s knowledge, up to now there are no results on the problem of robust finite time stabilization for a uncertain neural networks with leakage delay and probabilistic time-varying delays. Thus, the main purpose of this manuscript is to linkage such a gap by making the first attempt of applying the Wirtinger-based integral inequality to study the robust finite-time stabilization problem for uncertain neural networks with leakage delay and probabilistic time-varying delays.

Motivated by the above points, in this paper, the leakage delay and probabilistic time-varying delays are taken into the finite time stabilization problem of uncertain neural networks. The parameter uncertainties are chosen as time-varying but norm bounded which appear in all the matrices in the state equation. By utilizing the Wirtinger-based integral inequality and Lyapunov–Krasovskii functional combined with the linear matrix inequalities (LMIs) technique, the general condition on the feedback control law for this system is derived, which guarantees the robust finite time stability of the resulting closed-loop system in the mean square sense. Finally, the numerical examples are given to illustrate the feasibility and effectiveness of the proposed technique.

The rest of this paper is organized as follows. Section 2 states the problem description and preliminaries. Section 3 include the robust finite time stability of uncertain neural networks with leakage delay and probabilistic time-varying delays. Numerical examples are presented in 4 Numerical examples, 5 Conclusion conclude the paper.

Notations: Throughout this paper, ${\mathbb{R}}^n$ and ${\mathbb{R}}^{n\times n}$ denote the n-dimensional Euclidean space with the scalar product $〈x,y〉\text{or}{x}^{T}y$ of two vectors $x,y$ and the set of all $n\times n$ real matrices, respectively. The superscript T denotes the transposition of a matrix. For symmetric matrices X and Y, $X\ge Y$ (similarly, $X>Y$) means that $X-Y$ is positive semi-definite (similarly, positive definite). $\parallel ·\parallel$ is the Euclidean norm in ${\mathbb{R}}^n$. $\mathbb{E}\left\{x\right\}$, means the expectation of stochastic variable x. Notation ⁎ always denotes the symmetric term in a matrix. ${\lambda }_{\max }\left(A\right)$ or ${\lambda }_{\min }\left(A\right)$ denotes the maximum eigenvalue or the minimum eigenvalue of matrix A, respectively. I denotes the identity matrix with appropriate dimensions. $x_t≔\left\{x\right(t+s):s\in [-{\tau }_2^{\star },0\left]\right\}$. The notation $\mathrm{diag}\{\dots \}$ stands for a block-diagonal matrix.