Extended dissipativity state estimation for generalized neural networks with time-varying delay via delay-product-type functionals and integral inequality

Abstract

In this paper, the problem of extended dissipativity state estimation for delayed generalized neural networks (GNNs) is investigated. Firstly, in order to facilitate the use of more information of time-varying delay, a class of delay-product-type Lyapunov-Krasovskii functional (LKF) is proposed. Secondly, in order to accurately estimate the upper bound of the time-derivative of the constructed LKF, a delay-product-type integral inequality is proposed, then some sufficient conditions are obtained to guarantee the extended dissipativity state estimation for delayed GNNs. Moreover, the extended dissipativity state estimation can be used to tackle the problem of $H_{\infty }$ performance state estimation, passivity performance state estimation, $L_2$-$L_{\infty }$ performance state estimation, and $(\mathcal{Q},\mathcal{S},\mathcal{R})$-$\gamma$-dissipativity state estimation for delayed GNNs. Finally, simulations are provided to illustrate the effectiveness of the proposed method.

Introduction

In recent years, neural networks (NNs) have gained great attention since more applications of NNs have been found in many areas, such as fault diagnosis, optimization problem, pattern recognition, image processing, multi-agent systems, and industrial automation [1], [2], [3], [4], [5], [6]. On the basis of the states of neurons, NNs are cast as local field neural networks (LFNNs) and static neural networks (SNNs), and the study on LFNNs and SNNs is conducted in separate sense, then, it is significance of combining LFNNs and SNNs into unified models which can be called as GNNs. Due to the finite switching speed of amplifiers, time delay is inevitable in GNNs [7], [8], [9], [10], [11]. As we all know, the information of neural states is often unknown and it often needs to be estimated based on available information in GNNs [12], [13], [14], [15]. Hence, it is necessary to investigate state estimation problem for delayed GNNs.

The dissipativity theory was initially introduced by Willems [16], involving storage functions and supply rates. Dissipativity means that the energy dissipated inside a system is less than the energy supplied from an outside source, which has been useful for dealing with synthesis and analysis problems of circuit systems [17], [18]. Since storage functions induced by dissipativity usually act as LKF candidates, the dissipativity technique has now been widely used to solve stability, stabilization, and state estimation problems for GNNs [19], [20]. Since the extended dissipativity contains $H_{\infty }$ performance, passivity performance, $L_2$-$L_{\infty }$ performance, and $(\mathcal{Q},\mathcal{S},\mathcal{R})$-$\gamma$-dissipativity, the study of extended dissipativity state estimation has obtained great interest for delayed GNNs [21], [22]. It is well known that the extended dissipativity contains $H_{\infty }$ performance, passivity performance, $L_2$-$L_{\infty }$ performance, and $(\mathcal{Q},\mathcal{S},\mathcal{R})$-$\gamma$-dissipativity [23], [24], [25], [26]. In order to obtain some sufficient conditions on extended dissipativity state estimation for GNNs, various techniques of integral inequality method have been used to tackle this issue. In [23], by employing the Jensen inequality, the analysis of extended dissipativity for NNs was investigated. In [24], by employing the Wirtinger-based inequality, some extended dissipativity criteria for GNNs were established. In [25], by utilizing the reciprocally convex approach and the free-matrix-based inequality, some extended dissipativity criteria for generalized Markovian switching NNs were derived. In [26], less conservative extended dissipativity state estimation conditions were provided by employing the reciprocally convex approach.

Among the above mentioned analysis, it is clear that although the state estimation issue for NNs has been widely studied, there exist limited results on NNs based on LKF method and the above mentioned integral inequality still remains room for improvement, which mainly motivates this paper. In order to facilitate the use of more information of time-varying delay, a class of delay-product-type LKF is proposed, in which not only the non-integral quadratic terms are included, but also the integral terms are contained, too. Based on the above analysis, by employing the proposed delay-product-type integral inequality, the problem of extended dissipativity state estimation is continuously studied for delayed GNNs via delay-product-type functionals.

The main contributions of this paper can be reflected as follows:

• In order to facilitate the use of more information of time-varying delay, a class of delay-product-type LKF is proposed. Compared with the delay-product-type terms in [27], [28], the delay-product-type terms developed in this paper contain not only the non-integral quadratic terms but also the integral terms. Compared with the delay-product-type terms in [29], the non-integral quadratic terms and the integral terms are connected by employing the third-order Bessel-Legendre inequality rather than the Wirtinger-based inequality.

• A delay-product-type integral inequality is proposed, which combines the reciprocally convex inequality and the third-order Bessel-Legendre inequality into a unified integral inequality, then the time-derivative of the constructed LKF is estimated with less conservatism. As a result, some less conservative extended dissipativity state estimation conditions for delayed GNNs can be obtained.

Notations: ${\mathbb{R}}^n$ represents the n-dimensional Euclidean space. $R>0$ denotes that R is positive definite. The superscript T and $-1$ stand for matrix transposition and matrix inverse, respectively. The space of square-integrable vector functions over $[0,\infty )$ can be denoted by $L_2[0,\infty )$. $\mathrm{diag}\{\cdots \}$ represents the block-diagonal matrix. ‘*’ in linear matrix inequalities (LMIs) stands for the symmetric term of the matrix. $\mathit{Co}\{\cdots \}$ denotes a point set. $\mathrm{col}\{\cdots \}$ represents a block-column matrix. $\mathrm{He}\left\{X\right\}$ stands for $X+X^T$.