Design of H∞ performance state estimator for static neural networks with time-varying delay

doi:10.1016/j.neucom.2019.07.018

Neurocomputing

Volume 364, 28 October 2019, Pages 203-208

https://doi.org/10.1016/j.neucom.2019.07.018 Get rights and content

Abstract

In this paper, an improved integral inequality is proposed to design H_∞ performance state estimator for delayed static neural networks (SNNs). A less conservative design criterion of H_∞ performance state estimator is proposed, which ensures the asymptotic stability of estimation error system. Then, the criterion is applied to present a method for designing suitable estimator gain matrices and optimal H_∞ performance index. A numerical example with simulation results is provided to verify the effectiveness of the proposed method.

Introduction

Since more and more applications of neural networks (NNs) have been found in a large number of fields, numerous scholars and researchers have focused on the investigation of NNs in the past decades, and many results have been reported in [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. Recalling existing results on NNs, many of which are about static neural networks (SNNs), such as static imagine processing, combinatorial optimization, associative memory, pattern recognition, classification, and other areas. Due to the fundamental work of [13], the study on design of H_∞ performance state estimator for SNNs has gained increasing attention and a large number of remarkable results of SNNs have been reported in [14], [15], [16], [17], [18], [19]. In addition, in [20], [21], an algorithm of state estimation was provided and much better performance was achieved. Therefore, it is significant to study the problem of design of H_∞ performance state estimator for SNNs.

In practice, time-varying delay is inevitable during the applications of SNNs [22], [23], [24], [25], [26], tremendous efforts have been devoted to study delayed SNNs. Therefore, the study on design of H_∞ performance state estimator for delayed SNNs is of great importance. Generally speaking, based on Lyapunov-Krasovskii functional method, together with reciprocally convex inequality, Wirtinger-based integral inequality, and some other integral inequalities, some sufficient conditions on design of H_∞ performance state estimator for SNNs are derived when noise disturbance is contained [27], [28], [29], [30], [31], [32]. In [27], [28], a traditional reciprocally convex inequality [33] and a double-integral inequality were employed to derive a less conservative design criterion of H_∞ performance state estimator, and better performance was achieved. In [29], [30], the problem of design of H_∞ performance state estimator for delayed SNNs was studied by a traditional reciprocally convex inequality [33] and Wirtinger-based integral inequality [34], and a delay-dependent criterion was proposed. In [31], [32], by using an improved reciprocally convex inequality [35] and some other integral inequalities, further results were obtained. Although the above integral inequalities can obtain good results, there is still room to improve the integral inequality. One of way is how to combine the Wirtinger-based integral inequality [34] and the improved reciprocally convex inequality [35]. Then, the new integral inequality can be used to analyse the problem of design of H_∞ performance state estimator for delayed SNNs, and much better performance can be achieved. In addition, in [27], [28], the estimator gain matrix was dependent on the inverse gain matrix of activation function, which implies that the activation function is strictly monotonically increasing. In fact, the condition restricts the selection of activation function and has great limitations. Therefore, in this paper, we design the estimator gain matrices which are independent of activation function and more general.

The main contributions of this paper are given as follows:

1.
An improved integral inequality based on Wirtinger-based integral inequality and the improved reciprocally convex inequality is proposed to derive a less conservative design criterion of H_∞ performance state estimator, and much better performance can be obtained.
2.
The designed estimator gain matrices are independent of activation function, which eliminates the limitation that the activation function is strictly monotonically increasing.

The rest of this paper is organized as follows. In Section 2, the problem of design of H_∞ performance state estimator for delayed SNNs with noise disturbance is described, and an improved integral inequality is proposed. In Section 3, based on the improved integral inequality, a less conservative criterion is derived to design H_∞ performance state estimator. In Section 4, a numerical example is provided to illustrate the effectiveness of the proposed method. In Section 5, the conclusion is drawn.

Notations: The notations used in this paper are standard. $R^{n}$ represents the n-dimensional Euclidean space. I is an identity matrix with appropriate dimensions. The superscripts T and -1, stand for the transpose and inverse of a matrix, respectively. diag{⋅⋅⋅} denotes a block-diagonal matrix. L₂[0, ∞) is the space of square-integrable vector functions over [0, ∞). P > 0 (P < 0) means that P is a positive definite (negative definite) symmetric matrix. ‘*’ in linear matrix inequalities (LMIs) represents the symmetric term of the matrix.

Section snippets

Problem description

Consider a delayed static neural network with noise disturbance described by ${\begin{matrix} \dot{x} (t) = - A x (t) + f (W x (t - τ (t)) + J) + B_{1} ω (t) \\ y (t) = C x (t) + D x (t - τ (t)) + B_{2} ω (t) \\ z (t) = H_{1} x (t) + H_{2} x (t - τ (t)) \\ x (t) = ϕ (t), t \in [- τ, 0] \end{matrix}$ where $x (t) = {[x_{1} (t), x_{2} (t), \dots, x_{n} (t)]}^{n} \in R^{n}$ is the neuron state vector with n neurons, $y \in R^{m}$ is the network measurement vector, $z (t) \in R^{p}$ is the linear combination of the states to be estimated, $ω (t) \in R^{q}$ is the noise disturbance vector belonging to L₂[0, ∞). $A = d i a g {a_{1}, a_{2}, \dots, a_{n}} > 0$ is a constant matrix, $W = [w_{1}, w_{2}, \dots, w_{n}] \in R^{n \times n}$ is a

Main result

In this section, a less conservative criterion for delayed SNNs is proposed to design H_∞ performance state estimator (4).

Theorem 1

For given scalars γ > 0, τ > 0, μ₁ and μ₂ satisfying (2), the problem of design of H_∞ performance state estimator is solvable if there exist real symmetric matrices P > 0, Q₁ > 0, Q₂ > 0, R > 0, S > 0, T > 0, $Λ = d i a g {λ_{1},$ λ₂, $\dots,$ λ_n} > 0, $M = [\begin{matrix} M_{11} & M_{12} \\ * & M_{22} \end{matrix}],$ $N = [\begin{matrix} N_{11} & N_{12} \\ * & N_{22} \end{matrix}],$ and appropriately dimensioned matrices $Y = [\begin{matrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix}],$ $U_{i} (i = 1, 2, \dots, 7),$ G₁, and G₂ such that: $\begin{matrix} [\begin{matrix} Φ_{1} (τ (t), \dot{τ} (t)) & τ Φ_{2} \end{matrix} \end{matrix}$

Numerical example

In this section, an example is provided to show the effectiveness of the proposed method and advantage over some recent results. Consider the system (1) with parameters from [28] and [32] as follows: $\begin{matrix} A & = d i a g {1.06, 1.42, 0.88}, H_{2} = 0 \\ H_{1} & = [\begin{matrix} 1 & 0 & 0.5 \\ 1 & 0 & 1 \\ 0 & - 1 & 1 \end{matrix}], W = [\begin{matrix} - 0.32 & 0.85 & - 1.36 \\ 1.1 & 0.41 & - 0.5 \\ 0.42 & 0.82 & - 0.95 \end{matrix}] \\ B_{1} & = {[0.2 0.2 0.2]}^{T}, B_{2} = {[0.4 - 0.3]}^{T} \\ C & = [\begin{matrix} 1 & 0.5 & 0 \\ 0 & - 0.5 & 0.6 \end{matrix}], D = [\begin{matrix} 0 & 1 & 0.2 \\ 0 & 0 & 0.5 \end{matrix}] . \end{matrix}$

Firstly, let $τ = 0.8$ and $μ_{2} = - μ_{1} = 0.6,$ for $L = λ I (λ > 0),$ the corresponding H_∞ performance index γ_min can be calculated by our method for different values of λ

Conclusion

In this paper, the issue of design of H_∞ performance state estimator has been studied for delayed SNNs. An improved integral inequality based on Wirtinger-based integral inequality and an improved reciprocally convex inequality is proposed to handle time-derivative of Lyapunov functional. Then, a less conservative criterion of H_∞ performance state estimator is derived, which ensures the asymptotic stability of estimation error system. As a result, the designed estimator gain matrices can be

Disclosure of conflicts of interest

The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (grant nos. 61433004 and 61627809), the Liaoning Revitalization Talents Program (grant no. XLYC1802010), and in part by SAPI Fundamental Research Funds (grant no. 2018ZCX22).

Guoqiang Tan received the B.S. degree in Shandong Jianzhu University, Jinan, China, in 2015 and received the M.S. degree in Qufu Normal University, Rizhao, China, in 2018. He is currently pursuing the Ph.D. degree in College of Information Science and Engineering, Northeastern University, Shenyang, China. His current research interests include neural network, state estimation, time delay system.

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Zhanshan Wang received the B.S. degrees in industry electric automation from Baotou Iron and Steel Institute (now Inner Mongolia University of Science & Technology), Baotou, China, in 1994, and received the M.S. degree in control theory and control engineering from Fushun Petroleum Institute (now Liaoning Shihua University), Fushun, China, in 2001. He received the Ph.D. degree in control theory and control engineering from Northeastern University, Shenyang, China, in 2006. His research interests include stability theory of dynamical systems with delays, neural networks theory, complex networks, learning control, fault diagnosis, fault tolerant control, intelligent automation and their applications in power systems and smart grid.

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Design of H∞ performance state estimator for static neural networks with time-varying delay

Abstract

Introduction

Section snippets

Problem description

Main result

Numerical example

Conclusion

Disclosure of conflicts of interest

Acknowledgments

Neural Netw.

Neurocomputing

Neurocomputing

Neural Netw.

Neurocomputing

Neurocomputing

Neurocomputing

Automatica

Neurocomputing

Neurocomputing

Automatica

Automatica

Automatica

Automatica

Automatica

J. Frankl. Inst.

Neurocomputing

Design of H_∞ performance state estimator for static neural networks with time-varying delay