A new result on H∞ performance state estimation for static neural networks with time-varying delays

doi:10.1016/j.amc.2020.125556

Applied Mathematics and Computation

Volume 388, 1 January 2021, 125556

https://doi.org/10.1016/j.amc.2020.125556 Get rights and content

Highlights

•
By dividing the estimation error of activation function into two parts, an improved Lyapunov-Krasovskii functional (LKF) is constructed, in which the slope information of activation function (SIAF) can be fully captured compared with those in [23, 28].
•
A parameter-dependent reciprocally convex inequality is proposed to estimate the derivative of the LKF, which encompasses some existing results as its special cases such as [29-31], in which the two parameters can be chosen freely and independently.
•
Compared with the existing methods such as in [23, 28, 29, 32], this paper fully considers the free structure of the introduced slack matrices, which directly lead to the reduction of conservativeness in the estimator solution.

Abstract

This paper investigates the H_∞ performance state estimation problem for static neural networks with time-varying delays. A parameter-dependent reciprocally convex inequality (PDRCI) is presented, which encompasses some existing results as its special cases. By dividing the estimation error of activation function into two parts, an improved Lyapunov-Krasovskii functional (LKF) is constructed, in which the slope information of activation function (SIAF) can be fully captured. Combining PDRCI and the improved LKF, a new criterion is obtained to ensure the estimation error system to be asymptotically stable with H_∞ performance. By using a decoupling principle, the estimator gain matrices are solved in terms of linear matrix inequalities (LMIs). Compared with some existing works, the restrictions on slack matrices are overcome, which directly leads to performance improvement and reduction of conservativeness in the estimator solution. Two examples are illustrated to verify the advantages of the developed criterion.

Introduction

During the past few decades, neural networks (NNs) have been applied in many fields, such as pattern recognition, signal processing, associative memories, optimization problems and so on [1], [2], [3], [4], [5], [6], [7]. Depending on the basic variables used in modeling, the recurrent NNs can be classified into two types [8]. One of them is local field NNs in which the local field states of neurons are adopted as the basic variables. Another is SNNs with the neuron states being the basic variables. The two types of RNNs are generally not equivalent except that special conditions are met. In [9], a practical example was presented to show that these conditions cannot be easily reached. Therefore, in recent years, SNNs have gradually become an active area of research, and many works have been reported such as static image processing, combinatorial optimization, associative memory, pattern recognition, classification, and other areas [10], [11]. Due to the fundamental work of [12], 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 [13], [14], [15].

Time delay is inevitable in practical systems and they may affect the performance of dynamic systems [16], [17], [18], [19], [20], [21], [22], which may cause oscillation, instability, or bad performance of dynamic systems. Thus, the research on state estimation of SNNs with time delay in very important. Based on the Lyapunov-Krasovskii functional (LKF) method and bounding techniques, many significant results on state estimation for SNNs have been reported in [23], [24], [25], [26], [27], [28], [29], [30], [31], [32]. In the case of bounding techniques, various reciprocally convex inequalities (RCIs) are popularly employed to handle the single integral term $- \int_{t - δ}^{t} \dot{υ} (s) d s$ . For example, the traditional reciprocally convex inequality (TRCI) is used to handle the state estimation problems for SNNs with time-varying delays, see [23], [24], [25], [26], [27]. Then, by introducing some slack matrices, the improved reciprocally convex inequality (IRCI) is proposed in [22], which have been widely applied to solve the state estimation problems for SNNs such as [28], [29], [30]. Further, an extended version of IRCI is proposed in [31], [32] to derive a less conservative design criterion of estimator. Although the improved reciprocally convex inequality in [29], [30], [31], [32] provide tighter bounds than some existing ones, there still deserves further investigation.

Recently, to facilitate the use of the slope information of the activation (SIAF), a novel LKF is constructed to derive an Arcak-type estimator design criterion for SNNs in [23]. Then, the method has been applied to design the non-fragile H_∞ performance state estimation for delayed SNNs in [28]. However, the SIAF has not been fully considered in the constructed LKFs in [23], [28], there still deserves further investigation. On the other hand, in some existing works such as in [23], [28], [29], [32], some slack matrices are introduced to avoid the cross-coupling of matrix product terms (such as X₁, X₂, X₃ in [29]). However, in order to obtain LMI-based estimator design conditions, these slack matrices require to be constrained (see $X_{2} = ε_{1} X_{1},$ $X_{3} = ε_{2} X_{1}$ in [29] for details), which may bring some conservativeness in estimator solution. Thus, how to avoid the restrictions is a meaningful topic for the reduction of conservativeness.

Summarize the above discussions, this paper considers the problem of H_∞ performance state estimation for delayed SNNs. The main contributions are as follows. (i) An improved LKF is constructed to make full use of the SIAF compared with those in [23], [28]. (ii) A parameter-dependent reciprocally convex inequality is proposed to estimate the derivative of the LKF, which encompasses some existing results as its special cases such as [29], [30], [31]. (iii) Compared with the existing methods such as in [23], [28], [29], [32], this paper fully considers the free structure of the introduced slack matrices, which directly lead to the reduction of conservativeness in the estimator solution.

Notation: Throughout this paper, $R^{n}$ represents the n-dimensional Euclidean space; He[X] means $X + X^{T}$ ; col[X, Y] represents [X^T, Y^T]^T.

Section snippets

System description and preliminaries

Consider the following SNNs with time-varying delays: ${\begin{matrix} \dot{τ} (t) = - A τ (t) + ζ (ϵ (t)) + B_{1} w (t), \\ ϵ (t) = W τ (t - δ (t)) + J, \\ j (t) = C_{0} τ (t) + C_{1} τ (t - δ (t)) + B_{2} w (t), \\ λ (t) = L_{0} τ (t) + L_{1} τ (t - δ (t)), \\ τ (t) = ϕ (t), t \in [- δ, 0], \end{matrix}$ where $τ (t) = c o l [τ_{1} (t), τ_{2} (t), \dots, τ_{n} (t)] \in R^{n_{u}},$ $j (t) \in R^{n_{j}},$ $λ (t) \in R^{n_{λ}},$ $w (t) \in R^{n_{w}}$ (belongs to $L_{2} [0, + \infty)$ ), ϕ(t) are the state vector, the network measurement vector, the estimated vector, the disturbance input, and the initial condition defined on $[- δ, 0],$ respectively. $W = [ψ_{1}, ψ_{2}, \dots, ψ_{n}]$ is the delayed connection weight matrix. A, B₁, B₂, C₀

Main results

In this section, a novel design method on H_∞ state estimation for SNNs (1) is given as follows.

Theorem 1

For given scalars α, σ, κ_n, γ, δ, and ν, system (5) is asymptotically stable and satisfies (7) if there exist diagonal positive matrices Λ_l, $S_{m} \in R^{n_{u} \times n_{u}},$ symmetric positive-definite matrices $P \in R^{5 n_{u} \times 5 n_{u}},$ $Q_{n} \in R^{3 n_{u} \times 3 n_{u}},$ $R_{n} \in R^{n_{u} \times n_{u}},$ matrices $V_{n} \in R^{n_{u} \times n_{u}},$ $Y_{n} \in R^{n_{u} \times n_{u}},$ $M \in R^{n_{u} \times n_{u}},$ $N \in R^{n_{u} \times n_{u}},$ $U \in R^{n_{u} \times n_{u}}$ such that the following inequalities hold for $l = 1, 2, \dots, 6,$ $m = 1, 2, 3, 4,$ and $n = 1, 2$ $[\begin{matrix} Ω (0, \dot{δ} (t)) |_{\dot{δ} (t) \in {- ν, ν}} σ M + G^{T} \\ * - σ U - σ U^{T} \end{matrix}] < 0,$ $[\begin{matrix} Ω \end{matrix}$

Numerical example

In this section, two examples are provided to show the proposed results by comparing some recent works.

Consider the SNNs (1) given in [23] with: $\begin{matrix} A = d i a g {1.06, 1.42, 0.88}, L_{1} = 0, \\ L_{0} = [\begin{matrix} 1 & 0 & 0.5 \\ 1 & 0 & 1 \\ 0 & - 1 & 1 \end{matrix}], W = [\begin{matrix} - 0.32 & 0.85 & - 1.36 \\ 1.10 & 0.41 & - 0.5 \\ 0.42 & 0.82 & - 0.95 \end{matrix}], \\ C_{0} = [\begin{matrix} 1 & 0.5 & 0 \\ 0 & - 0.5 & 0.6 \end{matrix}], C_{1} = [\begin{matrix} 0 & 1 & 2 \\ 0 & 0 & 0.5 \end{matrix}], \\ B_{1} = {[\begin{matrix} 0.2 & 0.2 & 0.2 \end{matrix}]}^{T}, B_{2} = {[\begin{matrix} 0.4 & - 0.3 \end{matrix}]}^{T} . \end{matrix}$ The advantages of the proposed method of present paper can be shown by the following three cases. (i) Let $σ = 0.5,$ $L = d i a g {1.2, 1.3, 1.4},$ $ν = 0.5,$ $κ_{1} = 0.02,$ and $κ_{2} = 0.01$ . For different upper bounds of δ, the

Conclusions

In this paper, the problem of H_∞ performance state estimation has been considered for delayed static neural networks by adopting an Arcak-type state estimator. A parameter-dependent reciprocally convex inequality (PDRCI) covering some existing results is proposed in this paper. To take advantages of the SIAF, an improved LKF has been constructed. Based on the PDRCI, the improved LKF, and a decoupling principle, the estimator gain matrices have been obtained by solving LMIs. Compared with some

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under No. 61973070, the Liaoning Revitalization Talents Program under Grant No. XLYC1802010, and in part by SAPI Fundamental Research Funds under Grant No. 2018ZCX22.

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A new result on H∞ performance state estimation for static neural networks with time-varying delays

Highlights

Abstract

Introduction

Section snippets

System description and preliminaries

Main results

Numerical example

Conclusions

Acknowledgments

Neural Netw.

Neural Netw.

J. Franklin Inst.

Appl. Math. Comput.

Inf. Sci.

Automatica

Appl. Math. Comput.

Appl. Math. Comput.

Neurocomputing

Neurocomputing

Syst. Control Lett.

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IEEE Trans. Fuzzy Syst.

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IEEE Trans. Neural Netw. Learn. Syst.

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IEEE Trans. Circuits Syst. II

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A new result on H_∞ performance state estimation for static neural networks with time-varying delays

h_∞ performance state estimation for static neural networks with time-varying delays via two improved inequalities