Anti-synchronization for periodic BAM neural networks with Markov scheduling protocol

doi:10.1016/j.neucom.2020.08.015

Neurocomputing

Volume 417, 5 December 2020, Pages 585-592

https://doi.org/10.1016/j.neucom.2020.08.015 Get rights and content

Abstract

This work addresses the problem of the mean square anti-synchronization for the periodic BAM neural networks (NNs) with time-varying delays. To overcome the channel capacity constraint, a new Markov scheduling protocol is introduced, and the scheduling protocol dependent controller is designed. An anti-synchronization error system (ASES) is obtained, and anti-synchronization for the BAM NNs is studied via analyzing the ASES, and a corresponding sufficient condition is established. Then the controller gain calculating method is given. Finally, a numerical simulation is given to shown the effectiveness of the derived approach.

Introduction

Neural networks (NNs) can model complex nonlinear systems and have been explored significantly in different fields, such as signal processing, pattern recognition, complex systems control and parallel computing due to their excellent performances [1], [2], [3]. In many situations, the periodic phenomenon is unavoidable, like geology, life sciences, celestial body movement and engineering project cycle, etc. The complex practical system makes it difficult to model by traditional method, therefore the periodic NNs are proposed to solve this problem which attracts many researchers’ attention [4], [5], [6]. In the first place, periodic NNs can model many physical dynamic systems with cyclic behavior and facilitate many problems in engineering [5], [7]. In the next place, the periodic phenomena addition of NNs leads to many control related issues, which motivates the researchers to study extensively [8]. The research of associative memory network is also an important branch of NNs, and the bidirectional associative memory (BAM) NNs proposed by scholar Kosko have the widest applications in all kinds of associative memory network models [9], [10]. BAM NNs are double-layer bidirectional networks, in which the neurons in one layer are fully connected with the neurons in the other layer, and there are no connections between neurons in the same layer. When the input signal is added to one layer, the output signal can be obtained from the other one, moreover, the initial model can effect any layer in networks to cause bidirectional transmission of information. Comparing with the auto-associative memory of Hopfiled networks, the BAM NNs can realize bidirectional different association which has many forms, for instances discrete, continuous and self-adaptive [11]. BAM NNs are regarded to have better performance than general NNs[12], hence, it is one of the work to study the stability of periodic BAM NNs.

As is known to all, the stability of the system is greatly affected by time delay. According to the different functions of time delay, they will have a negative or positive impact on the system stability, and they have been studied by some scholars [13]. It is worth mentioning that many contributions have been made regarding the stability analysis of fuzzy BAM NNs with delay terms [14], the global exponential stability for discrete BAM NNs with time-varying delays [15], the general decay synchronization for memristor-based BAM NNs under the condition of hybrid time delays [16], and the anti-synchronization of master–slave NNs with time delay in a finite time [17]. For the BAM NNs, the transmission time-varying delays among two lay neurons always different, which implies that study the BAM NNs with asymmetric time-varying delays is an important research topic.

Sharing communication channel is a characteristic of the networked control systems, over which connects spatially distributed elements of control systems, for instance, controllers, actuators and sensors, etc. [18], [19], [20]. Shared communication channel has been used in a broad range of areas, such as vehicle control [21], [22], collaborative virtual environments [23], networked mobile robots [24], and so on, because it is beneficial to improve the system structure flexibility and reduce the installation cost [25]. In some distributed control problems for linear time-invariant plants, feedback-control loops are completed by shared digital network. It is necessary to ensure distributed stable of control loops by formulating scheduling strategy. Networked control for a group of continuous linear time-invariant systems were considered in [26] and a scheduling strategy meeting above conditions were designed. In [27], the data scheduling policy for limited controller-plant communication was constructed to investigate optimal control of the scalar systems. In [28], based on measurement delays and actuation delays, a new delay scheduled impulsive controller was built and the robust $L_{2}$ stability analysis of the closed-loop system was proceeded. In terms of networked control security, researchers promoted the impact of attacks on system performance via optimizing attack scheduling in [29], and the stability of system under the optimal attack was researched. This work aims to further consider a Markov scheduling protocol to overcome communication channel capacity limited.

The synchronization of NNs has been concerned by a lot of scholars and many conclusions have been drawn. On the one hand, global exponential synchronization and robust synchronization have been explored [30], [31]. On the other hand, finite-time synchronization gradually comes into the field of vision of scholars [32], [33]. With the further exploration of system synchronization, the concept of synchronization has been extended to generalized synchronization [34] and even anti-synchronization [35], [36], which means that the state vectors of the response system and the drive system have the same amplitude and opposite signs. As a result, the sum of the two signals converges to zero in the case of anti-synchronization where enhances the realtime performance and confidentiality of the signal [37]. The performances mentioned above are particularly important in some engineering applications such as composite optimization, hardware implementation and communication systems, etc. Now there are many conclusions about anti-synchronization control. In [38], the drive-response finite-time anti-synchronization for BAM NNs was investigated, an appropriate Lyapunov function was employed to obtain some anti-synchronization criteria. In [39], the fixed-time anti-synchronization for master–slave systems with time-varying delays was studied, note that the design of the feedback controller only depends on the current state of the system, not on the delay state. Based on the existing research, this work is devoted to anti-synchronization research for the BAM NNs.

In this paper, the problem of anti-synchronization for the periodic BAM NNs is considered with time-varying delays. A sufficient condition of the periodic BAM NNs anti-synchronization is established. The derived results are illustrated by a numerical example. The main contributions of this work are stated as follows.

•
The BAM NNs with the periodic parameters and the asymmetric time-varying delays are considered, which are more general and can be applied in more complex situations.
•
A new Markov scheduling protocol is proposed to overcome the channel capacity constraint, where the transmission dimensions determined by the channel capacity, and which also depends on the mode of the periodic BAM NNs.
•
The anti-synchronization method for the periodic Markov systems is proposed, which can be directly used to deal with periodic systems and the Markov systems, respectively.

Notations: The symbols $R^{n}$ and $R^{m \times n}$ express the n dimensional vectors and the $m \times n$ real matrices respectively. The symbol ${diag}_{n} {\dots}$ denotes n-dimensional diagonal matrix, and $‖ \cdot ‖_{2}$ denotes the Euclidean norm. For a matrix Q, its transpose, largest eigenvalue and smallest eigenvalue, are expressed as $Q^{T}, λ_{\max} (Q)$ and $λ_{\min} (Q)$ , respectively, and the positive definite of Q is denoted by $Q > 0$ . For a stochastic variable $α (k)$ , its probability and expectation are represented by $P {α (k)}$ and $E {α (k)}$ , respectively.

Section snippets

System description

This work considers the drive periodic BAM NNs with time-varying delays, $\{\begin{matrix} x (k + 1) = & A_{ϑ (k)} x (k) + B_{ϑ (k)} g (y (k - d_{x} (k))) \\ y (k + 1) = & C_{ϑ (k)} y (k) + D_{ϑ (k)} g (x (k - τ_{y} (k))) \end{matrix}$ where the vectors $x (k) \in R^{n}$ and $y (k) \in R^{n}$ are states, the initial states $x (ι_{1}) = φ_{1} (ι_{1})$ for $ι_{1} \in {- τ_{2}, - τ_{2} + 1, \dots, 0}$ and $y (ι_{2}) = φ_{2} (ι_{2})$ for $ι_{2} \in {- d_{2}, - d_{2} + 1, \dots, 0}$ are known. The parameter matrices $A_{ϑ (k)} \in R^{n \times n}, B_{ϑ (k)} \in R^{n \times n}, C_{ϑ (k)} \in R^{n \times n}$ , and $D_{ϑ (k)} \in R^{n \times n}$ are known. The odd nonlinear function $g (\cdot) = {[g_{1} (\cdot), g_{2} (\cdot), \dots, g_{n} (\cdot)]}^{T}$ is the neuron activation function and satisfies Assumption 1.

Results

A sufficient condition of periodic BAM NNs anti-synchronization is given as follows. Before the presentation we define $\begin{matrix} ψ_{1} = {diag}_{n} {{\underset{̲}{l}}_{1} {\overline{l}}_{1}, {\underset{̲}{l}}_{2} {\overline{l}}_{2}, \dots, {\underset{̲}{l}}_{n} {\overline{l}}_{n}} \\ ψ_{2} = {diag}_{n} \{\frac{{\underset{̲}{l}}_{1} + {\overline{l}}_{1}}{2}, \frac{{\underset{̲}{l}}_{2} + {\overline{l}}_{2}}{2}, \dots, \frac{{\underset{̲}{l}}_{n} + {\overline{l}}_{n}}{2}\} . \end{matrix}$

Theorem 1

The periodic BAM NNs (1), (4) reach anti-synchronization, if there exist positive definite matrices $P_{ı, q}, Q_{ı, q}, P_{1}$ and $Q_{1}$ , matrices $K_{x, q}$ and $K_{y, q}$ , positive scalars $∊_{1}, ∊_{2}, ε_{1}$ and $ε_{2}$ , such that the following conditions hold for $ı, j \in M, q, r \in K$ : $[\begin{matrix} Ψ_{11} & Ψ_{12} & 0 & 0 & Ψ_{15} & 0 \\ * & Ψ_{22} & 0 & 0 & 0 & Ψ_{26} \\ * & * & Ψ_{33} & Ψ_{34} & 0 & Ψ_{36} \\ * & * & * & Ψ_{44} & Ψ_{45} & 0 \\ * & * & * & * & - {\bar{P}}_{j, r}^{- 1} & 0 \\ * & * & * & * & * & - {\bar{Q}}_{j, r}^{- 1} \end{matrix}] < 0$ where $\begin{matrix} Ψ \end{matrix}$

Simulation

In this section, a numerical example is used to prove the validity of the above results for anti-synchronization.

Without loss of generality, we assume that the state vectors $x (k) \in R^{3}$ and $y (k) \in R^{3}, m = 4$ dimensions state information can be stochastically sent at each instant, and the periodic $T = 2$ . From the Eq. (5), give the scalar $κ = 9$ , the variable $δ (k) \in K = {1, 2, \dots, 9}$ , and the transition probability matrices are assumed as $\begin{matrix} Π^{1} = [\begin{matrix} 0.10 & 0.05 & 0.10 & 0.20 & 0.10 & 0.04 & 0.20 & 0.16 & 0.05 \\ 0.05 & 0.05 & 0.15 & 0.10 & 0.20 & 0.04 & 0.20 & 0.06 & 0.15 \\ 0.16 & 0.10 \end{matrix}] \end{matrix}$

Conclusions

The anti-synchronization of the periodic BAM NNs with time-varying delays has been studied in this paper. A new Markov scheduling protocol has been introduced to overcome the channel capacity constraint, and a corresponding scheduling protocol depended controller has been designed. Then, sufficient condition has been established to ensure the anti-synchronization of the periodic BAM NNs (1), (4). Finally, a numerical example has been given to prove the effectiveness of the obtained results. How

CRediT authorship contribution statement

Yiting Gan: Methodology, Investigation. Chang Liu: Writing - original draft. Hui Peng: Investigation. Fen Liu: Supervision. Hongxia Rao: Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This work was supported in part by the Local Innovative and Research Teams Project of Guangdong Special Support Program (2019BT02X353), the National Natural Science Foundation of China under Grants (61906047, 61875040, 61903093), and the Science and Technology Program of Guangzhou (201904020006).

Yiting Gan was born in Guangdong province, China. She is currently working toward the B.S. degree in Electrical engineering and automation at Guangdong University of Technology, Guangzhou, China. Her research interest includes neural networks, synchronization.

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Chang Liu was born in Xiangcheng, China, in 1992. He received the B.S. degree from Henan University, Kaifeng, China, in 2016, and the M.S. degree in control science and engineering with the Guangdong University of Technology, Guangzhou, China, in 2019, where he is currently pursuing the PhD. degree in control science and engineering. His current research interests include networked control systems and finite time control and estimation.

Hui Peng received the B.S. degree in automation and the Ph.D. degree in control science and engineering from Hangzhou Dianzi University, Hangzhou, China, in 2013 and 2018, respectively. She is currently a Lecturer with School of Automation, Guangdong University of Technology, Guangzhou, China. Her current research interests include networked control systems, Markov jump systems, and coupled large-scale systems.

Fen Liu was born in Hunan province, China. She received the B.S. degree from Changsha University of Science & Technology, Changsha, China, in 2017. She is currently working toward the M.S. degree in Control Engineering at Guangdong University of Technology, Guangzhou, China. Her research interest includes synchronization of neural network and impulse control.

Hong-Xia Rao was born in Jiangsu Province, China, in 1986. She received the B.S. degree from Nanchang Hangkong University, Nanchang, China, in 2007, the M.S. degree from Nanjing University of Science and Technology, NanJing, China, in 2009, and the Ph.D. degree in control science and engineering from Guangdong University of Technology, China, in 2019. Now she is a lecturer with School of Automation, at Guangdong University of Technology, Guangzhou, China. Her research interests include networked control systems, Markov jump systems and neural networks.

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Anti-synchronization for periodic BAM neural networks with Markov scheduling protocol

Abstract

Introduction

Section snippets

System description

Results

Simulation

Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgment

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IEEE Transactions on Cybernetics

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Chaos Solitons & Fractals

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Robust exponential stability of uncertain delayed neural networks with stochastic perturbation and impulse effects

IEEE Transactions on Neural Networks