Finite-time synchronization of time-delayed neural networks with unknown parameters via adaptive control

doi:10.1016/j.neucom.2018.04.053

Neurocomputing

Volume 308, 25 September 2018, Pages 65-74

https://doi.org/10.1016/j.neucom.2018.04.053 Get rights and content

Abstract

In this paper, the problem of finite-time adaptive synchronization is investigated for two different delayed neural networks with unknown parameters. Two adaptive control approaches are designed in order to synchronize the neural networks in finite time. The first controller fully involves the information of time-varying delay and the second one is delay-independent under the case that time-varying delay is unknown. By utilizing the Lyapunov stability theory, sufficient conditions are proposed to guarantee the finite-time synchronization of the addressed neural networks. In addition, the settling time for synchronization is estimated. Finally, two numerical simulations are used to illustrate the correctness and effectiveness of the proposed methods.

Introduction

In recent years, the neural networks have been widely used in different areas, such as combinatorial optimization, signal processing, machine learning, image encryption and so on [1], [2]. Particularly, the synchronization of neural networks, as a typical collective dynamical behavior, has also received massive attention due to its potential applications in engineering, such as synchronization-based secret communication [3], [4].

It is well known that the time delay cannot be avoided during the hardware implementation of neural networks because of the finite switching speed of the neuron amplifiers and the finite signal propagation speed. Moreover, the time delay may also be the source of instability, oscillation or other undesirable dynamical performance [5], [6], [7], [8], [9]. Note that the neural networks with time delays can exhibit complicated dynamics and even chaotic behaviors. Therefore, there have been an increasing research interest concerning on the synchronization of time-delayed neural networks in recent years [10]. For example, the global synchronization has been discussed in [11] for multiple recurrent neural networks with time delays. In [12], [13], the synchronization problems have been studied for the memristive neural networks with time-varying delays. The adaptive synchronization issues of Cohen–Crossberg neural networks with mixed time-varying delays have been studied in [14] and the exponential synchronization of chaotic neural networks with delays has been investigated in [15], [16].

It is worth mentioning that most of existing results including those mentioned above are focused on the asymptotic or exponential synchronization of time-delayed neural networks, which means that the state trajectories of the response system can keep track of those of the drive system over an infinite time interval. In reality, a requirement is for more precise time specifications of the synchronization in practical process of engineering. Therefore, the finite-time synchronization for various types of neural networks has been attracted increasing attention in order to attain faster convergence speed of the dynamical system of synchronization error [17], [18], [19]. Moreover, the finite-time synchronization control have demonstrated better robustness or disturbance rejection properties [20], [21], [22]. Recently, some methods to investigate finite-time synchronization of time-delayed networks are given by designing the state feedback controller [23], [24], [25], the pining controller [26], [27], the sliding mode controller [28], [29] and the periodically intermittent controller [29], [30].

One should note that all aforementioned finite-time synchronization schemes of time-delayed networks are only applicable to the drive-response systems with same structure, i.e., the system parameter matrices and activation functions of both systems are all identical. However, the structure of both neural networks can not be exactly identical due to external disturbances and modeling errors in practical applications. On the other hand, most studies have been available under the assumption that the system parameters are limited to be known in advance. However, under some circumstances, it is difficult to determine the values of parameters. Therefore, the time-delayed neural networks with different structure and unknown parameters are more general and reasonable in the real world. Furthermore, most of the existing finite-time synchronization control approaches for time-delayed neural networks rely on the assumption that the time delay of systems is known. However, to our best knowledge, very little work has been done in the literature on the finite-time synchronization problem for neural networks with unknown time-varying delay.

Motivated by the above discussions, in this paper, we discuss the problem of finite-time synchronization for two different delayed neural networks with unknown parameters. The main contributions of this paper can be listed in the following aspects. (1) The considered two neural networks, which possess different structure, the unknown parameters and time-varying delays, are more complicated. (2) Compared with the existing control methods, the adaptive control approach is utilized to realize the finite-time synchronization of the delayed neural networks. Furthermore, the adaptive controller can be used to identify the unknown parameters. (3) A delay-dependent adaptive controller is designed when the time-varying delay of the neural networks is known. When the time-varying delay is unknown, a delay-independent adaptive controller design method is given. (4) Several finite-time synchronization criteria are derived based on Lyapunov stability theory and the settling time for synchronization is estimated, which has an important significance to guaranteeing fast convergence speed.

Section snippets

Problem formulation

In this paper, we consider the following class of delayed neural networks described as: $\begin{matrix} \dot{x} (t) = - C x (t) + A g (x (t)) + B g (x (t - τ (t))) + J (t) \end{matrix}$ where $x (t) = {[x_{1} (t), x_{2} (t), \dots, x_{n} (t)]}^{T} \in R^{n}$ is the state vector at time t; n represents the number of neurons in the neural networks; $C = diag {c_{1}, c_{2}, \dots, c_{n}}$ is the self-connection of neurons with c_i > 0; $A = {[a_{i j}]}_{n \times n}$ and $B = {[b_{i j}]}_{n \times n}$ are real matrices standing for the connection strength matrix and the delayed connection strength matrix, respectively; $g (x (t)) = [g_{1} (x_{1} (t)), g_{2} (x_{2} (t)), \dots, g$

Main results

In this section, theoretical results can be given to ensure that the drive system (1) and response system (2) realize finite-time synchronization based on adaptive control protocol. To proceed, we assume that the systems (1) and (2) satisfy the following conditions:

Assumption 1

The unknown system parameter matrices C, A, B, D, M and N are norm bounded, i.e. $\begin{matrix} \begin{matrix} ∥ C ∥ \leq θ_{C}, & ∥ A ∥ \leq θ_{A}, & ∥ B ∥ \leq θ_{B}, \\ ∥ D ∥ \leq θ_{D}, & ∥ M ∥ \leq θ_{M}, & ∥ N ∥ \leq θ_{N}, \end{matrix} \end{matrix}$ where θ_C, θ_A, θ_B, θ_D, θ_M and θ_N are positive constants; ‖ · ‖ is the Euclidean norm denoted by $∥ A ∥ = (\sum_{i = 1}^{n} \sum_{j = 1}^{n}$

Illustrative examples

In this section, we provide two examples to illustrate the usefulness of main results.

Example 1

Consider the delayed neural networks (1) and (2) with the following parameters: $\begin{matrix} C & = & [\begin{matrix} 0.1 & 0 \\ 0 & 0.5 \end{matrix}], A = [\begin{matrix} 2 & - 0.1 \\ - 5 & 4.5 \end{matrix}], \\ B & = & [\begin{matrix} - 1.5 & - 0.1 \\ - 0.2 & - 4 \end{matrix}], D = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}], \\ M & = & [\begin{matrix} 1.8 & 1.5 \\ 0.1 & 1.8 \end{matrix}], N = [\begin{matrix} - 1.4 & 0.1 \\ 0.1 & - 1.4 \end{matrix}] . \end{matrix}$ Thus, we can derive that $∥ C ∥ \leq θ_{C} = 1.5,$ $∥ A ∥ \leq θ_{A} = 8,$ $∥ B ∥ \leq θ_{B} = 5,$ $∥ D ∥ \leq θ_{D} = 2,$ $∥ M ∥ \leq θ_{M} = 4$ and $∥ N ∥ \leq θ_{N} = 3$ . Take activation functions as follows: $\begin{matrix} g (x (t)) & = & [\begin{matrix} 0.5 (| x_{1} (t) + 1 | - | x_{1} (t) - 1 |) \\ 0.5 (| x_{2} (t) + 1 | - | x_{2} (t) - 1 |) \end{matrix}], \\ f (y (t)) & = & [\begin{matrix} tanh (y_{1} (t)) \\ tanh (y_{2} (t)) \end{matrix}] . \end{matrix}$

The time-varying

Conclusion

In this paper, the finite-time synchronization problem has been studied for neural networks with time-varying delay. Two adaptive control approaches have been proposed to ensure the synchronization for two neural networks with different structure and unknown parameter matrices in finite time interval. Based on Lyapunov stability theory, theoretical results have been obtained by choosing some appropriate Lyapunov functionals. The obtained theoretical results have been successfully verified

Acknowledgments

This work was supported by the Heilongjiang Postdoctoral Scientific Research Developmental Fund of China under Grant LBH-Q16121, the National Natural Science Foundation of China under Grant nos. 61673141, 11371115, the Fok Ying Tung Education Foundation of China under Grant 151004, and the Science Research Foundation of Heilongjiang Province Educational Committee of China under Grant 12541175.

Shanqiang Li received the M.Sc. Degree in Applied Mathematics from Harbin University of Science and Technology, China in 2010. He is currently associate professor with the Harbin University of Science and Technology, China and working toward the Ph.D. degree with Harbin Engineering University, China. His research interests include finite-time control and its applications.

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Xiuyan Peng received the Ph.D. degrees in Control Science and Engineering from the Harbin Engineering University, China in 1995. She is currently a professor with the Harbin Engineering University, China. Her research interests include stochastic estimation and control, ship movement prediction and control.

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Yujing Shi received the Ph.D. degree in Control Science and Engineering from Northeastern University, China in 2009. She is currently associate professor with the Harbin University of Science and Technology, China. Her research interests include finite-time synchronization control and distributed control.

View full text

Finite-time synchronization of time-delayed neural networks with unknown parameters via adaptive control

Abstract

Introduction

Section snippets

Problem formulation

Main results

Illustrative examples

Conclusion

Acknowledgments

Neurocomputing

Automatica

Appl. Math. Comput.

Neurocomputing

Neurocomputing

Neural Netw.

Neural Netw.

Automatica

Neurocomputing

Neural Netw.

Int. J. Mach. Learn. Cybern.

Neurocomputing

Neurocomputing

Neurocomputing

Inf. Fusion

Syst. Control Lett.

A unified framework for chaotic neural network approaches to combinatorial optimization

IEEE Trans. Neural Netw.

Adaptive synchronization of chaos for secure communication

Phys. Rev. Lett.

State estimation for a class of discrete nonlinear systems with randomly occurring uncertainties and distributed sensor delays

Int. J. Gen. Syst.