Advances in Mathematical Physics

Volume 2018, Article ID 7535628, 13 pages

https://doi.org/10.1155/2018/7535628

## Adaptive Fuzzy Synchronization of Fractional-Order Chaotic Neural Networks with Backlash-Like Hysteresis

^{1}School of Science, Xi’an Technological University, Xi’an 710021, China^{2}Department of Applied Mathematics, Huainan Normal University, Huainan 2320389, China

Correspondence should be addressed to Heng Liu; moc.liamg@221gnehuil

Received 6 January 2018; Revised 11 April 2018; Accepted 26 April 2018; Published 3 June 2018

Academic Editor: Christos Volos

Copyright © 2018 Wenqing Fu and Heng Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An adaptive fuzzy synchronization controller is designed for a class of fractional-order neural networks (FONNs) subject to backlash-like hysteresis input. Fuzzy logic systems are used to approximate the system uncertainties as well as the unknown terms of the backlash-like hysteresis. An adaptive fuzzy controller, which can guarantee the synchronization errors tend to an arbitrary small region, is given. The stability of the closed-loop system is rigorously analyzed based on fractional Lyapunov stability criterion. Fractional adaptation laws are established to update the fuzzy parameters. Finally, some simulation examples are provided to indicate the effectiveness and the robust of the proposed control method.

#### 1. Introduction

In the past two decades, study results of fractional calculus have received more and more attention because, compared with the classical integer-order calculus, the fractional-order one has many interesting and special properties. It has also been proven that a lot kinds of actual systems, ranging from life science and engineering to secret communication and system control, can be better modeled by using fractional-order differential equations (FDE) [1–10]. The nonlinear system, which is described by FDE, has memory. This advantage makes it possible to describe the hereditary as well as memory characters of many systems and processes. On this account, a lot of scholars employed the fractional-order derivative to replace the integer-order one in neural networks to get the FONNs [11–18]. It is known that the fractional model equips the neurons with more powerful computation ability, and these abilities could be used in information processing, frequency-independent phase shifts of oscillatory neuronal firing, and stimulus anticipation [13, 19]. By far, lots of methods have been given to synchronize FONNs [5, 12, 13, 20–22]. It should be mentioned that, in above works, the model of the master FONN should be known in advance. How to design synchronization controller when the master system’s model is unknown is a challenging but interesting work.

It is well known that hysteresis can be found in a great mount of physical systems or devices, for instance, biology optics, mechanical actuators, electromagnetism, and electronic circuits [6, 23–26]. Hysteresis can damage the control performance or even lead to the instability of the controlled system. How to construct proper controller for these kinds of systems is an interesting work. With respect to integer-order systems subject to hysteresis, a lot of results have been given. In [27], a feedback controller was introduced to control nonlinear systems with hysteresis. The control of systems subject to Prandtl-Ishlinskii hysteresis was studied in [28]. To see more results on the control of integer-order systems with hysteresis please refer to [29–33]. However, with respect to fractional-order nonlinear systems with hysteresis, the related literatures are very few.

Up to now, fuzzy control methods have been studied extensively [34–42]. Specially, this approach has been particularly used to synchronize or control integer-order neural networks (IONNs) [43–47]. In above literature, fuzzy logic systems were employed to approximate the uncertain functions. To enhance the approximation ability of the fuzzy system, some robust terms, for example, sliding mode control, control should be used together with the main fuzzy adaptive control term. It should be pointed out that the above results are limited to uncertain IONNs. It is advisable to discuss the synchronization problem for uncertain FONNs.

In our paper, an adaptive fuzzy control approach is introduced for synchronizing two uncertain FONNs. Based on some fractional Lyapunov stability theorems, the stability analysis and the controller implement are given. To show the effectiveness of the proposed synchronization method, some illustrative examples are presented. Bearing the results of aforementioned works in mind, the main contributions of our study consist of the following: by designing an adaptive fuzzy controller, a practical synchronization is proposed for a class of uncertain FONNs. To the best of our knowledge, how to construct fuzzy adaptive control for FONNs has not been previously investigated up to now, except some preliminaries works in [8, 46]. It should be pointed out that, in these works, the integer-order stability analysis method is used. However, in this paper, we will use the fractional stability analysis approach, and the stability of the closed-loop system is proved rigorously. The models of the FONNs are assumed to be fully unknown (i.e., the controller designed is free of the models of both master and slave systems). The control of fractional-order nonlinear systems with backlash-like hysteresis input is studied.

#### 2. Preliminaries

##### 2.1. Some Basic Results of Fractional Calculus

The th fractional integral is defined bywhere . The th fractional-order derivative is given aswhere (). The Laplace transform of the Caputo fractional derivative iswhere . For convenience, we always assume that in the rest of this paper.

The following results on fractional calculus will help us to facilitate the synchronization controller design as well as the stability analysis.

*Definition 1 (see [1]). *The Mittag-Leffler function is defined bywhere , and .

The Laplace transform of (4) is as follows [1]:

Lemma 2 (see [1]). *Let with ; then one has*

Lemma 3 (see [1]). *Let and to constants satisfying andand then the following equality holds:*

Lemma 4 (see [1]). *Let and . If there exists a positive constant such that , then one has where is a positive real constant, and .*

Lemma 5 (see [2]). *Let be an equilibrium point of the following fractional-order nonlinear system:If one can find a Lyapunov function as well as three class- functions such thatthen system (10) will be asymptotically stable.*

Lemma 6 (see [4]). *Let be a continuous and derivable function. Then, for any ,*

Lemma 7 (see [3]). *Let be a continuous and derivable function. Then, for any ,where is a positive definite constant matrix.*

##### 2.2. Description of a Fuzzy System

A fuzzy logic system consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine working on the fuzzy rules, and the defuzzifier [34–40, 48]. Usually, a fuzzy logic system is modeled bywhere (a Lipschitz-continuous mapping from a compact subset to the real line ) is called the output of the fuzzy logic system, (the set of all continuous mappings from to which have continuous derivatives) is called the input vector, , consists of fuzzy sets , (a mapping from to the closed unit interval ) is called the membership function of rule , and (a mapping from to ) is called the centroid of the th consequent set (); we may identify with for the sake of convenience. Write and , where (called the th fuzzy basis function, ) is a continuous mapping (and thus is continuous) defined byThen system (14) can be rewritten as

#### 3. Main Results

##### 3.1. Problem Description

Consider a class of FONNs described aswhere is the state variable, and are constants, represents the external input, and is a smooth nonlinear function.

Write , , , , ,then (17) can be written into the following compact form:

To guarantee the existence and uniqueness of the solutions of the fractional-order neural network (19) (see, [3]), we assume that the functions are Lipschitz-continuous, i.e., for all ,where is a positive constant.

The slave system is expressed bywhere is the state vector of the slave system, is a positive definite control gain matrix, is an unknown external disturbance, and represents hysteresis type of nonlinear control input which is described bywhere is the control input, and are three constants satisfying and . One can rewrite (22) as

When , , , and , the behavior of the backlash-like hysteresis is depicted in Figure 1.