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Advances in Mathematical Physics
Volume 2018, Article ID 7535628, 13 pages
https://doi.org/10.1155/2018/7535628
Research Article

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

1School of Science, Xi’an Technological University, Xi’an 710021, China
2Department 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) [110]. 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 [1118]. 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, 2022]. 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, 2326]. 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 [2933]. 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 [3442]. Specially, this approach has been particularly used to synchronize or control integer-order neural networks (IONNs) [4347]. 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 [3440, 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.

Figure 1: Hysteresis curve.

The objective of this paper is to construct an adaptive fuzzy controller such that the slave system (21) synchronizes the master system (19). To proceed, the following assumption is needed.

Assumption 8. The external disturbance is bounded, i.e., where is an unknown positive constant.

3.2. Synchronization Controller Implement

In this part, we will give the detailed procedure of the adaptive fuzzy controller design as well as the stability analysis. Let the synchronization error be . It follows from (19), (21), and (23) that

Denote ; (24) implies thatwhere is an unknown nonlinear function.

Remark 9. In fact, it is easy to know that is a function of , , , and . In this paper, the synchronization error will be used as the input of the fuzzy logic systems. In addition, the synchronization error can be seen as a bridge between the signals and . Consequently, we will use the denotation for convenience.

Firstly, let us consider an ideal condition. Suppose that is known (i.e., , , and are all known in advance). Thus, the ideal controller can be given bywhere with , . Then, we can give the following theorem.

Theorem 10. If the models of the master and slave systems are known, and the synchronization controller is given by (26), then we have that the synchronization error converges to zero asymptotically.

Proof. Substituting (26) into (25), one hasDefine the following Lyapunov function:According to Lemma 7, the th derivative of is bounded bywhere . Thus, according to Lemma 5 and (29) that the synchronization error will converge to zero asymptotically.

Since is an unknown function, the above controller (26) may be inapplicable. In such a case, by using the fuzzy logic system, let us approximate (16) aswhere is a fuzzy basis function, and is an adjustable parameter of the fuzzy system which drive small enough. Let the parameter estimation error with respect to beand the fuzzy approximation error with respect to be

According to the universal approximation theorem, one can know that the fuzzy systems do not violate the universal approximator property. Consequently, one can make the following assumption.

Assumption 11. There exists an unknown constant such that

Based on the above analysis, one has

Denote and = ,, then one can rewrite (34) aswhere ( represents the amount of the fuzzy rules).

To simplify the stability analysis, we give the following results first.

Theorem 12. Suppose that is a positive definite smooth function. are two adjustable parameters. If it holds thatthen will be small enough eventually if proper parameters are chosen.

Proof. According to (36), one can find a function satisfying andIt follows from (37) thatwhere and . One solves (38) according to (5) aswhere represents the convolution operator. Since that and are all nonnegative, one has . Consequently, one hasIt is easy to know that , and ; thus, by using Lemma 4, one obtains thatwhere , i.e.,Thus, by using Lemma 3, one has for every and large enough time thatThat is to say, if the design parameters are chosen as , then according to (40) and (43) one hasThis completes the proof of Theorem 12.

Then, one can obtain the following theorem.

Theorem 13. Under Assumptions 8 and 11 and proper control parameters, if is the estimation of , is implemented asand and are, respectively, updated bywith , , , being positive design parameters, then one has that the synchronization error eventually converges to an arbitrary small region of zero.

Proof. It follows from (25), (35), and (45) thatMultiplying to both sides of (48) giveswhereis the estimation error of .
It is known that the where is an arbitrary constant. Thus, it follows from (31) and (50) that and .
Define the Lyapunov function asUsing (46), (47), (49), and Lemma 6 one haswith and being two positive constants.
Thus, based on (52) and Theorem 12, one knows that the synchronization error eventually converges to an arbitrary small region of zero if proper control parameters are chosen. This completes the proof of Theorem 13.

Remark 14. It should be pointed out that the fractional-order adaptation law was also introduced in [3, 5, 49, 50]. However, the above adaptation laws only contain a positive term (i.e., the adaptation law is designed as ). Despite using this kind of adaptation law, the asymptotical stability of the system can be guaranteed. Yet, the boundedness of the control parameters cannot be ensured. The proposed adaptation law contains a negative term (for example, in (46)) which will drive the updated parameter tends to a small neighborhood of the origin eventually (see the proof of Theorem 13).

Remark 15. To enforce the synchronization error tending to a region as small as possible, one must make small enough. To meet this objective, one should choose large , and choose small .

Remark 16. It is worth mentioning that, in [51], to discuss the stability of the fractional-order nonlinear systems, a very complicated boundary conditionis assumed to be known. The above condition was proven in [51]. Yet, how to get the exact value of is a challenging work. But in our paper, by using the quadratic Lyapunov functions, the aforementioned problem is solved.

Remark 17. It should be pointed out that the proposed control method does not need the prior knowledge of systems models. Therefore, the control method can be easily extended to the following domains: control of fractional-order nonlinear systems, synchronization of fractional-order chaotic system, and secret communication, and so on. And relating out control method to a potential application is one of our future research directions.

4. Simulation Studies

In (19), letting , , andthe FONN (19) shows chaotic behavior, which is depicted in Figure 2.

Figure 2: Chaotic dynamic of the FONN (19) in (a) 3D space; (b) plane; (c) plane; (d) plane.
4.1. Synchronization of FONN with Constant System Parameters

Let the initial condition of the slave FONN (21) be , and . The external disturbance is defined as . The control design parameters are given as . . In the simulation, the noncontinuous sign function in (45) is replaced with arctan .

With respect to the fuzzy logic systems, its input variable is chosen as the synchronization error . For each input, we give seven Gaussian membership functions on . The parameters of the proposed membership functions, which are defined as , are given in Table 1. These functions are depicted in Figure 3. The initial conditions of the fuzzy parameters are given as .

Table 1: Fuzzy parameters.
Figure 3: Membership functions.

The simulation results are depicted in Figures 46. The results that the signals , , and , respectively, track , , and , as well as the time response of the synchronization errors are presented in Figure 4. The smoothness and the boundedness of the control inputs are given in Figure 5. The fuzzy parameters, which can be concluded to be bounded according to the proposed adaptation law (46), are shown in Figure 6. From these figures one knows that the proposed controller works well and has good synchronization performance.

Figure 4: Simulation results in (a) and ; (b) and ; (c) and ; (d) synchronization errors.
Figure 5: Control inputs.
Figure 6: Fuzzy parameters.

It is well known that the conventional systems usually suffer from discontented performance resulting from modeled errors, parametric uncertainties, input nonlinearities, and external disturbances, because it is impossible to provide accurate mathematical models of practical systems. These system uncertainties can damage the control performance or even lead to unstable of the controlled system if they are not well handled. To show the robust of the proposed method, let us consider the condition that the master system (19) suffers from time-varying system parameters and uncertainties. Suppose that the model of the master FONN (19) is replaced withwhere , , and is an unknown external input signal. and are two unknown matrices. That is, represents unknown system parameter perturbations, and denotes unknown input nonlinearities.

For convenience, let us suppose that , where and are known matrices, and is an unknown matrix.

In the simulation, let , , , and . It is easy to know that and are time-varying matrices.

The simulations are presented in Figure 7. It should be pointed out that, in the simulation, the slave system and the controller are chosen to be the same as those in Section 4.1. From the simulation results we can see that good synchronization performance has been achieved even the master system suffers from time-varying parameters and input nonlinearities. That is, the proposed method has good robustness.

Figure 7: Simulation results when = in (a) and ; (b) and ; (c) and ; (d) synchronization errors.

To indicate the effectiveness of our methods, the simulation results when where represents the random function produced in MATLAB software are shown in Figure 8.

Figure 8: Simulation results when in (a) and ; (b) and ; (c) and ; (d) synchronization errors.

5. Conclusions

In this paper, an synchronization method was proposed for a class of FONNs subject to backlash-like hysteresis by means of adaptive fuzzy control. We showed that fuzzy logic systems can be employed to estimate nonlinear functions in fractional-order nonlinear systems. Based on the fractional stability theorems, an adaptive fuzzy synchronization controller, which can guarantee the synchronization error tends to an arbitrary small region of zero, was constructed. How to combine the proposed method with other control method, such as fractional-order sliding mode control, is one of our future research directions.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors do not have a direct financial relation with any commercial identity mentioned in their paper that might lead to conflicts of interest for any of the authors.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant 11771263 and the Natural Science Foundation of Anhui Province of China under Grant 1808085MF181.

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