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

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.


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 fractionalorder differential equations (FDE) [1][2][3][4][5][6][7][8][9][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][12][13][14][15][16][17][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][21][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][24][25][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][30][31][32][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][35][36][37][38][39][40][41][42].Specially, this approach has been particularly used to synchronize or control integer-order neural networks (IONNs) [43][44][45][46][47].In above literature, fuzzy logic systems were employed to approximate the uncertain 2 Advances in Mathematical Physics 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: (1) 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.(2) 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).(3) The control of fractional-order nonlinear systems with backlashlike hysteresis input is studied.

Preliminaries
2.1.Some Basic Results of Fractional Calculus.The th fractional integral is defined by where The th fractional-order derivative is given as where  − 1 ≤  <  ( ∈ N).The Laplace transform of the Caputo fractional derivative is where () = L{()}.For convenience, we always assume that 0 <  ≤ 1 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.
Lemma 5 (see [2]).Let () = 0 be an equilibrium point of the following fractional-order nonlinear system: If one can find a Lyapunov function (, ()) as well as three class- functions   ,  = 1, 2, 3 such that then system (10) will be asymptotically stable.

Problem Description. Consider a class of FONNs described as
where  = 1, . . ., ,   () is the state variable,   > 0 and   ,  = 1, 2, . . .,  are constants,   represents the external input, and 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 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., |  ()| ≤   ,  = 1, 2, . . .,  where   is an unknown positive constant.
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.
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.
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.
To simplify the stability analysis, we give the following results first.
Theorem 12. Suppose that ℎ() ∈ R is a positive definite smooth function. 1 > 0,  2 ∈ R are two adjustable parameters.If it holds that then ℎ() will be small enough eventually if proper parameters are chosen.
Then, one can obtain the following theorem.
Define the Lyapunov function as Advances in Mathematical Physics Using ( 46), ( 47), (49), and Lemma 6 one has with 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 fractionalorder 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 D  () =   ()  ()).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, − 1  2   () 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  2 / 1 small enough.To meet this objective, one should choose large   ,  1 ,  3 and choose small  2 ,  4 .
Remark 16.It is worth mentioning that, in [51], to discuss the stability of the fractional-order nonlinear systems, a very complicated boundary condition is 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.

Simulation Studies
In (19), letting () ∈ R 3 , ( the FONN (19) shows chaotic behavior, which is depicted in Figure 2.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 [− 5 5].The parameters of the proposed membership functions, which are  defined as  −(  ()−) 2 /2 2 , are given in Table 1.These functions are depicted in Figure 3.The initial conditions of the fuzzy parameters are given as The simulation results are depicted in Figures 4-6.The results that the signals  1 (),  2 (), and  3 (), respectively, track  1 (),  2 (), and  3 (), 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.
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)     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.

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.