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.
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) [
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 [
Up to now, fuzzy control methods have been studied extensively [
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:
The
The following results on fractional calculus will help us to facilitate the synchronization controller design as well as the stability analysis.
The Mittag-Leffler function is defined by
The Laplace transform of (
Let
Let
Let
Let
Let
Let
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 [
Consider a class of FONNs described as
Write
To guarantee the existence and uniqueness of the solutions of the fractional-order neural network (
The slave system is expressed by
When
Hysteresis curve.
The objective of this paper is to construct an adaptive fuzzy controller such that the slave system (
The external disturbance is bounded, i.e.,
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
Denote
In fact, it is easy to know that
Firstly, let us consider an ideal condition. Suppose that
If the models of the master and slave systems are known, and the synchronization controller is given by (
Substituting (
According to Lemma
Since
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.
There exists an unknown constant
Based on the above analysis, one has
Denote
To simplify the stability analysis, we give the following results first.
Suppose that
According to (
It is easy to know that
Thus, by using Lemma
That is to say, if the design parameters are chosen as
Then, one can obtain the following theorem.
Under Assumptions
It follows from (
Multiplying
It is known that the
Define the Lyapunov function as
Using (
Thus, based on (
It should be pointed out that the fractional-order adaptation law was also introduced in [
To enforce the synchronization error tending to a region as small as possible, one must make
It is worth mentioning that, in [
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.
In (
Chaotic dynamic of the FONN (
Let the initial condition of the slave FONN (
With respect to the fuzzy logic systems, its input variable is chosen as the synchronization error
Fuzzy parameters.
Membership functions | Parameter | Parameter |
---|---|---|
Membership function 1 | 1 | |
Membership function 2 | 0.8 | |
Membership function 3 | 0.2 | |
Membership function 4 | 0.09 | 0 |
Membership function 5 | 0.2 | 0.3 |
Membership function 6 | 0.8 | 2 |
Membership function 7 | 1 | 5 |
Membership functions.
The simulation results are depicted in Figures
Simulation results in (a)
Control inputs.
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 (
For convenience, let us suppose that
In the simulation, let
The simulations are presented in Figure
Simulation results when
To indicate the effectiveness of our methods, the simulation results when
Simulation results when
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.
The data used to support the findings of this study are available from the corresponding author upon request.
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.
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.