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This paper presents an adaptive fuzzy synchronization control strategy for a class of different uncertain fractional-order chaotic/hyperchaotic systems with unknown external disturbances via T-S fuzzy systems, where the parallel distributed compensation technology is provided to design adaptive controller with fractional adaptation laws. T-S fuzzy models are employed to approximate the unknown nonlinear systems and tracking error signals are used to update the parametric estimates. The asymptotic stability of the closed-loop system and the boundedness of the states and parameters are guaranteed by fractional Lyapunov theory. This approach is also valid for synchronization of fractional-order chaotic systems with the same system structure. One constructive example is given to verify the feasibility and superiority of the proposed method.

Fractional calculus is a mathematical topic being more than 300 years old, which can be traced back to the birth of integer-order calculus. The fundamentals results of fractional calculus were concluded in [

The conventional nonlinear systems control approaches suffer from discontented performance resulting from structure and parametric uncertainties, external disturbances. Usually, it is very hard to provide accurate mathematical models [

This work investigates the chaos synchronization of fractional-order chaotic systems with different structures based on T-S fuzzy systems, where external disturbances in slaves system are considered. T-S fuzzy systems with random rule consequents are introduced to model controlled systems, whereas T-S fuzzy systems that have the same rule consequents with Mamdani fuzzy systems are used to approximate unknown nonlinear functions. The asymptotic stability of closed-loop system is proofed based on fractional Lyapunov stability theory. Compared to previous literature, the main contributions of this paper are as follows:

There are two frequently used definitions for fractional integration and differentiation: Riemann-Liouville (denote R-L) and Caputo definitions. In this paper, we will consider Caputo’s definition, whose initial conditions are as the same form of the integer-order one [

Some useful properties of fractional calculus that will be used in the controller design are listed as follows.

Caputo’s fractional derivative and integral are linear operations with

Let

The Laplace transform of (

The two-parameter Mittag-Leffler function was defined by [

In the subsequent paper, we only consider the case that

Unlike the Mamdani fuzzy logic systems, the ith rule of a Multi-Input and Multioutput general fractional-order Takagi-Sugeno (T-S) fuzzy systems can be expressed as follows

with

Depending on the above statements, a main difference of Mamdani fuzzy logic systems and T-S fuzzy systems is that the rule consequents are functions for T-S fuzzy system whereas the rule consequents are fuzzy sets for Mamdani fuzzy logic systems. Moreover, the T-S fuzzy logic systems are also universal approximators [

Consider the following fractional-order chaotic system as the master system via T-S type fuzzy systems. The

where

Consider the following fractional-order chaotic system with external disturbances in the equation as the slave system based on T-S fuzzy models. The

where

The control objective of this work is to design a proper adaptive controller

The structure of master system (

The unknown disturbances

It is worth pointing out that Assumptions

The synchronization error dynamic equation can be obtained from (

Based on T-S fuzzy logic system universal approximation theorem, T-S fuzzy systems

As shown in [

Based on above discussion, the controller is designed with the fuzzy system

Let us denote

In order to update parametric estimates, the fractional adaptation laws are designed as

Here, fractional Lyapunov’s theory is used to analyze the stability in closed-loop system. The following Lemmas are proposed to simplify the stability analysis.

If

If

We only consider the front part. If

Let

According to Lemma

We will proof that

From above discussion, the boundedness of all signals in closed-loop system and the convergence of tracking error based on adaptive fuzzy control scheme via T-S fuzzy logic systems is presented in the following theorem.

For the master system (

Define the following Lyapunov function:

In this section, in order to further illustrate the effectiveness of the proposed control method designed in previous sections, one example about the synchronization for two different uncertain fractional-order chaotic system is given. The master system of a fractional-order chaotic system via T-S fuzzy model is given as

The upper system is formulated to the alike form in (

Master system.

Two fuzzy sets are defined for the state

Two fuzzy sets are defined for the state

Two fuzzy sets are defined for the state

The slave system of a fractional-order chaotic system with unknown disturbances via T-S fuzzy model is given as

The upper system is formulated to the alike form in (

Figure

Slave system.

Two fuzzy sets are defined for the state

Two fuzzy sets are defined for the state

Two fuzzy sets are defined for the state

In the simulation, the initial conditions of master system and slave system are selected as

The controller is designed as

Let

The fractional adaptation laws of

The simulation results of the proposed adaptive control approach are shown in Figure

(a) Synchronization error and (b) controller.

In this paper, synchronization of different fractional-order chaotic or hyperchaotic systems with unknown disturbances and parametric uncertainties is addressed with adaptive fuzzy control algorithm based on T-S fuzzy models. The distinctive features of the proposed control approach are that T-S fuzzy logic systems are introduced to approximate the unknown disturbances and to model the unknown controlled systems; both adaptive fuzzy controller and fractional adaptation laws are developed based on combined fractional Lyapunov stability theory and parallel distributed compensation technique. It is shown that the proposed control method can guarantee that all the signals in the closed-loop system remain bounded and the synchronization error converges towards an arbitrary small neighbourhood of the origin asymptotically. A simulation example is used for verifying the effectiveness of the proposed control strategy. Further works would focus on chaos synchronization control of different uncertain fractional-order chaotic systems with time delay and input saturation.

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 Natural Science Foundation of Anhui Province of China under Grant 1808085MF181.