Synchronization of Different Uncertain Fractional-Order Chaotic Systems with External Disturbances via TS Fuzzy Model

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 systemswith the same system structure. One constructive example is given to verify the feasibility and superiority of the proposed method.


Introduction
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 [1].At present, researchers found that fractional differential equations not only improve the veracity in modeling physical systems but also generate a lot of applications in physics, electrical engineering, robotics, control systems, and chemical mixing [2][3][4][5][6][7][8][9][10][11].In addition, the chaotic behavior has been discovered in many fractionalorder systems, for instance, the fractional-order Chen's system, the fractional-order Chua's system, and the fractionalorder Liu system.In view of chaotic potential value in control systems and secure communication [12], chaos synchronization was studied by more and more researches [13,14].
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 [15][16][17][18][19][20][21][22][23][24][25].To control these uncertain systems, adaptive fuzzy/neural-network control was proposed [26,27].This method is effective and superior for handling parametric and structure uncertainties, external disturbances in integerorder nonlinear systems [28,29], where tracking error is developed to update adjusted parameters and fuzzy logic systems or neural networks are introduced to model unknown physical systems as well as to approximate unknown nonlinear functions.There are two types of fuzzy logic systems: Mamdani type and T-S type.T-S fuzzy logic system is first proposed by Takagi and Sugeno [30].Subsequently, many works found that T-S fuzzy systems can uniformly approximate any continuous functions on a compact set with random accuracy based on the Weierstrass approximation theorem [31].Moreover, it was also shown that the approximation ability of T-S fuzzy systems was better than the Mamdani fuzzy systems [32].Therefore, many studies focused on the chaos synchronization of fractional-order chaotic systems via T-S fuzzy models.For example, synchronization of fractional-order modified chaotic system via new linear control, backstepping control, and T-S fuzzy approaches was investigated in [33].Impulsive control for fractional-order chaotic system was presented in [34].Other results about the synchronization of a fractional-order chaotic system via T-S fuzzy approaches can be found in [35,36].However, only chaos synchronization of fractional-order nonlinear systems with same structure based on T-S fuzzy systems is considered in above previous works.
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: (1) This paper first considers the chaos synchronization of the master system and slave system with different structure based on T-S fuzzy systems, and the external disturbances are assumed to be unknown.The required knowledge of the disturbances is weaker than above previous works, for example, in [34][35][36].In these works, the external disturbances are assumed to be bounded with known upper bounds.However, in our control method, we do not need to know the exact value of the upper bounds of external disturbances.
(2) T-S fuzzy logic systems are used to model the controlled system and the final outputs of system can be obtained.By combining the adaptive fuzzy control method and parallel distributed compensation technique, an adaptive controller with fractional-order laws is designed.The proposed method is superior to some works based on linear matrix inequality (LMI) and modified LMI [37].

Fundamentals of Fractional Calculus and
Fuzzy Logic Systems

Fractional Calculus.
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 [38][39][40].The fractional integral is designed as [1]  0 where  > 0,  − 1 ≤  < , and Γ(⋅) is Euler's Gamma function, which is defined as The fractional derivative operator is given as Some useful properties of fractional calculus that will be used in the controller design are listed as follows.

Takagi-Sugeno
where =1   (x()) = 1 and   (x()) ≥ 0. 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 [31].

Problem Statement.
Consider the following fractionalorder chaotic system as the master system via T-S type fuzzy systems.The th rule can be expressed as where   is a constant matrix, x() ∈  1 ⊆   is the state vector ( 1 is a compact set), b 1 is a constant vector, and    ,  = 1, 2, ⋅ ⋅ ⋅ , , are fuzzy sets.Hence, the final output of master system can be rewritten as with Consider the following fractional-order chaotic system with external disturbances in the equation as the slave system based on T-S fuzzy models.The th rule can be written in the following form ( = 1, ⋅ ⋅ ⋅ , ): where   is a constant matrix, y() ∈  2 ⊆   is the state vector ( 2 is a compact set), b 2 is a constant vector, F  ,  = 1, 2, ⋅ ⋅ ⋅ , , are fuzzy sets, () ∈  is control input, and d  (, y) ∈   are unknown external disturbances.Hence, the final output of slave system can be obtained as with The control objective of this work is to design a proper adaptive controller () to synchronize the above chaotic systems ( 9) and (10) with the tracking error signal asymptotically converging to zero with random accuracy, that is, lim →+∞ ‖e()‖ = 0.The norm adopts Euclid norm in this paper.In addition, all states and parameters in the closedloop system are bounded.The following assumptions are necessary.
Assumption 5.The structure of master system (9) and slave system (10) is different.The parameters and the structure of the master system are complete unknown or partial unknown, but the parameters and structure of the slave system are known.Remark 7. It is worth pointing out that Assumptions 5 and 6 are rational.Due to the boundedness of chaos systems, we assume that  1 and  2 are compact sets.Since d  (, y) are unknown external disturbances and may be not continuous, they are assumed to be unknown measurable nonlinear functions.The slave systems and the controller lie on the receiving terminal; hence, the parameters and the structure of the master system may be complete unknown or partial unknown, but the parameters and structure of the slave system are known.

Control Design.
The synchronization error dynamic equation can be obtained from (11) as Then we obtain the optimal parameter vector as is the ideal approximator of   (y).The minimum approximation errors and the ideal parameter errors of the fuzzy systems are defined as According to [29,51,52], the approximation errors  ( Remark 8.As shown in [53], if the rule consequences of T-S fuzzy systems have the same form with the rule consequences of Mamdani type logic systems, then T-S type is equivalent to Mamdani type fuzzy system. Based on above discussion, the controller is designed with the fuzzy system ρ (y,   ()) as well as the estimate value ε *   (y) as where the th rule of u  () and ũ() can be written as follows, respectively, ( = 1, 2, ⋅ ⋅ ⋅ , ): with   ,   > 0 being adaptation rates which are constant parameters.Taking the control law ( 17) into (12) and letting  = diag[ 1 , ⋅ ⋅ ⋅ ,   ] (  > 0), we have  (21)

Stability Analysis.
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.
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.
Theorem 12.For the master system (9) and slave system (10) under the known initial conditions, if Assumptions 5 and 6 are satisfied and the adaptive controller is given as (17) with the fractional adaptation laws (18) and (19), then all signals in the closed-loop system are bounded and the tracking error signal tends to zero asymptotically.
Proof.Define the following Lyapunov function: with θ () =   () −  *  and ε *  (y) = ε *  (y) −  *  .Hence, using the Lemma 9, the -order derivative of () with respect to time  is obtained as Substituting ( 21) into (35), one gets Taking ( 18) and ( 19) into (36), one gets the following inequality: where   is the least eigenvalue of the positive definite matrix .According to Lemma 11 and above discussion, we know that the tracking error signal e() tends to 0 asymptotically (that is, lim →∞ ‖e()‖ = 0) and θ () and ε *  (y) are bounded.Further, it means that   () and ε *  (y) are bounded.Because of the boundedness of e() and x(), we know that y() is bounded.Based on the control design, u() is bounded.Therefore, we know that all signals in the closedloop system are bounded.

Simulation Example
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 ith rule of master system is given by The upper system is formulated to the alike form in (9) with Figure 1 depicts the simulation results of the master system with the parameters  = 2,  = 0.8 with time step ℎ = 0.005.Figure 1 shows  1 (), Two fuzzy sets are defined for the state  2 over the interval [−10, 10.6] with the membership functions as Two fuzzy sets are defined for the state  3 over the interval [0.2, 9.05] with the membership functions as (42) The slave system of a fractional-order chaotic system with unknown disturbances via T-S fuzzy model is given as The ith rule of slave system is given by  , y).The upper system is formulated to the alike form in (10) with Figure 2 with u() = 0 and without the external disturbance is depicted the simulation results of the slave system with the parameters below:  = 2,  = 0.8, for time step ℎ = 0.005.Moreover, Chaos was found in system (43) with  = 0.8.Two fuzzy sets are defined for the state  1 over the interval [−29.21,21.52] with the membership functions as follows: Two fuzzy sets are defined for the state  2 over the interval [−35.6,26.5] with the membership functions as follows: Two fuzzy sets are defined for the state  3 over the interval [0, 53.6] with the membership functions as follows:  In the simulation, the initial conditions of master system and slave system are selected as x(0) = (1, 1, −2)  and y(0) = (−2, −3, 3)  .The parameters relating the synchronization problem are set to  =  and The controller is designed as Let ρ(y, ()) = ( ρ1 , ρ2 , ρ3 )  ,  = (ε * 1 (y), ε * 2 (y), and ε * 3 (y))  ; then The simulation results of the proposed adaptive control approach are shown in Figure 3, where subgraph (a) denotes the tracking error trajectory and subgraph (b) denotes the control trajectory.Define the initial conditions of the approximation errors as ε * 1 (0) = 0, ε * 2 (0) = 0, ε * 3 (0) = 0.In reducing the computation of the numerical simulation, x() and y() are replaced by e().Four fuzzy sets are defined for the tracking errors  1 (),  2 (),  3 () over the interval [−3, 3] with the Gaussian membership functions, where the first parameters are 1.1 and the second parameters are −3, −1, 1, 3, respectively.Comparing the conventional control method with the proposed method, we can see that the proposed approach can synchronize two chaotic plants to desired high accuracy and improve the performance as shown in Figure 3.

Conclusions
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.