Robust Exponential Synchronization of a Class of Chaotic Systems with Variable Convergence Rates via the Saturation Control

-is article is concerned with the exponential synchronization of a class of the chaotic systems with external disturbance via the saturation control. -rough appropriate coordinate transformation, the exponential synchronization is translated into the asymptotic stability of the error system. By using the Lyapunov stability theory, a novel sufficient condition which possesses the exponential convergence rate λ is presented.-e rich choices of the exponential convergence rate λ turn our scheme more general than some existing approaches. Numerical simulations are employed to the Genesio chaotic system and the Coullet chaotic system to illustrate the ability and effectiveness of the presented approach.


Introduction
Synchronization exists in many systems, such as chaotic system, complex network system, and neural network system. Since Pecora and Carroll [1] proposed the drive-respond synchronization scenario in 1990, chaos synchronization has turned out to be a hot topic. So far, many kinds of synchronization schemes have been proposed by the experts, such as the complete synchronization [1], lag synchronization [2], phase synchronization [3], projective synchronization [4], and combination synchronization [5]. Chaos synchronization, owing to its great application in engineering science, medicine, secure communication, and telecommunications, has attracted widespread concern in a variety of areas and has been studied extensively during the last decades [6][7][8][9][10][11][12]. However, most of the synchronization schemes are based on asymptotic stability. From a practical point of view, chaos systems are required not only to be synchronized but also with a fast synchronizing rate. us, the exponential synchronization, which can quantify the rate of convergence and possesses the faster convergence speed than that of general asymptotic stability, has received much attention in many research fields. For example, the study in [13] investigated the exponential synchronization of the chaotic Lur'e systems via the stochastic sampled-data controller. e authors in paper [14] discussed the exponential synchronization of a class of fractional-order chaotic systems based on the discontinuous input. e exponential synchronization of a special chaotic system which has no linear term was considered in paper [15] by using the exponential stability theorem. e study in [16] discussed the exponential synchronization of a class of fractional-order chaotic systems with uncertainty. A new criterion was proposed by using the linear matrix inequalities approach. e exponential synchronization between two identical xian chaotic systems was considered in paper [17]. By using algebraic Riccati equation, a linear feedback controller was presented.
In the literature, most of the published papers concerned the exponential synchronization (see [13][14][15][16][17] for example), and the convergence rate is fixed which means that the convergence speed is constant. From a practical point of view, it is to be hoped that the exponential synchronization can be achieved as soon as possible. On the other hand, it is well known that every physical actuator is subject to saturation. When the actuator is saturated, the performance of the designed control system will be deteriorated seriously. In order to improve the control performance, the effect of saturation should be incorporated into the design of the controlled system.
Motivated by such circumstances, in this paper, we investigate the exponential drive-response synchronization of a class of chaotic systems with plant uncertainties via the saturation control. A novel synchronization controller, in which a variable convergence rate is incorporated into the control law, is proposed. Numerical studies are provided to verify the effectiveness of the given scheme. e remainder of this paper is organized as follows. In Section 2 and Section 3, the problem formulation and driveresponse synchronization schemes are proposed, respectively. e numerical example is provided in Section 4 to demonstrate the effectiveness and the benefit of the proposed control scheme. Finally, the concluding remarks are drawn in Section 5. Theorem 1. If lim t⟶∞ z i � 0, then there exists α > 0 and λ > 0 such that |e i | ≤ αe − λt which means that system (1) and system (2) can reach exponential synchronization, where i � 1, 2, · · · , n.
Proof. Based on eorem 2, we know that if then system (1) and system (2) can reach exponential synchronization. In order to obtain in the first step, we choose the Lyapnov function V 1 as Its derivative is Let us suppose that lim t⟶∞ (C 0 1 (λ + β)z 1 + C 1 1 z 2 ) � 0. In this case, we can show that z 1 is bounded. In fact, If z 1 is unbounded, i.e., lim t⟶∞ z 1 � ∞. en, there exists a finite time T, such that when t > T, we have In light of the Lyapunov stability theory, we obtain lim t⟶∞ z 1 � 0 which is contradicted with lim t⟶∞ z 1 � ∞.
erefore, z 1 is bounded; then, lim t⟶∞ z 1 (C 0 Based on the above analysis, one can derive that if Complexity 3 Similarly, according to Lemma 1, if In view of step 1, we have Suppose that in the ith step, we have proven that if lim t⟶∞ ρ � 0, then lim t⟶∞ z 1 � 0, where Taking Using the mathematical formula we obtain en, by Lemma 1, we know that if In the above equation, if we set i � n − 2, then one can conclude that which implies that where Choose the following Lyapunov function: us, we have us, we have 4 Complexity Note that By substituting (15), (41), and (42) into (40), one has According to Lyapunov's stability theory, we obtain lim t⟶∞ ρ 1 � 0.
(44) erefore, lim t⟶∞ z 1 � 0. According to eorem 1, we know that system (1) and system (2) can reach exponential synchronization. □ Remark 2. From eorem 1, one can see that the convergence rate of system (6) is λ. Since λ is variable and can be chosen freely by the controller, eorem 3 ensures that system (1) and system (2) can reach exponential synchronization with variable convergence rates via the saturation control.

Remark 3.
e convergence rate of many published papers [13][14][15][16][17] concerned that the exponential synchronization is fixed which means that the convergence speed is constant. However, from eorem 3, it is easy to see that the convergence rate of the exponential synchronization is variable. In addition, the control schemes proposed in papers [13][14][15][16][17] have not taken into consideration the effect of saturation. Note that in reality, the physical actuator is usually subject to saturation; therefore, our control strategy is applicable to the practical systems.
In view of that most of the chaotic systems are 3-or 4-dimension systems, now we discuss two special cases.

Numerical Simulations
In the sequel, the Genesio chaotic system and the Coullet chaotic system are used to test the effectiveness of the proposed method. e Genesio chaotic system, proposed by Genesio and Tesi [19], is given as where x 1 , x 2 , and x 3 are state variables and a 1 , b 1 , and c 1 (c 1 b 1 < a 1 ) are the positive real parameters. d m is the external disturbance. When d m � 0, system (47) is chaotic and the chaos attractor is shown in Figure 1 with c 1 � 1.2, b 1 � 2.92, and a 1 � 6.
In the synchronization scheme, we suppose that system (47) is the drive system and the Coullet system [20] is the response system which is described by where y 1 , y 2 , and y 3 are state variables anda 2 , b 2 , and c 2 are positive constants. d r is the external disturbance, and u is the controller. When d r � 0 and u � 0 and a 2 � 5.5, b 2 � 3.5, and c 2 � 1.0, system (48) is chaotic and the chaos attractor is depicted in Figure 2.
e time response of synchronization error variables e 1 , e 2 , and e 3 are shown in Figures 3-5, respectively. e time response of input signal u with λ � 1 and λ � 3 are exhibited in Figures 6 and 7, respectively. From Figures 3-5, one can easily see that the synchronization between systems (47) and (48) is realized. Meanwhile, one can also observe from Figures 3-5 that the synchronization speed of λ � 3 is faster than that of λ � 1.

Conclusions
In this paper, the exponential synchronization of a class of n D chaotic systems with external disturbances has been investigated via the coordinates transformation method. Based on the Lyapunov stability theory, a new saturation controller is presented to ensure that the coupled chaotic systems can achieve synchronization exponentially. e proposed controller contains the convergence rate λ which can be used to control the convergence speed of the synchronization. By selecting different values of λ, the exponential synchronization will be reached with any prespecified exponential convergence rate. Numerical examples are proposed to demonstrate the usefulness and merits of our presented scheme.

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

Conflicts of Interest
e authors declare that they have no conflicts of interest. 8 Complexity