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

Authors: Meichun Huang; Runzi Luo; Jiaojiao Fu; Haipeng Su
Journal: Complexity
 (2020)

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## Abstract

This article is concerned with the exponential synchronization of a class of the chaotic systems with external disturbance via the saturation control. Through 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 rateλ is presented. The 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.


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## Body

## 1. 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 [6–12]. 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. Thus, 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. The authors in paper [14] discussed the exponential synchronization of a class of fractional-order chaotic systems based on the discontinuous input. The exponential synchronization of a special chaotic system which has no linear term was considered in paper [15] by using the exponential stability theorem. The 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. The 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 [13–17] 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.The remainder of this paper is organized as follows. In Section2 and Section 3, the problem formulation and drive-response synchronization schemes are proposed, respectively. The 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.


## 2. Problem Formulation

In order to observe the chaotic synchronization phenomenon, in this paper, the following system is considered as the drive system:(1)x˙1=x2,x˙2=x3,⋯x˙n−1=xn,x˙n=fx+dm,where x=x1,x2,…,xnT is the state variable, fx is a continuous function, and dm is the external disturbance.Based on system (1), the response system is given as(2)y˙1=y2,y˙2=y3,⋯y˙n−1=yn,y˙n=gy+dr+u,where y=y1,y2,…,ynT is the state variable, gy is a continuous function, and dr is the external disturbance. u is the controller which is defined as(3)u=u0,v>u0,v,v≤u0,−u0,v<−u0,where u0 is a constant which can be designed by the controller.For the purpose of facilitating the analysis and design, the controlleru is represented as(4)u=v−ϕv,where(5)ϕv=v−u0,v>u0,0,v≤u0,v+u0,v<−u0.The error variable is defined ase=e1,e2,…,enT=y1−x1,y2−x2,…,yn−xnT. By subtracting system (1) from system (2), the following error system is obtained:(6)e˙1=e2,e˙2=e3,⋯e˙n−1=en,e˙n=gy−fx+dr−dm+u.Assumption 1.
Supposedr,dm, and ϕv are all bounded which means that there is a constant M>0 such that(7)dr−dm−ϕv≤M,where M is known in advance.Remark 1.
It is well known that the chaos attractor is bounded which means thatϕv is also bounded. In addition, the external disturbances dr and dm are bounded, and thus, Assumption 1 is reasonable.Definition 1.
System (1) and system (2) are said to be globally exponentially synchronized if there exist constants α>0andλ>0 such that ei≤αe−λt hold for any initial values and t≥0, where λ is called as the exponential convergence rate, i=1,2,…,n.


## 3. Synchronization Schemes

Lemma 1.
(see [18]). Suppose x˙=−δx+ϕt. If δ>0 and limt⟶∞ϕt=0, then limt⟶∞xt=0, where x∈R is the state variable and ϕt is a continuous function.Now, we introduce the new variablezi which is expressed as(8)zi=eλtei,i=1,2,…,n.According to system (6), one obtains(9)z˙i=λeλtei+eλte˙i,i=1,2,…,n−1.Thus, system (6) can be converted into the following system (10):(10)z˙1=λz1+z2,z˙2=λz2+z3,⋮⋮z˙n−1=λzn−1+zn,z˙n=λzn+eλtgy−fx+dr−dm+v−ϕv.Theorem 1.
Iflimt⟶∞zi=0, then there exists α>0 and λ>0 such that ei|≤αe−λt which means that system (1) and system (2) can reach exponential synchronization, where i=1,2,⋯,n.Proof.
Iflimt⟶∞zit=0, then for any ϵ>0, there exists time T>0 such that for t>T, we have zi≤ε. Since the state variables of chaotic systems are bounded, therefore for t∈0,T, there exists ϵ1>0 such that zi≤ε1. Let α=maxε1,ε; then, for any t≥0, we obtain zi≤α. Note that(11)ei=e−λtzi≤αe−λt.
Thus, based on Definition1, we conclude that system (1) and system (2) can reach exponential synchronization.Theorem 2.
Iflimt⟶∞z1=0, then limt⟶∞zi=0 which means that system (1) and system (2) can reach exponential synchronization, where i=1,2,…,n.Proof.
Supposelimt⟶∞z1=0, then limt⟶∞z˙1t=0. In view of that(12)z˙1=λz1+z2,we have(13)limt⟶∞z˙1=limt⟶∞λz1+limt⟶∞z2=0,which implies that(14)limt⟶∞z2=0.
In the same way, we can obtainlimt⟶∞zi=0,i=3,4,…,n. According to Theorem 1, we know that system (1) and system (2) can reach exponential synchronization.Theorem 3.
If(15)v=fx−gy−λζ−ξ−Msigneλtζ,ξ=Cn0λ+βne1+Cn1λ+βn−1e2+Cn2λ+βn−2e3+⋯+Cnn−1λ+βen,ζ=Cn−10λ+βn−1e1+Cn−11λ+βn−2e2+Cn−12λ+βn−3e3+⋯+Cn−1n−1en,then system (1) and system (2) can reach exponential synchronization, where β>0.Proof.
Based on Theorem2, we know that if(16)limt⟶∞z1=0,then system (1) and system (2) can reach exponential synchronization. In order to obtain(17)limt⟶∞z1=0,in the first step, we choose the Lyapnov function V1 as(18)V1=12z12.
Its derivative is(19)V˙1=z1z˙1=z1λz1+z2=−βz12+z1λ+βz1+z2=−βz12+z1C10λ+βz1+C11z2.
Let us suppose thatlimt⟶∞C10λ+βz1+C11z2=0. In this case, we can show that z1 is bounded. In fact, If z1 is unbounded, i.e., limt⟶∞z1=∞. Then, there exists a finite time T, such that when t>T, we have(20)V˙1=−βz12+z1C10λ+βz1+C11z2<0.
In light of the Lyapunov stability theory, we obtainlimt⟶∞z1=0 which is contradicted with limt⟶∞z1=∞. Therefore, z1 is bounded; then, limt⟶∞z1C10λ+βz1+C11z2=0. According to Lemma 1, we know that limt⟶∞z1=0.
Based on the above analysis, one can derive that iflimt⟶∞C10λ+βz1+C11z2=0, then limt⟶∞z1=0. Now, in order to obtain limt⟶∞C10λ+βz1+C11z2=0, in the second step, we choose the Lyapnov function V2 as(21)V2=12C10λ+βz1+C11z22.
The derivative ofV2 is(22)V˙2=λ+βz1+z2λ+βz˙1+z˙2=λ+βz1+z2λ+βλz1+z2+λz2+z3=−βλ+βz1+z22+λ+βz1+z2C20λ+β2z1+C21λ+βz2+C22z3.
Similarly, according to Lemma1, if(23)limt⟶∞C20λ+β2z1+C21λ+βz2+C22z3=0,then(24)limt⟶∞C10λ+βz1+C11z2=0.
In view of step 1, we have(25)limt⟶∞z1=0.
Suppose that in theith step, we have proven that if limt⟶∞ρ=0, then limt⟶∞z1=0, where(26)ρ=Ci0λ+βiz1+Ci1λ+βi−1z2+Ci2λ+βi−2z3+Ci3λ+βi−3z4+⋯+Cii−1λ+βzi+Ciizi+1.
Taking(27)Vi+1=12ρ2,then we have V˙i+1=ρρ˙=−βρ2+ρρ˙+βρ.
Thus, we have(28)V˙i+1=−βρ2+ρCi0λ+βiλz1+z2+Ci1λ+βi−1λz2+z3+Ci2λ+βi−2λz3+z4+Ci3λ+βi−3λz4+z5+⋯+Cii−1λ+βλzi+zi+1+Ciiλzi+1+zi+2+βCi0λ+βiz1+Ci1λ+βi−1z2+Ci2λ+βi−2z3+Ci3λ+βi−3z4+⋯+Cii−1λ+βzi+Ciizi+1=−βρ2+ρCi0λ+βi+1z1+λ+βiCi0+Ci1z2+λ+βi−1Ci1+Ci2z3+⋯+λ+βCii−1+Ciizi+1+Ciizi+2.
Using the mathematical formula(29)Cnm+Cnm−1=Cn+1m,we obtain(30)V˙i+1=−βρ2+ρCi+10λ+βi+1z1+Ci+11λ+βiz2+Ci+12λ+βi−1z3+⋯+Ci+1iλ+βzi+1+Ci+1i+1zi+2,i≤n−2.
Then, by Lemma1, we know that if(31)limt⟶∞Ci+10λ+βi+1z1+Ci+11λ+βiz2+Ci+12λ+βi−1z3+⋯+Ci+1iλ+βzi+1+Ci+1i+1zi+2=0,then(32)limt⟶∞z1=0.
In the above equation, if we seti=n−2, then one can conclude that(33)limt⟶∞ρ1=0,which implies that(34)limt⟶∞z1=0,where(35)ρ1=Cn−10λ+βn−1z1+Cn−11λ+βn−2z2+Cn−12λ+βn−3z3+⋯+Cn−1n−2λ+βzn−1+Cn−1n−1zn.
The derivative ofρ1 is(36)ρ˙1=Cn−10λ+βn−1λz1+z2+Cn−11λ+βn−2λz2+z3+Cn−12λ+βn−3λz3+z4+⋯+Cn−1n−2λ+βλzn−1+zn+Cn−1n−1λzn+eλtgy−fx+dr−dm−ϕv+v.
Choose the following Lyapunov function:(37)Vn=12ρ12.
The derivative ofVn is(38)V˙n=ρ1ρ1˙=−βρ12+ρ1ρ˙1+βρ1.
Thus, we have(39)V˙n=−βρ12+ρ1Cn−10λ+βn−1λz1+z2+Cn−11λ+βn−2λz2+z3+Cn−12λ+βn−3λz3+z4+⋯+Cn−1n−2λ+βλzn−1+zn+Cn−1n−1λzn+eλtgy−fx+dr−dm−ϕv+v+βCn−10λ+βn−1z1+Cn−11λ+βn−2z2+Cn−12λ+βn−3z3.
Thus, we have(40)V˙n=−βρ12+ρ1Cn−10λ+βn−1λz1+z2+Cn−11λ+βn−2λz2+z3+Cn−12λ+βn−3λz3+z4+⋯+Cn−1n−2λ+βλzn−1+zn+Cn−1n−1λzn+eλtgy−fx+dr−dm−ϕv+v+βCn−10λ+βn−1z1+Cn−11λ+βn−2z2+Cn−12λ+βn−3z3+⋯+Cn−1n−2λ+βzn−1+Cn−1n−1zn=−βρ12+ρ1Cn0λ+βnz1+Cn1λ+βn−1z2+Cn2λ+βn−2z3+⋯+Cnn−2λ+β2zn−1+Cnn−1λ+βzn+Cnneλtgy−fx+dr−dm−ϕv+v.
Note that(41)eλtξ=Cn0λ+βnz1+Cn1λ+βn−1z2+Cn2λ+βn−2z3+⋯+Cnn−2λ+β2zn−1+Cnn−1λ+βzn,eλtζ=ρ1,(42)ρ1eλtgy−fx+dr−dm−ϕv+v=ρ1eλtdr−dm−ϕv−λζ−ξ−Msigneλtζ≤−ρ1eλtξ−λρ12+eλtρ1M−eλtρ1Msigneλtζ≤−ρ1eλtξ−λρ12.
By substituting (15), (41), and (42) into (40), one has(43)V˙n≤−λ+βρ12.
According to Lyapunov’s stability theory, we obtain(44)limt⟶∞ρ1=0.
Therefore,limt⟶∞z1=0. According to Theorem 1, we know that system (1) and system (2) can reach exponential synchronization.Remark 2.
From Theorem1, one can see that the convergence rate of system (6) is λ. Since λ is variable and can be chosen freely by the controller, Theorem 3 ensures that system (1) and system (2) can reach exponential synchronization with variable convergence rates via the saturation control.Remark 3.
The convergence rate of many published papers [13–17] concerned that the exponential synchronization is fixed which means that the convergence speed is constant. However, from Theorem 3, it is easy to see that the convergence rate of the exponential synchronization is variable. In addition, the control schemes proposed in papers [13–17] 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.Case 1.
The chaotic system is a 3-dimension system, i.e.,n=3. Based on Theorem 2, in this case, we obtain(45)v=−gy+fx−λ+β3e1+3λ+β2e2+3λ+βe3−λλ+β2e1+2λ+βe2+e3−Msigneλtλ+β2e1+2λ+βe2+e3.Case 2.
The chaotic system is a 4-dimension system, i.e.,n=4. Based on Theorem 2, in this case, we have(46)v=−gy+fx−λ+β4e1+4λ+β3e2+6λ+β2e3+4λ+βe4−λλ+β3e1+3λ+β2e2+3λ+βe3+e4−Msigneλtλ+β3e1+3λ+β2e2+3λ+βe3+e4.


## 4. Numerical Simulations

In the sequel, the Genesio chaotic system and the Coullet chaotic system are used to test the effectiveness of the proposed method.The Genesio chaotic system, proposed by Genesio and Tesi [19], is given as(47)x˙1=x2,x˙2=x3,x˙3=−a1x1−b1x2−c1x3+x12+dm,where x1,x2,andx3 are state variables and a1,b1, and c1c1b1<a1 are the positive real parameters. dm is the external disturbance. When dm=0, system (47) is chaotic and the chaos attractor is shown in Figure 1 with c1=1.2,b1=2.92,anda1=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(48)y˙1=y2,y˙2=y3,y˙3=a2y1−b2y2−c2y3−y13+dr+u,where y1,y2,andy3 are state variables anda2,b2, and c2 are positive constants. dr is the external disturbance, and u is the controller. When dr=0andu=0 and a2=5.5,b2=3.5,andc2=1.0, system (48) is chaotic and the chaos attractor is depicted in Figure 2.Figure 1
The chaos attractor of system (47) with x10=2,x20=1,andx30=−4.Figure 2
The chaos attractor of system (48) with y10=4,y20=−1,andy30=1.Based on (45), the v can be chosen as(49)v=−a2y1−b2y2−c2y3−y13+−a1x1−b1x2−c1x3+x12−λ+β3e1+3λ+β2e2+3λ+βe3−λλ+β2e1+2λ+βe2+e3−Msigneλtλ+β2e1+2λ+βe2+e3,where e1=y1−x1,e2=y2−x2,ande3=y3−x3. According to Theorem 3, the exponential synchronization between system (47) and system (48) will be reached.In the simulation process, we setc1=1.2,b1=2.92,anda1=6 and a2=5.5,b2=3.5,andc2=1 such that the two systems are chaotic. In addition, for simplicity, we take β=1, dr=y12+0.2cost, and dm=x1+0.2sint. The M and u0 are selected as M=30 and u0=30. The initial conditions of the drive and response systems are chosen as x10,x20,x30=2,1,−4 and y10,y20,y30=4,−1,1, respectively.The simulation graphs withλ=1 and λ=3 are depicted in Figures 3–7.Figure 3
The time response of synchronization error variablee1.Figure 4
The time response of synchronization error variablee2.Figure 5
The time response of synchronization error variablee3.Figure 6
The time response of input signalu with λ=1.Figure 7
The time response of input signalu with λ=3.The time response of synchronization error variablese1,e2, and e3 are shown in Figures 3–5, respectively. The 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.


## 5. Conclusions

In this paper, the exponential synchronization of a class ofnD 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. The 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.



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