# Exponential Synchronization of Chaotic Xian System Using Linear Feedback Control

Authors: J. Humberto Pérez-Cruz; Pedro A. Tamayo-Meza; Maricela Figueroa; Ramón Silva-Ortigoza; Mario Ponce-Silva; R. Rivera-Blas; Mario Aldape-Pérez
Journal: Complexity
 (2019)

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## Abstract

In this paper, a new linear feedback controller for synchronization of two identical chaotic systems in a master-slave configuration is presented. This controller requires knowing a priori Lipschitz constant of the nonlinear function of the chaotic system on its attractor. The controller development is based on an algebraic Riccati equation. If the gain matrix and the matrices of Riccati equation are selected in such a way that a unique positive definite solution is obtained for this equation, then, with respect to previous works, a stronger result can be guaranteed here: the exponential convergence to zero of the synchronization error. Additionally, the nonideal case is also studied, that is, when unmodeled dynamics and/or disturbances are present in both master system and slave system. On this new condition, the synchronization error does not converge to zero anymore. However, it is still possible to guarantee the exponential convergence to a bounded zone. Numerical simulation confirms the satisfactory performance of the suggested approach.


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## Body

## 1. Introduction

The problem of the unidirectional synchronization of chaotic systems consists of finding an appropriate control law such that when this is applied to a system with coupled inputs called “slave” or “response,” such system follows the dynamics of an autonomous chaotic system called “master” or “drive” [1–11]. This proper control action is necessary because, without it, two identical autonomous chaotic systems could never be synchronized due to their high sensitivity to initial conditions [12–18].During the last three decades, several strategies have been proposed to solve the synchronization problem when the structure and the parameters of both chaotic systems are known. One of these approaches is active control [19–31]. In this approach, the controller is selected based on the synchronization error dynamics in such a way that the nonlinearities are compensated and the dynamics equations are decoupled [32]. Another approach is nonlinear control [33–40]. In this technique, given a Lyapunov function candidate V, the control law is selected considering that the first derivative of V must be compelled to be negative definite [41]. Hence, the asymptotic convergence to zero of the synchronization error can be guaranteed. The aforementioned strategies present two drawbacks: (a) their practical implementation can be difficult, particularly for analog systems; (b) the controller can be expensive with respect to the consumption of energy. These problems can be avoided if a linear feedback controller is used [42–45]. This kind of controller is formed by the product of a gain matrix K and the negative synchronization error, -e, that is, u=-Ke. In [46], Wang et al. proposed an approach for synchronization of Chen system using linear feedback control. Given a proper Lyapunov function candidate, they could express the first time derivative of this function as a quadratic form of the synchronization error absolute value. Thus, by taking into account the conditions on which this quadratic form is negative definite, they could guarantee the asymptotic convergence to zero. This result was obtained by using only a unique control input. Based on this technique, the synchronization of the unified chaotic system and Lü system was presented in [47] and [48], respectively. It is important to mention that both Chen system and unified system are formed by five linear terms and two cross-product terms. Lü system is formed by four linear terms and two cross-product terms. Both cross-product terms are exactly the same expression in the three cases. By using Wang’s technique, Yassen synchronized a four-scroll chaotic system formed by three linear terms and three cross-product terms [49]. However, to be able to guarantee the asymptotic convergence, Yassen had to use three control inputs. Likewise, Liu chaotic system was synchronized in [50] with three control inputs and using the same procedure but applied from a more general perspective. Based on the same technique but with a unique control input, hyperchaotic Lorenz–Stenflo system formed by eight linear terms and two cross-product terms, and Lü hyperchaotic system formed by seven linear terms and two cross-product terms were synchronized in [51] and [52], respectively. In spite of the aforementioned successful applications, this technique has two main disadvantages: (a) the design procedure must be particularized to each system; (b) the technique is valid only for a very constrained class of systems. On the other hand, by using linearization and Lyapunov’s direct methods and linear feedback control, a sufficient synchronization criterion was presented in [42] for the case of generalized Lorenz systems. A wider class (but still restrictive) of systems were considered in [44] for synchronization based on linear feedback control. In all the aforementioned works, only the asymptotic convergence of the synchronization error can be guaranteed. To overcome this situation and the limitations of Wang's technique, in this paper, a new linear feedback controller based on a matrix Riccati equation is presented. This procedure requires knowing a priori Lipschitz constant of nonlinear function of the chaotic system on its attractor. If the gain matrix K and the matrices of the Riccati equation are selected in such a way that this equation has a unique positive definite solution, then a stronger result can here be established: the exponential convergence to zero of the synchronization error. The proposed strategy can be applied to a very wide class of systems. However, for simplicity, the attention is focused on a system recently reported by Xian et al in [53]. Despite the similarity between this system and the system synchronized by Yassen in [49] (the main difference is only a unique cross-product term), Wang’s technique is not able to handle this new system.Throughout this paper, given a vectorw∈Rn,w denotes the Euclidean norm of w, that is, w=wTw; given a matrix C∈Rn×n,λminC and λmaxC denote the minimum and maximum eigenvalues of C, respectively; C denotes the two-norm of C, that is, C=λmaxCTC.


## 2. System Description

The system proposed by Xian [53] is a new generalized third-order Lü chaotic system formed by three linear terms and four cross-product terms. The system can be described as (1)x˙=-ax+xz+yzy˙=by-xzz˙=-cz+xywhere x,y, and z are the system states and a,b, and c are constant parameters. This system shows chaotic behavior for the values a=5,b=7,c=3 and the initial condition x(0)=-1,y(0)=-1, and z(0)=-2. In Figure 1, the corresponding attractor is shown. The time series for the states x,y, and z during the first twenty seconds of numerical simulation are plotted in Figure 2.Figure 1
Attractor for Xian system: (a)x-y phase plane, (b) y-z phase plane, (c) x-z phase plane, and (d) phase space.
(a) (b) (c) (d)Figure 2
Time evolution of states of Xian system for first 20 seconds: (a) statex, (b) state y, and (c) state z.
(a) (b) (c)It can be noted that, by using vector notation, system (1) can succinctly be represented as follows: (2)w˙=Aw+fwwhere(3)w≔xyzT(4)A≔-a000b000-c(5)fw≔xz+yz-xzxy


## 3. Problem Formulation

The simpler structure for synchronization of chaotic systems is master-slave configuration. In this one, a slave chaotic system with control inputs must follow the dynamic behavior of an autonomous master chaotic system. For system (2), the corresponding master system can be represented simply as(6)w˙m=Awm+fwmwhere wm:=xmymzmTand subscriptm denotes “master.” The corresponding slave system for system (2) is given by(7)w˙s=Aws+fws+uwhere ws:=xsyszsT,u:=u1u2u3T,u1, u2, and u3 are control inputs, and the subscripts denotes “slave.” Define the synchronization error [54] as (8)e≔ws-wmThus, the problem of synchronization based on linear feedback control for systems (7) and (6) consists of finding an appropriate control law of the form(9)u=-Kewhere (10)K=k1000k2000k3and k1, k2, and k3 are real constants selectable by the designer in such a way that limt→∞e=0.


## 4. Background Results

In this section, some basic definitions and results are reviewed briefly.Definition 1 (see [55–57]).
A functionfw:Rn→Rn is said to be locally Lipschitz on D⊂Rn if there exists a constant γ (known as Lipschitz constant) such that, for all w1,w2∈D, the following inequality holds:(11)fw1-fw2≤γw1-w2Finally, f is said to be globally Lipschitz if it satisfies (11) with D=Rn.Lemma 2 (see [56, 57]).
If a functionf:Rn→Rn is continuously differentiable on a set D⊂Rn, then it is locally Lipschitz on D.Based on Lemma2, a procedure can be established in order to calculate the Lipschitz constant γ [56]. For example, consider the function (12)gw=w1w22-w2w1w2on the set D=w1,w2∣-4≤w1≤2,-1≤w2≤3. Let us define w1,max and w2,max such that w1≤w1,max and w2≤w2,max. Then, on D, w1,max=4 and w2,max=3. On the other hand, the Jacobian matrix of g is given by (13)∂g∂w=w222w1w2-1w2w1Let us define the matrix G as(14)G≔∂g∂wmaxThat is, G is formed by the maximum absolute values of each corresponding component in (13). Consequently, (15)G=92534Finally, a Lipschitz constant for g (12) on the set D can be taken as(16)γ=G=26.9984Lemma 3 (see [58]).
For any vectorsw1,w2∈Rn and any positive definite matrix ΛT=Λ>0,Λ∈Rn×n, the following inequality holds: (17)w1Tw2+w2Tw1≤w1TΛw1+w2TΛ-1w2Lemma 4 (see [58]).
The matrix Riccati equation(18)A0TP+PA0+PRP+Q=0with known constant matrices A0,R,Q∈Rn×n has a unique positive definite solution PT=P>0,P∈Rn×n if the following conditions are satisfied:(a)
RT=R>0,QT=Q>0(b)
A0 is Hurwitz(c)
The pairA0,R1/2 is controllable(d)
The pairQ1/2,A0 is observable(e)
The following matrix inequality holds:(19)14A0TR-1-R-1A0RA0TR-1-R-1A0T≤A0TR-1A0-Q


## 5. Main Results

In this section, the conditions under which the control law (9) can synchronize system (7) with respect to system (6) are found. First, the dynamics of the synchronization error is determined. By taking the first derivative of (8), we obtain(20)e˙=w˙s-w˙mBy substituting (7) and (6) into (20) and taking into account that the control law has the form given in (9), we get (21)e˙=A-Ke+fws-fwmLet us define(22)A0≔A-KThus, (21) becomes(23)e˙=A0e+fws-fwmTo evaluate the stability of the synchronization error dynamics (23), the following Lyapunov function candidate is proposed:(24)V=eTPewhere P is a positive definite matrix to be found. The first time derivative of (24) is calculated as(25)V˙=e˙TPe+eTPe˙By substituting (23) into (25) and after some operations, we get(26)V˙=eTA0TPe+eTPA0e+fws-fwmTPe+eTPfws-fwmNow, let us consider the last two terms of (26):(27)fws-fwmTPe+eTPfws-fwmBy using Lemma 3, (27) can be bounded as(28)fws-fwmTPe+eTPfws-fwm≤eTPΛPe+fws-fwmTΛ-1fws-fwmwhere Λ is a definite positive matrix selectable by the designer. Besides, considering that f is locally Lipschitz on the attractor of the autonomous system (6), with Lipschitz constant γ, then the following can be established: (29)fws-fwm≤γws-wmand(30)fws-fwmTΛ-1fws-fwm≤γ2eTΛ-1eBy substituting (30) into (28) and the corresponding result into (26), the first time derivative of Vcan be bounded by (31)V˙≤eTA0TPe+eTPA0e+eTPΛPe+γ2eTΛ-1eBy adding and subtracting the term eTQ0e, where Q0 is a positive definite matrix, into the right hand side of (31), we obtain(32)V˙≤eTA0TPe+eTPA0e+eTPΛPe+γ2eTΛ-1e+eTQ0e-eTQ0eNow, if we define (33)R≔Λ,Q≔γ2Λ-1+Q0,and the following matrix Riccati equation is formed:(34)A0TP+PA0+PRP+Q=0and if the constant matrices A0,Λ,Q0 are selected in such a way that the conditions of the Lemma 4 are satisfied, then (34) has a unique positive definite solution P and (32) becomes simply(35)V˙≤-eTQ0eFrom (35) and from second Lyapunov method, the asymptotic convergence to zero of e can be concluded. However, a stronger result can still be obtained. Let us consider that(36)-eTQ0e=-eTP1/2P-1/2Q0P-1/2P1/2eBy using Rayleigh inequality into (36), we can claim that(37)-eTQ0e≤-λminP-1/2Q0P-1/2eTPeFrom the last inequality and by taking into account (35),(38)V˙≤-λminP-1/2Q0P-1/2eTPeLet us define (39)ζ≔λminP-1/2Q0P-1/2And from (24), (38) becomes(40)V˙≤-ζVThis implies that(41)V≤V0exp⁡-ζtBy using twice Rayleigh inequality into (24), (41) becomes(42)λminPeTe≤λmaxPeT0e0exp⁡-ζtor (43)e2≤λmaxPλminPe02exp⁡-ζtBy taking square root of both sides of inequality (43), we get(44)e≤λmaxPλminPe0exp⁡-ζ2tFinally, based on (44), we can conclude the exponential convergence to zero of the synchronization error e. Hence, the following theorem has been proven.Theorem 5.
If the functionf in equations (6) and (7) is locally Lipschitz on the attractor of the autonomous system (6) with Lipschitz constant γ and the gain K and the matrices Λ,Q0,A0=A-K, R=Λ, Q=γ2Λ-1+Q0 are selected according to Lemma 4 in such a way that the matrix Riccati equation A0TP+PA0+PRP+Q=0 has a unique positive definite solution P and the control law u=-Ke is applied to the slave system (7), then the synchronization error e=ws-wm converges exponentially to zero.Now, a more realistic case can be considered when unmodeled dynamics and/or disturbances are present in both master system and slave system, that is,(45)w˙m=Awm+fwm+ψm(46)w˙s=Aws+fws+u+ψsAssumption 6.
The termsψm and ψs represent unknown unmodeled dynamics and/or disturbances. Although these terms are unknown, they must be bounded. Besides, it is not necessary to know the specific value for each bound.Consider a functionψ defined as(47)ψ≔ψs-ψmwhich satisfies (48)ψTΩψ≤Ψ-where Ω is a definite positive matrix selectable by the designer and Ψ- is a positive constant not necessarily a priori known.Theorem 7.
If the functionf in equations (45) and (46) is locally Lipschitz on the attractor of the autonomous system (45) with Lipschitz constant γ and the gain K and the matrices Λ,Ω,Q0,A0=A-K, R-=Λ+Ω-1, Q=γ2Λ-1+Q0 are selected according to Lemma 4 in such a way that the matrix Riccati equation A0TP1+P1A0+P1R-P1+Q=0 has a unique positive definite solution P1 and the control law u=-Ke is applied to the slave system (46), then the norm of the synchronization error e=ws-wm converges exponentially to a zone bounded by Ψ-/αλminP1 where α=λminP1-1/2Q0P1-1/2.Proof.
As the proof of this theorem is very similar to the proof of Theorem5, only the main points will be presented. Given e=ws-wm, the synchronization error dynamics is given by(49)e˙=A0e+fws-fwm+ψTo analyze this dynamics, the following Lyapunov function candidate is proposed:(50)V=eTP1eThe first time derivative of (50) can be expressed as(51)V˙≤eTA0TP1e+eTP1A0e+eTP1ΛP1e+γ2eTΛ-1e+ψTP1e+eTP1ψBy using Lemma 3 into ψTP1e+eTP1ψ and inequality (48), it can be obtained that(52)ψTP1e+eTP1ψ≤eTP1Ω-1P1e+Ψ-By substituting (52) into (51), we get(53)V˙≤eTA0TP1e+eTP1A0e+eTP1ΛP1e+eTP1Ω-1P1e+γ2eTΛ-1e+Ψ-By adding and subtracting the term eTQ0e, where Q0 is a positive definite matrix, into the right hand side of (53), we obtain(54)V˙≤eTA0TP1e+eTP1A0e+eTP1ΛP1e+eTP1Ω-1P1e+γ2eTΛ-1e+eTQ0e-eTQ0e+Ψ-By defining(55)R-≔Λ+Ω-1,Q≔γ2Λ-1+Q0,the following matrix Riccati equation is formed:(56)A0TP1+P1A0+P1R-P1+Q=0and if the constant matrices A0,Λ,Ω,Q0 are selected using Lemma 4, then (56) has a unique positive definite solution P1 and (54) becomes simply(57)V˙≤-eTQ0e+Ψ-Now, it can be shown that(58)-eTQ0e≤-αeTP1ewhere α=λminP1-1/2Q0P1-1/2. By substituting (58) into (57) and (50) into the resulting expression, we get (59)V˙≤-αV+Ψ-According to [59, 60], (59) implies that(60)V≤V0exp⁡-αt+Ψ-α1-exp⁡-αtThis means that(61)e≤V0λminP1exp⁡-αt+Ψ-αλminP11-exp⁡-αtThe boundedness of e can be concluded from (61). Finally, by taking the limit on both sides of (61) as time tends to infinity, we can conclude the exponential convergence of e to a zone bounded by Ψ-/αλminP1.


## 6. Numerical Simulation

In order to use the result of Theorem5, first, by numerical simulation, the maximum absolute value of each state of the autonomous system (6) is estimated as xmax=40,ymax=66.54, and zmax=74.1. Next, the Lipschitz constant of function f on the attractor of the autonomous system (6) is determined. The Jacobian matrix of f is given by (62)∂f∂w=zzx+y-z0-xyx0The matrix F formed by the maximum absolute values of each element of the corresponding Jacobian matrix on the attractor of the autonomous system (6) is calculated as(63)F=zmaxzmaxxmax+ymaxzmax0xmaxymaxxmax0The two-norm of F is 174.88. Thus, the Lipschitz constant of f can be estimated as γ=175. Given the matrix A (as in (4)) with the nominal values a=5,b=7,c=3, in order to guarantee a positive definite solution for Riccati matrix equation (34), the constant matrices Λ,Q0,K,R=Λ,Q=γ2Λ-1+Q0 are selected in such a way that the conditions of Lemma 4 are satisfied. Thus, R=Λ=diag102.0833,102.0833,102.0833,Q0=I, K=diag(170.30,182.30,172.30), Q=diag301,301,301. With these matrices, the solution for Riccati matrix equation (34) is P=diag1.7002,1.7002,1.7002, and according to Theorem 5, the synchronization error e converges exponentially to zero.The performance of the control lawu=-Ke is verified by simulation. First, the Xian master system (6) with the initial condition xm(0)=-1,ym(0)=-1, and zm(0)=-2 and the Xian slave system (7) with the initial condition xs(0)=-4,ys(0)=1, and zs(0)=2 are built on Simulink®. Once the error signal e is obtained, the controller u=-Ke is applied to the slave system (7). The results of the simulation using the method ode23tb (stiff/TR-BDF2) with relative tolerance=1e-6 and absolute tolerance=1e-7 are presented in Figures 3–6. As can be appreciated in Figures 3, 4, and 5, the states of the slave system (7) do follow the corresponding states of the master system (6) in spite of the difference between the initial conditions. The exponential convergence of the error signal e is shown in Figure 6. For practical purposes, the convergence to zero is attained in less than 0.1 seconds.Figure 3
Synchronization process between slave statexs and master state xm.Figure 4
Synchronization process between slave stateys and master state ym.Figure 5
Synchronization process between slave statezs and master state zm.Figure 6
Time evolution of synchronization error signalse1, e2, and e3.Finally, a comparison is accomplished between the proposed technique and active control. As mentioned in Introduction, active control is based on the compensation of the nonlinearities and the decoupling of the synchronization error dynamics. Given the slave system (7) and the master system (6) for Xian system (1), the synchronization error dynamics can be determined as (64)e˙1=-ae1+xszs-xmzm+yszs-ymzm+u1e˙2=be2-xszs+xmzm+u2e˙3=-ce3+xsys-xmym+u3However, it should be taken into account that(65)xszs-xmzm=e1e3+zme1+xme3(66)yszs-ymzm=e2e3+zme2+yme3(67)xsys-xmym=e1e2+yme1+xme2By substituting (65), (66), and (67) into (64), the synchronization error dynamics can be expressed as(68)e˙1=-ae1+e1e3+zme1+xme3+e2e3+zme2+yme3+u1e˙2=be2-e1e3-zme1-xme3+u2e˙3=-ce3+e1e2+yme1+xme2+u3Thus, the corresponding active control is given by(69)u1=-e1e3-zme1-xme3-e2e3-zme2-yme3-k1e1u2=e1e3+zme1+xme3-k2e2u3=-e1e2-yme1-xme2-k3e3With the objective of accomplishing a systematic comparison between both techniques, the following performance index is used: (70)J=∫0T1l1e12+l2e22+l3e32+l4u12+l5u22+l6u32dtwhere l1,…,l6 are weighting factors and T1 is final time. Considering a standard weighting where each term has the same importance and the signals are normalized, the following values should be used: l1=1/54,l2=1/24,l3=1/96,l4=1/1,566,726,l5=1/799,350,l6=1/2,856,600, and T1=0.1 sec. To realize a fair comparison, the same values for the gains k1, k2, and k3 are used in both cases, that is, k1=170.30, k2=182.30, and k3=172.30. The results are presented in Figures 7 and 8.Figure 7
Performance index for linear feedback control and active control.Figure 8
Zoom of Figure7.As can be appreciated in Figure7, the performance is almost the same for both cases. In Figure 8, it can be seen that linear feedback control produces slightly lower values for performance index. The apparent similarity in the performance of both techniques is due to the high values used for the gains. However, although the performances were similar in both cases, the implementation of linear feedback control is considerably easier with respect to active control.


## 7. Conclusion

The main attractiveness of the linear feedback is its simple structure being very convenient for practical purposes. In this paper, a linear feedback controller based on an algebraic Riccati equation was presented. To use this controller, first, the Lipschitz constant of nonlinear function of the chaotic system on its attractor must be determined. To achieve this objective, it is necessary to find the maximum values of the states of chaotic system. Although this could be calculated analytically, for simplicity, such values are determined by simulation. Next, values for the gain matrixK and the matrices of the Riccati equation, Q0,Q, and R, are selected in such a way that this equation has a unique positive definite solution. Consequently, according to Theorem 5, it is possible to guarantee the exponential convergence to zero of the synchronization error. The strategy is tested on two identical chaotic systems, master and slave, based on Xian system. Numerical simulation confirms the satisfactory performance of the suggested approach. In a future work, a design procedure to obtain systematically the controller gains will be provided.



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