Exponential Synchronization of Chaotic Xian System Using Linear Feedback Control

1Sección de Estudios de Posgrado e Investigación, Escuela Superior de Ingenierı́a Mecánica y Eléctrica, Unidad Azcapotzalco, Instituto Politécnico Nacional, Ciudad de México 02250, Mexico 2Área de Mecatrónica, Centro de Innovación y Desarrollo Tecnológico en Cómputo, Instituto Politécnico Nacional, Ciudad de México 07700, Mexico 3Departamento de Ingenieŕıa Electrónica, Tecnológico Nacional de México, CENIDET, Cuernavaca 62490, Mexico

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][20][21][22][23][24][25][26][27][28][29][30][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][34][35][36][37][38][39][40].In this technique, given a Lyapunov function candidate , the control law is selected considering that the first derivative of  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][43][44][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

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 where , , and  are the system states and , , and  are constant parameters.This system shows chaotic behavior for the values  = 5,  = 7,  = 3 and the initial condition (0) = −1, (0) = −1, and (0) = −2.In Figure 1, the corresponding attractor is shown.The time series for the states , , and  during the first twenty seconds of numerical simulation are plotted in Figure 2.
It can be noted that, by using vector notation, system (1) can succinctly be represented as follows: where

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 where w  := [      ]  and subscript m denotes "master."The corresponding slave system for system (2) is given by where Thus, the problem of synchronization based on linear feedback control for systems (7) and ( 6) consists of finding an appropriate control law of the form  where and  1 ,  2 , and  3 are real constants selectable by the designer in such a way that lim →∞ e = 0.

Background Results
In this section, some basic definitions and results are reviewed briefly.
Definition 1 (see [55][56][57]).A function f(w) : R  → R  is said to be locally Lipschitz on  ⊂ R  if there exists a constant  (known as Lipschitz constant) such that, for all w 1 , w 2 ∈ , the following inequality holds: Finally, f is said to be globally Lipschitz if it satisfies (11) with  = R  .
Lemma 2 (see [56,57]).If a function f : R  → R  is continuously differentiable on a set  ⊂ R  , then it is locally Lipschitz on .
Based on Lemma 2, a procedure can be established in order to calculate the Lipschitz constant  [56].For example, consider the function Let us define  1,max and  2,max such that | 1 | ≤  1,max and | 2 | ≤  2,max .Then, on ,  1,max = 4 and  2,max = 3.On the other hand, the Jacobian matrix of g is given by Let us define the matrix G as That is, G is formed by the maximum absolute values of each corresponding component in (13).Consequently, Finally, a Lipschitz constant for g (12) on the set  can be taken as Lemma 3 (see [58]).For any vectors w 1 , w 2 ∈ R  and any positive definite matrix Λ  = Λ > 0, Λ ∈ R × , the following inequality holds: Lemma 4 (see [58]).The matrix Riccati equation with known constant matrices A 0 , R, Q ∈ R × has a unique positive definite solution P  = P > 0, P ∈ R × if the following conditions are satisfied: The following matrix inequality holds:

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 By substituting ( 7) and ( 6) into (20) and taking into account that the control law has the form given in (9), we get Let us define Thus, (21) becomes To evaluate the stability of the synchronization error dynamics ( 23), the following Lyapunov function candidate is proposed: where P is a positive definite matrix to be found.The first time derivative of ( 24) is calculated as By substituting ( 23) into (25) and after some operations, we get Now, let us consider the last two terms of ( 26): By using Lemma 3, ( 27) can be bounded as where 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 , then the following can be established: and By substituting (30) into (28) and the corresponding result into (26), the first time derivative of can be bounded by and the following matrix Riccati equation is formed: and if the constant matrices A 0 , Λ, Q 0 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 From (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 By using Rayleigh inequality into (36), we can claim that From the last inequality and by taking into account (35), Let us define And from ( 24), (38) becomes This implies that By using twice Rayleigh inequality into ( 24), (41) becomes or By taking square root of both sides of inequality (43), we get Finally, 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 function f in equations ( 6) and ( 7) is locally Lipschitz on the attractor of the autonomous system (6) with Lipschitz constant  and the gain K and the matrices Lemma 4 in such a way that the matrix Riccati equation A  0 P + PA 0 + 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 = w  −w  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, Assumption 6.The terms  m and  s 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 function  defined as which satisfies 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 function f in equations ( 45) and ( 46) is locally Lipschitz on the attractor of the autonomous system (45) with Lipschitz constant  and the gain K and the matrices Λ, Ω, Q 0 , A 0 = A−K, R = Λ+Ω −1 , Q =  2 Λ −1 +Q 0 are selected according to Lemma 4 in such a way that the matrix Riccati equation A  0 P 1 + P 1 A 0 + P 1 RP 1 + Q = 0 has a unique positive definite solution P 1 and the control law u = −Ke is applied to the slave system (46), then the norm of the synchronization error ‖e‖ = ‖w  − w  ‖ converges exponentially to a zone bounded by √ Ψ/( min ( 1 )) where  =  min (P −1/2 1 Proof.As the proof of this theorem is very similar to the proof of Theorem 5, only the main points will be presented.Given e = w  − w  , the synchronization error dynamics is given by To analyze this dynamics, the following Lyapunov function candidate is proposed: The first time derivative of (50) can be expressed as By defining the following matrix Riccati equation is formed: and if the constant matrices A 0 , Λ, Ω, Q 0 are selected using Lemma 4, then (56) has a unique positive definite solution P 1 and (54) becomes simply Now, it can be shown that where  =  min (P −1/2 1 ).By substituting ( 58) into ( 57) and ( 50) into the resulting expression, we get According to [59,60], (59) implies that This means that The 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/( min ( 1 )).

Numerical Simulation
In order to use the result of Theorem 5, first, by numerical simulation, the maximum absolute value of each state of the autonomous system ( 6) is estimated as  max = 40,  max = 66.54, and  max = 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 The 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 The two-norm of F is 174.88.Thus, the Lipschitz constant of f can be estimated as  = 175.Given the matrix A (as in ( 4)) with the nominal values  = 5, = 7,  = 3, in order to guarantee a positive definite solution for Riccati matrix equation ( 34), the constant matrices With these matrices, the solution for Riccati matrix equation ( 34) is P = diag(1.7002,1.7002, 1.7002), and according to Theorem 5, the synchronization error e converges exponentially to zero.The performance of the control law u = −Ke is verified by simulation.First, the Xian master system (6) with the initial condition   (0) = −1,   (0) = −1, and   (0) = −2 and the Xian slave system (7) with the initial condition   (0) = −4,   (0) = 1, and   (0) = 2 are built on Simulink5.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.
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 However, it should be taken into account that By substituting (65), (66), and (67) into (64), the synchronization error dynamics can be expressed as Thus, the corresponding active control is given by With the objective of accomplishing a systematic comparison between both techniques, the following performance index is used:  As can be appreciated in Figure 7, 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.

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 matrix K and the matrices of the Riccati equation, Q 0 , 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.

Figure 2 :
Figure 2: Time evolution of states of Xian system for first 20 seconds: (a) state , (b) state , and (c) state .

Figure 3 :Figure 4 :
Figure 3: Synchronization process between slave state   and master state   .

Figure 5 :Figure 6 :
Figure 5: Synchronization process between slave state   and master state   .