THE SLIDING MODE CONTROLLER DESIGN FOR HYBRID SYNCHRONIZATION OF PAN SYSTEMS

This paper investigates the hybrid synchronization of identical Pan systems (Pan, Xu and Zhou, 2010) by sliding mode control. In hybrid synchronization of chaotic systems, one part of the master and slave systems is completely synchronized, while the other part is anti-synchronized. The co-existence of complete and anti-synchronization enhances the security of the communication devices using chaotic systems. The stability results derived in this paper for the hybrid synchronization of identical Pan systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve hybrid synchronization of the identical Pan systems. Numerical simulations are shown to illustrate and validate the hybrid synchronization schemes derived in this paper for the identical Pan systems.


INTRODUCTION
Nonlinear dynamical systems, which are highly sensitive to initial conditions, are called as chaotic systems.The sensitive nature of chaotic systems is known as the butterfly effect [1].Chaos theory has been applied in many scientific disciplines such as Mathematics, Computer Science, Microbiology, Biology, Ecology, Economics, Population Dynamics and Robotics.
Synchronization of chaotic systems is a phenomenon which may occur when two or more chaotic oscillators are coupled or when a chaotic oscillator drives another chaotic oscillator.Because of the butterfly effect, this is a challenging problem in the control literature.
In most of the chaos synchronization approaches, the master-slave or drive-response formalism is used.If a particular chaotic system is called the master or drive system and another chaotic system is called the slave or response system, then the idea of the anti-synchronization is to use the output of the master system to control the slave system so that the states of the slave system have the same amplitude but opposite signs as the states of the master system asymptotically.
In this paper, we derive new results based on the sliding mode control [24][25][26][27][28] for the hybrid synchronization of identical Pan systems ( [29], Pan, 2010).In robust control systems, the sliding mode control method is often adopted due to its inherent advantages of easy realization, fast response and good transient performance as well as its insensitivity to parameter uncertainties and external disturbances.
This paper has been organized as follows.In Section 2, we describe the problem statement and our methodology using sliding mode control (SMC).In Section 3, we discuss the hybrid synchronization of identical Pan systems.In Section 4, we summarize the main results obtained in this paper.

PROBLEM STATEMENT AND OUR METHODOLOGY USING SMC
In this section, we describe the problem statement for the hybrid synchronization for identical chaotic systems and our methodology using sliding mode control (SMC).Consider the chaotic system described by ( ) where n x ∈ R is the state of the system, A is the n n × matrix of the system parameters and : R is the nonlinear part of the system.We consider the system (1) as the master or drive system.
As the slave or response system, we consider the following chaotic system described by the dynamics ( ) where n y ∈ R is the state of the system and m u ∈ R is the controller to be designed.
We define the hybrid synchronization error as , if is odd , if is even then the error dynamics is obtained as ( , ) , e Ae x y u The objective of the global chaos synchronization problem is to find a controller u such that lim ( ) 0 for all (0) .
To solve this problem, we first define the control u as ( , ) u x y Bv where B is a constant gain vector selected such that ( , ) A B is controllable.
Substituting ( 6) into ( 4), the error dynamics simplifies to e Ae Bv = +  (7) which is a linear time-invariant control system with single input .
v Thus, the original hybrid synchronization problem can be replaced by an equivalent problem of stabilizing the zero solution 0 e = of the system (7) by a suitable choice of the sliding mode control.
In the sliding mode control, we define the variable where [ ] is a constant row vector to be determined.
In the sliding mode control, we constrain the motion of the system (7) to the sliding manifold defined by which is required to be invariant under the flow of the error dynamics (7).
When in sliding manifold , S the system (7) satisfies the following conditions: ( ) 0 s e = (9) which is the defining equation for the manifold S and ( ) 0 s e =  (10) which is the necessary condition for the state trajectory ( ) e t of (7) to stay on the sliding manifold .S Using ( 7) and ( 8), the equation ( 10) can be rewritten as [ ] Solving (11) for , v we obtain the equivalent control law where C is chosen such that 0. CB ≠ Substituting (12) into the error dynamics (7), we obtain the closed-loop dynamics as The row vector C is selected such that the system matrix of the controlled dynamics 1 ( ) . it has all eigenvalues with negative real parts.Then the controlled system ( 13) is globally asymptotically stable.
To design the sliding mode controller for (7), we apply the constant plus proportional rate reaching law sgn( ) where sgn( ) ⋅ denotes the sign function and the gains 0, q > 0 k > are determined such that the sliding condition is satisfied and sliding motion will occur.
From equations ( 11) and ( 14), we can obtain the control ( ) which yields Theorem 1.The master system (1) and the slave system (2) are globally and asymptotically hybrid synchronized for all initial conditions (0), (0) where ( ) v t is defined by ( 15) and B is a column vector such that ( , ) A B is controllable.Also, the sliding mode gains , k q are positive.
Proof.First, we note that substituting (17) and ( 15) into the error dynamics (4), we obtain the closed-loop error dynamics as

( ) e Ae B CB C kI A e q s
To prove that the error dynamics ( 18) is globally asymptotically stable, we consider the candidate Lyapunov function defined by the equation which is a negative definite function on .

n R
This calculation shows that V is a globally defined, positive definite, Lyapunov function for the error dynamics (18), which has a globally defined, negative definite time derivative .V  Thus, by Lyapunov stability theory [37], it is immediate that the error dynamics ( 18) is globally asymptotically stable for all initial conditions (0) .Hence, it follows that the master system (1) and the slave system (2) are globally and asymptotically hybrid-synchronized for all initial conditions (0), (0) .
This completes the proof.

Theoretical Results
In this section, we apply the sliding mode control results derived in Section 2 for the hybrid synchronization of identical Pan systems ([29], Pan, 2010).
Thus, the master system is described by the Pan dynamics where 1 2 3 , , x x x are state variables and , , a b c are positive, constant parameters of the system.
The slave system is also described by the Pan dynamics , , y y y are state variables and 1 2 3 , , u u u are the controllers to be designed.The Pan systems ( 21) and ( 22  The hybrid synchronization error is defined by The error dynamics is easily obtained as e a e e ax u e ce cx y y x x u e be y y x x u We write the error dynamics (24) in the matrix notation as ( , ) e Ae x y u The sliding mode controller design is carried out as detailed in Section 2.
First, we set u as ( , ) u x y Bv where B is chosen such that ( , ) A B is controllable.
We take B as 1 1 1 In the chaotic case, the parameter values are chosen as which makes the sliding mode state equation asymptotically stable.
We choose the sliding mode gains as 6 k = and 0.1.q = We note that a large value of k can cause chattering and an appropriate value of q is chosen to speed up the time taken to reach the sliding manifold as well as to reduce the system chattering.
By Theorem 2.1, we obtain the following result.
Theorem 3.1.The identical Pan systems ( 21) and ( 22) are globally and asymptotically hybrid synchronized for all initial conditions with the sliding mode controller u defined by (31).

Numerical Results
In this section For the numerical simulations, the fourth-order Runge-Kutta method with timestep

CONCLUSIONS
In this paper, we have developed a sliding mode controller (SMC) to achieve hybrid synchronization for the identical Pan systems (2010).Our hybrid synchronization results for the identical Pan systems have been proved using sliding control theory and Lyapunov stability theory.Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve hybrid synchronization for the identical Pan systems.Numerical simulations are also shown to illustrate the effectiveness of the hybrid synchronization results derived in this paper using the sliding mode control.

2 (
follows that V is a positive definite function on .n R Differentiating V along the trajectories of (18) or the equivalent dynamics (14), we get

Figure 1
Figure 1 illustrates the strange chaotic attractor of the chaotic Pan system (21).

Figure 1 .
Figure 1.Strange Chaotic Attractor of the Pan System

Figure 2
Figure 2 illustrates the hybrid synchronization of the identical Pan systems (21) and (22).

Figure 3
Figure 3 illustrates the time-history of the anti-synchronization errors 1 2 3 , , .e e e

Figure 2 .
Figure 2. Hybrid Synchronization of Identical Pan Systems