Active Sliding Mode Control Antisynchronization of Chaotic Systems with Uncertainties and External Disturbances

The antisynchronization behavior of chaotic systems with parametric uncertainties and external disturbances is explored by using robust active sliding mode control method. The sufficient conditions for achieving robust antisynchronization of two identical chaotic systemswith different initial conditions and two different chaotic systems with terms of uncertainties and external disturbances are derived based on the Lyapunov stability theory. Analysis and numerical simulations are shown for validation purposes.


Introduction
Chaotic nonlinear systems appear ubiquitously in nature and can occur inman-made systems.These systems are recognized by their great sensitivity to initial conditions.Many scientists who are interested in this field have struggled to achieve the synchronization or antisynchronization of different chaotic systems, mainly due to its potential applications especially in chemical reactions, power converters, biological systems, information processing, secure communications, and so forth 1-7 .But due to its complexities, such tasks are always difficult to achieve.It is much more attractive and challenging to realize the synchronization or antisynchronization of two different chaotic systems especially if terms of uncertainty are considered.A wide variety of approaches have been proposed for the synchronization or antisynchronization of chaotic systems, which include generalized active control 8-10 , nonlinear control 11, 12 , adaptive control 13-17 , and sliding mode control 18, 19 .Most of the above-mentioned works did not consider the uncertainty of parameters and its effects on the systems.The aim of this paper is to further develop the state observer method for

Active Sliding Mode Controller Design
Consider a chaotic system described by the following nonlinear differential equation: where x t ∈ R n denotes the system's n-dimensional state vector, A 1 ∈ R n×n represents the linear part of the system dynamics, and f 1 : R n → R n is the nonlinear part of the system, ΔA 1 ∈ R n×n is the matrix of uncertainties, and D 1 t ∈ R n is a vector representing external disturbances.Relation 2.1 represents the master system.The controller u t ∈ R n is added into the slave system, so it is given by where y t ∈ R n is the slave system's n-dimensional state vector, A 2 ∈ R n×n and f 2 : R n → R n play similar roles as A 1 and f 1 for the master system, ΔA 2 ∈ R n×n is the matrix uncertainties, and D 2 t ∈ R n is a vector representing external disturbances.If A 1 A 2 and f 1 • f 2 • , then x and y are the states of two identical chaotic systems.Otherwise they represent the states of two different chaotic systems.The antisynchronization problem is to design the controller u t ∈ R n which antisynchronizes the states of the master and slave systems.The dy-nam-ics of the antisynchronization errors can be expressed as where e y x and F x, y where • is the Euclidean norm.As the trajectories of the chaotic systems are always bounded, then one can assume uncertainties to be bounded, so, in general, where ρ, μ, δ, and are positive constants.According to the active control design procedure, one uses the control input u t to eliminate the nonlinear part of the error dynamics.In other words, the input vector is considered as The error system 2. In what follows, the appropriate sliding mode controller will be designed according to the sliding mode control theory.

Sliding Surface Design
The sliding surface can be defined as follows: where C c 1 , c 2 , c 3 is a constant vector.The equivalent control approach is found by the fact that ṡ e 0 is a necessary condition for the state trajectory to stay on the switching surface s e 0. Hence, when in sliding mode, the controlled system satisfies the following conditions: where the existence of CK −1 is a necessary condition.Replacing for w t in 2.10 from w eq t of 2.15 , the state equation in the sliding mode is determined as follows:

2.16
As long as the system 2.10 has all eigenvalues with negative real parts, it is asymptotically stable.

Design of the Sliding Mode Controller
We assume that the constant plus proportional rate reaching law is applied.The reaching law can be chosen such that where γ is a positive real number.The gains q > 0 and r > 0 are determined such that the sliding condition is satisfied and the sliding mode motion occurred.From 2.10 and 2.11 , it can be found that Now, from 2.17 and 2.18 , the control input is determined as

Stability Analysis
To check the stability of the controlled system, one can consider the following Lyapunov candidate function: The time derivative of 2.21 is Since s 2 / |s| γ > 0, r > 0 and q > 0, we have V ṡs < 0; therefore, V e is negative definite.This property implies boundedness of the sliding surface s.The error dynamics can be obtained using 2.19 in 2.10 : As a linear system with bounded input −K CK −1 q for s ≥ 0 and K CK −1 q for s < 0 , the error system is asymptotically stable if and only if 2 ΔA 1 has negative eigenvalues.Because of the special structure for matrix A 2 in the given chaotic systems, one of the eigenvalues is always −r and therefore is stable.The two other eigenvalues are independent from r and determined by the other control parameters that is, K and C. The latter two eigenvalues can be negative or positive depending on K and C values.By appropriate choices of r, K, and C, one is able not only to stabilize the error system but also to adjust the rate of the error convergence.The parameter q can be used to enhance the robust property expected from a sliding mode controller.

Systems Description
The L ü attractor, which connects the Lorenz attractor 20 andChen and Ueta attractor 21, 22 and represents the transition from one to another, was proposed and analyzed by l ü et al. 23 .The L ü chaotic system is described by the following system of differential equations:

Active Sliding Mode Antisynchronization between Two Identical Systems
In order to observe synchronization behavior between two identical chaotic systems via active sliding mode control, we consider two examples, the first one is L ü system in 3.1 and the second example is Genesio system which is described in 3.2 .

Active Sliding Mode Antisynchronization between Two Identical L ü Systems
For the L ü system, let us consider that 4.1 The master system and the slave system can be written as the following respectively: and the slave system can be written as where u 1 , u 2 , u 3 are three control functions to be designed in order to determine the control functions and to realize the active sliding mode antisynchronization between the systems in 4.2 and 4.3 .We add 4.2 and 4.3 to get ė1 a e 2 − e 1 e 2 0.2 cos 50t u 1 ,

4.7
Applying controller 4.7 to the slave system then the slave system in 4.3 can synchronize master system equation 4.2 asymptotically.To verify and demonstrate the effectiveness of the proposed method, we discuss the simulation result for the active sliding mode antisynchronization between two identical L ü systems.For these simulations, the fourth-order Runge-Kutta method is used to solve the systems with time step size 0.001.We assumed that the initial conditions, x 1 0 , y 1 0 , z 1 0 2, 2, 20 and x 2 0 , y 2 0 , z 2 0 −3, −1, −1 .Hence the error system has the initial values e 1 0 −1, e 2 0 1, and e 3 0 19.The systems parameters are chosen as a 36, b 3,c 20 in the simulations such that L ü system exhibit chaotic behavior.Antisynchronization of the systems equations 4.   The master system and the the slave system can be written as the following; respectively, ẋ1 y 1 0.5x 1 0.1 cos 100t , ẏ1 z 1 − 0.3y 1 0.1 sin 100t , ż1 −cx 1 − by 1 − az 1 x 2 1 − 0.5x 1 0.1 sin 100t , 4.9 and the slave system can be written as ẋ2 y 2 0.5x 2 0.1 cos 100t u 1 ,

4.10
where u 1 , u 2 , u 3 are three control functions to be designed in order to determine the control functions and to realize the active sliding mode antisynchronization between the systems in 4.9 and 4.10 .We add 4.9 and 4.10 to get ė1 e 2 0.5e 1 0.2 cos 100t u 1 ,

4.11
where e 1 x 2 x 1 , e 2 y 2 y 1 and e 3 z 2 z 1 .Our goal is to find proper control functions u i i 1, 2, 3 such that system equation 4.10 globally antisynchronizes system equation 4.9 asymptotically, that is, where e e 1 , e 2 , e 3 T .Without the controls u i 0, i 1, 2, 3 , the trajectories of the two systems, 4.9 and 4.10 , will quickly separate with each other and become irrelevant.However, when controls are applied, the two systems will approach synchronization for any initial conditions by appropriate control functions.if the control parameters are chosen as C 6, 0, −1 , K 1, 0, 0 T and r 1, q 60, and γ 0.01, then the switching surface 13 so the controllers are

10
Journal of Applied Mathematics Applying controller 4.14 to the slave system then the slave system, in 4.10 can synchronize master system equation 4.9 asymptotically.In the following, we discuss the simulation results of the proposed method.For these simulations, the fourth-order Runge-Kutta method is used to solve the systems with time step size 0.001.We assumed that the initial conditions, x 1 0 , y 1 0 , z 1 0 2, 2, 2 and x 2 0 , y 2 0 , z 2 0 −2, −3, −3 .Hence the error system has the initial values e 1 0 0, e 2 0 −1 and e 3 0 −1.The systems parameters are chosen as a 1.2, b 2.92, and c 6 in the simulations such that both systems exhibit chaotic behavior.Antisynchronization of the systems equations 4.10 and 4.9 via active sliding mode controllers in 4.14 are shown in Figure 2. Figures 2 a -2 c display the state trajectories of master system 4.9 and slave system 4.10 .Figure 2 d display the error signals e 1 , e 2 , e 3 of Genesio system under the controller equations 4.14 .

Active Sliding Mode Antisynchronization between Two Different Systems
In this section the antisynchronization behavior between two different chaotic systems via active sliding mode control is investigated, the L ü system in 3.1 is assumed as the master system and the Genesio system in 3.2 is taken as the slave system.If we take:

5.3
We add 5.2 to 5.3 to get  where e 1 x 2 x 1 , e 2 y 2 y 1 , and e 3 z 2 z 1 .So we are aiming to find proper control functions u i i 1, 2, 3 such that system equation 5.3 globally antisynchronizes system equation 5.2 asymptotically; that is, lim t → ∞ e 0, 5.5 where e e 1 , e 2 , e 3 T .

5.8
Applying controllers in 5.8 , slave system equation 5.3 can antisynchronize master system equation 5.2 asymptotically.To verify and demonstrate the effectiveness of the proposed method, we discuss the simulation result for the active sliding mode antisynchronization between the L ü system and the Genesio system.In the numerical simulations, the fourthorder Runge-Kutta method is used to solve the systems with time step size 0.001.For this numerical simulation, we assumed that the initial conditions x 1 0 , y 1 0 , z 1 0 −3, 2, −8 and x 2 0 , y 2 0 , z 2 0 2, 3, −2 .Hence the error system has the initial values e 1 0 −1, e 2 0 5, and e 3 0 −10.The system parameters are chosen as a 1 36, b 1 3, c 1 20, and a 2 1.2, b 2 2.92, c 2 6 in the simulations such that both systems exhibit chaotic behavior.Antisynchronization of the systems equations 5.2 and 5.3 via active sliding mode control law in 5.8 are shown in Figure 3. Figures 3 a -3 c display the state trajectories of master system 5.2 and slave system 5.3 .Figure 3 d displays the error signals e 1 , e 2 , e 3 of the L ü system and the Genesio system under the controller equations 5.8 .

Concluding Remark
In this paper, we have applied the antisynchronization to some chaotic systems with parametric uncertainties and external disturbances via active sliding mode control.We have proposed a novel robust active sliding mode control scheme for asymptotic chaos antisynchronization by using the Lyapunov stability theory.Finally, the numerical simulations proved the robustness and effectiveness of our method.
ẋ a y − x , ẏ −xz cy, ż xy − bz, 3.1 where x, y, and z are state variables and a, b, and c are positive parameters.This system is dissipative for c < a b and has a chaotic attractor when a 36, b 3, and c 20.The Genesio system was introduced by Genesio and Tesi in 24 .It is given by ẋ y, ẏ z, ż −cx − by − az x 2 , 3.2 where x, y, and z are state variables, and a, b, and c are the positive real constants satisfying ab < c.Throughout this paper, we set a 1.2, b 2.92, and c 6 such that the system exhibits chaotic behavior.
3 and 4.2 via active sliding mode controllers in 4.7 is shown in Figure 1.Figures 1 a -1 c display the state trajectories of master system 4.2 and slave system 4.3 .Figure 1 d displays the error signals e 1 , e 2 , e 3 of the L ü system under the controller equations 4.7 .

3 dFigure 1 :
Figure 1: State trajectories of drive system 4.2 and response system 4.3 , a signals x 1 and x 2 ; b signals y 1 and y 2 ; c signals z 1 and z 2 ; d the error signals e 1 , e 2 , e 3 between two identical L ü systems with different initial conditions under the controller 4.7 .

3 dFigure 2 :
Figure 2: State trajectories of drive system 4.9 and response system 4.10 , a signals x 1 and x 2 ; b signals y 1 and y 2 ; c signals z 1 and z 2 ; d the error signals e 1 , e 2 , e 3 between two identical Genesio systems with different initial conditions under the controller 4.14 .

3 dFigure 3 :
Figure 3: State trajectories of drive system 5.2 and response system 5.3 , a signals x 1 and x 2 ; b signals y 1 and y 2 ; c signals z 1 and z 2 ; d the error signals e 1 , e 2 , e 3 between the L ü system and the Genesio system under the controller 5.8 .
2 y 2 x 1 y 1 0.2 sin 50t u 3 ,4.4wheree 1 x 2 x 1 , e 2 y 2 y 1 , and e 3 z 2 z 1 .Our goal is to find proper control functions u i i 1, 2, 3 such that system equation 4.3 globally antisynchronizes system equation 4., e 2 , e 3 T .Without the controls u i 0, i 1, 2, 3 , the trajectories of the two systems equations 4.2 and 4.3 , will quickly separate with each other and become irrelevant.However, when controls are applied, the two systems will approach synchronization for any initial conditions by appropriate control functions.if the control parameters are chosen as T, and r 1, q 30, and γ 0.01, then the switching surfaceu 3 −x 1 y 1 − x 2 y 2 3e 3 .