150 Chaos suppression and stabilization of generalized Liu chaotic control system

In this paper, t he concept of generalized stabilization for nonlinear systems is introduced and the stabilization of the generalized Liu chaotic contro l system is explored. Based on the time approach with differential inequalities, a suitable control is presented s uch that the generalized stabilization for a class of Liu chaotic system can be achieved. Meanwhile, not only the guaranteed exponential convergence rate can be arbitrarily pre specified but also the critical time can be correct ly estimated. Finally, some numerical simulations are given to demonstrate the feasibility and effectiven ess of the obtained results.


INTRODUCTION
In recent years, chaotic dynamic systems have been widely investigated by researchers; see, for instance, [1][2][3][4][5][6][7][8][9][10][11][12] and the references therein. Very often, chaos in many dynamic systems is an origin of the generation of oscillation and an origin of instability. control system, it is important to design a controller that has both good transient and steady-state response. Furthermore, suppressing the occurrence of chaos plays an important role in the controller design of a nonlinear system.
In the past decades, various methodologies in control design of chaotic system have been presented variable structure control approach, approach, adaptive control approach, adaptive sliding mode control approach, back stepping control approach, and others.

International Journal of Trend in Scientific Research and Development (IJTSRD)
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Chaos Suppression and Stabilization of Generalized Liu Chaotic Control System
Yeong-Jeu Sun 1 , Jer-Guang Hsieh 2 1 Professor, 2 Chair Professor Electrical Engineering, I-Shou University, Kaohsiung he concept of generalized stabilization for nonlinear systems is introduced and the stabilization of the generalized Liu chaotic control system is explored. Based on the time-domain approach with differential inequalities, a suitable uch that the generalized stabilization for a class of Liu chaotic system can be achieved. Meanwhile, not only the guaranteed exponential convergence rate can be arbitrarily prespecified but also the critical time can be correctly numerical simulations are given to demonstrate the feasibility and effectiveness Generalized synchronization, Liu chaotic system, critical time, exponential convergence rate tic dynamic systems have been ; see, for instance, ] and the references therein. Very often, chaos in an origin of the generation an origin of instability. For a chaotic control system, it is important to design a controller state response. Furthermore, suppressing the occurrence of chaos oller design of a In the past decades, various methodologies in control presented, such as time-domain approach, adaptive control approach, adaptive sliding stepping control In this paper, the concept of generalized for nonlinear dynamic systems is introduced and the stabilizability of generalized Liu chaotic system will be investigated domain approach with differential inequality, a suitable control will be offered generalized stabilization can be achieved for a class of Liu chaotic system. Not only correctly estimated, but also exponential convergence rate can be arbitrarily pre specified. Several numerical simulations will also be provided to illustrate the use of the main results.
The layout of the rest of this paper is organized as follows. The problem formulation, mai controller design procedure are presented Numerical simulations are given in Section 3 the effectiveness of the developed results. conclusion remarks are drawn In this paper, we explore the following generalized Liu chaotic system: Kaohsiung, Taiwan generalized stabilizability dynamic systems is introduced and the generalized Liu chaotic control investigated. Based on the timedomain approach with differential inequality, a offered such that the stabilization can be achieved for a class of system. Not only the critical time can be , but also the guaranteed exponential convergence rate can be arbitrarily pre-. Several numerical simulations will also be provided to illustrate the use of the main results.

PROBLEM FORMULATION AND MAIN RESULTS Nomenclature
The layout of the rest of this paper is organized as The problem formulation, main result, and controller design procedure are presented in Section 2.
given in Section 3 to show developed results. Finally, in Section 4.

ROBLEM FORMULATION AND MAIN
dimensional real space the modulus of a complex number a the transport of the matrix A the Euclidean norm of the vector ] , 0 0 0 30 20 10 is the initial value, and a i , the parameters of the system. The original system is a special case of system (1) with . It is well known that the system (1) without any control (i.e., ( ) 0 = t u ) displays chaotic behavior for certain values of the parameters [1]. The paper is to search a novel control for the system (1) such that the generalized stability of the controlled system can be guaranteed. In this concept of generalized stabilization will be introduced. Motivated by time-domain approach with differential inequality, a suitable control be established. Our goal is to design a control such that the generalized stabilization of system achieved.
Let us introduce a definition which will be used in subsequent main results. 1 There exist two positive numbers k

Definition 1
The system (1) is said to realize the stabilization, provided that there exist a suitable control u such that the conditons (i) and (ii) are satisfied. In this case, the positive number the exponential convergence rate and number c t is called the critical time.
Now we present the main result for the stabilization of the system (1) via approach with differential inequalities.

Theorem 1
The system (1) realizes the generalized under the following control of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 is the state vector, is the system control, indicate . The original Liu chaotic is a special case of system (1) with ( ) , , and It is well known that the system (1) without ) displays chaotic behavior for he aim of this for the system (1) of the feedback-In this paper, the stabilization will be domain approach with suitable control strategy will a control such stabilization of system (1) can be Let us introduce a definition which will be used in There exists a positive number c t such that The system (1) The time derivative of ( ) ( ) t x W feedback-controlled system is given by (6) and (7), it can be readily obtained that ( ) ( ), It is easy to deduce that Hence, from (6), (7), and (8), we have Consquently, we conclude that in view of (5) with above condition (i) completes the proof. □

NUMERICAL SIMULATIONS
Consider the generalized Liu chaotic system with Consequently, by Theorem 1, we conclude that system (1) achieves generalized stabilization parameters of , and feedback control law of ( Furthermore, the exponential convergence rate guaranteed critical time are given by The typical state trajectories of uncontrolled systems and controlled systems are depicted in Fig  Figure 2 Our objective, in this example, is to design a feedback control such that the stabilization with with the guaranteed exponential convergence rate , 2 we deduce , we conclude that the achieves generalized stabilization with , 1 − , 0 6 5 = = a a feedback control law of (9). exponential convergence rate and the iven by and of uncontrolled systems and controlled systems are depicted in Figure 1 and , respectively. From the foregoing simulations results, it is seen that the dynamic system of ( achieves the generalized stabilization under the control law of (9).

CONCLUSION
In this paper, the concept of generalized for nonlinear systems has been introduced and the stabilization of generalized Liu chaotic has been studied. Based on the time with differential inequalities, a s been presented such that the generalized for a class of Liu chaotic system Besides, not only the guaranteed exponential convergence rate can be arbitrarily pre also the critical time can be correctly estimated. Finally, some numerical simulations have been offered to show the feasibility and effectiveness of the obtained results.  , "Fault tolerant hronization of chaotic systems with time triggered sampled Applied Mathematics and Computation, Typical state trajectories of the feedbackcontrolled system of (1) with (9).