Simple Exponential Observer Design for the Generalized Liu Chaotic System

In this paper, the generalized Liu chaotic system is firstly introduced and the state observation problem of such a system is investigated. Based on the time domain approach with differential and integral equalities, a novel state observer for the generalized Liu chaotic system is constructed to ensure the global exponential stability of the resulting error system. Besides, the guaranteed exponential convergence rate can be precisely calculated. Finally, numerical simulations are presented to exhibit the effectiveness and feasibility of the obtained results.


INTRODUCTION
In recent years, numerous chaotic systems have been generally explored; see, for example, [1 references therein. Regularly, chaos in many dynamic systems is an origin of the generation of oscillation and an origin of instability. Chaos commonly various fields of application; for instance, ecological systems, secure communication, and system identification. Based on the practical considerations, it is either impossible or inappropriate to measure all the elements of the state vector. Furthermore, a state observer can be used to take the place of sensor signals, in the event of sensor failures. Undoubtedly, the state observer design of dynamical systems with chaos is not as easy as that without chaos. Owing to the foregoing reasons, the observer design of chaotic systems is really essential and significant.
In this paper, the observability problem for the generalized Liu chaotic system is investigated. By @ IJTSRD | Available Online @ www.ijtsrd.com | Volume -2 | Issue -1 | Nov-Dec 2017 In this paper, the generalized Liu chaotic system is firstly introduced and the state observation problem of such a system is investigated. Based on the timeh differential and integral equalities, a novel state observer for the generalized Liu chaotic system is constructed to ensure the global exponential stability of the resulting error system. Besides, the guaranteed exponential convergence rate sely calculated. Finally, numerical simulations are presented to exhibit the effectiveness Generalized Liu chaotic system, observer design, chaotic system, exponential convergence rate In recent years, numerous chaotic systems have been generally explored; see, for example, [1][2][3][4][5][6][7][8][9][10] and the references therein. Regularly, chaos in many dynamic systems is an origin of the generation of oscillation and an origin of instability. Chaos commonly appeared in various fields of application; for instance, ecological systems, secure communication, and system identification. Based on the practical considerations, it is either impossible or inappropriate to measure all the Furthermore, a state observer can be used to take the place of sensor signals, in the event of sensor failures. Undoubtedly, the state observer design of dynamical systems with chaos is not as easy as that without chaos. Owing to the foregoing observer design of chaotic systems is really In this paper, the observability problem for the generalized Liu chaotic system is investigated. By using the time-domain approach with differential and integral equalities, a new generalized Liu chaotic system will be provided to ensure the global exponential stability of the resulting error system. In addition, the guaranteed exponential convergence rate can be precisely calculated. Finally, some numerical simulations will be given to demonstrate the effectiveness of the main result.

PROBLEM FORMULATION AND MAIN RESULTS
Nomenclature n  the n-dimensional real space x the Euclidean norm of the vector In this paper, we explore the following generalized Liu chaotic system: Simple Exponential Observer Design for the domain approach with differential and integral equalities, a new state observer for the generalized Liu chaotic system will be provided to ensure the global exponential stability of the resulting error system. In addition, the guaranteed exponential convergence rate can be precisely calculated. Finally, imulations will be given to demonstrate the effectiveness of the main result.   . It is a well-known fact that since states are not always available for direct measurement, particularly in the event of sensor failures, states must be estimated. The aim of this paper is to search a novel state observer for the system (1) such that the global exponential stability of the resulting error systems can be guaranteed.

ROBLEM FORMULATION AND MAIN
Before presenting the main result, the state reconstructibility is provided as follows.

Definition 1
The system (1) represents the reconstructed state of the system (1). In this case, the positive number  is called the exponential convergence rate. Now, we are in a position to present the main results for the exponential state observer of system (1).

Theorem 1
The system (1) is exponentially state reconstructible. Besides, a suitable state observer is given by    , In this case, the guaranteed exponentia convergence rate is given by 7 : a    .