Nonlinear Observer Design of the Generalized Rössler Hyperchaotic Systems via DIL Methodology

The generalized Rössler hyperchaotic systems are presented, and the state observation problem of such systems is investigated. Based on the differential inequality with Lyapunov methodology DIL methodology , a nonlinear observer design for the generalized Rössler hyperchaotic systems is developed to guarantee the global exponential stability of the resulting error system.Meanwhile, the guaranteed exponential decay rate can be accurately estimated. Finally, numerical simulations are provided to illustrate the feasibility and effectiveness of proposed approach.


Introduction
In recent decades, several kinds of chaotic systems have been widely explored; see, for instance, 1-11 and the references therein.This is due to theoretical interests as well as to an efficient tool for chaos synchronization and chaos control design.As a rule, chaos in many systems is a source of the generation of oscillation and a source of instability.Chaotic systems frequently exist in various fields of application, such as system identification, master-slave chaotic systems, secure communication, and ecological systems.
Form practical considerations, it is either impossible or inappropriate to measure all the elements of the state vector.The state observer has come to take its pride of place in system identification, filter theory, and control design.As we know, the tasks of observerbased control systems with or without chaos can be divided into two categories: tracking or synchronization and observer-based stabilization or regulation .The state observer can be skillfully applied in observer-based stabilization, synchronization of master-slave chaotic systems, and secure communication.For more detailed knowledge, one can refer to 1, 2, 7-9, 11-14 .However, the state observer design of dynamic systems with chaos is in general not as easy as that without chaos.Motivated by the above reasons, the observer design of chaotic systems is actually crucial and meaningful.On the other hand, a variety of methods have been proposed for the observer design of systems, such as Chebyshev neural network CNN , sliding-mode approach, passivation of error dynamics, separation principle, and frequency domain analysis; see, for instance, 15-20 and the references therein.
In this paper, the nonlinear state reconstructor of the generalized R össler hyperchaotic systems is investigated.Using the DIL methodology, a nonlinear observer for such systems is provided to guarantee the global exponential stability of the resulting error system.Furthermore, the guaranteed exponential decay rate can be correctly estimated.Finally, numerical simulations are given to verify the effectiveness of proposed approach.

Problem Formulation and Main Result
In this paper, we consider the generalized R össler hyperchaotic systems as follows: 2.1 where x t : x 1 t x 2 t x 3 t x 4 t T ∈ 4 is the state vector, y t ∈ is the system output, r 1 , r 2 , and α i , for all i ∈ {1, 2, 3, 4} are the system parameters with r 1 r 2 / 0. For the existence and uniqueness of system 2.1 , we assume that all the functions g i • , for all i ∈ {1, 2, 3, 4}, are sufficiently smooth.
The following assumption is made on system 2.1 throughout this paper.
A1 There exists a constant h 1 such that It is noted that the R össler hyperchaotic system 21 is the special cases of system 2.1 with

2.3
The objective of this paper is to search a nonlinear observer for system 2.1 such that the global exponential stability of the resulting error systems can be guaranteed.Before presenting the main result, let us introduce a definition which will be used in the main theorem.

2.5
In this case, the guaranteed exponential decay rate is given by α : 1/λ max P , where P > 0 is the unique solution to the following Lyapunov equation: Proof.From 2.1 , 2.5 with e i t : it can be readily obtained that Mathematical Problems in Engineering

Numerical Simulations
Consider the generalized hyperchaotic system:

3.1
Case 1 a 1, b 0.25 or, equivalently, the Rössler hyperchaotic system .It can be verified that condition A1 is satisfied with h 1 1.2.By Theorem 2.3, we conclude that system 3.1 with a 1 and b 0.25 is exponentially state reconstructible by the nonlinear observer: with the guaranteed exponential decay rate α 0.164.
Case 2 a −20, b −50 .It can be verified that condition A1 is satisfied with h 1 10.By Theorem 2.3, we conclude that system 3.1 with a −20 and b −50 is exponentially state reconstructible by the nonlinear observer: with the guaranteed exponential decay rate α 8.47.
Case 3 a 30, b −40 .It can be verified that condition A1 is satisfied with h 1 5.By Theorem 2.3, we conclude that system 3.1 with a 30 and b −40 is exponentially state reconstructible by the nonlinear observer: with the guaranteed exponential decay rate α 3.79.
The time response of error states for system 3.1 with Case 1-Case 3 is depicted in Figures 1, 2, and 3, respectively.From the foregoing simulations results, it is seen that system 3.1 with Case 1-Case 3, regardless of chaotic system or nonchaotic system, is exponentially state reconstructible by the nonlinear observers 3.2 -3.4 , respectively.

Conclusion
In this paper, the generalized R össler hyperchaotic systems have been presented, and the state observation problem of such systems has been investigated.Based on the DIL methodology, a nonlinear state reconstructor of the generalized R össler hyperchaotic systems has been developed to guarantee the global exponential stability of the resulting error system.Besides, the guaranteed exponential decay rate can be accurately estimated.However, the state observation design for more general uncertain hyperchaotic system still remains unanswered.This constitutes an interesting future research problem.The modulus of a real number a x : The Euclidean norm of the vector x ∈ n A : The induced Euclidean norm of the matrix A A T : The transpose of the matrix A σ A : The set of all eigenvalues of the matrix A P > 0: The symmetric matrix P is positive definite λ max P : The maximum eigenvalue of the symmetric matrix P with real eigenvalues λ min P : The minimum eigenvalue of the symmetric matrix P with real eigenvalues.

1 Figure 1 :
Figure 1: The time response of error states, with a 1 and b 0.25.

e 1 Figure 2 :
Figure 2: The time response of error states, with a −20 and b −50.

e 1 Figure 3 :
Figure 3: The time response of error states, with a 30 and b −40.
real space C − : Thesetof{a bj | a < 0, b ∈ } |a|: Definition 2.2.System 2.1 is exponentially state reconstructible if there exist an observer E ˙ x t g x t , y t and positive numbers k and α such that e t : x t − x t ≤ k exp −αt , ∀t ≥ 0, 2.4 where x t expresses the reconstructed state of system 2.1 .In this case, the positive number α is called the exponential decay rate.Now we present the main result for the state observer of system 2.1 .