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.

In recent decades, several kinds of chaotic systems have been widely explored; see, for instance, [

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 observer-based 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 [

In this paper, the nonlinear state reconstructor of the generalized Rössler 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.

In this paper, we consider the generalized Rössler hyperchaotic systems as follows:

The following assumption is made on system (

There exists a constant

It is noted that the Rössler hyperchaotic system [

The objective of this paper is to search a nonlinear observer for system (

System (

Now we present the main result for the state observer of system (

System (

From (

Let

Consider the generalized hyperchaotic system:

It can be verified that condition (A1) is satisfied with

It can be verified that condition (A1) is satisfied with

It can be verified that condition (A1) is satisfied with

The time response of error states for system (

The time response of error states, with

The time response of error states, with

The time response of error states, with

In this paper, the generalized Rössler 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 Rössler 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

The set of

The modulus of a real number

The Euclidean norm of the vector

The induced Euclidean norm of the matrix

The transpose of the matrix

The set of all eigenvalues of the matrix

The symmetric matrix

The maximum eigenvalue of the symmetric matrix

The minimum eigenvalue of the symmetric matrix

The author thanks the National Science Council of Republic of China for supporting this work under Grant NSC-100-2221-E-214-015. The author also wishes to thank the anonymous reviewers for providing constructive suggestions.