Abstract
The generalized Liu system is firstly introduced and the state observation problem of such a system is explored. A simple state estimator for the generalized Liu system is developed to guarantee the global exponential stability of the resulting error system. Applications of proposed state estimator strategy to chaotic secure communication, circuit implementation, and numerical simulations are provided to show the effectiveness and feasibility of the obtained results. Besides, the guaranteed exponential convergence rate of the proposed state estimator and that of the proposed chaotic secure communication can be precisely calculated.
1. Introduction
Frequently, it is either inappropriate or impossible to measure all the elements of the state vector. In particular, states estimator is more intricate when system is chaotic, nonlinear, or stochastic in model or parameters. Estimating system states has come to take its pride of place in system identification, control design, and filter theory, which has taken up engineers’ attention from early 1960s. Recently, a wide variety of methodologies have been proposed for the estimator design of systems, such as sliding-mode observer (SMO), Chebyshev neural network (CNN), separation principle, frequency domain analysis, and passivation of error dynamics. For more detailed knowledge, one can refer to [1–9] and the references therein. As usual, most of the observer designs focus on high tracking accuracy and the fast response. For the fast response, the use of a sigmoid function in a boundary layer is particularly popular. Nevertheless, the observer error cannot be guaranteed to converge to zero within the boundary layer [1].
Because chaotic system is highly sensitive to initial conditions and the output behaves like a random signal, several kinds of chaotic systems have been widely applied in various applications such as secure communication, ecological systems, system identification, master-slave chaotic systems, chemical reactions, and biological systems; see, for instance, [10–21] and the references therein. Recently, various problems of chaotic Liu system have been investigated; see, for example, [11–21]. In [12], by geometric analysis, it has been shown that the basin of attraction of the Liu attractor has riddled property, leading to the conclusion that it has a strange attractor in the sense of Milnor. In [21], by means of Routh-Hurwitz criteria, a feedback strategy has been proposed for stabilizing the unstable equilibrium points of Liu chaotic system. Besides, based on Lyapunov stability theory, an adaptive control law has been offered in [20] to achieve the antisynchronization between Liu system and Rössler system with unknown parameters.
In this paper, the state estimator of the generalized Liu system is firstly introduced and studied. A simple state estimator for such systems is provided to guarantee the global exponential stability of the resulting error system. Moreover, the guaranteed exponential convergence rate can be accurately estimated. Applications of proposed state observer scheme to secure communication, circuit implementation, and numerical simulations are offered to evidence the utility and effectiveness of the main results.
This rest of the paper is organized as follows. The problem formulation and main result are presented in Section 2. Several numerical simulations and implementation of electronic circuits are given in Section 3 to demonstrate the presented schemes. Finally, some conclusions are drawn in Section 4.
2. Problem Formulation and Main Result
In this paper, we consider the following generalized Liu system: where is the state vector, is the system output, and and are the system parameters with and . For the existence and uniqueness of the system ((1a), (1b), (1c), and (1d)), we assume that all of functions , for all , are sufficiently smooth with being an invertible function. It goes without saying that since states are not always available for direct measurement, states must be estimated. The aim of this paper is to search a state estimator for the system ((1a), (1b), (1c), and (1d)) such that the global exponential stability of the resulting error systems can be guaranteed. In what follows, is used to denote the transpose for a matrix , denotes the Euclidean norm of the column vector , and denotes the absolute value of a real number .
Before presenting the main result, let us introduce a definition which will be used in the main theorem.
Definition 1. The system ( (1a), (1b), (1c), and (1d)) is exponentially state reconstructible if there exist a state estimator and positive numbers and such that where expresses the reconstructed state of the system ((1a), (1b), (1c), and(1d)). In this case, the positive number is called the exponential convergence rate.
Remark 2. It is noted that the chaotic Liu system, is the special cases of system ((1a), (1b), (1c), and (1d)) with
Now we present the main result for the state estimator of system ((1a), (1b), (1c), and (1d)).
Theorem 3. The system ((1a), (1b), (1c), and (1d)) is exponentially state reconstructible. Besides, a suitable state estimator is given bywhere denotes the inverse function of . In this case, the guaranteed exponential convergence rate is given by .
Proof. From ((1a), (1b), (1c), (1d), (3a), (3b), (3c), and (3d)) with it can be readily obtained that It follows that As a result, we conclude that This completes the proof.
By Theorem 3 with ((3a), (3b), (3c), and (3d)), it is straightforward to obtain the following result.
Corollary 4. The chaotic Liu system ((3a), (3b), (3c), and (3d)) is exponentially state reconstructible. Besides, a suitable state estimator is given byIn this case, the guaranteed exponential convergence rate is given by .
3. Application with Numerical Simulations
For any information vector in the transmitter system, the objective of secure communication system is to recover the message in the receiver system. Let us consider the following chaotic secure communication system and the proposed scheme is illustrated in Figure 1.
Transmitter. One has Receiver. One has where , , is the information vector, , is the signal recovered from , with and . By Theorem 3 and Corollary 4 with ((5a), (5b), (5c)–(10a), (10b), and (10c)), one can see that This implies that one can recover the message in the receiver system, with the guaranteed exponential convergence rate . In brief, the synchronization of signals and for the proposed chaotic secure communication (11a), (11b), (11c), (11d), and (11e) and (12a), (12b), (12c), and (12d) can always be achieved with the guaranteed convergence rate
With, for example, , , two electronic circuits, shown in Figures 2 and 3, have been designed and built to realize the system ((3a), (3b), (3c), and (3d)) and ((10a), (10b), and (10c)), respectively. Besides, the recovered message and the error signal are depicted in Figures 4 and 5, respectively, which clearly indicates that the real message is recovered after 2.5 seconds.
4. Conclusion
In this paper, the generalized Liu system has been firstly introduced and the state observation problem of such a system has been investigated. A simple state estimator for the generalized Liu system has been developed to guarantee the global exponential stability of the resulting error system. Applications of proposed state estimator scheme to chaotic secure communication, implementation of electronic circuits, and numerical simulations have also been provided to illustrate the practicability and effectiveness of the main results. Meanwhile, we have shown that the guaranteed exponential convergence rate of the proposed state estimator and that of the proposed chaotic secure communication can be precisely calculated. However, the state estimator design for more general uncertain Liu system still remains unanswered. This constitutes an interesting future research problem.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author thanks the National Science Council of Republic of China for supporting this work under Grant NSC-102-2221-E-214-043. The author also thanks the I-Shou University for supporting this work under Grant ISU102-04-07. Besides, the author wishes to thank the anonymous reviewers for providing constructive suggestions.