synchronization of time-delayed chaotic systems
Introduction
During the last decades, chaotic systems have attracted a lot of interests. Many fundamental characteristics have been found in a chaotic system, such as excessive sensitivity to initial conditions and fractal properties of the motion in phase space. In particular, chaos synchronization, first proposed by Fujisaka and Yamada in 1983 [1], did not received great attention until 1990 [2]. From then on, chaos synchronization has been extensively investigated due to the potential application in various fields such as biology, economics, signal generator design, secure communication, and so on [3], [4], [5], [6]. Based on various control theories, a number of synchronization schemes have been proposed in the literature [7], [8], [9], [10], [11], [12], [13], [14], [15].
In real physical systems, some noise or disturbances always exist that may cause instability and poor performance. Therefore, the effect of the noises or disturbances must be also reduced in synchronization process for chaotic systems. In this regards, recently, Hou et al. [16] firstly adopted the control concept [17], [18] to reduce the effect of the disturbance for chaotic synchronization problem of a class of chaotic systems. On the other hand, since Mackey and Glass [19] first found chaos in time-delay system, there has been increasing interest in time-delay chaotic systems [20], [21]. The synchronization problem for time-delayed chaotic systems is also investigated by some researchers [22], [23], [24]. However, the synchronization for time-delayed chaotic systems with disturbances has not been studied yet.
In this paper, we consider the problem of chaos synchronization for time-delayed chaotic system with disturbance. Dynamic feedback controller for the synchronization between derive and response systems is proposed. By the feedback control scheme, the closed-loop error system is asymptotically stable and the -norm from the disturbance to controlled output is reduced to a prescribed level. Based on the Lyapunov method and LMI framework, an existence criterion for such controller is represented in terms of LMIs. The LMIs can be easily solved by various convex optimization algorithms developed recently [25].
Notation: denotes the n-dimensional Euclidean space, and is the set of real matrix. represents the absolute value. refers to the Euclidean vector norm and the induced matrix norm. I denotes the identity matrix with appropriate dimension. means the transpose of the matrix A. We denote the positive (non-negative) definiteness of by and the negative (non-positive) definiteness of A by . means is a positive definite matrix. denotes the block diagonal matrix. represents the elements below the main diagonal of a symmetric matrix. denotes the smallest eigenvalue of A.
Section snippets
Problem formulation
Consider a class of time-delayed chaotic systems described bywhere is the state variable, is the output, the matrices and are known constant matrices, and is a nonlinear function vector satisfying the global Lipschitz condition: i.e,for a known constant matrix F.
The synchronization problem of system (1) is considered using the drive-response configuration [2]. This is, if
Main result
The main result for achieving synchronization is stated in the following theorem. Theorem 1 For given , , , and , there exist a dynamic feedback controller (6) for the error system (5), if there exist any matrices , a positive scalar , and positive-definite matrices , S and Y satisfying the following LMIs:andwhere
Conclusions
The problem of synchronization for a class of time-delayed chaotic systems with disturbances has been presented. Based on Lyapunov theory and LMI formulation, A dynamic feedback control scheme has proposed to guarantee synchronization for drive and response systems and reduce the -norm from the disturbance input to the output error within a prescribed level. Furthermore, a model of Hopfield neural network is given to illustrate the effectiveness of the proposed control scheme.
References (26)
- et al.
Phys. Lett. A
(1991) - et al.
Chaos, Solitons Fractals
(2004) - et al.
Chaos, Solitons Fractals
(2002) - et al.
Chaos, Solitons Fractals
(2005) - et al.
Phys. Lett. A
(2007) Chaos, Solitons Fractals
(2005)Chaos, Solitons Fractals
(2005)- et al.
Chaos, Solitons Fractals
(2007) - et al.
Phyisca A
(2007) - et al.
Physica D
(1998)
Chaos, Solitons Fractals
Phys. Lett. A
Phys. Lett. A
Cited by (90)
Chaotic attitude synchronization and anti-synchronization of master-slave satellites using a robust fixed-time adaptive controller
2022, Chaos, Solitons and FractalsAsynchronous mixed H<inf>∞</inf> and passive control for fuzzy singular delayed Markovian jump system via hidden Markovian model mechanism
2022, Applied Mathematics and ComputationNeural adaptive learning synchronization of second-order uncertain chaotic systems with prescribed performance guarantees
2021, Chaos, Solitons and FractalsSynchronization of second-order chaotic systems with uncertainties and disturbances using fixed-time adaptive sliding mode control
2021, Chaos, Solitons and FractalsAdaptive lag-synchronization of two nonidentical time-delayed chaotic systems in the presence of external disturbances subjected to input nonlinearity
2020, Control Strategy for Time-Delay Systems: Part II: Engineering Applications