Static and adaptive feedback control for synchronization of different chaotic oscillators with mutually Lipschitz nonlinearities

doi:10.1088/1674-1056/23/11/110502

中国物理B ›› 2014, Vol. 23 ›› Issue (11): 110502-110502.doi: 10.1088/1674-1056/23/11/110502

Static and adaptive feedback control for synchronization of different chaotic oscillators with mutually Lipschitz nonlinearities

Muhammad Riaz^{a b}, Muhammad Rehan^a, Keum-Shik Hong^c, Muhammad Ashraf^b, Haroon Ur Rasheed^a

a Department of Electrical Engineering, Pakistan Institute of Engineering and Applied Sciences (PIEAS), Islamabad, Pakistan;
b Department of Electronics Engineering, Mohammad Ali Jinnah University, Islamabad, Pakistan;
c Department of Cogno-Mechatronics Engineering and School of Mechanical Engineering, Pusan National University; 2 Busandaehak-ro, Geumjeong-gu, Busan 609-735, Republic of Korea

收稿日期:2014-03-18 修回日期:2014-05-05 出版日期:2014-11-15 发布日期:2014-11-15
基金资助:
Project supported by the Higher Education Commission of Pakistan through the Indigenous 5000 Ph.D. Fellowship Program (Phase Ⅱ, Batch Ⅱ).

Static and adaptive feedback control for synchronization of different chaotic oscillators with mutually Lipschitz nonlinearities

Muhammad Riaz^{a b}, Muhammad Rehan^a, Keum-Shik Hong^c, Muhammad Ashraf^b, Haroon Ur Rasheed^a

a Department of Electrical Engineering, Pakistan Institute of Engineering and Applied Sciences (PIEAS), Islamabad, Pakistan;
b Department of Electronics Engineering, Mohammad Ali Jinnah University, Islamabad, Pakistan;
c Department of Cogno-Mechatronics Engineering and School of Mechanical Engineering, Pusan National University; 2 Busandaehak-ro, Geumjeong-gu, Busan 609-735, Republic of Korea

Received:2014-03-18 Revised:2014-05-05 Online:2014-11-15 Published:2014-11-15
Contact: Muhammad Rehan E-mail:rehanqau@gmail.com
Supported by:
Project supported by the Higher Education Commission of Pakistan through the Indigenous 5000 Ph.D. Fellowship Program (Phase Ⅱ, Batch Ⅱ).

摘要/Abstract

摘要： This paper addresses the control law design for synchronization of two different chaotic oscillators with mutually Lipschitz nonlinearities. For analysis of the properties of two different nonlinearities, an advanced mutually Lipschitz condition is proposed. This mutually Lipschitz condition is more general than the traditional Lipschitz condition. Unlike the latter, it can be used for the design of a feedback controller for synchronization of chaotic oscillators of different dynamics. It is shown that any two different Lipschitz nonlinearities always satisfy the mutually Lipschitz condition. Applying the mutually Lipschitz condition, a quadratic Lyapunov function and uniformly ultimately bounded stability, easily designable and implementable robust control strategies utilizing algebraic Riccati equation and linear matrix inequalities, are derived for synchronization of two distinct chaotic oscillators. Furthermore, a novel adaptive control scheme for mutually Lipschitz chaotic systems is established by addressing the issue of adaptive cancellation of unknown mismatch between the dynamics of different chaotic systems. The proposed control technique is numerically tested for synchronization of two different chaotic Chua's circuits and for obtaining identical behavior between the modified Chua's circuit and the Rössler system.

关键词: synchronization, mutually Lipschitz nonlinearity, adaptive control system, control theory and feedback

Abstract: This paper addresses the control law design for synchronization of two different chaotic oscillators with mutually Lipschitz nonlinearities. For analysis of the properties of two different nonlinearities, an advanced mutually Lipschitz condition is proposed. This mutually Lipschitz condition is more general than the traditional Lipschitz condition. Unlike the latter, it can be used for the design of a feedback controller for synchronization of chaotic oscillators of different dynamics. It is shown that any two different Lipschitz nonlinearities always satisfy the mutually Lipschitz condition. Applying the mutually Lipschitz condition, a quadratic Lyapunov function and uniformly ultimately bounded stability, easily designable and implementable robust control strategies utilizing algebraic Riccati equation and linear matrix inequalities, are derived for synchronization of two distinct chaotic oscillators. Furthermore, a novel adaptive control scheme for mutually Lipschitz chaotic systems is established by addressing the issue of adaptive cancellation of unknown mismatch between the dynamics of different chaotic systems. The proposed control technique is numerically tested for synchronization of two different chaotic Chua's circuits and for obtaining identical behavior between the modified Chua's circuit and the Rössler system.

Key words: control theory and feedback, synchronization, mutually Lipschitz nonlinearity, adaptive control system

中图分类号: (Nonlinear dynamics and chaos)

05.45.-a

05.45.Gg (Control of chaos, applications of chaos) 05.45.Xt (Synchronization; coupled oscillators)

Muhammad Riaz, Muhammad Rehan, Keum-Shik Hong, Muhammad Ashraf, Haroon Ur Rasheed. Static and adaptive feedback control for synchronization of different chaotic oscillators with mutually Lipschitz nonlinearities[J]. 中国物理B, 2014, 23(11): 110502-110502.

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Static and adaptive feedback control for synchronization of different chaotic oscillators with mutually Lipschitz nonlinearities

Static and adaptive feedback control for synchronization of different chaotic oscillators with mutually Lipschitz nonlinearities

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