Synchronization Between Two Novel Different Chaotic Systems with Unknown Parameters
Considered the assumption that the control direction is known, a kind of chaotic system with different driven and response structures is studied. Based on Lyapunov stability theorem, a kind of double integral sliding mode synchronization controller is des
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Synchronization Between Two Novel Different Chaotic Systems with Unknown Parameters Hui Wang, Jinyong Yu, and Junwei Lei
Abstract Considered the assumption that the control direction is known, a kind of chaotic system with different driven and response structures is studied. Based on Lyapunov stability theorem, a kind of double integral sliding mode synchronization controller is designed. Also, the adaptive strategy is introduced to solve the full unknown parameters and uncertain nonlinear functions. Furthermore, the situation for chaotic system with input nonlinearity is researched. Finally, a kind of super chaotic system is taken as an example and numerical simulation is done to testify the rightness of the proposed method. Keywords Double integral • Sliding mode • Chaos • Synchronization • Uncertainty
159.1
Introduction
Input nonlinearity with known control direction is a simple situation for the study of input nonlinearity. But it has being researched widely by scientists because this question is very meaningful and it appears in real control practice very frequently [1–8].
H. Wang Naval Equipment Department of PLA, Equipment Purchase Center, Yantai 264001, China e-mail: [email protected] J. Yu • J. Lei (*) Department of Control Engineering, Naval Aeronautical and Astronautical University, Yantai 264001, China e-mail: [email protected]; [email protected] S. Zhong (ed.), Proceedings of the 2012 International Conference on Cybernetics 1249 and Informatics, Lecture Notes in Electrical Engineering 163, DOI 10.1007/978-1-4614-3872-4_159, # Springer Science+Business Media New York 2014
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H. Wang et al.
A three order system with single input and single output is researched in [1] and the control direction is assumed to be known and the absolute value is uncertain. So a virtual coefficient is designed with adaptive method. In this paper, some improvements have been made such that it can be applied in the synchronization of multi-input and multi-output chaotic systems.
159.2
Problem Description
Consider the below driven system and response system, where both the driven system and response system contains unknown parameters and uncertain nonlinear function. The structure of driven system is different from response system. Without loss of generality, assume the dimension of driven system is higher than its of response system and assume the dimension of response system is N and the dimension of driven system is N + R, so it can be written as follows: For driven system, it holds x_p ¼ fpx ðxÞ þ Fpx ðxÞθpx þ Δpx ðx; tÞ
(159.1)
For the else R dimension system, it has x_ r ¼ frx ðxÞ þ Frx ðxÞθrx þ Δrx ðx; tÞ
(159.2)
For the response system y_ ¼ fy ðyÞ þ Fy ðyÞθy þ Δy ðy; tÞ þ bðuÞ
(159.3)
Where θ is unknown parameter, Δ is uncertain nonlinear functions and bðuÞ is uncertain nonlinear input function, dðtÞ is outer disturbance. _ The control objective is to design a control u ¼ uðx; y; ^θpx ; q^px ; θ^y ; q^y ; b^i Þ, θ^px ¼ _ _ f ðx; y; ^θpx Þ, q^_ px ¼ f ðx; y; q^px Þ, ^ θy ¼ f ðx; y; ^ θy Þ, q^_ y ¼
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