新分数阶混沌系统的电路设计和同步控制

New fractional chaotic system circuit design and synchronization control

针对一个新的仅含一个非线性项的三维分数阶混沌系统,首先通过分析系统的相轨迹图、李雅普诺夫指数谱、功率谱以及庞加莱截面,验证了系统的混沌特性;其次研究了系统平衡点的稳定性以及系统关于分数阶次和参数的分岔图,结果表明系统具有丰富的混沌特性;然后设计出系统的分数阶混沌电路并在软件Multisim中进行模拟实验,实验结果验证了新分数阶混沌系统实现的可能性;最后研究了带有扰动的、不确定参数的分数阶系统的同步问题,设计了基于自适应的滑模控制器、未知扰动和不确定参数的自适应律,实现了在未知扰动上界未知和参数不确定情况下的系统的同步控制.

For a new three-dimensional fractional chaotic system with only one nonlinear term,chaotic characteristics of the system are verified firstly by analyzing phase trajectory,Lyapunov exponent spectrum,power spectrum and Poincare section.Secondly,the stability of the equilibrium point of the system and the bifurcation diagram of the system with respect to fractional order and parameters are studied.The bifurcation diagram shows that the system has abundant chaotic characteristics.Then,a fractional-order chaotic circuit of the system is designed and simulated in software Multisim.The simulation results verify the possibility of realizing the new fractional chaotic system.Finally,the synchronization problem of fractional order systems with disturbances and uncertain parameters is studied.A sliding mode controller based on adaptive control is then designed,and the adaptive laws of unknown disturbances and uncertain parameters are explained reasonably.The synchronization of the system can be realized in the case of both unknown upper bound of the disturbances and parameter uncertainty.