两个非线性系统Hopf分岔的状态反馈控制

THE STATE FEEDBACK HOPF BIFURCATION CONTROL OF TWO NONLINEAR SYSTEM

本文简单介绍了Hopf分岔理论 ,对于Vanderpol振子和一个发生亚临界Hopf分岔的二阶非线性系统分别设计了具有可变增益的状态反馈控制器 ,利用Lyapunov稳定性定理证明了在该控制器作用下 ,可以消除非线性系统的Hopf分岔现象 ,保证系统的渐近稳定 ,控制器与系统参数的变化范围无关 ,数值仿真也表明了这种控制方法的有效性。因此 ,对于发生Hopf分岔的非线性系统 ,可以通过设计合适的状态反馈控制器去消除Hopf分岔 ,维持系统的渐近稳定 ,而不仅是延迟分岔的出现 。

Hopf bifurcation theory was briefly presented. The state feedback Hopf bifurcation controllers with variable gains were designed for a Van der pol oscillator and a 2-D nonlinear system giving rise to sub-critical Hopf bifurcations. Lyapunov stability theorem is used to prove the validity of the controllers. The maximal characteristic of this state feedback Hopf bifurcation controller is that it is irrelevant to the parameter of the system and the controller can eliminate Hopf bifurcation and guarantee asymptotical stability. Simulation results show that the nonlinear system can obtain asymptotical stability. As a result, the state feedback can be designed to eliminate inherent Hopf bifurcation of a nonlinear system, yet not delaying bifurcation, modifying the shape or type of a bifurcation chain, or introducing a new bifurcation at a preferable parameter value to enlarge the system's stability range.