物理学报2016，Vol.65Issue(6)：41-51,11.DOI:10.7498/aps.65.060502
一类Hopf分岔系统的通用鲁棒稳定控制器设计方法∗
General robust stability controller design metho d for a class of Hopf bifurcation systems
摘要
Abstract
For the nonlinear Hopf bifurcation system, the change of bifurcation parameter has an important influence on the state of the system. In order to control the Hopf bifurcations of the nonlinear dynamic system, the parameter values of bifurcation points in the system need to be found out before controller designing. However, due to uncertainties of the system structure and parameters in the nonlinear system, or disturbance, it is difficult to determine the bifurcation point precisely. So it is a good way of designing a robust controller near the bifurcation point. Although, lots of works have discussed the robust control of a Hopf bifurcation in a nonlinear dynamic system, the solutions are not satisfactory and there are still many problems. The controller is always designed for some special system. Its structure is usually complex, not general, and the design process is complicated. And before controller design, the value of bifurcation point must be solved accurately. In this paper, a parametric robust stability controller design method is proposed for a class of polynomial form Hopf bifurcation systems. Using this method, it is not necessary to solve the exact values of the bifurcation parameter, it is only needed to determine the bifurcation parameter range. The designed controller includes a system state polynomial; its structure is general, simple and keeps the equilibrium of the original system unchanged. By using the Hurwitz criterion, the system stability constraints for bifurcation parameter boundaries are obtained at equilibrium, and they are described by algebraic inequalities. Cylindrical algebraic decomposition is applied to calculate the stability region of the controller parameters. And then, in the region, parameters of the robust controller can be calculated to make the dynamic system stable. In this paper, the Lorenz system without disturbance is used as an example to show the designing process of the method, and then the controller of the van der Pol oscillator system with disturbance is designed by this method as an engineering application. Simulations of the two systems are given to demonstrate that the proposed controller designing method can be effectively applied to the robust stability control of the Hopf bifurcation systems.关键词
鲁棒稳定控制器/Hopf分岔/参数化/柱形代数剖分Key words
robust stabilization controller/Hopf bifurcation/parametric/cylindrical algebraic decompo-sition引用本文复制引用
陆金波,侯晓荣,罗敏..一类Hopf分岔系统的通用鲁棒稳定控制器设计方法∗[J].物理学报,2016,65(6):41-51,11.基金项目
国家自然科学基金(批准号:61374001)资助的课题.* Project supported by the National Natural Science Foundation of China (Grant No.61374001) （批准号:61374001）
