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General robust stability controller design method for a class of hopf bifurcation systems

Lu Jin-Bo Hou Xiao-Rong Luo Min

General robust stability controller design method for a class of hopf bifurcation systems

Lu Jin-Bo, Hou Xiao-Rong, Luo Min
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  • Received Date:  24 October 2015
  • Accepted Date:  10 December 2015
  • Published Online:  05 March 2016

General robust stability controller design method for a class of hopf bifurcation systems

    Corresponding author: Hou Xiao-Rong,
  • 1. School of Energy Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant No. 61374001).

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.

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