Anti-controlling Hopf bifurcation in a type of centrifugal governor system

Abstract

Anti-controlling Hopf bifurcation is considered as one way to design Hopf limit circle into a dynamical system, and the oscillatory behavior of Hopf limit circle can be beneficial in many practical applications such as mixing, low-energy navigation control and fault diagnosis. In this paper, the feedback control problem of designing Hopf bifurcation in a centrifugal governor system is addressed. A feedback control method is proposed to achieve three aspects of controlling problem including existence, stability, and adjusting amplitude and frequency of the limit cycle to be designed. An explicit criterion of Hopf bifurcation including eigenvalue assignment and transversality conditions, without using eigenvalue computation, is utilized to derive the linear gains responsible for control of the bifurcation existence. The center manifold theory and normal form reduction is utilized to derive the nonlinear gains responsible for control of the stability of the created limit circle. The expressions of the approximate amplitude and frequency of the limit cycle are developed to derive the nonlinear gains responsible for controls of amplitude and frequency of the limit cycle. Numerical simulations for a centrifugal governor system show that Hopf limit cycle with desired properties can be created at any a pre-specified parameter point.