Hopf bifurcations in time-delayed nonlinear feedback control systems

摘要

The Hopf bifurcation in a time-delayed nonlinear closed-loop control system is studied, where the nonlinear system has a linear dynamic plant represented by a feedforward transfer function and a nonlinear feedback controller represented by a Taylor series about an equilibrium operating point. The time delays occurred in the system are contained not only in the linear plant but also in the nonlinear feedback controller. The effective harmonic balance method is modified and applied to calculating the amplitude and frequency of the periodic solution emerging from the Hopf bifurcation of the dynamic system. Some new formulas containing up to an eighth-order harmonic balance approximation are derived, and two interesting examples are analyzed using the new results.