Feedback control of limit cycle amplitudes

A bifurcation control problem of modifying the amplitudes of limit cycles via feedback is studied. A graphical approach reminiscent to the familiar describing function method is developed for validating the harmonic balance approximations of both the amplitude and the frequency of the system oscillatory outputs, starting from the Hopf bifurcation mechanism. The second, fourth, and sixth-order harmonic balance approximations provide a sequential graphical testing for the convergence of the oscillatory outputs, thereby yielding an accurate approximation of the desired limit cycles of small amplitudes. The knowledge of degenerate Hopf bifurcations and the associate Poincare normal forms are useful for formulating the control objective: to capture small-amplitude oscillatory system outputs and to avoid unstable equilibria or other complicated limit sets. A power system example is included for illustration.