Monitoring and Control of Bifurcations Using Probe Signals

Abstract

Systems undergoing Hopf bifurcation are known to amplify nearly resonant perturbation signals. A lesser known fact is that such probe signals tend to also produce a shift in the parameter value where bifurcation occurs. In this paper, these rarely used phenomena are used as a basis for stability monitoring of systems that are susceptible to loss of stability through a Hopf bifurcation. The fact that the perturbation signals delay supercritical bifurcations and advance subcritical bifurcations is noted, and the amount of this shift is quantified analytically. This analysis is based on work by Gross that employs second order averaging. A monitoring system is developed that provides an early warning signal for subcritical Hopf bifurcation. Since subcritical bifurcations lead to large departures from normal operation, detection of an impending subcritical bifurcation is a valuable goal. The results are tested numerically on a second order system of van der Pol type. The monitoring system is further used to trigger a preventive control action moving the system away from the stability boundary and catastrophic bifurcation. The results are also applied to a power system model where the search for impending subcritical bifurcation is performed in a two dimensional bifurcation parameter space.