Stability monitoring and control of nonlinear systems susceptible to oscillatory bifurcations

Stability monitoring and control of nonlinear systems subject to oscillatory bifurcations are addressed in this thesis. The two subjects of monitoring and control are viewed and treated as closely related but different problems.;Stability monitoring in the context of this dissertation entails techniques for automatic detection of impending instability in uncertain or otherwise complex systems. Two approaches are taken to address this problem. In the first approach sensor and actuator placement is optimized to give the best possible monitoring performance for a dynamical mode of interest. This is applied to detect destabilization of modes that are likely to cause bifurcation.;In the second approach systems that are susceptible to subcritical bifurcation are considered and periodic excitation signals are used. A monitoring system that detects impending subcritical bifurcation without the need for modal estimation is designed. Moreover, the shift in the parameter value where the bifurcation occurs is computed by using the method of averaging. The results are tested numerically on a second order system of van der Pol type. The monitoring system is then used to trigger a preventive control action moving the system away from the stability boundary and a catastrophic bifurcation.;Next, local feedback control of the Neimark-Sacker bifurcation is considered. The thesis develops the stability and amplitude equations for the bifurcated invariant curves, and proceeds to apply these relationships to the design of nonlinear feedbacks. The results are tested numerically on the classical delayed logistic map undergoing supercritical Neimark-Sacker bifurcation to improve the stability of bifurcated closed curves. The nonlinear feedback control is shown to change direction of Neimark-Sacker bifurcation from subcritical to supercritical for a range of system parameters and for linear, quadratic and cubic un-modeled terms. (Abstract shortened by UMI.).