Control of Bifurcations and Routes to Chaos in Dynamical Systems

Abstract

This dissertation addresses issues in the control of nonlinear instabilities in high performance engineering systems. Specifically, we consider nonlinear control systems near the limits of their operating envelope. These are highly nonlinear situations occurring in the high-performance operation of a wide variety of systems. Such systems tend to exhibit nonlinear stabilities in terms of a jump to a new low-performance operating point, oscillatory behavior, chaotic behavior or system collapse in the absence of appropriate control action. Such situations necessitate the study of controlling nonlinear phenomena such as bifurcations and chaos.<P>A new approach to the control of chaotic dynamical systems is introduced, namely control of routes to chaos. The theme is to design feedback control laws which ensure a sufficient degree of stability for a primary bifurcation in the routes to chaos. A thermal convection loop is used as a vehicle to illustrate the idea. Moreover, as the period doubling cascade is one of the most famous routes to chaos, the stabilization of period doubling bifurcations for general n- dimensional discrete-time nonlinear systems is investigated. The technique presented here affords considerable flexibility in terms of achievable behavior of the nonlinear system over a range of parameter values.<P>One contribution is the modeling, analysis and control of voltage collapse in electric power systems. a new mechanism of voltage collapse is suggested based on the framework of catastrophic bifurcations. Bifurcation control laws are designed to control these nonlinear phenomena at the inception of voltage collapse. The control laws are shown to result in improved performance of the system for a greater range of parameter values.<P>Another important application considered is the stall phenomenon in axial flow compressors. A combination of bifurcation analysis and nonlinear control is used to study the dynamics and active control of rotating stall in an axial flow compressor model. Both smooth and nonsmooth feedbacks are considered. Successful experimental verifications have been reported on these results.