Robust Control of Bifurcating Nonlinear Systems with Applications

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This dissertation addresses issues in the robust control of nonlinear dynamic systems near points of bifurcation, with application to the feedback control of aircraft high angle-of- attack flight dynamics. Specifically, we consider nonlinear control systems for which a nominal equilibrium point loses stability with slight variation of a distinguished system parameter (the "bifurcation parameter"). At such a loss of stability, various static and dynamic bifurcations may occur. These bifurcations often entail the emergence from the nominal equilibrium of new equilibrium points or of periodic solutions. The control laws sought in this work are intended to achieve certain goals related to the stability and/or amplitude of the bifurcated solutions. An important contribution of this dissertation is the introduction of the so-called "washout filters" into the control of systems undergoing bifurcations. These filters have been used for some time in certain practical control systems. They facilitate attainment of a degree of robustness of the system operating point to control actions and to uncertainty. Here, washout filter-aided feedback stabilization of nonlinear systems is studied in a general framework. Moreover, washout filters are employed in the feedback control of bifurcating systems. Several critical cases associated with bifurcations are considered. These include cases in which stability is lost through a zero eigenvalue, a pair of pure imaginary eigenvalues, two zero eigenvalues, and two pairs of pure imaginary eigenvalues. Robustness estimates are given for the achieved stabilization.

The foregoing analytical work is complemented with a thorough control study of nonlinear models for the high angle-of-attack lingitudinal flight dynamics of an F-8 Crusader aircraft. In this application, we demonstrate the superiority of washout filters in extending the stable high angle-of-attach flight regime. Also, we demonstrate the robustness of the control algorithm by using a fixed controller to stabilize twelve different Hopf bifurcation points in six different aircraft dynamic models. The numerical work employs state-of-the-art software packages for bifurcation analysis.