Topics in Stabilization of Nonlinear Control Systems

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9411145 Schwartz The control of systems modeled by nonlinear differential equations is an important topic to study because of the prevalence of these models in manufacturing, chemical process, automotive, power, and aerospace industries. Normal form techniques have been successfully applied to simplify the state space expression of nonlinear ordinary differential equations. Here we propose to study the stabilization of nonlinear control systems using normal form techniques. Thus, we are interested in systems whose linearizations possess at least one uncontrollable critical mode. From some preliminary analysis, it can be inferred that there is a great advantage to using the normal form as a method to reduce the dimension of the system whose stability is to be analyzed. We propose to study this dimensional advantage in detail as it may be applied to various problems. One question of interest is to understand and characterize the decoupling of the critical and stable modes of a nonlinear control system in normal form. Another line of proposed study is to examine how the normal form characterizes the Lyapunov function associated with the nonlinear system. We would also estimate the region of attraction of stability for nonlinear systems using the normal form. We will study the relationship between the normal form and the center manifold techniques which have been previously used in stabilization analyses. Full and Partial feedback linearization, and approximate feedback linearization will also be studied, to see if these objectives can be achieved more simply using the normal form. Time scale techniques have been applied to control systems in order to simplify analysis and obtain stability results. A further direction of the research herein proposed will be to derive a deeper understanding of the relationship between stability and time scale, and the relevance of time scale techniques to existing center manifold and normal form techniques in stabilization of nonlinear control systems. ***