Some Techniques in the Control of Dynamic Systems with Periodically Varying Coefficients
Type of DegreeDissertation
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The goal of this dissertation is to develop control system design methods for dynamic systems with time periodic coefficients. These systems arise naturally in many branches of science and engineering. As a rule, time periodic ordinary differential equations can not be integrated explicitly in general cases. Therefore, even for the linear time periodic systems, the controller design problem is much more challenging than its time invariant counterpart. In this dissertation, several strategies are developed for the controller design problem associated with dynamic systems with time periodic coefficients. First, a linear feedback control system design techniques is developed for stabilizing the linear systems. The Floquet multipliers of the closed loop system can be assigned to desired locations in the complex plane via this method. This method is also used to design robust controllers for linear time periodic systems with parameter uncertainty. For the nonlinear time periodic system, we design a linear feedback controller such that the local stability of the closed system is guaranteed. Second, the feedback linearization problem of nonlinear time periodic systems is addressed. The classical feedback linearization theory is generalized to the time periodic case. Two approaches are developed for the feedback linearization of such systems. The first approach is based upon the time dependent Lie derivative and the second method utilizes the time dependent Poincaré normal form theory. Finally, observer design methods for time periodic systems are developed. It is shown that observer and feedback control design problems are dual to each other. The symbolic control design method is applied to design identity observers for linear time periodic systems. For nonlinear time periodic systems, an observer design method is developed based on the time invariant manifold theory. A fast computation method is also suggested to obtain the Fourier-Taylor series expression for the invariant manifold of the nonlinear time periodic system. Then the identity observers for nonlinear time periodic systems are constructed.