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This thesis focuses on practical methods for constructing robust nonlinear control systems. In general, the development of such control systems is characterized by the solution to one or more Hamilton-Jacobi partial differential equations (HJE). However, no general analytical solution has yet been obtained to solve this optimization problem. Solutions have thus far only been obtained under certain conditions. Therefore, the first significant contribution of this thesis is a method for obtaining analytical expressions for approximate solutions to a common form of HJE (under certain assumptions regarding the class of nonlinear systems used). Additionally, modern state space controller synthesis techniques typically result in state estimators of equal or greater dimension than the plant model. However, it is often desirable, or even necessary, to approximate these controllers by models of lower state dimension. Presently, methods for developing nonlinear state balancing transformations are not very well understood. Therefore, the second significant contribution of this thesis is a proper algorithm for the application of state balancing techniques to nonlinear control systems and the subsequent reduction of the number of control states. The method to be developed for state balancing is based on the above framework for constructing analytical solutions to the HJE. In this thesis we will make use of three existing robust nonlinear control methods from the literature. These three methods have the advantage that they can all be constructed from solutions to a single form of HJE. Thus, by developing a method for obtaining analytical expressions for the solution to a single form of HJE, we are able to develop explicit polynomial solutions for each of these three control methods. Due to the difficulties associated with quantifying robustness and performance properties for nonlinear systems, the effectiveness of the three control methods considered shall be demonstrated via numerical simulations. The particular applications of interest to us here are space systems. First, we will consider the attitude control of a single spacecraft. Second, we examine the problem of formation flying control for a pair of spacecraft. The third and final problem we consider is the control of a nonlinear mass-spring chain.

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