| There are a variety of tools for computing stabilizing feedback control laws for non-linear systems. The difficulty is that these tools usually do not take into account the performance of the closed-loop systems. On the other hand, optimal control theory gives guaranteed closed loop performance but the resulting problem is difficult to solve for general nonlinear systems. While there may be many feedback control laws that provide adequate performance, optimal control theory insists on the one control that provides peak performance. In this thesis we bypass the difficulties inherent in the optimal control problem by developing a design algorithm to improve the closed-loop performance of arbitrary, stabilizing feedback control laws.;The problem of improving the closed-loop performance of a stabilizing control reduces to solving a first-order, linear partial differential equation called the Generalized-Hamilton-Jacobi-Bellman (GHJB) equation. An interesting fact is that when the process is iterated, the solution to the GHJB equation converges uniformly to the solution of the Hamilton-Jacobi-Bellman equation which solves the optimal control problem. The main contribution of the thesis is to show that Galerkin's method can be used to find a solution to the GHJB equation and that the resulting control laws are stable, and that when the process is iterated, it still converges to the optimal control. The thesis therefore solves an important problem that has remained open for over thirty years, i.e., it shows how to find a uniform approximation to the Hamilton-Jacobi-Bellman equation such that the approximate controls are still stable on a specified set.;The method developed in the thesis is a practical, off-line algorithm that computes closed-form, feedback control laws with guaranteed performance. Our algorithm is the first practical method that computes arbitrarily close uniform approximations to the optimal control, while simultaneously guaranteeing closed-loop asymptotic stability. |
