Control of nonlinear systems

Abstract

Dynamic programming is employed to obtain a solution to the problem of controlling a nonlinear system in an optimal fashion, subject to a quadratic performance index. The technique sued is similar to that given by Merriam and Kalman for linear systems. For some special nonlinear systems, the solution can be computed by direct application of this technique. As an example, the optimal control system for a freely spinning body is determined. For more general nonlinear systems, the solution cannot be obtained directly. However, it is possible to obtain a solution indirectly. this is done by first linearizing the vector-state equations representing the nonlinear system. Next, dynamic programming is used to obtain an approximate solution based on the linearized state equations. Then an iterative procedure for improving the solution is presented. It can be shown that if the iterative procedure converges, it converges to the exact solution of the optimal nonlinear control problem. Computer example problems are given to illustrate the method, and to indicate the convergence that is usually achieved. In addition, the performance of the optimal control system is compared with the performance of a simple sub-optimal control system for some of the example problems given.