Optimal Control

Introduction

Optimal control is one of the main techniques of modern control design, as it has been for many years. The linear-quadratic theory of optimal control design is well established and has various forms including the receding horizon approach for a robust, easily implementable variation of the theory. It is also useful in H ^∞ control in the well-known state-space game theoretic formulation. Obtaining the ‘best’ controller in any given circumstance is clearly important, but for general nonlinear systems, one is led to the solution of an extremely difficult (in general, non-smooth) partial differential equation. This makes the existing general nonlinear theory very difficult to apply.

In this chapter we shall show how to use the iteration technique developed above to solve nonlinear optimal control problems. In the next section we shall outline the classical linear quadratic regulator theory and derive the optimal feedback control in terms of the solution of a Riccati equation.We shall also indicate the modifications necessary to solve the linear tracking problem. The generalisation to nonlinear systems will be given in Section 6.3 and some examples will be presented in Section 6.4. Some comments on viscosity solutions of the Hamilton-Jacobi-Bellman (HJB) equation and the optimality of the solution will be given in Section 6.5.