Closed-loop optimal control of discrete systems

Abstract

In this chapter, first in Section 7.1, a method is described to analyse the singularly perturbed nonlinear difference equations for initial and boundary value problems. The approximate solution is obtained in the form of an outer series and a correction series. It is seen that considerable care has to be taken in formulating the equations for the boundary-layer correction series in the case of nonlinear equations. Then, in Section 7.2, the closed-loop optimal control problem is formulated, resulting in the singularly perturbed nonlinear matrix Riccati difference equation. It is seen that the degeneration (the process of suppressing a small parameter) affects some of the final conditions of the Riccati equation. In Section 7.3, a method is given to obtain approximate solutions in terms of an outer series and a terminal boundary-layer correction series. A method is also discussed in Section 7.4 for the important case of the steady-state solution of the matrix Riccati equation. The time-scale analysis of the regulator problem is also given. It is found that these methods, with the special feature of order reduction, offer considerable computational simpli city in evaluating the inverse of a matrix associated with the solution of the Riccati equation. Examples are given to illustrate these methods.