Discrete-time Regulator of an Infinite-dimensional System

Abstract

The recurrence relations which are satisfied by optimal controllers and state estimators are given in Part I. These relations are extensions of the work of Tou ( 1, 2 . The extensions could be heuristically achieved by changing finite-dimensional matrices with infinite-dimensional ones (operators) and conjugate matrices with normed conjugate operators. The definitions of Falb ( 3 ) are used as the covariance operator of infinite- dimensional random vectors. By using the functional analysis approach, it is shown that the relations obtained heuristically are correct. In Part II, the infinite-dimensional system is approximated by the finite-dimensional system in order to calculate the suboptimal regulator. If the infinite-dimensional system is approximated by the n-dimensional system, the optimal regulator of the latter can be easily calculated by Tou's formulae. The solution of this may not be necessarily a suboptimal regulator, because there remains an important question as to whether this regulator tends to the optimal one of the infinite-dimensional system when n tends to infinity. This convergence is shown in the operator topology by making use of the facts that the suboptimal feedback controllers and state estimators are uniformly bounded in n. By this convergence the suboptimal regulator can be obtained as exactly as desired by choosing n appropriately large.