A New Approach to the Solution of Linear Optimal Control Problems

A simplified approach to the solution of linear optimal control problems with quadratic performance indexes is described in this paper. The phase-variable canonical form is used to develop a new type of optimal system equivalence. This concept leads to a substantial simplification of the matrix Riccati equation. The simplified matrix Riccati equation is of the same form for any problem of a given order, say, n, and contains only n nonzero forcing functions. That is, it always corresponds to a set of constant-coefficient scalar differential equations; in various nth-order problems the n nonzero forcing functions and the terminal conditions simply assume different forms. In a very strong sense, this simplified matrix Riccati equation is the simplest possible Riccati equation arising from optimization problems. The method is developed for general time-varying systems with finite terminal time. It is developed also for the important special case of time-invariant systems with infinite terminal time.