Optimal regulator for linear systems with delays in state and control — spectrum decomposition and prediction approach

Abstract

For systems with delays in state and control, a generalized linear-quadratic optimal control problem is formulated. The generalization exists in the state quadratic form of the cost functional, and the optimal solution is given as a state feedback form which requires a solution of the correspondingly generalized Riccati partial differential equation. It is shown first that the optimal closed-loop system satisfies the circle condition. Second it is shown that the generalized cost functional contains a special class of cost functionals for which the optimal control can be realized by calculating a finite number of open-loop poles of the system and solving a finite dimensional Riccati equation; it is also clarified that the optimal control law for such a special cost functional is composed of spectrum decomposition part and prediction part. Finally it is shown that the generalization of the cost functional makes it possible to discuss a pole shifting problem within the framework of the linear-quadratic optimal control problem.