On the Optimal Control Systems with Multiple Controllers

Abstract

The synthesis problem of a control system with multiple controllers is one of interesting design problems because the larger the dimension of the plant becomes the more difficult it becomes to control the plant with a single controller. In this paper we consider the problem of finding a set of internally stabilizing controllers which is globally equivalent to a single internally stabilizing controller such that the state of the plant x(t) is driven to the origin as t → ∞ and a certain performance index on the state is minimized. As the simplest case of control problems with multiple controllers, the control problem for the system with 2 controllers is considered. Under the assumptions of reachability and observability, it is shown that the decoupling problem of a Linear Quadratic Regulator (LQR) can be formulated and the solvability condition is derived for a set of admissible controllers. For the more general system which includes the system does not satisfy the solvability condition of the decoupling problem of a Linear Quadratic Regulator, we will introduce a set of internally stabilizing controller which is almost equivalent to a single internally stabilizing controller via internally balanced state space representation.