Decoupling Optimal Controllers

Decoupling is a way to decompose a complex system into non-interacting subsystems. In fact, certain applications necessitate controlling independently different parts of the system. Even if this is not required, the absence of interaction can significantly simplify the synthesis of the desired control laws. The basic form of decoupling into single-input single-output subsystems, often referred to as the diagonal decoupling, was posed by Voznesenskij in terms of transfer function matrices. A more general form of decoupling into multi-input multi-output subsystems is known as the block decoupling and was introduced by Wonham and Morse. Using the state space approach, they solved the problem by static as well as dynamic state feedback. A comeback of the transfer function methods in the study of block decoupling is due to Hautus and Heymann, Kučera, Desoer and Gündeş, and Lee and Bongiorno. A dynamic state feedback was shown to be equivalent with combined dynamic output feedback and feedforward reference compensation, often referred to as a two-degree-of-freedom controller. To address stability issues, the Youla-Kučera parameterization of all stabilizing controllers was invoked. The two-degree-of-freedom controller structure is ideally suited to decoupling since stability and non-interaction can be treated as two independent constraints. This is not true for a pure feedback, or a one-degree-of-freedom controller. This paper adopts the two-degree-of-freedom controller structure. The class of all such controllers that decouple and stabilize the system is determined in parametric form and the parameter is used to obtain the H2-optimal controller. The solution is simple and direct.