Closed-form solutions to a class of H∞-optimization problems

Abstract

A closed-form solution is presented to the H ∞ sub-optimal control problem via state or output feedback. The problem formulation is general and does not impose any restrictions on feedthrough terms between the control input and the controlled output or between the disturbance input and the measurement output variables. It is, however, required that both the transfer function from the disturbance input to the measurement output and the transfer function from the control input to the controlled output have no invariant zeros on the jw axis and that they satisfy certain geometric conditions. For the same class of systems, the conditions under which the H ∞ optimal control problem, via state feedback, has a solution are given. Moreover, explicit expressions for the optimal solutions are provided. Finally, the pole/zero cancellations in the closed-loop system resulting from the H ∞ optimal or sub-optimal state or output feedback are examined.