H₂-optimal control of linear systems with tracking/disturbance rejection constraints

Abstract

This paper considers the design of two-degree-of-freedom, linear multivariable controllers for four-block matrix transfer functions with objectives pertaining to both random and deterministic signals. The former are taken into account in the cost functional whereas the latter entail that the controller is constrained to solve a servomechanism problem involving a class of reference signals and two classes of disturbances. The cost functional is set up by adding quadratic terms to the usual LQG/Wiener-Hopf cost in such a way as to ensure the existence of (unique) solutions even when the signals to be tracked/rejected, as well as models, have poles on the imaginary axis. The solutions of the servomechanism problem are explicitly parametrized by free, stable, rational and proper parameters under two sets of assumptions: one which is necessary and sufficient for the existence of solutions and another, stronger set which avoids Kronecker products and the associated higher dimensional factorizations involved in the first case. On the basis of this parametrization the servomechanism constraint is incorporated into an unconstrained problem which is then quite explicitly solved.