Robust frequency-shaped LQ control

Abstract

Certain frequency-shaped standard linear quadratic (LQ) controllers involving state feedback have attractive loop robustness properties. These robustness properties may vanish in passing to an output feedback scheme, as when state estimates are used instead of states. The known exception to date is when the plants are minimum phase and state estimation with loop recovery is used, as in LQG/LTR designs. In this paper, generalized frequency-shaped LQ theory is developed for plants with matrix fraction descriptions. For the case of minimum phase plants, a spectral factorization based construction procedure is given which leads to stable output feedback controllers that are optimal in an LQ sense. Among the class of such optimal LQ controllers, these achieve the attractive loop robustness properties of standard LQ state feedback designs (and of LQG/LTR output feedback designs). Perhaps surprisingly the controllers can be expressed in terms of the solution of a transfer function matrix Riccati equation. The LQ/Riccati theory of the paper specializes to known theory in the case when the plant outputs are the states and the indices are not frequency shaped.