Simultaneous stabilization with near-optimal H∞ performance

In this paper, we consider the use of linear periodically time-varying (LPTV) controllers for simultaneous stabilization and disturbance rejection. Here, we consider the case in which the control signal u does not appear directly in the output to be controlled (z) and the disturbance w does not appear directly in the measured output y. we prove that for every finite set of such plants for which the high-frequency gains (from u to y) satisfy a linear independence condition, we can design an LPTV controller which provides not only closed-loop stability but also near-optimal disturbance rejection in the Hinfinity. sense. The approach works as follows. We first design a (near-optimal) linear time-invariant (LTI) controller for each possible model. We then propose the following periodic control mechanism: during the first part of every period, we (linearly) estimate what the control signal would be if we knew the plant and if the corresponding LTI controller were applied; during the last part of the period, we apply this (suitably scaled) estimate. We end up with a stable LPTV controller, parametrized by the period T, which will provide simultaneous stability for modest values of T, and which will recover the near-optimal performance as T --> 0.