Some conditions for convexifying static H_∞ control problems

In this paper, we show that a set of static controllers satisfying a certain level of H_∞ performance becomes convex when the underlying generalized plant satisfy several structural conditions. More precisely, we characterize such static H_∞ controllers by an LMI with the controller parameters being kept directly as decision variables. The conditions on the generalized plant are not too strict as illustrated by the fact that a sort of mixed sensitivity problems indeed satisfies these conditions. In addition, for the generalized plant of interest, we prove that full-order dynamical H_∞ controllers can be characterized by an LMI with simple change of variables. In stark contrast to known LMI formulations, the change of variables does not involve coefficient matrices of the generalized plant. This property is promising when dealing with a whole variety of robust control problems. As an illustration, the real μ synthesis problem is discussed.