Computing optimal fixed order H∞-synthesis values by matrix sum of squares relaxations

The computation of optimal H_∞ controllers with a prescribed order is important for their real-time implementation. This problem is well-known to be non-convex, and only algorithms that compute upper bounds on the global optimal value are known. We consider the applicability of matrixvalued sum-of-squares (sos) techniques for the computation of lower bounds. First we show that the size of the Linear Matrix Inequality (LMI) relaxations grow exponentially with the state dimension if we apply the sos-technique directly on the bounded-real lemma bilinear matrix inequality. We overcome this deficiency by two-fold sequential dualization: first we dualize in the variables that grow with the state dimension, which leads to a re-formulation of the fixed order synthesis problem into a robust analysis problem with the controller variables as parametric uncertainty. Second we dualize in the controller variables by relaxing the robust LMI problem. We present two approaches to solve robust LMI problems based on sos matrix decompositions, a direct approach and one based on the S-procedure. Both lead to an asymptotically exact family of LMI relaxations for computing lower bounds on the optimal fixed-order H_∞-norm whose size only grow quadratically in the dimension of the system state. The method is applied to an academic 4th-order example and to the tuning of two controller parameters of a 4-block H_∞ design of an active suspension system, with a Mc-Millan degree of the weighted plant of 27.