Higher-order relaxations for robust LMI problems with verifications for exactness

Robust semi-definite programming problems are know to have a wide range of applications, in particular in robust control. For rational uncertainty dependence, the full block S-procedure allows to systematically construct relaxations for the computation of guaranteed bounds. Typically these relaxations are conservative (causing a gap between actual and computed optimal values) since they involve the approximation of a set of so-called multipliers. The main purpose of this paper is to suggest a novel sequence of multiplier approximations which can be exploited in computations and which can be proved to be asymptotically exact. The second goal is to provide a numerical test for checking whether relaxations are exact (thus guaranteeing the absence of conservatism) which extends a recently formulated general principle to synthesis problems. We discuss the practical relevance of our results for LPV synthesis, and we illustrate them in terms of a numerical example.