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Relaxations for Robust Linear Matrix Inequality Problems with Verifications for Exactness

Robust semidefinite programming problems with rational dependence on uncertainties are known to have a wide range of applications, in particular in robust control. It is well established how to systematically construct relaxations on the basis of the full block S-procedure. In general, such relaxations are expected to be conservative, but for concrete problem instances they are often observed to be tight. The main purpose of this paper is to suggest novel computationally verifiable conditions for when general classes of linear matrix inequality relaxations do not involve any conservatism. If the convex set of uncertainties is finitely generated, we suggest a novel sequence of relaxations which can be proved to be asymptotically exact. Finally, our results are applied to the particularly relevant robustness analysis problem for linear time-invariant dynamical systems affected by uncertainties that are full ellipsoidal or repeated and contained in intersections of disks or circles. This leads to extensions of known results on relaxation exactness for small block structures, including an elementary proof for tightness of standard structured singular value computations for three full complex uncertainty blocks.