LMI Relaxations in Robust Control

The purpose of this tutorial paper is to discuss the important role of robust linear matrix inequalities with rational dependence on uncertainties in robust control. We review how various classical relaxations based on the S-procedure can be subsumed to a unified framework. Based on Lagrange duality for semi-definite programs, we put particular emphasis on a clear understanding under which conditions such relaxations can be verified to be exact. We finally address the systematic construction of families of relaxations which can be shown to be asymptotically exact, based on recent results on the sum-of-squares representation of polynomial matrices.