On the non-conservatism of a novel LMI relaxation for robust analysis of polytopic systems

It is known that robust analysis of polytopic systems, i.e. linear system with polytopic state space uncertainty, is a difficult problem. Recently we have proposed for this problem, in both continuous-time and discrete-time cases, the use of homogeneous polynomially parameter-dependent quadratic Lyapunov functions (HPD–QLFs) by introducing a novel LMI relaxation which provides a sufficient condition via a convex optimization without requiring the use of auxiliary multipliers of unknown degree. In this paper we show that this condition is not only sufficient but it is also necessary.