Asymptotic exactness of dual LMI approach for robust performance analysis of uncertain LTI systems

In the preceding studies we proposed a dual LMI approach for robust performance analysis problems of LTI systems that are affected by parametric uncertainties. By starting from a dual LMI that characterizes a dissipation performance of uncertainty-free LTI systems, we showed that the robust dissipation performance analysis problem can be reduced into a feasibility problem of a polynomial matrix inequality (PMI). Moreover, by applying a linearization to the PMI, we derived an infinite sequence of LMI relaxation problems that allows us to reduce the relaxation gap gradually. Nevertheless, the asymptotic behaviour of this infinite sequence has been open, and this motivates us to study mutual relationship among the dual LMI approach and existing approaches. As the main result of this paper, we prove that our dual LMI approach corresponds to the dual of the polynomial parameter-dependent Lyapunov function approach with matrix sum-of-squares (SOS) relaxations, which is known to be asymptotically exact. Thus we clarify a close relationship between these two approaches that are seemingly very different. This relationship readily leads us to the desired conclusion that the proposed dual LMI approach is asymptotically exact as well.