Robust H∞ Performance Analysis of LTI Systems with a Single Uncertain Parameter

This paper addresses the robust H∞ performance analysis problem of linear time-invariant (LTI) systems whose state-space coefficient matrices depend polynomially on a single uncertain parameter. By means of a dual LMI that characterizes the H∞ performance of uncertainty-free LTI systems, we firstly formulate this analysis problem as a polynomial matrix inequality (PMI) optimization problem. However, this PMI problem is non-convex and hence intractable in general. Therefore, we apply linearization and construct an infinite sequence of relaxation problems, represented by SDPs, with theoretical guarantee of asymptotic exactness in the limit. In order to detect whether an arbitrary relaxation problem in the sequence is “exact” in the sense that it provides the same optimal value as that of the original problem, we derive a rank condition on the SDP solution under which we can conclude the exactness.