Robust performance analysis of uncertain LTI systems: Dual LMI approach and verifications for exactness

This paper addresses robust performance analysis problems of LTI systems affected by real parametric uncertainties. These problems, known also as a special class of structured singular value computation problems, are inherently intractable (NP-hard problems). As such intensive research effort has been made to obtain computationally tractable and less conservative analysis conditions, where linear matrix inequality (LMI) plays an important role. Nevertheless, since LMI-based conditions are expected to be conservative in general, it is often the case that we cannot conclude anything directly if the LMI at hand turns out to be infeasible. This motivates us to consider the dual of the LMI and examine the structure of the dual solution, which does exist if the primal LMI is infeasible. By pursuing this direction, in this paper, we provide a rank condition on the dual solution matrix under which we can conclude that the underlying robust performance is never attained. In particular, a set of uncertain parameters that violates the specified performance can readily be obtained. The key idea to derive these results comes from simultaneous diagonalizability property of commuting diagonalizable matrices. The block-moment matrix structure of the dual variable plays an essential role to make good use of this property.