In this paper, we study a robust performance analysis problemof linear time-invariant (LTI) systems whose state space matrices depend polynomially on an uncertain parameter. Most of existing approaches to this problem is such thatwe recast the problem into a robust SDP by means of standard LMIs that characterize the performance. On the other hand, in the present paper, we firstly consider the dual of those LMIs and show that the robust performance analysis problem can be naturally reduced to an polynomial optimization problem with matrix variables. For this problem, we can readily construct a hierarchy of LMI relaxations whose infeasibility imply the fulfillment of the robust performance. In addition, in the case where the LMI relaxation has a feasible solution, we derive a rank condition with respect to the solution under which the robust performance is never attained.