In this paper, we address robust <i>H</i><sub>∞</sub> performance analysis problems of linear time-invariant polytopic-type uncertain systems. We employ polynomially parameter-dependent Lyapunov functions to assess the robust <i>H</i><sub>∞</sub> performance and derive sufficient conditions for the existence of those Lyapunov functions in terms of numerically verifiable finitely many linear matrix inequalities (LMIs). To this end, we first consider to analyze the <i>H</i><sub>∞</sub> performance of uncertainty-free systems by means of Lyapunov functions of a particular form, and explore the existence condition of such Lyapunov functions. We show that, by considering a suitable redundant system description, the existence condition of such Lyapunov functions can be reduced into constrained inequality conditions to which Finsler's Lemma can be applied. It turns that we can readily obtain novel LMI conditions for the <i>H</i><sub>∞</sub> performance analysis of uncertainty-free systems. The LMI-based conditions that enable us to assess the robust <i>H</i><sub>∞</sub> performance of uncertain systems by means of polynomially parameter-dependent Lyapunov functions follow immediately from these newly obtained LMIs. The LMI conditions obtained in this paper can be regarded as a natural extension of those known as extended or dilated LMIs in the literature.