In this paper, we propose new LMI-based conditions for robust stability/performance analysis of linear time-invariant (LTI) uncertain systems. To get around the conservatism of existing conditions resulting from Lyapunov's stability theory, we first consider to employ Lyapunov functions that can be associated with higher-order derivatives of the state vectors. This motivates us to introduce a redundant system description so that we can take the behavior of the higher-order derivatives of the state into consideration. Indeed, by considering suitable redundant system descriptions, the existence conditions of those Lyapunov functions can be reduced into constrained inequality conditions, to which we can apply Finsler's Lemma. Thus we can readily obtain new LMI-based conditions for (robust) stability/performance analysis of LTI systems in a constructive way. It turns out that the proposed LMI conditions can be regarded as a natural extension of those known as extended or dilated LMIs in the literature.