The problem of robust energy-to-peak filtering for linear systems with convex bounded uncertainties is investigated in this paper. The main purpose is to design a full order stable linear filter that minimizes the worst-case peak value of the filtering error output signal with respect to all bounded energy inputs, in such a way that the filtering error system remains quadratically stable. Necessary and sufficient conditions are formulated in terms of linear Matrix Inequalities - LMIs, for both continuous- and discrete-time cases.