Robust strict positive realness via polynomial positivity

This paper attempts to convert the strictly positive real (SPR) conditions for rational functions and matrices to conditions involving only positivity of polynomials. The new polynomial formulation provides efficient SPR criteria for functions and matrices with uncertain parameters. To establish the robust SPR property it suffices to test positivity of only two uncertain polynomials for functions and three for matrices. The most interesting feature of the proposed polynomial approach is that all coefficients of the uncertain functions and matrices can have polynomial uncertainty structures. This generality is easily handled in numerical computations by applying the Bernstein expansion algorithm.