SPR criteria for uncertain rational matrices via polynomial positivity and Bernstein's expansions

The main purpose of this brief is to convert the strict positive real (SPR) conditions for rational matrices to conditions involving only positivity of polynomials. The polynomial formulation provides efficient SPR criteria for matrices with uncertain interval parameters. To establish the robust SPR property, it suffices to test positivity of only three uncertain polynomials regardless of the order of the matrix. The most interesting feature of the proposed polynomial formulation is that the coefficients of uncertain matrices are allowed to have polynomic uncertainty structure. This generality is easily handled by using the Bernstein expansion algorithm. The efficiency of the proposed polynomial approach is illustrated by testing absolute stability of a MIMO Lur'e-Postnikov system having interval parameters.