Synthesis of robust strictly positive real systems with l_2 parametric uncertainty

The problem of designing filters ensuring strict positive realness of a family of uncertain polynomials over an assigned region of the complex plane is frequently investigated issue in the analysis of absolute stability of nonlinear Lur'e systems and the design of adaptive schemes. This paper addresses the problem of designing a continuous-time rational filter when the uncertain polynomial family is assumed to be an ellipsoid in coefficient space. It is shown that the stability of all the polynomials of such a family is a necessary and sufficient condition for the existence of the filter. More importantly, contrary to the results available for the case of a polyhedral uncertainty set in coefficient space, it turns out that the filter is a proper rational function with degree smaller than twice the degree of the uncertain polynomials. Furthermore, a closed form solution to the filter synthesis problem based on polynomial factorization is derived.