Synthesis of robust strictly positive real systems with l2 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 a longly 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.