Design criteria for robust strict positive realness in adaptive schemes

Design of filters ensuring convergence of recursive estimation algorithms in the presence of uncertainty in the plant model is a key problem in the area of identification and adaptive control. This paper addresses the problem of designing filters ensuring strict positive realness of a family of uncertain polynomials over an assigned region of the complex plane. The uncertainty is assumed to be both structured and unstructured. When the structured uncertainty of the family is represented through regions of root location of a certain shape (for instance, circles centered on the real axis), an optimal solution of the problem is provided. For more general uncertainty regions, a simple procedure for constructing sub-optimal solutions to the filter design problem is proposed. A numerical example is fully developed to show the effectiveness of the proposed approach.