The Covariance Extension Equation: A Riccati-Type Approach to Analytic Interpolation

Analytic interpolation problems with rationality and derivative constraints are ubiquitous in systems and control. This article provides a new method for such problems, both in the scalar and matrix case, based on a nonstandard Riccati-type equation. The rank of the solution matrix is the same as the degree of the interpolant, thus providing a natural approach to model reduction. A homotopy continuation method is presented and applied to some problems in modeling and robust control. We also address a question on the positive degree of a covariance sequence originally posed by Kalman.