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

TL;DR

This paper introduces a Riccati-type approach to analytic interpolation problems with rationality and derivative constraints, offering a new method applicable to systems and control, with implications for model reduction and covariance analysis.

Contribution

It presents a novel Riccati-type equation method for scalar and matrix interpolation problems, connecting solution rank to model degree and addressing covariance sequence positivity.

Findings

The method effectively solves interpolation problems with derivative constraints.

It provides a natural framework for model reduction in control systems.

The approach is demonstrated through applications in modeling and robust control.

Abstract

Analytic interpolation problems with rationality and derivative constraints are ubiquitous in systems and control. This paper provides a new method for such problems, both in the scalar and matrix case, based on a non-standard 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.