Composition of rational functions: state-space realization and applications

TL;DR

This paper develops formulas for composing matrix-valued rational functions in state-space form, with applications to electrical circuits and Stieltjes functions, enhancing understanding of their structure and interconnections.

Contribution

It introduces two new composition methods for matrix-valued rational functions and provides explicit state-space realization formulas, linking them to practical applications.

Findings

Formulas for state-space realization of composed functions

Application to electrical circuit feedback networks

Connection to Stieltjes functions

Abstract

We define two versions of compositions of matrix-valued rational functions of appropriate sizes and whenever analytic at infinity, offer a set of formulas for the corresponding state-space realization, in terms of the realizations of the original functions. Focusing on positive real functions, the first composition is applied to electrical circuits theory along with introducing a connection to networks of feedback loops. The second composition is applied to Stieltjes functions.