Optimal H2 order-one reduction by solving eigenproblems for polynomial equations

TL;DR

This paper presents a constructive algebraic method for optimal H2 order-one reduction of SISO LTI systems, guaranteeing a global optimum via eigenproblem solutions without gradient search, especially effective for systems with distinct poles.

Contribution

It introduces a new algebraic approach that simplifies and improves the efficiency of solving the H2 optimal reduction problem, avoiding local minima and reducing computational resources.

Findings

Method guarantees global optimality

Reduces computational time and memory usage

Applicable to higher-order systems with distinct poles

Abstract

A method is given for solving an optimal H2 approximation problem for SISO linear time-invariant stable systems. The method, based on constructive algebra, guarantees that the global optimum is found; it does not involve any gradient-based search, and hence avoids the usual problems of local minima. We examine mostly the case when the model order is reduced by one, and when the original system has distinct poles. This case exhibits special structure which allows us to provide a complete solution. The problem is converted into linear algebra by exhibiting a finite-dimensional basis for a certain space, and can then be solved by eigenvalue calculations, following the methods developed by Stetter and Moeller. The use of Buchberger's algorithm is avoided by writing the first-order optimality conditions in a special form, from which a Groebner basis is immediately available. Compared with our previous work the method presented here has much smaller time and memory requirements, and can therefore be applied to systems of significantly higher McMillan degree. In addition, some hypotheses which were required in the previous work have been removed. Some examples are included.