# H2 model reduction of linear network systems by moment matching and optimization

## TL;DR

This paper develops a novel model reduction method for linear network systems using moment matching and optimization to produce stable, reduced models that preserve structure and minimize H2 norm error.

## Contribution

It introduces an optimization-based framework for model reduction that combines moment matching with structural constraints, including two solution approaches for nonconvex problems.

## Key findings

- The first method uses semidefinite programming with a block diagonal Gramian assumption.
- The second method employs a gradient projection technique for local optimality.
- Application to a power network demonstrates the effectiveness of the proposed methods.

## Abstract

In this paper we study the problem of model reduction of linear network systems. We aim at computing a reduced order stable approximation of the network with the same topology and optimal w.r.t. H2 norm error approximation. Our approach is based on time-domain moment matching framework, where we optimize over families of parameterized reduced order models matching a set of moments at arbitrary interpolation points. The parameterization of the low order models is in terms of the free parameters and of the interpolation points. For this family of parameterized models we formulate an optimization-based model reduction problem with the H2 norm of error approximation as objective function while the preservation of some structural and physical properties yields the constraints. This problem is nonconvex and we write it in terms of the Gramians of a minimal realization of the error system. We propose two solutions for this problem. The first solution assumes that the error system admits a block diagonal observability Gramian, allowing for a simple convex reformulation as semidefinite programming, but at the cost of some performance loss. We also derive sufficient conditions to guarantee block diagonalization of the Gramian. The second solution employs a gradient projection method for a smooth reformulation yielding (locally) optimal interpolation points and free parameters. The potential of the methods is illustrated on a power network.