Full state approximation by Galerkin projection reduced order models for stochastic and bilinear systems

TL;DR

This paper introduces a model reduction method for stochastic and bilinear systems using Galerkin projection based on dominant subspaces, ensuring stability and providing error bounds, with demonstrated effectiveness through numerical experiments.

Contribution

The paper presents a novel approach for full state approximation in stochastic and bilinear systems using reachability Gramian-based subspace identification and Galerkin projection, with stability guarantees and error analysis.

Findings

Method preserves mean square stability in stochastic systems.

Error bounds depend on neglected eigenvalues of the reachability Gramian.

Performs well compared to balanced truncation in numerical tests.

Abstract

In this paper, the problem of full state approximation by model reduction is studied for stochastic and bilinear systems. Our proposed approach relies on identifying the dominant subspaces based on the reachability Gramian of a system. Once the desired subspace is computed, the reduced order model is then obtained by a Galerkin projection. We prove that, in the stochastic case, this approach either preserves mean square asymptotic stability or leads to reduced models whose minimal realization is mean square asymptotically stable. This stability preservation guarantees the existence of the reduced system reachability Gramian which is the basis for the full state error bounds that we derive. This error bound depends on the neglected eigenvalues of the reachability Gramian and hence shows that these values are a good indicator for the expected error in the dimension reduction procedure. Subsequently, we establish the stability preservation result and the error bound for a full state approximation to bilinear systems in a similar manner. These latter results are based on a recently proved link between stochastic and bilinear systems. We conclude the paper by numerical experiments using a benchmark problem. We compare this approach with balanced truncation and show that it performs well in reproducing the full state of the system. \end{abstract}