Model Reduction by Balanced Truncation

Abstract

Model reduction by balanced truncation for bounded real and positive real input-stateoutputsystems, known as bounded real balanced truncation and positive real balancedtruncation respectively, is addressed. Results for finite-dimensional systems were establishedin the mid to late 1980s and we consider two extensions of this work. Firstly,using a more behavioral framework we consider the notion of a finite-dimensional dissipativesystem, of which bounded real and positive real input-state-output systems areparticular instances. Specifically, we work in a framework where we make no a prioridistinction between inputs and outputs. We derive model reduction by dissipative balancedtruncation, where a gap metric error bound is obtained, and demonstrate thatthe aforementioned bounded real and positive real balanced truncation can be seen asspecial cases.In the second part we generalise bounded real and positive real balanced truncationto classes of bounded real and positive real systems respectively that have non-rationaltransfer functions, so called infinite-dimensional systems. Here we work in the contextof well-posed linear systems. We derive approximate transfer functions, which weprove are rational and preserve the relevant dissipativity property. We also obtain errorbounds for the difference of the original transfer function and its reduced order transferfunction, in the H-infinity norm and gap metric for the bounded real and positive realcases respectively. This extension to bounded real and positive real balanced truncationrequires new results for Lyapunov balanced truncation in the infinite dimensional case,which we also describe. We conclude by highlighting possible future research.

Date of Award	26 Jun 2012
Original language	English
Awarding Institution	University of Bath
Supervisor	Mark Opmeer (Supervisor)