Model Reduction by Balanced Truncation

Abstract

Model reduction by balanced truncation for bounded real and positive real input-stateoutput systems, known as bounded real balanced truncation and positive real balanced truncation respectively, is addressed. Results for finite-dimensional systems were established in 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 dissipative system, of which bounded real and positive real input-state-output systems are particular instances. Specifically, we work in a framework where we make no a priori distinction between inputs and outputs. We derive model reduction by dissipative balanced truncation, where a gap metric error bound is obtained, and demonstrate that the aforementioned bounded real and positive real balanced truncation can be seen as special cases. In the second part we generalise bounded real and positive real balanced truncation to classes of bounded real and positive real systems respectively that have non-rational transfer functions, so called infinite-dimensional systems. Here we work in the context of well-posed linear systems. We derive approximate transfer functions, which we prove are rational and preserve the relevant dissipativity property. We also obtain error bounds for the difference of the original transfer function and its reduced order transfer function, in the H-infinity norm and gap metric for the bounded real and positive real cases respectively. This extension to bounded real and positive real balanced truncation requires new results for Lyapunov balanced truncation in the infinite dimensional case, which we also describe. We conclude by highlighting possible future research.EThOS - Electronic Theses Online ServiceGBUnited Kingdo