Model Reduction by Balanced Truncation

Research output: ThesisDoctoral Thesis

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.
LanguageEnglish
QualificationPh.D.
Awarding Institution
  • University of Bath
Supervisors/Advisors
  • Opmeer, Mark, Supervisor
Award date26 Jun 2012
StatusPublished - 2012

Fingerprint

Balanced Truncation
Model Reduction
Transfer Function
Gap Metric
Error Bounds
Dissipativity
Infinite-dimensional Systems
Output
Dissipative Systems
Lyapunov
Linear Systems
Infinity
Norm
Generalise

Cite this

Model Reduction by Balanced Truncation. / Guiver, Christopher.

2012. 203 p.

Research output: ThesisDoctoral Thesis

Guiver, C 2012, 'Model Reduction by Balanced Truncation', Ph.D., University of Bath.
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title = "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.",
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year = "2012",
language = "English",
school = "University of Bath",

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T1 - Model Reduction by Balanced Truncation

AU - Guiver,Christopher

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N2 - 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.

AB - 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.

M3 - Doctoral Thesis

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