{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/107999"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/107999","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"On the Tightness of the Balanced Truncation Error Bound with an Application to Arrowhead Systems","abstract":"Balanced truncation model reduction for linear systems yields reduced-order models that satisfy a well-known error bound in terms of a system's Hankel singular values. This bound is known to hold with equality under certain conditions, such as when the full-order system is state-space symmetric. In this work, we derive more general conditions in which the balanced truncation error bound holds with equality. We show that this holds for single-input, single-output systems that exhibit a generalized type of state-space symmetry based on the sign parameters corresponding to a system's Hankel singular values. We prove an additional result that shows how to determine this state-space symmetry from the arrowhead realization of a system, if available. In particular, we provide a formula for the sign parameters of an arrowhead system in terms of the off-diagonal entries of its arrowhead realization. We then illustrate these results with an example of an arrowhead system arising naturally in power systems modeling that motivated our study.","abstract_html":"Balanced truncation model reduction for linear systems yields reduced-order models that satisfy a well-known error bound in terms of a system's Hankel singular values. This bound is known to hold with equality under certain conditions, such as when the full-order system is state-space symmetric. In this work, we derive more general conditions in which the balanced truncation error bound holds with equality. We show that this holds for single-input, single-output systems that exhibit a generalized type of state-space symmetry based on the sign parameters corresponding to a system's Hankel singular values. We prove an additional result that shows how to determine this state-space symmetry from the arrowhead realization of a system, if available. In particular, we provide a formula for the sign parameters of an arrowhead system in terms of the off-diagonal entries of its arrowhead realization. We then illustrate these results with an example of an arrowhead system arising naturally in power systems modeling that motivated our study.","abstract_has_math":false,"creators":["Reiter, Sean Joseph"],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Embree, Mark P."],"committee_members":["Beattie, Christopher A.","Gugercin, Serkan","Kekatos, Vasileios"],"year":2022,"date_issued":"2022-01-28","date_published":"2022-01-28","updated_at":"2026-07-22T22:19:10Z","subjects":["Model Reduction","Balanced Truncation","Error Bound","Arrowhead Systems","Power Systems"],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:33886"],"render_values":[{"text":"vt_gsexam:33886","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/107999","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Embree, Mark P."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Beattie, Christopher A.","Gugercin, Serkan","Kekatos, Vasileios"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Reiter, Sean Joseph"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2022-01-29T09:00:40Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-01-29T09:00:40Z"]},{"key":"dc:date.issued","label":"Date","values":["2022-01-28"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Model Reduction","Balanced Truncation","Error Bound","Arrowhead Systems","Power Systems"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:33886"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/107999"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Balanced truncation model reduction for linear systems yields reduced-order models that satisfy a well-known error bound in terms of a system's Hankel singular values. This bound is known to hold with equality under certain conditions, such as when the full-order system is state-space symmetric. In this work, we derive more general conditions in which the balanced truncation error bound holds with equality. We show that this holds for single-input, single-output systems that exhibit a generalized type of state-space symmetry based on the sign parameters corresponding to a system's Hankel singular values. We prove an additional result that shows how to determine this state-space symmetry from the arrowhead realization of a system, if available. In particular, we provide a formula for the sign parameters of an arrowhead system in terms of the off-diagonal entries of its arrowhead realization. We then illustrate these results with an example of an arrowhead system arising naturally in power systems modeling that motivated our study."]},{"key":"dc:description.abstractgeneral","label":"General Abstract","values":["Mathematical modeling of dynamical systems provides a powerful means for studying physical phenomena. Due the complexities of real-world problems, many mathematical models face computational difficulties due to the costs of accurate modeling. Model-order reduction of large-scale dynamical systems circumvents this by approximating the large-scale model with a ``smaller'' one that still accurately describes the problem of interest. Balanced truncation model reduction for linear systems is one such example, yielding reduced-order models that satisfy a tractable upper bound on the approximation error. This work investigates conditions in which this bound is known to hold with equality, becoming an exact formula for the error in reduction. We additionally show how to determine these conditions for a special class of linear dynamical systems known as arrowhead systems, which arise in special applications of network modeling. We provide an example of one such system from power systems modeling that motivated our study."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["ETD"]},{"key":"dc:title","label":"Title","values":["On the Tightness of the Balanced Truncation Error Bound with an Application to Arrowhead Systems"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Embree, Mark P."],"dc:contributor.committeemember":["Beattie, Christopher A.","Gugercin, Serkan","Kekatos, Vasileios"],"dc:contributor.department":["Mathematics"],"dc:creator":["Reiter, Sean Joseph"],"dc:date.accessioned":["2022-01-29T09:00:40Z"],"dc:date.available":["2022-01-29T09:00:40Z"],"dc:date.issued":["2022-01-28"],"dc:description.abstract":["Balanced truncation model reduction for linear systems yields reduced-order models that satisfy a well-known error bound in terms of a system's Hankel singular values. This bound is known to hold with equality under certain conditions, such as when the full-order system is state-space symmetric. In this work, we derive more general conditions in which the balanced truncation error bound holds with equality. We show that this holds for single-input, single-output systems that exhibit a generalized type of state-space symmetry based on the sign parameters corresponding to a system's Hankel singular values. We prove an additional result that shows how to determine this state-space symmetry from the arrowhead realization of a system, if available. In particular, we provide a formula for the sign parameters of an arrowhead system in terms of the off-diagonal entries of its arrowhead realization. We then illustrate these results with an example of an arrowhead system arising naturally in power systems modeling that motivated our study."],"dc:description.abstractgeneral":["Mathematical modeling of dynamical systems provides a powerful means for studying physical phenomena. Due the complexities of real-world problems, many mathematical models face computational difficulties due to the costs of accurate modeling. Model-order reduction of large-scale dynamical systems circumvents this by approximating the large-scale model with a ``smaller'' one that still accurately describes the problem of interest. Balanced truncation model reduction for linear systems is one such example, yielding reduced-order models that satisfy a tractable upper bound on the approximation error. This work investigates conditions in which this bound is known to hold with equality, becoming an exact formula for the error in reduction. We additionally show how to determine these conditions for a special class of linear dynamical systems known as arrowhead systems, which arise in special applications of network modeling. We provide an example of one such system from power systems modeling that motivated our study."],"dc:description.degree":["Master of Science"],"dc:format.medium":["ETD"],"dc:identifier.other":["vt_gsexam:33886"],"dc:identifier.uri":["http://hdl.handle.net/10919/107999"],"dc:language.iso":["en"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Model Reduction","Balanced Truncation","Error Bound","Arrowhead Systems","Power Systems"],"dc:title":["On the Tightness of the Balanced Truncation Error Bound with an Application to Arrowhead Systems"],"dc:type":["Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:10Z"}