{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/44939"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/44939","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Homotopy algorithms for the H² and the combined H²/H<sup>∞</sup> model order reduction problems","abstract":"The problem of finding a reduced order model, optimal in the H² sense, to a given system model is a fundamental one in control system analysis and design. The addition of a H<sup>∞</sup> constraint to the H² optimal model reduction problem results in a more practical yet computationally more difficult problem. Without the global convergence of homotopy methods, both the H² optimal and the combined H²/H<sup>∞</sup> model reduction problems are very difficult. For both problems homotopy algorithms based on several formulations input normal form; Ly, Bryson, and Cannon's 2 X 2 block parametrization; a new nonminimal parametrization are developed and compared here. For the H² optimal model order reduction problem, these numerical algorithms are also compared with that based on Hyland and Bernstein's optimal projection equations. Both the input normal form and Ly form are very efficient compared to the over parametrization formulation and the optimal projection equations approach, since they utilize the minimal number of possible degrees of freedom. However, they can fail to exist or be very ill conditioned. The conditions under which the input normal form and the Ly form become ill conditioned are examined. The over-parametrization formulation solves the ill conditioning issue, and usually is more efficient than the approach based on solving the optimal projection equations for the H² optimal model reduction problem. However, the over-parametrization formulation introduces a very high order singularity at the solution, and it is doubtful whether this singularity can be overcome by using interpolation or other existing methods.","abstract_html":"The problem of finding a reduced order model, optimal in the H² sense, to a given system model is a fundamental one in control system analysis and design. The addition of a H&lt;sup&gt;∞&lt;/sup&gt; constraint to the H² optimal model reduction problem results in a more practical yet computationally more difficult problem. Without the global convergence of homotopy methods, both the H² optimal and the combined H²/H&lt;sup&gt;∞&lt;/sup&gt; model reduction problems are very difficult. For both problems homotopy algorithms based on several formulations input normal form; Ly, Bryson, and Cannon&#x27;s 2 X 2 block parametrization; a new nonminimal parametrization are developed and compared here. For the H² optimal model order reduction problem, these numerical algorithms are also compared with that based on Hyland and Bernstein&#x27;s optimal projection equations. Both the input normal form and Ly form are very efficient compared to the over parametrization formulation and the optimal projection equations approach, since they utilize the minimal number of possible degrees of freedom. However, they can fail to exist or be very ill conditioned. The conditions under which the input normal form and the Ly form become ill conditioned are examined. The over-parametrization formulation solves the ill conditioning issue, and usually is more efficient than the approach based on solving the optimal projection equations for the H² optimal model reduction problem. However, the over-parametrization formulation introduces a very high order singularity at the solution, and it is doubtful whether this singularity can be overcome by using interpolation or other existing methods.","abstract_has_math":false,"creators":["Ge, Yuzhen"],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Computer Science and Applications","degree_department":"Computer Science and Applications","school":null,"contributors":[],"advisors":[],"committee_chairs":["Watson, Layne T."],"committee_members":["Ball, Joseph A.","Beattie, Christopher A."],"year":1993,"date_issued":"1993-04-04","date_published":"1993-04-04","updated_at":"2026-07-22T22:19:37Z","subjects":[],"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":["etd-09292009-020303"],"render_values":[{"text":"etd-09292009-020303","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/44939","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Watson, Layne T."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Ball, Joseph A.","Beattie, Christopher A."]},{"key":"dc:contributor.department","label":"Department","values":["Computer Science and Applications"]},{"key":"dc:creator","label":"Author","values":["Ge, Yuzhen"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T21:46:37Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T21:46:37Z","2009-09-29"]},{"key":"dc:date.issued","label":"Date","values":["1993-04-04"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Computer Science and Applications"]},{"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":"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":["etd-09292009-020303"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/44939"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The problem of finding a reduced order model, optimal in the H² sense, to a given system model is a fundamental one in control system analysis and design. The addition of a H<sup>∞</sup> constraint to the H² optimal model reduction problem results in a more practical yet computationally more difficult problem. Without the global convergence of homotopy methods, both the H² optimal and the combined H²/H<sup>∞</sup> model reduction problems are very difficult. For both problems homotopy algorithms based on several formulations input normal form; Ly, Bryson, and Cannon's 2 X 2 block parametrization; a new nonminimal parametrization are developed and compared here. For the H² optimal model order reduction problem, these numerical algorithms are also compared with that based on Hyland and Bernstein's optimal projection equations. Both the input normal form and Ly form are very efficient compared to the over parametrization formulation and the optimal projection equations approach, since they utilize the minimal number of possible degrees of freedom. However, they can fail to exist or be very ill conditioned. The conditions under which the input normal form and the Ly form become ill conditioned are examined. The over-parametrization formulation solves the ill conditioning issue, and usually is more efficient than the approach based on solving the optimal projection equations for the H² optimal model reduction problem. However, the over-parametrization formulation introduces a very high order singularity at the solution, and it is doubtful whether this singularity can be overcome by using interpolation or other existing methods."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["BTD"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Homotopy algorithms for the H² and the combined H²/H<sup>∞</sup> model order reduction problems"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Watson, Layne T."],"dc:contributor.committeemember":["Ball, Joseph A.","Beattie, Christopher A."],"dc:contributor.department":["Computer Science and Applications"],"dc:creator":["Ge, Yuzhen"],"dc:date.accessioned":["2014-03-14T21:46:37Z"],"dc:date.available":["2014-03-14T21:46:37Z","2009-09-29"],"dc:date.issued":["1993-04-04"],"dc:description.abstract":["The problem of finding a reduced order model, optimal in the H² sense, to a given system model is a fundamental one in control system analysis and design. The addition of a H<sup>∞</sup> constraint to the H² optimal model reduction problem results in a more practical yet computationally more difficult problem. Without the global convergence of homotopy methods, both the H² optimal and the combined H²/H<sup>∞</sup> model reduction problems are very difficult. For both problems homotopy algorithms based on several formulations input normal form; Ly, Bryson, and Cannon's 2 X 2 block parametrization; a new nonminimal parametrization are developed and compared here. For the H² optimal model order reduction problem, these numerical algorithms are also compared with that based on Hyland and Bernstein's optimal projection equations. Both the input normal form and Ly form are very efficient compared to the over parametrization formulation and the optimal projection equations approach, since they utilize the minimal number of possible degrees of freedom. However, they can fail to exist or be very ill conditioned. The conditions under which the input normal form and the Ly form become ill conditioned are examined. The over-parametrization formulation solves the ill conditioning issue, and usually is more efficient than the approach based on solving the optimal projection equations for the H² optimal model reduction problem. However, the over-parametrization formulation introduces a very high order singularity at the solution, and it is doubtful whether this singularity can be overcome by using interpolation or other existing methods."],"dc:description.degree":["Master of Science"],"dc:format.medium":["BTD"],"dc:format.mimetype":["application/pdf"],"dc:identifier.other":["etd-09292009-020303"],"dc:identifier.uri":["http://hdl.handle.net/10919/44939"],"dc:language.iso":["en"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Homotopy algorithms for the H² and the combined H²/H<sup>∞</sup> model order reduction problems"],"dc:type":["Thesis"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Computer Science and Applications"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:37Z"}