{"id":"https://openalex.org/W2972470817","doi":"https://doi.org/10.1109/tac.2021.3086319","title":"Gramians, Energy Functionals, and Balanced Truncation for Linear Dynamical Systems With Quadratic Outputs","display_name":"Gramians, Energy Functionals, and Balanced Truncation for Linear Dynamical Systems With Quadratic Outputs","publication_year":2021,"publication_date":"2021-06-03","ids":{"openalex":"https://openalex.org/W2972470817","doi":"https://doi.org/10.1109/tac.2021.3086319","mag":"2972470817"},"language":"en","primary_location":{"id":"doi:10.1109/tac.2021.3086319","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tac.2021.3086319","pdf_url":null,"source":{"id":"https://openalex.org/S184954342","display_name":"IEEE Transactions on Automatic Control","issn_l":"0018-9286","issn":["0018-9286","1558-2523","2334-3303"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Automatic Control","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1909.04597","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5060985095","display_name":"Peter Benner","orcid":"https://orcid.org/0000-0003-3362-4103"},"institutions":[{"id":"https://openalex.org/I149899117","display_name":"Max Planck Society","ror":"https://ror.org/01hhn8329","country_code":"DE","type":"nonprofit","lineage":["https://openalex.org/I149899117"]},{"id":"https://openalex.org/I4210127568","display_name":"Max Planck Institute for Dynamics of Complex Technical Systems","ror":"https://ror.org/030h7k016","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210127568"]},{"id":"https://openalex.org/I95793202","display_name":"Otto-von-Guericke-Universit\u00e4t Magdeburg","ror":"https://ror.org/00ggpsq73","country_code":"DE","type":"education","lineage":["https://openalex.org/I95793202"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Peter Benner","raw_affiliation_strings":["Fakult\u00e4t f\u00fcr Mathematik, Otto-von-Guericke-Universit\u00e4t Magdeburg, Magdeburg, Germany","Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany","Max-Planck-Society"],"affiliations":[{"raw_affiliation_string":"Fakult\u00e4t f\u00fcr Mathematik, Otto-von-Guericke-Universit\u00e4t Magdeburg, Magdeburg, Germany","institution_ids":["https://openalex.org/I95793202"]},{"raw_affiliation_string":"Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany","institution_ids":["https://openalex.org/I4210127568"]},{"raw_affiliation_string":"Max-Planck-Society","institution_ids":["https://openalex.org/I149899117"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5080481874","display_name":"Pawan Goyal","orcid":"https://orcid.org/0000-0003-3072-7780"},"institutions":[{"id":"https://openalex.org/I149899117","display_name":"Max Planck Society","ror":"https://ror.org/01hhn8329","country_code":"DE","type":"nonprofit","lineage":["https://openalex.org/I149899117"]},{"id":"https://openalex.org/I4210127568","display_name":"Max Planck Institute for Dynamics of Complex Technical Systems","ror":"https://ror.org/030h7k016","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210127568"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Pawan Goyal","raw_affiliation_strings":["Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany","Max-Planck-Society"],"affiliations":[{"raw_affiliation_string":"Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany","institution_ids":["https://openalex.org/I4210127568"]},{"raw_affiliation_string":"Max-Planck-Society","institution_ids":["https://openalex.org/I149899117"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5007136568","display_name":"Igor Pontes Duff","orcid":"https://orcid.org/0000-0001-6433-6142"},"institutions":[{"id":"https://openalex.org/I149899117","display_name":"Max Planck Society","ror":"https://ror.org/01hhn8329","country_code":"DE","type":"nonprofit","lineage":["https://openalex.org/I149899117"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Igor PontesDuffPereira","raw_affiliation_strings":["Max-Planck-Society"],"affiliations":[{"raw_affiliation_string":"Max-Planck-Society","institution_ids":["https://openalex.org/I149899117"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5060985095"],"corresponding_institution_ids":["https://openalex.org/I149899117","https://openalex.org/I4210127568","https://openalex.org/I95793202"],"apc_list":null,"apc_paid":null,"fwci":0.2625,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.50525104,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"67","issue":"2","first_page":"886","last_page":"893"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":1.0,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":1.0,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.9914000034332275,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9843000173568726,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/observability","display_name":"Observability","score":0.7994424700737},{"id":"https://openalex.org/keywords/controllability","display_name":"Controllability","score":0.7724838852882385},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.683785617351532},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5879768133163452},{"id":"https://openalex.org/keywords/dynamical-systems-theory","display_name":"Dynamical systems theory","score":0.5550654530525208},{"id":"https://openalex.org/keywords/lyapunov-function","display_name":"Lyapunov function","score":0.5457850694656372},{"id":"https://openalex.org/keywords/model-order-reduction","display_name":"Model order reduction","score":0.4703802168369293},{"id":"https://openalex.org/keywords/linear-dynamical-system","display_name":"Linear dynamical system","score":0.4684370160102844},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.45675453543663025},{"id":"https://openalex.org/keywords/reduction","display_name":"Reduction (mathematics)","score":0.4536532461643219},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.4508506953716278},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.430083304643631},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.4252411127090454},{"id":"https://openalex.org/keywords/truncation-error","display_name":"Truncation error","score":0.41249850392341614},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2586253881454468},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.10996285080909729},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.08453091979026794}],"concepts":[{"id":"https://openalex.org/C36299963","wikidata":"https://www.wikidata.org/wiki/Q1369844","display_name":"Observability","level":2,"score":0.7994424700737},{"id":"https://openalex.org/C48209547","wikidata":"https://www.wikidata.org/wiki/Q1331104","display_name":"Controllability","level":2,"score":0.7724838852882385},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.683785617351532},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5879768133163452},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.5550654530525208},{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.5457850694656372},{"id":"https://openalex.org/C2779277453","wikidata":"https://www.wikidata.org/wiki/Q12202921","display_name":"Model order reduction","level":3,"score":0.4703802168369293},{"id":"https://openalex.org/C114275822","wikidata":"https://www.wikidata.org/wiki/Q621512","display_name":"Linear dynamical system","level":3,"score":0.4684370160102844},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.45675453543663025},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.4536532461643219},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.4508506953716278},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.430083304643631},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.4252411127090454},{"id":"https://openalex.org/C104942944","wikidata":"https://www.wikidata.org/wiki/Q3434686","display_name":"Truncation error","level":2,"score":0.41249850392341614},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2586253881454468},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.10996285080909729},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.08453091979026794},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1109/tac.2021.3086319","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tac.2021.3086319","pdf_url":null,"source":{"id":"https://openalex.org/S184954342","display_name":"IEEE Transactions on Automatic Control","issn_l":"0018-9286","issn":["0018-9286","1558-2523","2334-3303"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Automatic Control","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1909.04597","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1909.04597","pdf_url":"https://arxiv.org/pdf/1909.04597","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:pure.mpg.de:item_3166037","is_oa":true,"landing_page_url":"http://hdl.handle.net/21.11116/0000-0004-BE82-3","pdf_url":null,"source":{"id":"https://openalex.org/S4306400654","display_name":"MPG.PuRe (Max Planck Society)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I149899117","host_organization_name":"Max Planck Society","host_organization_lineage":["https://openalex.org/I149899117"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"submitted to IEEE Trans. Autom. Control","raw_type":"info:eu-repo/semantics/workingPaper"},{"id":"doi:10.48550/arxiv.1909.04597","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1909.04597","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"},{"id":"mag:2972470817","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1909.04597","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1909.04597","pdf_url":"https://arxiv.org/pdf/1909.04597","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/7","display_name":"Affordable and clean energy","score":0.8999999761581421}],"awards":[{"id":"https://openalex.org/G8912274555","display_name":null,"funder_award_id":"2297","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"},{"id":"https://openalex.org/F4320335828","display_name":"Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg","ror":"https://ror.org/030h7k016"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2972470817.pdf","grobid_xml":"https://content.openalex.org/works/W2972470817.grobid-xml"},"referenced_works_count":35,"referenced_works":["https://openalex.org/W151970475","https://openalex.org/W297328598","https://openalex.org/W1551480811","https://openalex.org/W1984714339","https://openalex.org/W1988376519","https://openalex.org/W1993134778","https://openalex.org/W2002858896","https://openalex.org/W2013232064","https://openalex.org/W2030863762","https://openalex.org/W2060219659","https://openalex.org/W2061433628","https://openalex.org/W2064733273","https://openalex.org/W2067070250","https://openalex.org/W2069049088","https://openalex.org/W2079665343","https://openalex.org/W2104744926","https://openalex.org/W2135187213","https://openalex.org/W2137002893","https://openalex.org/W2138190945","https://openalex.org/W2155711822","https://openalex.org/W2159808517","https://openalex.org/W2169328438","https://openalex.org/W2226462370","https://openalex.org/W2480611454","https://openalex.org/W2492791808","https://openalex.org/W2507193350","https://openalex.org/W2530996620","https://openalex.org/W2551493491","https://openalex.org/W2555046119","https://openalex.org/W2886662961","https://openalex.org/W2962706012","https://openalex.org/W3003105325","https://openalex.org/W3124293288","https://openalex.org/W3193903964","https://openalex.org/W6622014286"],"related_works":["https://openalex.org/W3170162203","https://openalex.org/W2610527164","https://openalex.org/W2963865788","https://openalex.org/W2896460776","https://openalex.org/W2172066780","https://openalex.org/W1543090246","https://openalex.org/W2569924641","https://openalex.org/W1775584924","https://openalex.org/W3134795379","https://openalex.org/W2057786973","https://openalex.org/W2068919677","https://openalex.org/W2923281596","https://openalex.org/W2765405823","https://openalex.org/W2181411401","https://openalex.org/W3178693814","https://openalex.org/W2006207750","https://openalex.org/W1993300095","https://openalex.org/W2095674434","https://openalex.org/W2806969755","https://openalex.org/W2043092713"],"abstract_inverted_index":{"In":[0],"this":[1,25],"article,":[2],"we":[3,27,60,104],"investigate":[4,61],"a":[5,15,20,29,52,123],"balancing-based":[6],"model":[7],"order":[8],"reduction":[9],"method":[10],"for":[11,34],"dynamical":[12,17],"systems":[13],"with":[14],"linear":[16],"equation":[18,58],"and":[19,59,78],"quadratic":[21],"output":[22],"function.":[23],"With":[24],"aim,":[26],"propose":[28],"new":[30],"algebraic":[31],"observability":[32],"Gramian":[33,50],"the":[35,39,48,110,116,119],"system":[36],"based":[37,95],"on":[38,96,109],"Hilbert":[40],"space":[41],"adjoint":[42],"theory.":[43],"We":[44,114],"then":[45],"show":[46],"that":[47,73],"proposed":[49,120],"satisfies":[51],"particular":[53],"type":[54],"of":[55,88,118],"generalized":[56],"Lyapunov":[57],"its":[62],"connection":[63],"to":[64,70,76,80],"energy":[65,102],"functionals.":[66],"It":[67],"allows":[68],"one":[69],"find":[71],"states":[72,90],"are":[74],"hard":[75,79],"control":[77],"observe":[81],"via":[82],"an":[83],"appropriate":[84],"balancing":[85],"transformation.":[86],"Truncation":[87],"such":[89],"yields":[91],"reduced-order":[92],"models.":[93],"Finally,":[94],"<inline-formula":[97],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[98],"xmlns:xlink=\"http://www.w3.org/1999/xlink\"><tex-math":[99],"notation=\"LaTeX\">$\\mathscr":[100],"{H}_2$</tex-math></inline-formula>":[101],"considerations,":[103],"derive":[105],"error":[106],"bounds,":[107],"depending":[108],"neglected":[111],"singular":[112],"values.":[113],"demonstrate":[115],"efficiency":[117],"methodology":[121],"using":[122],"numerical":[124],"example.":[125]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":1}],"updated_date":"2026-04-09T08:11:56.329763","created_date":"2025-10-10T00:00:00"}