{"corpus_id":123269721,"paper_sha":"cb1632784ec7e34806735a6a3bdd3d3a073ca9ff","doi":"10.1137/S0895479801392016","arxiv_id":null,"pmid":null,"pmcid":null,"mag_id":2106387545,"dblp_id":null,"acl_id":null,"title":"Reduction to Versal Deformations of Matrix Pencils and Matrix Pairs with Application to Control Theory","year":2002,"publication_date":"2002-04-01","venue":"","journal":{"name":"SIAM Journal on Matrix Analysis and Applications","pages":"943-962","volume":"24"},"journal_issn":null,"journal_title":null,"publication_types":[],"pubmed_pub_types":null,"s2_fields_of_study":["Mathematics","Engineering"],"reference_count":17,"citation_count":15,"influential_citation_count":0,"is_open_access":true,"arxiv_categories":null,"arxiv_license":null,"arxiv_journal_ref":null,"mesh_headings":null,"chemicals":null,"comments_corrections":null,"source_flags":1,"s2_open_access_pdf_url":"https://upcommons.upc.edu/bitstream/2117/1195/4/prep.pdf","s2_open_access_landing_url":"https://www.semanticscholar.org/paper/cb1632784ec7e34806735a6a3bdd3d3a073ca9ff","s2_open_access_license":"CCBYNCND","s2_open_access_status":"GREEN","pmc_open_access_pdf_url":null,"pmc_open_access_landing_url":null,"pmc_open_access_license":null,"pmc_open_access_status":null,"unpaywall_open_access_pdf_url":null,"unpaywall_open_access_landing_url":null,"unpaywall_open_access_license":null,"unpaywall_open_access_status":null,"abstract":"Matrix pencils under the strict equivalence and matrix pairs under the state feedback equivalence are considered. It is known that a matrix pencil (or a matrix pair) smoothly dependent on parameters can be reduced locally to a special typically more simple form, called the versal deformation, by a smooth change of parameters and a strict equivalence (or feedback equivalence) transformation. We suggest an explicit recurrent procedure for finding the change of parameters and equivalence transformation in the reduction of a given family of matrix pencils (or matrix pairs) to the versal deformation. As an application, this procedure is applied to the analysis of the uncontrollability set in the space of parameters for a one-input linear dynamical system. Explicit formulae for a tangent plane to the uncontrollability set at its regular point and the perturbation of the uncontrollable mode are derived. A physical example is given and studied in detail.","claims":[{"public_id":"cl_68f062a4abac785c24cc431211f31cad","status":"active","text":"An explicit recurrent procedure is provided for computing the parameter change and equivalence transformation that reduce a given family of matrix pencils or matrix pairs to a versal deformation.","confidence":0.97,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_68f062a4abac785c24cc431211f31cad"},{"public_id":"cl_de3bdcc3c9ee73ac4823c2b3b495c5cb","status":"active","text":"Explicit formulae are derived for the tangent plane to the uncontrollability set at a regular point and for the perturbation of the uncontrollable mode.","confidence":0.95,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_de3bdcc3c9ee73ac4823c2b3b495c5cb"},{"public_id":"cl_cae0e95f531f6298289b7fceae102f0b","status":"active","text":"The reduction procedure is applied to the analysis of the uncontrollability set in parameter space for a one-input linear dynamical system.","confidence":0.93,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_cae0e95f531f6298289b7fceae102f0b"}],"concepts":[{"public_id":"co_268fef1b9b7a4fa951bcf6be03a150fd","status":"active","name":"matrix pencils","description":"Families of matrices of the form A - λB considered under strict equivalence.","types":["mathematical object"],"aliases":["matrix pencil"],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_268fef1b9b7a4fa951bcf6be03a150fd"},{"public_id":"co_39caa3994326f7a553cb5f7858fffd34","status":"active","name":"one-input linear dynamical system","description":"A linear control system with a single input channel.","types":["system"],"aliases":["single-input linear dynamical system"],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_39caa3994326f7a553cb5f7858fffd34"},{"public_id":"co_3b5d418b2b178685015ad56f49f44f4b","status":"active","name":"matrix pairs","description":"Pairs of matrices considered under state feedback equivalence.","types":["mathematical object"],"aliases":["matrix pair"],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_3b5d418b2b178685015ad56f49f44f4b"},{"public_id":"co_3cf03ca9457097af88d54d0d246cb8bf","status":"active","name":"uncontrollability set","description":"The set of parameter values for which a control system is uncontrollable.","types":["set","control-theoretic object"],"aliases":[],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_3cf03ca9457097af88d54d0d246cb8bf"},{"public_id":"co_43674e09cd92520da928489cfe4be82b","status":"active","name":"state feedback equivalence","description":"An equivalence relation on matrix pairs arising from state feedback transformations in control theory.","types":["equivalence relation"],"aliases":[],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_43674e09cd92520da928489cfe4be82b"},{"public_id":"co_c3f9c56cbee08371191a5fce61a9ee14","status":"active","name":"equivalence transformation","description":"A transformation applied to preserve equivalence while simplifying the family of matrices.","types":["transformation"],"aliases":[],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_c3f9c56cbee08371191a5fce61a9ee14"},{"public_id":"co_cd0b3c18ec86b38ca90ff781786a7b4f","status":"active","name":"uncontrollable mode","description":"A mode of a dynamical system that cannot be influenced by the input.","types":["system mode"],"aliases":[],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_cd0b3c18ec86b38ca90ff781786a7b4f"},{"public_id":"co_d83a788f7d6cb5e7720dddbb572c360e","status":"active","name":"versal deformation","description":"A locally simplified parameter-dependent normal form obtained by smooth parameter changes and equivalence transformations.","types":["normal form","deformation"],"aliases":["deformation versal"],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_d83a788f7d6cb5e7720dddbb572c360e"},{"public_id":"co_e3cca24177635af9e673a4269f293458","status":"active","name":"smooth change of parameters","description":"A differentiable reparameterization used to transform a family into a simpler local form.","types":["transformation"],"aliases":[],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_e3cca24177635af9e673a4269f293458"},{"public_id":"co_eaa4bcdc26a8d92463db603d5eb5bfce","status":"active","name":"strict equivalence","description":"An equivalence relation on matrix pencils used to classify them up to invertible left and right transformations.","types":["equivalence relation"],"aliases":[],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_eaa4bcdc26a8d92463db603d5eb5bfce"},{"public_id":"co_fd58c0de7929d8c51b597391bd12040a","status":"active","name":"tangent plane","description":"A linear approximation to a surface at a regular point.","types":["geometric object"],"aliases":[],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_fd58c0de7929d8c51b597391bd12040a"}],"external_ids":{"DOI":"10.1137/S0895479801392016","ArXiv":null,"PubMed":null,"PubMedCentral":null,"MAG":2106387545,"DBLP":null,"ACL":null},"open_access":{"is_open_access":true,"pdf_url":"https://upcommons.upc.edu/bitstream/2117/1195/4/prep.pdf","landing_url":"https://www.semanticscholar.org/paper/cb1632784ec7e34806735a6a3bdd3d3a073ca9ff","source":"semantic_scholar","pdf_url_source":"semantic_scholar_open_access_pdf","license":"CCBYNCND","status":"GREEN","reason":null},"reference_availability":{"status":"available","references_indexed":true,"full_text_available":false,"full_text_source":null,"count_basis":"semantic_scholar_metadata","extraction_status":"not_applicable","reason":null},"source":{"provider":"episteme2","base_corpus":"semantic_scholar_dump","freshness_mode":"unknown","basis":["semantic_scholar_metadata","postgres_metadata"],"limits":["paper metadata is based on indexed upstream scholarly datasets","claims and concepts are available only for extracted papers","absence of claims or concepts means no extracted graph data is available in this response"],"status":"available","degraded":false,"degraded_reasons":[],"diagnostics":{"status":"available","degraded":false,"degraded_reasons":[],"metadata_status":"available","graph_status":"available","abstract_status":"available"},"source_flags":1},"paper_id":635250,"paper_uid":"a7522e02-69e8-4354-882d-09d7bcd68726","canonical_identity":{"paper_id":635250,"paper_uid":"a7522e02-69e8-4354-882d-09d7bcd68726","identity_status":"available","lookup_basis":"semantic_scholar_external_id","compatibility_path":"corpus_id"},"url":"https://sah.borca.ai/papers/123269721"}