{"title":"Rank resilience of pattern matrices against structured perturbations with applications","authors":"Yuan Zhang, Yuanqing Xia, Gang Wang","doi":"10.1016/j.automatica.2025.112236","DOIUrl":null,"url":null,"abstract":"<div><div>In structured system theory, a pattern matrix is a matrix with entries either fixed to zero or free to take arbitrary numbers. The (generic) rank of a pattern matrix is the rank of almost all its realizations. The resilience of various system properties is closely tied to the rank resilience of the corresponding pattern matrices. Yet, existing literature predominantly explores the latter aspect by focusing on perturbations that change the zero–nonzero structure of pattern matrices, corresponding to link additions/deletions. In this paper, we consider the rank resilience of pattern matrices against structured perturbations that can arbitrarily alter the values of a prescribed set of entries, corresponding to link weight variations. We say a pattern matrix is structurally rank <span><math><mi>r</mi></math></span> resilient against a perturbation pattern if almost all realizations of this pattern matrix have a rank not less than <span><math><mi>r</mi></math></span> under arbitrary complex-valued realizations of the perturbation pattern. We establish a generic property in this concept. We then present combinatorial necessary and sufficient conditions for a rectangular pattern matrix to lose full rank under given perturbation patterns. We also generalize them to obtain a sufficient condition and a necessary one for losing an arbitrarily prescribed rank. We finally show our results can be applied to the resilience analysis of various properties of structured (descriptor) systems, including controllability and input-state observability, as well as characterizing zero structurally fixed modes.</div></div>","PeriodicalId":55413,"journal":{"name":"Automatica","volume":"176 ","pages":"Article 112236"},"PeriodicalIF":4.8000,"publicationDate":"2025-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Automatica","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0005109825001281","RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
引用次数: 0
Abstract
In structured system theory, a pattern matrix is a matrix with entries either fixed to zero or free to take arbitrary numbers. The (generic) rank of a pattern matrix is the rank of almost all its realizations. The resilience of various system properties is closely tied to the rank resilience of the corresponding pattern matrices. Yet, existing literature predominantly explores the latter aspect by focusing on perturbations that change the zero–nonzero structure of pattern matrices, corresponding to link additions/deletions. In this paper, we consider the rank resilience of pattern matrices against structured perturbations that can arbitrarily alter the values of a prescribed set of entries, corresponding to link weight variations. We say a pattern matrix is structurally rank resilient against a perturbation pattern if almost all realizations of this pattern matrix have a rank not less than under arbitrary complex-valued realizations of the perturbation pattern. We establish a generic property in this concept. We then present combinatorial necessary and sufficient conditions for a rectangular pattern matrix to lose full rank under given perturbation patterns. We also generalize them to obtain a sufficient condition and a necessary one for losing an arbitrarily prescribed rank. We finally show our results can be applied to the resilience analysis of various properties of structured (descriptor) systems, including controllability and input-state observability, as well as characterizing zero structurally fixed modes.
期刊介绍：
Automatica is a leading archival publication in the field of systems and control. The field encompasses today a broad set of areas and topics, and is thriving not only within itself but also in terms of its impact on other fields, such as communications, computers, biology, energy and economics. Since its inception in 1963, Automatica has kept abreast with the evolution of the field over the years, and has emerged as a leading publication driving the trends in the field.
After being founded in 1963, Automatica became a journal of the International Federation of Automatic Control (IFAC) in 1969. It features a characteristic blend of theoretical and applied papers of archival, lasting value, reporting cutting edge research results by authors across the globe. It features articles in distinct categories, including regular, brief and survey papers, technical communiqués, correspondence items, as well as reviews on published books of interest to the readership. It occasionally publishes special issues on emerging new topics or established mature topics of interest to a broad audience.
Automatica solicits original high-quality contributions in all the categories listed above, and in all areas of systems and control interpreted in a broad sense and evolving constantly. They may be submitted directly to a subject editor or to the Editor-in-Chief if not sure about the subject area. Editorial procedures in place assure careful, fair, and prompt handling of all submitted articles. Accepted papers appear in the journal in the shortest time feasible given production time constraints.
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