{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:16:40Z","timestamp":1760239000079,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,9,15]],"date-time":"2020-09-15T00:00:00Z","timestamp":1600128000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This article presents a stability analysis of linear time invariant systems arising in system theory. The computation of upper bounds of structured singular values confer the stability analysis, robustness and performance of feedback systems in system theory. The computation of the bounds of structured singular values of Toeplitz and symmetric Toeplitz matrices for linear time invariant systems is presented by means of low rank ordinary differential equations (ODE\u2019s) based methodology. The proposed methodology is based upon the inner-outer algorithm. The inner algorithm constructs and solves a gradient system of ODE\u2019s while the outer algorithm adjusts the perturbation level with fast Newton\u2019s iteration. The comparison of bounds of structured singular values approximated by low rank ODE\u2019s based methodology results tighter bounds when compared with well-known MATLAB routine mussv, available in MATLAB control toolbox.<\/jats:p>","DOI":"10.3390\/sym12091518","type":"journal-article","created":{"date-parts":[[2020,9,15]],"date-time":"2020-09-15T10:24:09Z","timestamp":1600165449000},"page":"1518","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Stability Analysis of Linear Feedback Systems in Control"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5828-3133","authenticated-orcid":false,"given":"Mutti-Ur","family":"Rehman","sequence":"first","affiliation":[{"name":"Department of Mathematics, Sukkur IBA University, Sukkur 65200, Pakistan"},{"name":"Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5262-1138","authenticated-orcid":false,"given":"Jehad","family":"Alzabut","sequence":"additional","affiliation":[{"name":"Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Muhammad Fazeel","family":"Anwar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sukkur IBA University, Sukkur 65200, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Vidyasagar, M. (2002). Nonlinear Systems Analysis, SIAM.","DOI":"10.1137\/1.9780898719185"},{"key":"ref_2","unstructured":"Thomsen, J.J. (2003). Vibrations and Stability: Advanced Theory, Analysis, and Tools, Springer Science & Business Media."},{"key":"ref_3","unstructured":"Doyle, J.C., Francis, B.A., and Tannenbaum, A.R. (2013). Feedback Control Theory, Courier Corporation."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1049\/ip-d.1982.0053","article-title":"Analysis of feedback systems with structured uncertainties","volume":"129","author":"Doyle","year":"1982","journal-title":"IEEE Proc. Control Theory Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1007\/s10208-015-9254-z","article-title":"Every matrix is a product of Toeplitz matrices","volume":"16","author":"Ye","year":"2016","journal-title":"Found. Comput. 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Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1201","DOI":"10.2514\/3.7211","article-title":"Simplified calculation of eigenvector derivatives","volume":"14","author":"Nelson","year":"1976","journal-title":"AIAA J."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"800","DOI":"10.1080\/00207721.2017.1419306","article-title":"Toeplitz matrices for LTI systems, an illustration of their application to Wiener filters and estimators","volume":"49","author":"Moir","year":"2018","journal-title":"Int. J. Syst. Sci."},{"key":"ref_11","unstructured":"Thomas, G.W., and Trefethen, L.N. (2020, July 24). Eigtool. Software. Available online: http:\/\/www.comlab.ox.ac.uk\/pseudospectra\/eigtool."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"714","DOI":"10.1109\/9.286248","article-title":"Robust stability with time-varying structured uncertainty","volume":"39","author":"Shamma","year":"1994","journal-title":"IEEE Trans. Autom. Control"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/9\/1518\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:10:04Z","timestamp":1760177404000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/9\/1518"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,15]]},"references-count":12,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2020,9]]}},"alternative-id":["sym12091518"],"URL":"https:\/\/doi.org\/10.3390\/sym12091518","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,9,15]]}}}