Document Zbl 1298.93264

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An MIR may exhibit different splitting behaviors and, to the best of the authors’ knowledge, their properties have not been fully investigated. In this paper, we focus on characterizing the invariance properties for MIRs with any multiplicity. Furthermore, the proposed methodology makes it possible to also cover some degenerate cases already encountered and discussed in the literature. In addition, we propose an easily implemented frequency-sweeping method, making it possible to derive the asymptotic behavior without invoking the Puiseux series.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6358530">Cited in <strong>2</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93D09" title="MSC2020">93D09</a> </td> <td class="space"> Robust stability </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93B60" title="MSC2020">93B60</a> </td> <td class="space"> Eigenvalue problems </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Atime-delay+systems">time-delay systems</a>; <a href="/?q=ut%3Astability">stability</a>; <a href="/?q=ut%3APuiseux+series">Puiseux series</a>; <a href="/?q=ut%3Ainvariance">invariance</a>; <a href="/?q=ut%3Afrequency-sweeping+method">frequency-sweeping method</a>; <a href="/?q=ut%3Amultiple+imaginary+roots+%28MIRs%29">multiple imaginary roots (MIRs)</a></div> <div class="software"> <h3>Software:</h3><a href="/software/2396">DDE-BIFTOOL</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1298.93264" data-ciurl="/ci/06358530" data-biburl="/bibtex/06358530.bib" data-amsurl="/amsrefs/06358530.bib" data-xmlurl="/xml/06358530.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06358530.pdf" title="Zbl 1298.93264 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.automatica.2013.12.007" aria-label="DOI for “Invariance properties for a class of quasipolynomials”" title="10.1016/j.automatica.2013.12.007">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Chen, J.; Fu, P.; Niculescu, S.-I.; Guan, Z., An eigenvalue perturbation approach to stability analysis, part II: when will zeros of time-delay systems cross imaginary axis?, SIAM Journal on Control and Optimization, 48, 5583-5605 (2010) · <a href="/1261.93061" class="nowrap">Zbl 1261.93061</a></td> </tr><tr> <td>[2]</td> <td class="space">Chen, J.; Latchman, H. A., Frequency sweeping tests for stability independent of delay, IEEE Transactions on Automatic Control, 40, 1640-1645 (1995) · <a href="/0834.93044" class="nowrap">Zbl 0834.93044</a></td> </tr><tr> <td>[3]</td> <td class="space">Cooke, K. L.; van den Driessche, P., On zeros of some transcendental equations, Funkcialaj Ekvacioj, 29, 77-90 (1986) · <a href="/0603.34069" class="nowrap">Zbl 0603.34069</a></td> </tr><tr> <td>[4]</td> <td class="space">Engelborghs, K.; Luzyanina, T.; Roose, D., Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Transactions on Mathematical Software, 28, 1-21 (2002) · <a href="/1070.65556" class="nowrap">Zbl 1070.65556</a></td> </tr><tr> <td>[5]</td> <td class="space">Gu, K.; Kharitonov, V. L.; Chen, J., Stability of time-delay systems (2003), Birkhäuser: Birkhäuser Boston, MA · <a href="/1039.34067" class="nowrap">Zbl 1039.34067</a></td> </tr><tr> <td>[6]</td> <td class="space">Gu, K.; Naghnaeian, M., Stability crossing set for systems with three delays, IEEE Transactions on Automatic Control, 56, 11-26 (2011) · <a href="/1368.34076" class="nowrap">Zbl 1368.34076</a></td> </tr><tr> <td>[7]</td> <td class="space">Jarlebring, E.; Michiels, W., Invariance properties in the root sensitivity of time-delay systems with double imaginary roots, Automatica, 46, 1112-1115 (2010) · <a href="/1191.93049" class="nowrap">Zbl 1191.93049</a></td> </tr><tr> <td>[8]</td> <td class="space">Kato, T., Perturbation theory for linear operators (1995), Springer-Verlag: Springer-Verlag Berlin · <a href="/0836.47009" class="nowrap">Zbl 0836.47009</a></td> </tr><tr> <td>[9]</td> <td class="space">Krantz, S. G., Function theory of several complex variables, second edition (1992), Wadsworth & Brooks/Cole: Wadsworth & Brooks/Cole Pacific Grove, CA · <a href="/0776.32001" class="nowrap">Zbl 0776.32001</a></td> </tr><tr> <td>[10]</td> <td class="space">Lee, M. S.; Hsu, C. S., On the $\tau $-decomposition method of stability analysis for retarded dynamical systems, SIAM Journal on Control, 7, 242-259 (1969) · <a href="/0182.12701" class="nowrap">Zbl 0182.12701</a></td> </tr><tr> <td>[14]</td> <td class="space">Li, X.-G.; Niculescu, S.-I.; Çela, A.; Wang, H.-H.; Cai, T.-Y., On computing Puiseux series for multiple imaginary characteristic roots of LTI delay systems with commensurate delays, IEEE Transactions on Automatic Control, 58, 1338-1343 (2013) · <a href="/1511.93099" class="nowrap">Zbl 1511.93099</a></td> </tr><tr> <td>[15]</td> <td class="space">Michiels, W.; Niculescu, S.-I., Stability and stabilization of time-delay systems: an eigenvalue-based approach (2007), SIAM: SIAM Philadelphia · <a href="/1140.93026" class="nowrap">Zbl 1140.93026</a></td> </tr><tr> <td>[16]</td> <td class="space">Niculescu, S.-I., Delay effects on stability: a robust control approach (2001), Springer-Verlag: Springer-Verlag Heidelberg · <a href="/0997.93001" class="nowrap">Zbl 0997.93001</a></td> </tr><tr> <td>[17]</td> <td class="space">Olgac, N.; Sipahi, R., An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems, IEEE Transactions on Automatic Control, 47, 793-797 (2002) · <a href="/1364.93576" class="nowrap">Zbl 1364.93576</a></td> </tr><tr> <td>[18]</td> <td class="space">Sipahi, R.; Delice, I. I., Advanced clustering with frequency sweeping methodology for the stability analysis of multiple time-delay systems, IEEE Transactions on Automatic Control, 56, 467-472 (2011) · <a href="/1368.93610" class="nowrap">Zbl 1368.93610</a></td> </tr><tr> <td>[19]</td> <td class="space">Sipahi, R.; Niculescu, S.-I.; Abdallah, C. T.; Michiels, W.; Gu, K., Stability and stabilization of systems with time delay, limitations and opportunities, IEEE Control Systems Magazine, 31, 38-65 (2011) · <a href="/1395.93271" class="nowrap">Zbl 1395.93271</a></td> </tr><tr> <td>[20]</td> <td class="space">Walton, K.; Marshall, J. E., Direct method for TDS stability analysis, Proceedings of the IEE, Part D, 134, 101-107 (1987) · <a href="/0636.93066" class="nowrap">Zbl 0636.93066</a></td> </tr><tr> <td>[21]</td> <td class="space">Welters, A., On explicit recursive formulas in the spectral perturbation analysis of a Jordan block, SIAM Journal on Control and Optimization, 32, 1-22 (2011) · <a href="/1229.15011" class="nowrap">Zbl 1229.15011</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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