[1705.00056] Stability analysis and stabilization of LPV systems with jumps and (piecewise) differentiable parameters using continuous and sampled-data controllers

<!DOCTYPE html> <html lang="en"> <head> <title>[1705.00056] Stability analysis and stabilization of LPV systems with jumps and (piecewise) differentiable parameters using continuous and sampled-data controllers</title> <meta name="viewport" content="width=device-width, initial-scale=1"> <link rel="apple-touch-icon" sizes="180x180" href="/static/browse/0.3.4/images/icons/apple-touch-icon.png"> <link rel="icon" type="image/png" sizes="32x32" href="/static/browse/0.3.4/images/icons/favicon-32x32.png"> <link rel="icon" type="image/png" sizes="16x16" href="/static/browse/0.3.4/images/icons/favicon-16x16.png"> <link rel="manifest" href="/static/browse/0.3.4/images/icons/site.webmanifest"> <link rel="mask-icon" href="/static/browse/0.3.4/images/icons/safari-pinned-tab.svg" color="#5bbad5"> <meta name="msapplication-TileColor" content="#da532c"> <meta name="theme-color" content="#ffffff"> <link rel="stylesheet" type="text/css" media="screen" href="/static/browse/0.3.4/css/arXiv.css?v=20241206" /> <link rel="stylesheet" type="text/css" media="print" href="/static/browse/0.3.4/css/arXiv-print.css?v=20200611" /> <link rel="stylesheet" type="text/css" media="screen" href="/static/browse/0.3.4/css/browse_search.css" /> <script language="javascript" src="/static/browse/0.3.4/js/accordion.js" /></script> <link rel="canonical" href="https://arxiv.org/abs/1705.00056"/> <meta name="description" content="Abstract page for arXiv paper 1705.00056: Stability analysis and stabilization of LPV systems with jumps and (piecewise) differentiable parameters using continuous and sampled-data controllers"><meta property="og:type" content="website" /> <meta property="og:site_name" content="arXiv.org" /> <meta property="og:title" content="Stability analysis and stabilization of LPV systems with jumps and (piecewise) differentiable parameters using continuous and sampled-data controllers" /> <meta property="og:url" content="https://arxiv.org/abs/1705.00056v5" /> <meta property="og:image" content="/static/browse/0.3.4/images/arxiv-logo-fb.png" /> <meta property="og:image:secure_url" content="/static/browse/0.3.4/images/arxiv-logo-fb.png" /> <meta property="og:image:width" content="1200" /> <meta property="og:image:height" content="700" /> <meta property="og:image:alt" content="arXiv logo"/> <meta property="og:description" content="Linear Parameter-Varying (LPV) systems with jumps and piecewise differentiable parameters is a class of hybrid LPV systems for which no tailored stability analysis and stabilization conditions have been obtained so far. We fill this gap here by proposing an approach based on a clock- and parameter-dependent Lyapunov function yielding stability conditions under both constant and minimum dwell-times. Interesting adaptations of the latter result consist of a minimum dwell-time stability condition for uncertain LPV systems and LPV switched impulsive systems. The minimum dwell-time stability condition is notably shown to naturally generalize and unify the well-known quadratic and robust stability criteria all together. Those conditions are then adapted to address the stabilization problem via timer-dependent and a timer- and/or parameter-independent (i.e. robust) state-feedback controllers, the latter being obtained from a relaxed minimum dwell-time stability condition involving slack-variables. Finally, the last part addresses the stability of LPV systems with jumps under a range dwell-time condition which is then used to provide stabilization conditions for LPV systems using a sampled-data state-feedback gain-scheduled controller. The obtained stability and stabilization conditions are all formulated as infinite-dimensional semidefinite programming problems which are then solved using sum of squares programming. 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We fill this gap here by proposing an approach based on a clock- and parameter-dependent Lyapunov function yielding stability conditions under both constant and minimum dwell-times. Interesting adaptations of the latter result consist of a minimum dwell-time stability condition for uncertain LPV systems and LPV switched impulsive systems. The minimum dwell-time stability condition is notably shown to naturally generalize and unify the well-known quadratic and robust stability criteria all together. Those conditions are then adapted to address the stabilization problem via timer-dependent and a timer- and/or parameter-independent (i.e. robust) state-feedback controllers, the latter being obtained from a relaxed minimum dwell-time stability condition involving slack-variables. Finally, the last part addresses the stability of LPV systems with jumps under a range dwell-time condition which is then used to provide stabilization conditions for LPV systems using a sampled-data state-feedback gain-scheduled controller. The obtained stability and stabilization conditions are all formulated as infinite-dimensional semidefinite programming problems which are then solved using sum of squares programming. 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<div class="authors"><span class="descriptor">Authors:</span><a href="https://arxiv.org/search/math?searchtype=author&query=Briat,+C" rel="nofollow">Corentin Briat</a></div> <div id="download-button-info" hidden>View a PDF of the paper titled Stability analysis and stabilization of LPV systems with jumps and (piecewise) differentiable parameters using continuous and sampled-data controllers, by Corentin Briat</div> <a class="mobile-submission-download" href="/pdf/1705.00056">View PDF</a> <blockquote class="abstract mathjax"> <span class="descriptor">Abstract:</span>Linear Parameter-Varying (LPV) systems with jumps and piecewise differentiable parameters is a class of hybrid LPV systems for which no tailored stability analysis and stabilization conditions have been obtained so far. We fill this gap here by proposing an approach based on a clock- and parameter-dependent Lyapunov function yielding stability conditions under both constant and minimum dwell-times. Interesting adaptations of the latter result consist of a minimum dwell-time stability condition for uncertain LPV systems and LPV switched impulsive systems. The minimum dwell-time stability condition is notably shown to naturally generalize and unify the well-known quadratic and robust stability criteria all together. Those conditions are then adapted to address the stabilization problem via timer-dependent and a timer- and/or parameter-independent (i.e. robust) state-feedback controllers, the latter being obtained from a relaxed minimum dwell-time stability condition involving slack-variables. Finally, the last part addresses the stability of LPV systems with jumps under a range dwell-time condition which is then used to provide stabilization conditions for LPV systems using a sampled-data state-feedback gain-scheduled controller. The obtained stability and stabilization conditions are all formulated as infinite-dimensional semidefinite programming problems which are then solved using sum of squares programming. Examples are given for illustration. </blockquote>  <div class="metatable"> <table summary="Additional metadata"> <tr> <td class="tablecell label">Comments:</td> <td class="tablecell comments mathjax">36 pages, 6 figures, 1 table</td> </tr> <tr> <td class="tablecell label">Subjects:</td> <td class="tablecell subjects"> <span class="primary-subject">Optimization and Control (math.OC)</span>; Systems and Control (eess.SY)</td> </tr><tr> <td class="tablecell label">Cite as:</td> <td class="tablecell arxivid"><span class="arxivid"><a href="https://arxiv.org/abs/1705.00056">arXiv:1705.00056</a> [math.OC]</span></td> </tr> <tr> <td class="tablecell label"> </td> <td class="tablecell arxividv">(or <span class="arxivid"> <a href="https://arxiv.org/abs/1705.00056v5">arXiv:1705.00056v5</a> [math.OC]</span> for this version) </td> </tr> <tr> <td class="tablecell label"> </td> <td class="tablecell arxivdoi"> <a href="https://doi.org/10.48550/arXiv.1705.00056" id="arxiv-doi-link">https://doi.org/10.48550/arXiv.1705.00056</a><div class="button-and-tooltip"> <button class="more-info" aria-describedby="more-info-desc-1"> <svg height="15" role="presentation" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M256 8C119.043 8 8 119.083 8 256c0 136.997 111.043 248 248 248s248-111.003 248-248C504 119.083 392.957 8 256 8zm0 110c23.196 0 42 18.804 42 42s-18.804 42-42 42-42-18.804-42-42 18.804-42 42-42zm56 254c0 6.627-5.373 12-12 12h-88c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h12v-64h-12c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h64c6.627 0 12 5.373 12 12v100h12c6.627 0 12 5.373 12 12v24z" class=""></path></svg> <span class="visually-hidden">Focus to learn more</span> </button>  <div role="tooltip" id="more-info-desc-1"> <span class="left-corner"></span> arXiv-issued DOI via DataCite</div> </div> </td> </tr></table> </div> </div> </div> <div class="submission-history"> <h2>Submission history</h2> From: Corentin Briat Dr [<a href="/show-email/a24479ab/1705.00056" rel="nofollow">view email</a>] <br/> <strong><a href="/abs/1705.00056v1" rel="nofollow">[v1]</a></strong> Fri, 28 Apr 2017 19:53:10 UTC (316 KB)<br/> <strong><a href="/abs/1705.00056v2" rel="nofollow">[v2]</a></strong> Wed, 10 Jan 2018 17:36:36 UTC (317 KB)<br/> <strong><a href="/abs/1705.00056v3" rel="nofollow">[v3]</a></strong> Fri, 12 Oct 2018 11:33:30 UTC (823 KB)<br/> <strong><a href="/abs/1705.00056v4" rel="nofollow">[v4]</a></strong> Tue, 21 Apr 2020 19:17:47 UTC (823 KB)<br/> <strong>[v5]</strong> Thu, 3 Dec 2020 23:48:09 UTC (826 KB)<br/> </div> </div>  <div class="extra-services"> <div class="full-text"> <a name="other"></a> <span class="descriptor">Full-text links:</span> <h2>Access Paper:</h2> <ul> <div id="download-button-info" hidden> View a PDF of the paper titled Stability analysis and stabilization of LPV systems with jumps and (piecewise) differentiable parameters using continuous and sampled-data 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[1705.00056] Stability analysis and stabilization of LPV systems with jumps and (piecewise) differentiable parameters using continuous and sampled-data controllers