Document Zbl 0987.93027

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Control Lett.</a> <a href="/?q=in%3A83748" title="Articles in this Issue">45, No. 2, 97-112 (2002)</a>. </div> <div class="abstract">Summary: We present a method to estimate the domain of attraction for a discrete-time linear system under a saturated linear feedback. A simple condition is derived in terms of an auxiliary feedback matrix for determining if a given ellipsoid is contractively invariant. Moreover, the condition can be expressed as linear matrix inequalities (LMIs) in terms of all the varying parameters and hence can easily be used for controller synthesis. The following surprising result is revealed for systems with single input: suppose that an ellipsoid is made invariant with a linear feedback, then it is invariant under the saturated linear feedback if and only if there exists a saturated (nonlinear) feedback that makes the ellipsoid invariant. Finally, the set invariance condition is extended to determine invariant sets for systems with persistent disturbances. LMI based methods are developed for constructing feedback laws that achieve disturbance rejection with guaranteed stability requirements.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A1715575">Cited in <strong>139</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93B50" title="MSC2020">93B50</a> </td> <td class="space"> Synthesis problems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C55" title="MSC2020">93C55</a> </td> <td class="space"> Discrete-time control/observation systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93D20" title="MSC2020">93D20</a> </td> <td class="space"> Asymptotic stability in control theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93B51" title="MSC2020">93B51</a> </td> <td class="space"> Design techniques (robust design, computer-aided design, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C10" title="MSC2020">93C10</a> </td> <td class="space"> Nonlinear systems in control theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C73" title="MSC2020">93C73</a> </td> <td class="space"> Perturbations in control/observation systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A15A39" title="MSC2020">15A39</a> </td> <td class="space"> Linear inequalities of matrices </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aactuator+saturation">actuator saturation</a>; <a href="/?q=ut%3Astability+analysis">stability analysis</a>; <a href="/?q=ut%3Adisturbance+rejection">disturbance rejection</a>; <a href="/?q=ut%3Aset+invariance">set invariance</a>; <a href="/?q=ut%3Ainvariant+ellipsoid">invariant ellipsoid</a>; <a href="/?q=ut%3Adiscrete-time+linear+system">discrete-time linear system</a>; <a href="/?q=ut%3Adomain+of+attraction">domain of attraction</a>; <a href="/?q=ut%3Asaturated+linear+feedback">saturated linear feedback</a>; <a href="/?q=ut%3Alinear+matrix+inequalities">linear matrix inequalities</a>; <a href="/?q=ut%3Acontroller+synthesis">controller synthesis</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 0987.93027" data-ciurl="/ci/01715575" data-biburl="/bibtex/01715575.bib" data-amsurl="/amsrefs/01715575.bib" data-xmlurl="/xml/01715575.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01715575.pdf" title="Zbl 0987.93027 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/S0167-6911(01)00168-2" aria-label="DOI for “Analysis and design for discrete-time linear systems subject to actuator saturation”" title="10.1016/S0167-6911(01)00168-2">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">Blanchini, F., Ultimate boundedness control for uncertain discrete-time systems via set-induced Lyapunov function, IEEE Trans. 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Control, 36, 1008-1020 (1991) · <a href="/0754.93030" class="nowrap">Zbl 0754.93030</a></td> </tr><tr> <td>[5]</td> <td class="space">H. Hindi, S. Boyd, Analysis of linear systems with saturation using convex optimization, Proceedings of the 37th IEEE CDC, Florida, 1998, pp. 903-908.; H. Hindi, S. Boyd, Analysis of linear systems with saturation using convex optimization, Proceedings of the 37th IEEE CDC, Florida, 1998, pp. 903-908.</td> </tr><tr> <td>[6]</td> <td class="space">Hu, T.; Lin, Z., Control Systems with Actuator Saturation: Analysis and Design (2001), Birkhäuser: Birkhäuser Boston · <a href="/1061.93003" class="nowrap">Zbl 1061.93003</a></td> </tr><tr> <td>[7]</td> <td class="space">T. Hu, Z. Lin, Exact characterization of invariant ellipsoids for linear systems with saturating actuators, IEEE Trans. Automat. Control, to appear.; T. Hu, Z. Lin, Exact characterization of invariant ellipsoids for linear systems with saturating actuators, IEEE Trans. Automat. Control, to appear.</td> </tr><tr> <td>[8]</td> <td class="space">T. Hu, Z. Lin, B.M. Chen, An analysis and design method for linear systems subject to actuator saturation and disturbance, Proceedings of the American Control Conferences, 2000, pp. 725-729; also in Automatica, to appear.; T. Hu, Z. Lin, B.M. Chen, An analysis and design method for linear systems subject to actuator saturation and disturbance, Proceedings of the American Control Conferences, 2000, pp. 725-729; also in Automatica, to appear.</td> </tr><tr> <td>[9]</td> <td class="space">Khalil, H., Nonlinear Systems (1996), Prentice-Hall: Prentice-Hall Upper Saddle River, NJ</td> </tr><tr> <td>[10]</td> <td class="space">Kosut, R. L., Design of linear systems with saturating linear control and bounded states, IEEE Trans. Automat. Control, 28, 1, 121-124 (1983) · <a href="/0501.93053" class="nowrap">Zbl 0501.93053</a></td> </tr><tr> <td>[11]</td> <td class="space">Loparo, K. A.; Blankenship, G. L., Estimating the domain of attraction of nonlinear feedback systems, IEEE Trans. Automat. Control, 23, 4, 602-607 (1978) · <a href="/0385.93023" class="nowrap">Zbl 0385.93023</a></td> </tr><tr> <td>[12]</td> <td class="space">C. Pittet, S. Tarbouriech, C. Burgat, Stability regions for linear systems with saturating controls via circle and Popov criteria, Proceedings of the 36th IEEE CDC, San Diego, 1997, pp. 4518-4523.; C. Pittet, S. Tarbouriech, C. Burgat, Stability regions for linear systems with saturating controls via circle and Popov criteria, Proceedings of the 36th IEEE CDC, San Diego, 1997, pp. 4518-4523.</td> </tr><tr> <td>[13]</td> <td class="space">Romanchuk, B. G., Computing regions for attraction with polytopes: planar case, Automatica, 32, 12, 1727-1732 (1996) · <a href="/0869.93036" class="nowrap">Zbl 0869.93036</a></td> </tr><tr> <td>[14]</td> <td class="space">Vanelli, A.; Vidyasagar, M., Maximal Lyapunov functions and domain of attraction for autonomous nonlinear systems, Automatica, 21, 1, 69-80 (1985) · <a href="/0559.34052" class="nowrap">Zbl 0559.34052</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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