Document Zbl 1189.93045

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Math. Appl.</a> <a href="/?q=in%3A259795" title="Articles in this Issue">59, No. 5, 1656-1666 (2010)</a>. </div> <div class="abstract">Summary: This paper tackles the problem of tuning a class of fractional controllers by using frequency specifications. These tuning strategies exhibit the advantage of allowing the design of robustness features in the closed-loop control system. Nowadays, relationships between time and frequency domain specifications are given by complicated formulas in the case of integer controllers, which become even more complicated if fractional controllers are used. The proposed strategy consists in finding the family of fractional controllers (in function of the fractional integrator) that makes the closed-loop system have the same time domain specifications: overshoot and settling time. Such family of controllers is obtained by minimizing a function based on these time domain specifications. Our method has been particularized to first order plants, controlled by $PIU^\alpha$ regulators, although the results can be easily extended to higher order plants and other fractional controllers. The obtained results provide simple expressions for tuning these controllers.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5727554">Cited in <strong>7</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></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afractional+PI+controller">fractional PI controller</a>; <a href="/?q=ut%3Afrequency+specifications">frequency specifications</a>; <a href="/?q=ut%3Atime+specification">time specification</a>; <a href="/?q=ut%3Acrossover+frequency">crossover frequency</a>; <a href="/?q=ut%3Aphase+margin">phase margin</a>; <a href="/?q=ut%3Aovershot">overshot</a>; <a href="/?q=ut%3Asettling+time">settling time</a></div> <div class="software"> <h3>Software:</h3><a href="/software/7382">CRONE</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 1189.93045" data-ciurl="/ci/05727554" data-biburl="/bibtex/05727554.bib" data-amsurl="/amsrefs/05727554.bib" data-xmlurl="/xml/05727554.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05727554.pdf" title="Zbl 1189.93045 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.camwa.2009.08.007" aria-label="DOI for “Design of a class of fractional controllers from frequency specifications with guaranteed time domain behavior”" title="10.1016/j.camwa.2009.08.007">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">Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations to Methods of their Solution and Some of their Applications (1999), Academic Press: Academic Press San Diego · <a href="/0924.34008" class="nowrap">Zbl 0924.34008</a></td> </tr><tr> <td>[2]</td> <td class="space">Vinagre, B. M.; Podlubny, I.; Hernández, I. A.; Feliu, V., Some approximations of fractional order operators used in control theory and applications, Fract. Calc. Appl. Anal., 3, 231-248 (2000) · <a href="/1111.93302" class="nowrap">Zbl 1111.93302</a></td> </tr><tr> <td>[3]</td> <td class="space">F. J. Castillo, R. Rivas, V. Feliu, Fractional IIa controller combined with a Smith predictor for effective water distribution in a main irrigation canal pool, in: 17th IFAC World Congress, Seoul, Korea, 2008; F. J. Castillo, R. Rivas, V. Feliu, Fractional IIa controller combined with a Smith predictor for effective water distribution in a main irrigation canal pool, in: 17th IFAC World Congress, Seoul, Korea, 2008</td> </tr><tr> <td>[4]</td> <td class="space">V. Feliu, R. Rivas, F. J. Castillo, L. Sánchez, Smith predictor based robust fractional order control. Application to water distribution in irrigation main canals, in: Process Control, Elsevier, 2008 (in press); V. Feliu, R. Rivas, F. J. Castillo, L. Sánchez, Smith predictor based robust fractional order control. Application to water distribution in irrigation main canals, in: Process Control, Elsevier, 2008 (in press)</td> </tr><tr> <td>[5]</td> <td class="space">Oustaloup, A., La commande CRONE: Commande robuste d’ordre non entire (1991), Editions Hermes · <a href="/0864.93003" class="nowrap">Zbl 0864.93003</a></td> </tr><tr> <td>[6]</td> <td class="space">Monje, C. A.; Calderon, A. J.; Vinagre, B. M.; Chen, Y.; Feliu, V., On fractional PI controllers: Some tuning rules for robustness to plant uncertainties, Nonlinear Dynam., 38, 1-2, 369-381 (2004) · <a href="/1134.93338" class="nowrap">Zbl 1134.93338</a></td> </tr><tr> <td>[7]</td> <td class="space">Y. Chen, H. Dou, B. M. Vinagre, C. A. Monje, A robust tuning method for fractional orderr PI controllers, in: 2nd IFAC Workshop on Fractional Differentiation and its Application, Porto, Portugal, 2006; Y. Chen, H. Dou, B. M. Vinagre, C. A. Monje, A robust tuning method for fractional orderr PI controllers, in: 2nd IFAC Workshop on Fractional Differentiation and its Application, Porto, Portugal, 2006</td> </tr><tr> <td>[8]</td> <td class="space">Oustaloup, A.; Moreau, X.; Nouillant, M., The CRONE suspension—Control engineering practice, J. IFAC, 4, 8, 1101-1108 (1996)</td> </tr><tr> <td>[9]</td> <td class="space">R. Caponetto, L. Fortuna, D. Porto, A new tuning strategy for a non integer order PID, in: 1st IFAC Workshop on Fractional Differentiation and its Application, Bordeaux, France, 2004; R. Caponetto, L. Fortuna, D. Porto, A new tuning strategy for a non integer order PID, in: 1st IFAC Workshop on Fractional Differentiation and its Application, Bordeaux, France, 2004</td> </tr><tr> <td>[10]</td> <td class="space">A. A. Oustaloup, O. Cois, P. Lanusse, P. Melchior, X. Moreau, J. Sabatier, The CRONE approach: Theoretical developments and major applications, in: Proceedings of the 2nd Workshop of the International Federation of Automatic Control on Fractional Differentiation and its Applications, Porto, Portugal, 2006; A. A. Oustaloup, O. Cois, P. Lanusse, P. Melchior, X. Moreau, J. Sabatier, The CRONE approach: Theoretical developments and major applications, in: Proceedings of the 2nd Workshop of the International Federation of Automatic Control on Fractional Differentiation and its Applications, Porto, Portugal, 2006</td> </tr><tr> <td>[11]</td> <td class="space">C. A. Monje, B. M. Vinagre, V. Feliu, Y. Chen, On auto-tuning of order PID controllers, in: 2nd IFAC Workshop on Fractional Differentiation and its Application, Porto, Portugal, 2006; C. A. Monje, B. M. Vinagre, V. Feliu, Y. 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M., Quantitative synthesis of uncertain multiple input-output feedback systems, Internat. J. Control, 30, 1, 81-106 (1979) · <a href="/0407.93022" class="nowrap">Zbl 0407.93022</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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