Document Zbl 1338.93348

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J. Robust Nonlinear Control</a> <a href="/?q=in%3A349975" title="Articles in this Issue">25, No. 17, 3422-3438 (2015)</a>. </div> <div class="abstract">Summary: The problem of $H_\infty$ control for networked Markovian jump system under event-triggered scheme is studied in this paper. In order to reduce the utilization of limited network bandwidth, a dynamic discrete event-triggered scheme to choose the transmitted data is designed. A Markovian jump time-delay system model is employed to describe the event-triggered scheme and the network related behavior, such as transmission delay, data package dropout, and disorder. Furthermore, a sufficient condition is derived to guarantee that the resulting closed-loop system is stable and has a prescribed performance index. A co-design method for the $H_\infty$ controller and the event-triggered scheme is then proposed. The effectiveness and potential of the theoretic results obtained are illustrated by a simulation example.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6575398">Cited in <strong>69</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93E03" title="MSC2020">93E03</a> </td> <td class="space"> Stochastic systems in control theory (general) </td> </tr><tr> <td class="mono"> 60J75 </td> <td class="space"> Jump processes (MSC2010) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C65" title="MSC2020">93C65</a> </td> <td class="space"> Discrete event control/observation systems </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Anetworked+control+systems">networked control systems</a>; <a href="/?q=ut%3Aevent-triggered+scheme">event-triggered scheme</a>; <a href="/?q=ut%3A%5C%28H_%5Cinfty%5C%29+control">$H_\infty$ control</a>; <a href="/?q=ut%3AMarkovian+jump+systems">Markovian jump systems</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 1338.93348" data-ciurl="/ci/06575398" data-biburl="/bibtex/06575398.bib" data-amsurl="/amsrefs/06575398.bib" data-xmlurl="/xml/06575398.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06575398.pdf" title="Zbl 1338.93348 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/10.1002/rnc.3273" aria-label="DOI for “Event-triggered control for networked Markovian jump systems”" title="10.1002/rnc.3273">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">Shi, P.; Boukas, EK; Agarwal, RK, Control of Markovian jump discrete‐time systems with norm bounded uncertainty and unknown delay, IEEE Transactions on Automatic Control, 44, 11, 2139-2144, 1999 · <a href="/1078.93575" class="nowrap">Zbl 1078.93575</a></td> </tr><tr> <td>[2]</td> <td class="space">Zhang, L., H_∞ estimation of piecewise homogeneous Markov jump linear systems, Automatica, 45, 11, 2570-2576, 2009 · <a href="/1180.93100" class="nowrap">Zbl 1180.93100</a></td> </tr><tr> <td>[3]</td> <td class="space">Wang, Y.; Wang, C.; Zuo, Z., Controller synthesis for Markovian jump systems with incomplete knowledge of transition probabilities and actuator saturation, Journal of the Franklin Institute, 348, 2417-2429, 2011 · <a href="/1239.93128" class="nowrap">Zbl 1239.93128</a></td> </tr><tr> <td>[4]</td> <td class="space">Wang, J.; Wang, H.; Lu, R.; Xue, A., Delay‐dependent, H_∞ control for singular Markovian jump systems with time delay, Nonlinear Analysis, Hybrid Systems, 8, 1-12, 2013 · <a href="/1257.93032" class="nowrap">Zbl 1257.93032</a></td> </tr><tr> <td>[5]</td> <td class="space">Yin, Y.; Shi, P.; Liu, F.; Teo, KL, Filtering for discrete‐time nonhomogeneous Markov jump systems with uncertainties, Information Sciences, 259, 118-127, 2014 · <a href="/1333.60078" class="nowrap">Zbl 1333.60078</a></td> </tr><tr> <td>[6]</td> <td class="space">Shi, P.; Yin, Y.; Liu, F.; Zhang, J., Robust control on saturated Markov jump systems with missing information, Information Sciences, 265, 123-138, 2014 · <a href="/1327.93400" class="nowrap">Zbl 1327.93400</a></td> </tr><tr> <td>[7]</td> <td class="space">Zhang, L.; Boukas, EK, H_∞ control for discrete‐time Markovian jump linear systems with partly unknown transition probabilities, International Journal of Robust and Nonlinear Control, 19, 5, 868-883, 2009 · <a href="/1166.93320" class="nowrap">Zbl 1166.93320</a></td> </tr><tr> <td>[8]</td> <td class="space">Li, B.; Teo, KL; Lim, CC; Duan, GR, An optimal PID controller design for nonlinear constrained optimal control problems, Discrete and Continuous Dynamical System ‐ Series B, 16, 4, 1101-1118, 2011 · <a href="/1234.49031" class="nowrap">Zbl 1234.49031</a></td> </tr><tr> <td>[9]</td> <td class="space">Hespanha, JP; Naghshtabrizi, P.; Xu, YG, A survey of recent results in networked control systems, Journal of the IEEE, 95, 138-162, 2007</td> </tr><tr> <td>[10]</td> <td class="space">Yang, R.; Shi, P.; Liu, G.; Gao, H., Network‐based feedback control for systems with mixed delays based on quantization and dropout compensation, Automatica, 7, 2805-2809, 2011 · <a href="/1235.93112" class="nowrap">Zbl 1235.93112</a></td> </tr><tr> <td>[11]</td> <td class="space">Xia, Y.; Yan, J.; Shi, P.; Fu, M., Stability analysis of discrete‐time systems with quantized feedback and measurements, IEEE Transactions on Industrial Informatics, 9, 313-324, 2013</td> </tr><tr> <td>[12]</td> <td class="space">Xu, Y.; Su, H.; Pan, Y.; Wu, Z., Robust H_∞ filtering for networked stochastic systems with randomly occurring sensor nonlinearities and packet dropouts, Signal Processing, 93, 1794-1803, 2013</td> </tr><tr> <td>[13]</td> <td class="space">Zhang, W.; Yu, L., A robust control approach to stabilization of networked control systems with time‐varying delays, Automatica, 45, 2440-2445, 2009 · <a href="/1183.93051" class="nowrap">Zbl 1183.93051</a></td> </tr><tr> <td>[14]</td> <td class="space">Vesely, V.; Rosinova, D.; Quang, TN, Networked output feedback robust predictive controller design, International Journal of Innovative Computing, Information and Control, 9, 10, 3941-3953, 2013</td> </tr><tr> <td>[15]</td> <td class="space">Zhang, L.; Gao, H.; Kaynak, O., Network‐induced constraints in networked control system – a survey, IEEE Transactions on Industrial Informatics, 9, 1, 403-416, 2013</td> </tr><tr> <td>[16]</td> <td class="space">Benitez‐Perez, H.; Benitez‐Perez, A.; Ortega‐Arjona, I., Networked control systems design considering scheduling restrictions and local faults using local state estimation, International Journal of Innovative Computing, Information and Control, 9, 8, 3225-3239, 2013</td> </tr><tr> <td>[17]</td> <td class="space">Liu, M.; Ho, DWC; Niu, Y., Stabilization of Markovian jump linear system over networks with random communication delay, Automatica, 45, 2, 416-421, 2009 · <a href="/1158.93412" class="nowrap">Zbl 1158.93412</a></td> </tr><tr> <td>[18]</td> <td class="space">Song, H.; Yu, L.; Zhang, W., H_∞ filtering of network‐based systems with random delay, Signal Processing, 89, 615-622, 2009 · <a href="/1157.93529" class="nowrap">Zbl 1157.93529</a></td> </tr><tr> <td>[19]</td> <td class="space">Goncalves, APC; Fioravanti, AR; Geromel, JC, Markov jump linear systems and filtering through network transmitted measurements, Signal Processing, 90, 2842-2850, 2010 · <a href="/1197.94056" class="nowrap">Zbl 1197.94056</a></td> </tr><tr> <td>[20]</td> <td class="space">Xiao, N.; Xie, L.; Fu, M., Stabilization of Markov jump linear systems using quantized state feedback, Automatica, 46, 1696-1702, 2010 · <a href="/1204.93127" class="nowrap">Zbl 1204.93127</a></td> </tr><tr> <td>[21]</td> <td class="space">Zhang, J.; Lam, J.; Xia, Y., Output feedback delay compensation control for networked control systems with random delays, Information Sciences, 265, 154-166, 2014 · <a href="/1327.93363" class="nowrap">Zbl 1327.93363</a></td> </tr><tr> <td>[22]</td> <td class="space">Tabuada, P., Event‐triggered real‐time scheduling of stabilizing control tasks, IEEE Transactions on Automatic Control, 52, 1680-1685, 2007 · <a href="/1366.90104" class="nowrap">Zbl 1366.90104</a></td> </tr><tr> <td>[23]</td> <td class="space">Donkers, MCF; Heemels, W., Output‐based event‐triggered control with guaranteed L_∞‐gain and improved and decentralized event‐triggering, IEEE Transactions on Automatic Control, 57, 1362-1376, 2012 · <a href="/1369.93362" class="nowrap">Zbl 1369.93362</a></td> </tr><tr> <td>[24]</td> <td class="space">Wang, X.; Lemmon, MD, Event‐triggering in distributed networked control systems, IEEE Transactions on Automatic Control, 56, 586-601, 2011 · <a href="/1368.93211" class="nowrap">Zbl 1368.93211</a></td> </tr><tr> <td>[25]</td> <td class="space">Guo, G.; Lu, Z.; Shi, P., Event‐driven actuators: to zero or to hold?, International Journal of Robust and Nonlinear Control, 2013 · <a href="/1305.93179" class="nowrap">Zbl 1305.93179</a></td> </tr><tr> <td>[26]</td> <td class="space">Yue, D.; Tian, E.; Han, QL, A delay system method for designing event‐triggered controllers of networked control systems, IEEE Transactions on Automatic Control, 58, 475-481, 2013 · <a href="/1369.93183" class="nowrap">Zbl 1369.93183</a></td> </tr><tr> <td>[27]</td> <td class="space">Peng, C.; Han, QL; Yue, D., To transmit or not to transmit: a discrete event‐triggered communication scheme for networked Takagi‐Sugeno fuzzy systems, IEEE Transactions on Fuzzy Systems, 21, 164-170, 2013</td> </tr><tr> <td>[28]</td> <td class="space">Peng, C.; Yang, TC, Event‐triggered communication and H_∞ control co‐design for networked control systems, Automatica, 49, 1326-1332, 2013 · <a href="/1319.93022" class="nowrap">Zbl 1319.93022</a></td> </tr><tr> <td>[29]</td> <td class="space">Hu, S.; Zhang, Y.; Du, Z., Network‐based H_∞ tracking control with event‐triggering sampling scheme, IET Control Theory and Applications, 6, 4, 533-544, 2012</td> </tr><tr> <td>[30]</td> <td class="space">Zhang, X.; Han, QL, Network‐based H_∞ filtering using a logic jumping‐like trigger, Automatica, 49, 1428-1435, 2013 · <a href="/1319.93076" class="nowrap">Zbl 1319.93076</a></td> </tr><tr> <td>[31]</td> <td class="space">Wang, Y.; Xie, L.; de Souza, CE, Robust control of a class of uncertain nonlinear systems, Systems and Control Letters, 19, 2, 139-149, 1992 · <a href="/0765.93015" class="nowrap">Zbl 0765.93015</a></td> </tr><tr> <td>[32]</td> <td class="space">Xiong, J.; Lam, L., Stabilization of networked control systems with a logic ZOH, IEEE Transactions on Automatic Control, 54, 2, 358-363, 2009 · <a href="/1367.93546" class="nowrap">Zbl 1367.93546</a></td> </tr><tr> <td>[33]</td> <td class="space">Oliveira, RCLF; Peres, PLD, LMI conditions for robust stability analysis based on polynomially parameter‐dependent Lyapunov functions, System & Control Letters, 55, 52-61, 2006 · <a href="/1129.93485" class="nowrap">Zbl 1129.93485</a></td> </tr><tr> <td>[34]</td> <td class="space">Ghaoui, LE; Oustry, F.; AitRami, M., A cone complementarity linearization algorithm for static output‐feedback and related problems, IEEE Transactions on Automatic Control, 42, 8, 1171-1176, 1997 · <a href="/0887.93017" class="nowrap">Zbl 0887.93017</a></td> </tr><tr> <td>[35]</td> <td class="space">Park, PG; Ko, JW; Jeong, C., Reciprocally convex approach to stability of systems with time‐varying delays, Automatica, 47, 235-238, 2011 · <a href="/1209.93076" class="nowrap">Zbl 1209.93076</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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