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Control Optim.</a> <a href="/?q=in%3A354067" title="Articles in this Issue">54, No. 2, 567-586 (2016)</a>. </div> <div class="abstract">Summary: This paper proposes a novel structure of Networked Control Systems (NCSs) with communication logic, which incorporates Model-Based Networked Control Systems (MB-NCSs), predictive control, and an event-triggered communication scheme into a unified framework to consider the bandwidth reduction of the network communications. Within this framework, first an event-triggered communication scheme at the sensor side is introduced to determine whether or not the sensor measurement should be transmitted to the controller through the imperfect forward paths. Second, at the controller, a model of the plant is used to predict future state behavior of the plant, and based on the predicted state between successful transmission instants, a novel predictive event-triggering scheme is proposed to compress the size of the packetized control signals transmitted from the controller side to the actuator side through feedback paths. Finally, a unified model of NCSs is established. Based on this model, a codesign condition of the two event-triggering schemes and state feedback controller gain in terms of linear matrix inequality will be derived, using a common quadratic Lyapunov approach. The advantage of the model presented in this paper in comparison to the existing event-based models is that it can compensate for the effect of data-dropout on the closed-loop system actively while requiring less bandwidth without sacrificing stability of the closed-loop system, which will be demonstrated by a practical example at the end of this article.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6559904">Cited in <strong>17</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><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><tr> <td class="mono"> 93B40 </td> <td class="space"> Computational methods in systems theory (MSC2010) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93A15" title="MSC2020">93A15</a> </td> <td class="space"> Large-scale systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C15" title="MSC2020">93C15</a> </td> <td class="space"> Control/observation systems governed by ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93D15" title="MSC2020">93D15</a> </td> <td class="space"> Stabilization of systems by feedback </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aevent-triggered+communication">event-triggered communication</a>; <a href="/?q=ut%3Anetworked+control+systems">networked control systems</a>; <a href="/?q=ut%3Amodel-based+predictive+control">model-based predictive control</a>; <a href="/?q=ut%3Adata-dropout">data-dropout</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <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;"> <!-- alert for connection errors etc --> </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"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1331.93149" data-ciurl="/ci/06559904" data-biburl="/bibtex/06559904.bib" data-amsurl="/amsrefs/06559904.bib" data-xmlurl="/xml/06559904.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06559904.pdf" title="Zbl 1331.93149 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/10.1137/130950951" aria-label="DOI for “Model-based event-triggered predictive control for networked systems with data dropout”" title="10.1137/130950951">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">D. Antunes, J. Hespanha, and C. Silvestre, {\it Stochastic hybrid systems with renewal transitions: Moment analysis with application to networked control systems with delays}, SIAM J. Control Optim., 51 (2013), pp. 1481-1499. · <a href="/1280.60054" class="nowrap">Zbl 1280.60054</a></td> </tr><tr> <td>[2]</td> <td class="space">J. Berglind, T. Gommans, and W. Heemels, {\it Self-triggered MPC for constrained linear systems and quadratic costs}, in Proceedings of the 4th IFAC Nonlinear Model Predictive Control Conference, 2012, pp. 342-348.</td> </tr><tr> <td>[3]</td> <td class="space">P. Cortes, J. Rodriguez, C. Silva, and A. Flores, {\it Delay compensation in model predictive current control of a three-phase inverter}, IEEE Trans. Indust. Electron., 59 (2012), pp. 1323-1325.</td> </tr><tr> <td>[4]</td> <td class="space">H. Gao, X. Meng, T. Chen, and J. Lam, {\it Stabilization of networked control systems via dynamic output-feedback controllers}, SIAM J. Control Optim., 48 (2010), pp. 3643-3658. · <a href="/1201.93102" class="nowrap">Zbl 1201.93102</a></td> </tr><tr> <td>[5]</td> <td class="space">E. Garcia and P. Antsaklis, {\it Model-based event-triggered control with time-varying network delays}, in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), 2011, pp. 1650-1655.</td> </tr><tr> <td>[6]</td> <td class="space">E. Garcia and P. Antsaklis, {\it Model-based event-triggered control for systems with quantization and time-varying network delays}, IEEE Trans. Automat. Control, 58 (2013), pp. 422-434. · <a href="/1369.93408" class="nowrap">Zbl 1369.93408</a></td> </tr><tr> <td>[7]</td> <td class="space">R. Gupta and M. Chow, {\it Networked control system: Overview and research trends}, IEEE Trans. Indust. Electron., 57 (2010), pp. 2527-2535.</td> </tr><tr> <td>[8]</td> <td class="space">W. Heemels and M. Donkers, {\it Model-based periodic event-triggered control for linear systems}, Automatica J. IFAC, 49 (2013), pp. 698-711. · <a href="/1267.93102" class="nowrap">Zbl 1267.93102</a></td> </tr><tr> <td>[9]</td> <td class="space">W. Heemels, M. Donkers, and A. Teel, {\it Periodic event-triggered control for linear systems}, IEEE Trans. Automat. Control, 58 (2013), pp. 847-861. · <a href="/1369.93363" class="nowrap">Zbl 1369.93363</a></td> </tr><tr> <td>[10]</td> <td class="space">W. Heemels, R. Postoyan, M. Donkers, T. Gommans, D. Antunes, P. Tabuada, D. Borgers, K. Johansson, A. Teel, A. Anta, and D. Nesic, {\it Event-triggered and self-triggered control}, Automatica J. IFAC, 50 (2014), pp. 1279-1287. · <a href="/1298.49052" class="nowrap">Zbl 1298.49052</a></td> </tr><tr> <td>[11]</td> <td class="space">S. Hu and D. Yue, {\it Event-triggered control design of linear networked systems with quantizations}, ISA Trans., 51 (2012), pp. 153-162.</td> </tr><tr> <td>[12]</td> <td class="space">S. Kouro, P. Cortés, R. Vargas, U. Ammann, and J. Rodríguez, {\it Model predictive control: A simple and powerful method to control power converters}, IEEE Trans. Indust. Electron., 56 (2009), pp. 1826-1838.</td> </tr><tr> <td>[13]</td> <td class="space">Y. Lan and Y. Zhou, {\it LMI-based robust control of fractional-order uncertain linear systems}, Comput. Math. Appl., 62 (2011), pp. 1460-1471. · <a href="/1228.93087" class="nowrap">Zbl 1228.93087</a></td> </tr><tr> <td>[14]</td> <td class="space">H. Li and Y. Shi, {\it Networked min-max model predictive control of constrained nonlinear systems with delays and packet dropouts}, Internat. J. Control, 86 (2013), pp. 610-624. · <a href="/1278.93115" class="nowrap">Zbl 1278.93115</a></td> </tr><tr> <td>[15]</td> <td class="space">F. Lin, {\it Control of networked discrete event systems: Dealing with communication delays and losses}, SIAM J. Control Optim., 52 (2014), pp. 1276-1298. · <a href="/1292.93083" class="nowrap">Zbl 1292.93083</a></td> </tr><tr> <td>[16]</td> <td class="space">G. Liu, {\it Predictive controller design of networked systems with communication delays and data loss}, IEEE Trans. Circuits Syst. II Express Briefs, 57 (2010), pp. 481-485.</td> </tr><tr> <td>[17]</td> <td class="space">D. Liu and F. Hao, {\it Decentralized event-triggered control strategy in distributed networked systems with delays}, Internat. J. Control Automat. Syst., 11 (2013), pp. 33-40.</td> </tr><tr> <td>[18]</td> <td class="space">B. Liu, Y. Xia, M. Mahmoud, H. Wu, and S. Cui, {\it New predictive control scheme for networked control systems}, Circuits Systems Signal Process., 31 (2012), pp. 945-960. · <a href="/1256.93098" class="nowrap">Zbl 1256.93098</a></td> </tr><tr> <td>[19]</td> <td class="space">L. Montestruque and P. Antsaklis, {\it State and output feedback control in model-based networked control systems}, in Proceedings of the 41st IEEE Conference on Decision and Control, Vol. 2, 2002, pp. 1620-1625.</td> </tr><tr> <td>[20]</td> <td class="space">L. Montestruque and P. Antsaklis, {\it On the model-based control of networked systems}, Automatica J. IFAC, 39 (2003), pp. 1837-1843. · <a href="/1027.93023" class="nowrap">Zbl 1027.93023</a></td> </tr><tr> <td>[21]</td> <td class="space">L. Montestruque and P. Antsaklis, {\it Stability of model-based networked control systems with time-varying transmission times}, IEEE Trans. Automat. Control, 49 (2004), pp. 1562-1572. · <a href="/1365.90039" class="nowrap">Zbl 1365.90039</a></td> </tr><tr> <td>[22]</td> <td class="space">M. Mazo and P. Tabuada, {\it Decentralized event-triggered control over wireless sensor/actuator networks}, IEEE Trans. Automat. Control, 56 (2011), pp. 2456-2461. · <a href="/1368.93358" class="nowrap">Zbl 1368.93358</a></td> </tr><tr> <td>[23]</td> <td class="space">V. Nguyen and Y. Suh, {\it Networked estimation for event-based sampling systems with packet dropouts}, Sensors, 9 (2009), pp. 3078-3089.</td> </tr><tr> <td>[24]</td> <td class="space">A. Onat, T. Naskali, E. Parlakay, and O. Mutluer, {\it Control over imperfect networks: Model-based predictive networked control systems}, IEEE Trans. Indust. Electron., 58 (2011), pp. 905-913.</td> </tr><tr> <td>[25]</td> <td class="space">C. Peng and T. Yang, {\it Event-triggered communication and \(H_∞\) control co-design for networked control systems}, Automatica J. IFAC, 49 (2013), pp. 1326-1332. · <a href="/1319.93022" class="nowrap">Zbl 1319.93022</a></td> </tr><tr> <td>[26]</td> <td class="space">D. Quevedo and D. Nesic, {\it Input-to-state stability of packetized predictive control over unreliable networks affected by packet-dropouts}, IEEE Trans. Automat. Control, 56 (2011), pp. 370-375. · <a href="/1368.93350" class="nowrap">Zbl 1368.93350</a></td> </tr><tr> <td>[27]</td> <td class="space">P. Shi, R. Yang, M. Basin, and H. Karimi, {\it Predictive control of networked systems with communication delays}, in Proceedings of the 2012 IEEE International Symposium on Intelligent Control (ISIC), 2012, pp. 1166-1171.</td> </tr><tr> <td>[28]</td> <td class="space">P. Tabuada, {\it Event-triggered real-time scheduling of stabilizing control tasks}, IEEE Trans. Automat. Control, 52 (2007), pp. 1680-1685. · <a href="/1366.90104" class="nowrap">Zbl 1366.90104</a></td> </tr><tr> <td>[29]</td> <td class="space">X. Wang and M. Lemmon, {\it Event-triggering in distributed networked control systems}, IEEE Trans. Automat. Control, 56 (2011), pp. 586-601. · <a href="/1368.93211" class="nowrap">Zbl 1368.93211</a></td> </tr><tr> <td>[30]</td> <td class="space">Y. Xia, G. Liu, M. Fu, and D. Rees, {\it Predictive control of networked systems with random delay and data dropout}, IET Control Theory Appl., 3 (2009), pp. 1476-1486.</td> </tr><tr> <td>[31]</td> <td class="space">X. Xie, H. Ma, Y. Zhao, D. Ding, and Y. Wang, {\it Control synthesis of discrete-time T-S fuzzy systems based on a novel non-PDC control scheme}, IEEE Trans. Fuzzy Syst., 21 (2013), pp. 147-157.</td> </tr><tr> <td>[32]</td> <td class="space">X. Yin, D. Yue, and S. Hu, {\it Model-based event-triggered predictive control for networked systems with communication delays compensation}, Internat. J. Robust Nonlinear Control, 25 (2015), pp. 3572-3595. · <a href="/1336.93065" class="nowrap">Zbl 1336.93065</a></td> </tr><tr> <td>[33]</td> <td class="space">D. Yue, E. Tian, and Q. Han, {\it A delay system method for designing event-triggered controllers of networked control systems}, IEEE Trans. Automat. Control, 58 (2013), pp. 475-481. · <a href="/1369.93183" class="nowrap">Zbl 1369.93183</a></td> </tr><tr> <td>[34]</td> <td class="space">L. Zhang, H. Gao, and O. Kaynak, {\it Network-induced constraints in networked control systems: A survey}, IEEE Trans. Indust. Inform., 9 (2013), pp. 403-416.</td> </tr><tr> <td>[35]</td> <td class="space">J. Zhang, Y. Xia, and P. Shi, {\it Design and stability analysis of networked predictive control systems}, IEEE Trans. Automat. Control, 21 (2013), pp. 1495-1501.</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. 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