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The delayed output measurements are treated as a special output disturbance, and the feedback channel delay is compensated with the aid of an extended functional observer. For the delay in the forward channel, the buffer and packet-based delay compensation approaches are presented, respectively. Then, the stability analysis is performed for the networked closed-loop systems. Finally, a servo motor control system is used to demonstrate the effectiveness of the proposed new design scheme.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6513936">Cited in <strong>27</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><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%3A93C05" title="MSC2020">93C05</a> </td> <td class="space"> Linear systems 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%3A93C95" title="MSC2020">93C95</a> </td> <td class="space"> Application models in control theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aoutput+tracking+control">output tracking control</a>; <a href="/?q=ut%3Anetwork+communication+delay">network communication delay</a>; <a href="/?q=ut%3Anetworked+control+systems">networked control systems</a></div> <div class="software"> <h3>Software:</h3><a href="/software/16358">LabNS2</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 1330.93149" data-ciurl="/ci/06513936" data-biburl="/bibtex/06513936.bib" data-amsurl="/amsrefs/06513936.bib" data-xmlurl="/xml/06513936.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06513936.pdf" title="Zbl 1330.93149 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/10.1016/j.automatica.2015.04.006" aria-label="DOI for “Output tracking control of networked control systems via delay compensation controllers”" title="10.1016/j.automatica.2015.04.006">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">Donkers, M. C.F.; Heemels, W. P. M. H.; Nathan, V. W., Stability analysis of networked control systems using a switched linear systems approach, IEEE Transactions on Automatic Control, 56, 9, 2101-2115 (2011) · <a href="/1368.93465" class="nowrap">Zbl 1368.93465</a></td> </tr><tr> <td>[2]</td> <td class="space">Fernando, T.; Trinh, H., Design of reduced-order state/unknown input observers based on a descriptor system approach, Asian journal of control, 9, 4, 458-465 (2007)</td> </tr><tr> <td>[3]</td> <td class="space">Gao, H.; Chen, T.; Lam, J., A new delay system approach to network-based control, Automatica, 44, 39-52 (2008) · <a href="/1138.93375" class="nowrap">Zbl 1138.93375</a></td> </tr><tr> <td>[4]</td> <td class="space">Gao, H.; Liu, X.; Lam, J., Stability analysis and stabilization for discrete-time fuzzy systems with time-varying delay, IEEE Transactions on Systems, Man and Cybernetics, Part B (Cybernetics), 39, 2, 306-317 (2009)</td> </tr><tr> <td>[5]</td> <td class="space">Guo, Y.; Li, S., A new networked predictive control approach for systems with random network delay in the forward channel, International Journal of Systems Science, 41, 5, 511-520 (2010) · <a href="/1302.93099" class="nowrap">Zbl 1302.93099</a></td> </tr><tr> <td>[6]</td> <td class="space">Ha, Q. P.; That, N. D.; Nam, P. T.; Trinh, H., Partial state estimation for linear systems with output and input time delays, ISA transactions, 53, 2, 327-334 (2014)</td> </tr><tr> <td>[7]</td> <td class="space">Ha, Q. P.; Trinh, H., State and input simultaneous estimation for a class of nonlinear systems, Automatica, 40, 10, 1779-1785 (2004) · <a href="/1088.93004" class="nowrap">Zbl 1088.93004</a></td> </tr><tr> <td>[8]</td> <td class="space">Hu, L.-S.; Bai, T.; Shi, P.; Wu, Z., Sampled-data control of networked linear control systems, Automatica, 43, 903-911 (2007) · <a href="/1117.93044" class="nowrap">Zbl 1117.93044</a></td> </tr><tr> <td>[9]</td> <td class="space">Hu, W.; Liu, G.-P.; Rees, D., Event-driven netowrked predictive control, IEEE Transactions on Industrial Electronics, 54, 3, 1603-1613 (2007)</td> </tr><tr> <td>[10]</td> <td class="space">Hu, S.; Zhu, Q., Stochastic optimal control and analysis of stability of networked control systems with long delay, Automatica, 39, 1877-1884 (2003) · <a href="/1175.93240" class="nowrap">Zbl 1175.93240</a></td> </tr><tr> <td>[11]</td> <td class="space">Imer, O. C.; Yuksel, S.; Basar, T., Optimal control of LTI systems over unreliable communication links, Automatica, 42, 9, 1429-1439 (2006) · <a href="/1128.93368" class="nowrap">Zbl 1128.93368</a></td> </tr><tr> <td>[12]</td> <td class="space">Lian, F.-L.; Moyne, J.; Tilbury, D., Modeling and optimal controller design of networked control systems with multiple delays, International Journal of Control, 76, 6, 591-606 (2003) · <a href="/1050.93038" class="nowrap">Zbl 1050.93038</a></td> </tr><tr> <td>[13]</td> <td class="space">Liu, G.-P.; Mu, J. X.; Rees, D.; Chai, C., Design and stability analysis of networked control systems with random communication time delay using the modified MPC, International Journal of Control, 79, 4, 288-297 (2006) · <a href="/1140.93394" class="nowrap">Zbl 1140.93394</a></td> </tr><tr> <td>[14]</td> <td class="space">Liu, G.-P.; Xia, Y.; Chen, J.; Rees, D.; Hu, W., Design and stability criteria of networked predictive control systems with random network delay in the feedback channel, IEEE Transactions on Systems, Man and Cybernatics C, 37, 2, 173-184 (2007)</td> </tr><tr> <td>[15]</td> <td class="space">Liu, G.-P.; Xia, Y.; Chen, J.; Rees, D.; Hu, W., Networked predictive control of systems with random network delays in both forward and feedback channels, IEEE Transactions on Industrial Electronics, 54, 3, 1282-1297 (2007)</td> </tr><tr> <td>[16]</td> <td class="space">Luan, X.; Shi, P.; Liu, F., Stabilization of networked control systems with random delays, IEEE Transactions on Industrial Electronics, 58, 9, 4323-4330 (2011)</td> </tr><tr> <td>[17]</td> <td class="space">Mkondweni, N. S.; Tzoneva, R., Labns2-co-simulation, co-emulation, and real-time control toolkit for investigation of network induced time delays and packet loss in networked control systems, International Journal of Innovative Computing, Information and Control, 10, 1, 1-18 (2014)</td> </tr><tr> <td>[18]</td> <td class="space">Nilsson, J.; Bernhardsson, B.; Wittenmark, B., Stochastic analysis and control of real-time systems with random time delays, Automatica, 34, 1, 57-64 (1998) · <a href="/0908.93073" class="nowrap">Zbl 0908.93073</a></td> </tr><tr> <td>[19]</td> <td class="space">Pin, G.; Parisini, T., Networked predictive control of uncertain constrained nonlinear systems: recursive feasibility and input-to-state stability analysis, IEEE Transactions on Automatic Control, 56, 1, 72-87 (2011) · <a href="/1368.93646" class="nowrap">Zbl 1368.93646</a></td> </tr><tr> <td>[20]</td> <td class="space">Qiu, J.; Feng, G.; Gao, H., Fuzzy-model-based piecewise \(H_\infty\) static output feedback controller design for networked nonlinear systems, IEEE Transactions on Fuzzy Systems, 18, 5, 919-934 (2010)</td> </tr><tr> <td>[21]</td> <td class="space">Sun, X. M.; Liu, G.-P.; Wang, W.; Rees, D., Stability analysis for networked control systems based on average dwell time method, International Journal of Robust and Nonlinear Control, 20, 15, 1774-1784 (2010) · <a href="/1204.93052" class="nowrap">Zbl 1204.93052</a></td> </tr><tr> <td>[22]</td> <td class="space">Trinh, H.; Fernando, T., Functional observers for dynamical systems (2012), Springer · <a href="/1244.93002" class="nowrap">Zbl 1244.93002</a></td> </tr><tr> <td>[23]</td> <td class="space">Vesely, V.; Rosinova, D.; Quang, T. N., Networked output feedback robust predictive controller design, International Journal of Innovative Computing, Information and Control, 9, 10, 3941-3953 (2013)</td> </tr><tr> <td>[24]</td> <td class="space">Wang, R.; Liu, G.-P.; Wang, W.; Rees, D.; Zhao, Y.-B., \(H_\infty\) control for networked predictive control systems based on the switched lyapunov function method, IEEE Transactions on Industrial Electronics, 57, 10, 3565-3571 (2010)</td> </tr><tr> <td>[25]</td> <td class="space">Wang, R.; Liu, G.-P.; Wang, B.; Wang, W.; Rees, D., \(l_2\)-gain analysis for networked predictive control systems based on switching method, International Journal of Control, 82, 6, 1148-1156 (2009) · <a href="/1168.93383" class="nowrap">Zbl 1168.93383</a></td> </tr><tr> <td>[26]</td> <td class="space">Xia, Y.; Fu, M.; Liu, G.-P., Analysis and synthesis of networked control systems (2011), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · <a href="/1228.93004" class="nowrap">Zbl 1228.93004</a></td> </tr><tr> <td>[27]</td> <td class="space">Xia, Y.; Shang, J.; Chen, J.; Liu, G.-P., Networked data fusion with packet losses and variable delays, IEEE Transactions on Systems, Man and Cybernetics, Part B (Cybernetics), 39, 5, 1107-1120 (2009)</td> </tr><tr> <td>[28]</td> <td class="space">Yue, D.; Han, Q.-L.; Lam, J., Network-based robust \(H_\infty\) control of systems with uncertainty, Automatica, 41, 6, 999-1007 (2005) · <a href="/1091.93007" class="nowrap">Zbl 1091.93007</a></td> </tr><tr> <td>[29]</td> <td class="space">Yue, D.; Tian, E.; Zhang, Y.; Peng, C., Delay-distribution-dependent stability and stabilization of T-S fuzzy systems with probabilistic interval delay, IEEE Transactions on Systems, Man and Cybernetics, Part B (Cybernetics), 39, 2, 503-516 (2009)</td> </tr><tr> <td>[30]</td> <td class="space">Zhang, J.; Lam, J.; Xia, Y., Observer-based output feedback control for discrete systems with quantized inputs, IET Control Theory & Applications, 5, 3, 478-485 (2011)</td> </tr><tr> <td>[31]</td> <td class="space">Zhang, J.; Shi, P.; Xia, Y., Fuzzy delay compensation control for T-S fuzzy systems over network, IEEE Transactions on Cybernetics, 43, 1, 259-268 (2013)</td> </tr><tr> <td>[32]</td> <td class="space">Zhang, J.; Xia, Y., Design of \(H_\infty\) fuzzy controllers for nonlinear systems with random data dropouts, Optimal Control, Applications and Methods, 32, 3, 328-349 (2011) · <a href="/1217.93054" class="nowrap">Zbl 1217.93054</a></td> </tr><tr> <td>[33]</td> <td class="space">Zhang, J.; Xia, Y.; Shi, P., Design and stability analysis of networked predictive control systems, IEEE Transactions on Control Systems Technology, 21, 4, 1495-1501 (2013)</td> </tr><tr> <td>[34]</td> <td class="space">Zhang, W.-A.; Yu, L., A robust control approach to stabilization of networked control systems with time-varying delays, Automatica, 45, 10, 2440-2445 (2009) · <a href="/1183.93051" class="nowrap">Zbl 1183.93051</a></td> </tr><tr> <td>[35]</td> <td class="space">Zhao, Y.-B.; Kim, J.; Liu, G.-P., Error bounded sensing for packet-based networked control systems, IEEE Transactions on Industrial Electronics, 58, 5, 1980-1989 (2011)</td> </tr><tr> <td>[36]</td> <td class="space">Zhao, Y.-B.; Liu, G.-P.; Rees, D., Networked predictive control systems based on hammerstein model, IEEE Transactions on Circuits and Systems II Express Briefs, 55, 5, 469-473 (2008)</td> </tr><tr> <td>[37]</td> <td class="space">Zhao, Y.-B.; Liu, G.-P.; Rees, D., A predictive control-based approach to networked hammerstein systems: Design and stability analysis, IEEE Transactions on Systems, Man and Cybernetics, Part B (Cybernetics), 38, 3, 700-708 (2008)</td> </tr><tr> <td>[38]</td> <td class="space">Zhao, Y.-B.; Liu, G.-P.; Rees, D., Modeling and stabilization of continuous-time packet-based networked control systems, IEEE Transactions on Systems, Man and Cybernetics, Part B (Cybernetics), 39, 6, 1646-1652 (2009)</td> </tr><tr> <td>[39]</td> <td class="space">Zhao, Y.-B.; Liu, G.-P.; Rees, D., Design of a packet-based control framework for networked control systems, IEEE Transactions on Control Systems Technology, 17, 4, 859-865 (2009)</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. 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