Document Zbl 1411.93191

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Franklin Inst.</a> <a href="/?q=in%3A432052" title="Articles in this Issue">356, No. 5, 2742-2758 (2019)</a>. </div> <div class="abstract">Summary: This paper investigates the non-fragile control for positive Markovian jump systems both in continuous-time and discrete-time cases with actuator uncertainty. It is assumed that the coefficient matrices of the non-fragile controller is unknown and bounded. The state-feedback controller gain consists of nominal controller gain and gain perturbation. First, a set of state-feedback controllers for the considered system are designed by using a stochastic co-positive Lyapunov function integrated with linear programming approach. Under the designed controllers, the resulting closed-loop systems are positive and stochastically stable. Then, the proposed controller design approach is extended to discrete-time systems. Through comparisons, it is shown that existing results are special cases of the presented ones in the paper. Finally, two examples are given to illustrate the effectiveness of the proposed design.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7046003">Cited in <strong>12</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93E15" title="MSC2020">93E15</a> </td> <td class="space"> Stochastic stability in control theory </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%3A93B52" title="MSC2020">93B52</a> </td> <td class="space"> Feedback control </td> </tr><tr> <td class="mono"> 60J75 </td> <td class="space"> Jump processes (MSC2010) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Anon-fragile+control">non-fragile control</a>; <a href="/?q=ut%3Apositive+markovian+jump+systems">positive markovian jump systems</a>; <a href="/?q=ut%3Astate-feedback+controller+gain">state-feedback controller gain</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 1411.93191" data-ciurl="/ci/07046003" data-biburl="/bibtex/07046003.bib" data-amsurl="/amsrefs/07046003.bib" data-xmlurl="/xml/07046003.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07046003.pdf" title="Zbl 1411.93191 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.jfranklin.2019.02.008" aria-label="DOI for “Non-fragile control of positive Markovian jump systems”" title="10.1016/j.jfranklin.2019.02.008">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">Farina, L.; Rinaldi, S., Positive Linear Systems: Theory and Applications (2000), Wiley: Wiley New York, USA · <a href="/0988.93002" class="nowrap">Zbl 0988.93002</a></td> </tr><tr> <td>[2]</td> <td class="space">Kaczorek, T., Positive 1D and 2D Systems (2002), Springer-Verlag: Springer-Verlag London, UK · <a href="/1005.68175" class="nowrap">Zbl 1005.68175</a></td> </tr><tr> <td>[3]</td> <td class="space">Rami, M. A.; Tadeo, F., Controller synthesis for positive linear systems with bounded controls, IEEE Trans. Circuits Syst. II Express Briefs, 54, 2, 151-155 (2007)</td> </tr><tr> <td>[4]</td> <td class="space">Liu, X.; Yu, W.; Wang, L., Stability analysis for continuous-time positive systems with time-varying delays, IEEE Trans. Autom. Control, 55, 4, 1024-1028 (2010) · <a href="/1368.93600" class="nowrap">Zbl 1368.93600</a></td> </tr><tr> <td>[5]</td> <td class="space">Zhao, X.; Yin, Y.; Zheng, X., State-dependent switching control of switched positive fractional-order systems, ISA Trans., 62, 103-108 (2016)</td> </tr><tr> <td>[6]</td> <td class="space">Shen, J.; Lam, J., Some extensions on the bounded real lemma for positive systems, IEEE Trans. Autom. Control, 62, 6, 3034-3038 (2017) · <a href="/1369.93180" class="nowrap">Zbl 1369.93180</a></td> </tr><tr> <td>[7]</td> <td class="space">Bolzern, P.; Colaneri, P.; De Nicolao, G., Stochastic stability of positive Markov jump linear systems, Automatica, 50, 4, 1181-1187 (2014) · <a href="/1298.93336" class="nowrap">Zbl 1298.93336</a></td> </tr><tr> <td>[8]</td> <td class="space">Bolzern, P.; Colaneri, P., Positive Markov jump linear systems, Found. Trends Syst. Control, 2, 3-4, 275-427 (2015)</td> </tr><tr> <td>[9]</td> <td class="space">Rami, M. A.; Shamma, J., Hybrid positive systems subject to Markovian switching, IFAC Proc., 42, 17, 138-143 (2009)</td> </tr><tr> <td>[10]</td> <td class="space">Zhang, J.; Han, Z.; Zhu, F., Stochastic stability and stabilization of positive systems with Markovian jump parameters, Nonlinear Anal. Hybrid Syst., 12, 1, 147-155 (2014) · <a href="/1291.93323" class="nowrap">Zbl 1291.93323</a></td> </tr><tr> <td>[11]</td> <td class="space">Zhu, S.; Han, Q.; Zhang, C., $L_1$-gain performance analysis and positive filter design for positive discrete-time Markov jump linear systems: A linear programming approach, Automatica, 50, 8, 2098-2107 (2014) · <a href="/1297.93168" class="nowrap">Zbl 1297.93168</a></td> </tr><tr> <td>[12]</td> <td class="space">Qi, W.; Gao, X., State feedback controller design for singular positive Markovian jump systems with partly known transition rates, Appl. Math. Lett., 46, 111-116 (2015) · <a href="/1319.93079" class="nowrap">Zbl 1319.93079</a></td> </tr><tr> <td>[13]</td> <td class="space">Park, I.; Kwon, N.; Park, P., A linear programming approach for stabilization of positive Markovian jump systems with a saturated single input, Nonlinear Anal. Hybrid Syst., 29, 322-332 (2018) · <a href="/06892053" class="nowrap">Zbl 06892053</a></td> </tr><tr> <td>[14]</td> <td class="space">Li, S.; Xiang, Z., Stochastic stability analysis and $L_∞$-gain controller design for positive Markov jump systems with time-varying delays, Nonlinear Anal. Hybrid Syst., 22, 31-42 (2016) · <a href="/1344.93107" class="nowrap">Zbl 1344.93107</a></td> </tr><tr> <td>[15]</td> <td class="space">Qi, W.; Gao, X., $L_1$ control for positive Markovian jump systems with time-varying delays and partly known transition rate, Circuits Syst. Signal Process., 34, 8, 2711-2716 (2015) · <a href="/1341.93101" class="nowrap">Zbl 1341.93101</a></td> </tr><tr> <td>[16]</td> <td class="space">Zhu, S.; Han, Q.; Zhang, C., $L_1$-stochastic stability and $L_1$-gain performance of positive Markov jump linear systems with time-delays: Necessary and sufficient conditions, IEEE Trans. Autom. Control, 62, 7, 3634-3639 (2017) · <a href="/1370.93311" class="nowrap">Zbl 1370.93311</a></td> </tr><tr> <td>[17]</td> <td class="space">Tao, G.; Chen, S.; Joshi, S., An adaptive control scheme for systems with unknown actuator failures, Automatica, 38, 6, 1027-1034 (2002) · <a href="/1015.93030" class="nowrap">Zbl 1015.93030</a></td> </tr><tr> <td>[18]</td> <td class="space">Li, X.; Yang, G., Robust adaptive fault-tolerant control for uncertain linear systems with actuator failures, IET Control Theory Appl., 6, 10, 1544-1551 (2012)</td> </tr><tr> <td>[19]</td> <td class="space">Wang, L.; Shao, C., Exponential stabilisation for time-varying delay system with actuator faults: An average dwell time method, Int. J. Syst. Sci., 41, 4, 435-445 (2010) · <a href="/1301.93137" class="nowrap">Zbl 1301.93137</a></td> </tr><tr> <td>[20]</td> <td class="space">Kaviarasan, B.; Sakthivel, R.; Kwon, O., Robust fault-tolerant control for power systems against mixed actuator failures, Nonlinear Anal. Hybrid Syst., 22, 249-261 (2016) · <a href="/1344.93039" class="nowrap">Zbl 1344.93039</a></td> </tr><tr> <td>[21]</td> <td class="space">Yu, L., An LMI approach to reliable guaranteed cost control of discrete-time systems with actuator failure, Appl. Math. Comput., 162, 3, 1325-1331 (2005) · <a href="/1125.93046" class="nowrap">Zbl 1125.93046</a></td> </tr><tr> <td>[22]</td> <td class="space">Hou, Y., Reliable stabilization of networked control systems with actuator faults, Nonlinear Dyn., 71, 3, 447-455 (2013)</td> </tr><tr> <td>[23]</td> <td class="space">Seo, C.; Kim, B., Robust and reliable $H_∞$ control for linear systems with parameter uncertainty and actuator failure, Automatica, 32, 3, 465-467 (1996) · <a href="/0850.93214" class="nowrap">Zbl 0850.93214</a></td> </tr><tr> <td>[24]</td> <td class="space">Hu, H.; Jiang, B.; Yang, H., Non-fragile $H_2$ reliable control for switched linear systems with actuator faults, Signal Process., 93, 7, 1804-1812 (2013)</td> </tr><tr> <td>[25]</td> <td class="space">Zhang, Y.; Shi, Y.; Shi, P., Robust and non-fragile finite-time $H_∞$ control for uncertain Markovian jump nonlinear systems, Appl. Math. Comput., 279, 125-138 (2016) · <a href="/1410.93142" class="nowrap">Zbl 1410.93142</a></td> </tr><tr> <td>[26]</td> <td class="space">Ma, Y.; Gu, N.; Zhang, Q., Non-fragile robust $H_∞$ control for uncertain discrete-time singular systems with time-varying delays, J. Frankl. Inst., 351, 6, 3163-3181 (2014) · <a href="/1290.93063" class="nowrap">Zbl 1290.93063</a></td> </tr><tr> <td>[27]</td> <td class="space">Xu, S.; Lam, J.; Wang, J.; Yang, G., Non-fragile positive real control for uncertain linear neutral delay systems, Syst. Control Lett., 52, 1, 59-74 (2004) · <a href="/1157.93418" class="nowrap">Zbl 1157.93418</a></td> </tr><tr> <td>[28]</td> <td class="space">Park, J., Robust non-fragile control for uncertain discrete-delay large-scale systems with a class of controller gain variations, Appl. Math. Comput., 149, 1, 147-164 (2004) · <a href="/1035.93021" class="nowrap">Zbl 1035.93021</a></td> </tr><tr> <td>[29]</td> <td class="space">Hom, R.; Johnson, C., Topics in Matrix Analysis (1991), Cambridge UP: Cambridge UP New York, USA · <a href="/0729.15001" class="nowrap">Zbl 0729.15001</a></td> </tr><tr> <td>[30]</td> <td class="space">Boukas, E., Stochastic Switching Systems: Analysis and Design (2005), Springer Science & Business Media: Springer Science & Business Media Berlin, Germany</td> </tr><tr> <td>[31]</td> <td class="space">Costa, O.; Fragoso, M.; Marques, R., Discrete-Time Markov Jump Linear Systems (2005), Springer: Springer London, UK · <a href="/1081.93001" class="nowrap">Zbl 1081.93001</a></td> </tr><tr> <td>[32]</td> <td class="space">Guo, Y., Stabilization of positive Markov jump systems, J. Frankl. Inst., 353, 14, 3428-3440 (2016) · <a href="/1347.93266" class="nowrap">Zbl 1347.93266</a></td> </tr><tr> <td>[33]</td> <td class="space">Zhang, J.; Zhao, X.; Zhu, F., $L_1/ℓ_1$-gain analysis and synthesis of Markovian jump positive systems with time delay, ISA Trans., 63, 93-102 (2016)</td> </tr><tr> <td>[34]</td> <td class="space">Yee, J.; Yang, G.; Wang, J., Non-fragile guaranteed cost control for discrete-time uncertain linear systems, Int. J. Syst. Sci., 32, 7, 845-853 (2001) · <a href="/1003.49024" class="nowrap">Zbl 1003.49024</a></td> </tr><tr> <td>[35]</td> <td class="space">Chen, G.; Yang, Y., Finite-time stability of switched positive linear systems, Int. J. Robust Nonlinear Control, 24, 1, 179-190 (2014) · <a href="/1278.93229" class="nowrap">Zbl 1278.93229</a></td> </tr><tr> <td>[36]</td> <td class="space">Shorten, R.; Leith, D.; Foy, R.; Kilduff, R., Towards an analysis and design framework for congestion control in communication networks, Proceedings of the Twelfth Yale Workshop on Adaptive Learning System, 1-15 (2003)</td> </tr><tr> <td>[37]</td> <td class="space">Zhang, J.; Han, Z.; Zhu, F., $L_1$-gain analysis and control synthesis of positive switched systems, Int. J. Syst. Sci., 46, 12, 2111-2121 (2015) · <a href="/1332.93127" class="nowrap">Zbl 1332.93127</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. 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