Document Zbl 1525.93135

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J. Robust Nonlinear Control</a> <a href="/?q=in%3A503802" title="Articles in this Issue">31, No. 3, 887-906 (2021)</a>. </div> <div class="abstract">Summary: In this article, the design problem of $\ell_2$-$\ell_\infty$ proportional-integral observer (PIO) is investigated for a class of discrete-time systems with mixed time-delays. The mixed time-delays comprise both the discrete time-varying delays and infinitely distributed delays. The round-robin protocol (RRP) is employed to schedule the data transmissions from the sensors to the observer so as to mitigate the communication burden and prevent the data collisions. A novel PIO is developed whose observer gain is dependent on the data transmission order as a reflection of the effects induced by the RRP scheduling. By resorting to the token-dependent Lyapunov functional and the matrix inequality technique, the desired PIO is designed with exponentially stable error dynamics of the state estimation and guaranteed $\ell_2$-$\ell_\infty$ disturbance attenuation/resistance capacity. Finally, a simulation example is exploited to verify the validity of the proposed observer design method.<br class="zbmathjax-paragraph">{© 2020 John Wiley & Sons Ltd}</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7757897">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93B53" title="MSC2020">93B53</a> </td> <td class="space"> Observers </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C43" title="MSC2020">93C43</a> </td> <td class="space"> Delay control/observation systems </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%3A93D23" title="MSC2020">93D23</a> </td> <td class="space"> Exponential stability </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aproportional-integral+observer">proportional-integral observer</a>; <a href="/?q=ut%3Around-robin+protocol">round-robin protocol</a>; <a href="/?q=ut%3Amixed+time-delays">mixed time-delays</a>; <a href="/?q=ut%3A%5C%28%5Cell_2%5C%29-%5C%28%5Cell_%5Cinfty%5C%29+performance">$\ell_2$-$\ell_\infty$ performance</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 1525.93135" data-ciurl="/ci/07757897" data-biburl="/bibtex/07757897.bib" data-amsurl="/amsrefs/07757897.bib" data-xmlurl="/xml/07757897.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07757897.pdf" title="Zbl 1525.93135 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.5328" aria-label="DOI for “$\ell_2$-$\ell_\infty$ proportional-integral observer design for systems with mixed time-delays under round-robin protocol”" title="10.1002/rnc.5328">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://bura.brunel.ac.uk/handle/2438/23534" title="Full Text Link">Link</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Chen, Y.; Zheng, WX, L_2‐L_∞ filtering for stochastic Markovian jump delay systems with nonlinear perturbations, Signal Process, 109, 154-164, 2015</td> </tr><tr> <td>[2]</td> <td class="space">Wilson, DA, Convolution and Hankel operator norms for linear systems, IEEE Trans Automat Control, 34, 1, 94-97, 1989 · <a href="/0661.93022" class="nowrap">Zbl 0661.93022</a></td> </tr><tr> <td>[3]</td> <td class="space">Yang, R.; Gao, H.; Shi, P., Delay‐dependent L_2‐L_∞ filter design for stochastic time‐delay systems, IET Control Theory Appl, 5, 1, 1-8, 2011</td> </tr><tr> <td>[4]</td> <td class="space">Wu, L.; Zheng, WX, L_2‐L_∞ control of nonlinear fuzzy Itô stochastic delay systems via dynamic output feedback, IEEE Trans Syst Man Cybern Part B (Cybern), 39, 5, 1308-1315, 2009</td> </tr><tr> <td>[5]</td> <td class="space">Grigoriiadis, KM; Watson, JT, Reduced order H_∞ and ℓ_2‐ℓ_∞ filtering via linear matrix inequaties, IEEE Trans Aerosp Electron Syst, 33, 4, 1326-1338, 1997</td> </tr><tr> <td>[6]</td> <td class="space">Qian, W.; Li, Y.; Zhao, Y.; Chen, Y., New optimal method for L_2‐L_∞ state estimation of delayed neural networks, Neurocomputing, 415, 258-265, 2020</td> </tr><tr> <td>[7]</td> <td class="space">Shen, Y.; Wang, Z.; Shen, B.; Alsaadi, FE; Dobaie, AM, ℓ_2‐ℓ_∞ state estimation for delayed artificial neural networks under high‐rate communication channels with Round‐Robin protocol, Neural Netw, 124, 170-179, 2020 · <a href="/1443.93131" class="nowrap">Zbl 1443.93131</a></td> </tr><tr> <td>[8]</td> <td class="space">Qian, W.; Li, Y.; Chen, Y.; Liu, W., L_2‐L_∞ filtering for stochastic delayed systems with randomly occurring nonlinearities and sensor saturation, Int J Syst Sci, 2360-2377, 2020 · <a href="/1483.93658" class="nowrap">Zbl 1483.93658</a></td> </tr><tr> <td>[9]</td> <td class="space">Zhao, D.; Wang, Z.; Ho, DWC; Wei, G., Observer‐based PID security control for discrete time‐delay systems under cyber‐attacks, IEEE Trans Syst Man Cybern Syst, 2019 · <a href="https://doi.org/10.1109/TSMC.2019.2952539" class="nowrap">doi:10.1109/TSMC.2019.2952539</a></td> </tr><tr> <td>[10]</td> <td class="space">Ang, KH; Chong, G.; Li, Y., PID control system analysis, design and technology, IEEE Trans Control Syst Technol, 3, 4, 559-576, 2005</td> </tr><tr> <td>[11]</td> <td class="space">Åström, KJ; Hägglund, T.; Hang, CC; Ho, WK, Automatic tuning and adaptation for PID controllers‐a survey, Control Eng Pract, 1, 4, 699-714, 1993</td> </tr><tr> <td>[12]</td> <td class="space">Rosinová, D.; Veselý, V., Robust PID decentralized controller design using LMI, Int J Comput Commun Control, 2, 2, 195-204, 2007</td> </tr><tr> <td>[13]</td> <td class="space">Zhao, D.; Wang, Z.; Wei, G.; Han, Q‐L, A dynamic event‐triggered approach to observer‐based PID security control subject to deception attacks, Automatica, 120, 2020 · <a href="/1448.93219" class="nowrap">Zbl 1448.93219</a></td> </tr><tr> <td>[14]</td> <td class="space">Beale, S.; Shafai, B., Robust control system design with a proportional integral observer, Int J Control, 50, 1, 97-111, 1989 · <a href="/0683.93022" class="nowrap">Zbl 0683.93022</a></td> </tr><tr> <td>[15]</td> <td class="space">Koenig, D., Unknown input proportional multiple‐integral observer design for linear descriptor systems: application to state and fault estimation, IEEE Trans Automat Control, 50, 2, 212-217, 2005 · <a href="/1365.93487" class="nowrap">Zbl 1365.93487</a></td> </tr><tr> <td>[16]</td> <td class="space">Söffker, D.; Yu, TJ; Müller, PC, State estimation of dynamical systems with nonlinearities by using proportional‐integral observer, Int J Syst Sci, 26, 9, 1571-1582, 1995 · <a href="/0830.93009" class="nowrap">Zbl 0830.93009</a></td> </tr><tr> <td>[17]</td> <td class="space">Chang, J‐L, Applying discrete‐time proportional integral observers for state and disturbance estimations, IEEE Trans Automat Control, 51, 5, 814-818, 2006 · <a href="/1366.93626" class="nowrap">Zbl 1366.93626</a></td> </tr><tr> <td>[18]</td> <td class="space">Youssef, T.; Chadli, M.; Karimi, HR; Wang, R., Actuator and sensor faults estimation based on proportional integral observer for TS fuzzy model, J Frankl Inst, 354, 6, 2524-2542, 2017 · <a href="/1398.93202" class="nowrap">Zbl 1398.93202</a></td> </tr><tr> <td>[19]</td> <td class="space">Liu, Y.; Shen, B.; Shu, H., Finite‐time resilient H_∞ state estimation for discrete‐time delayed neural networks under dynamic event‐triggered mechanism, Neural Netw, 121, 356-365, 2020 · <a href="/1443.93081" class="nowrap">Zbl 1443.93081</a></td> </tr><tr> <td>[20]</td> <td class="space">Zou, L.; Wang, Z.; Gao, H.; Liu, X., State estimation for discrete‐time dynamical networks with time‐varying delays and stochastic disturbances under the Round‐Robin protocol, IEEE Trans Neural Netw Learn Syst, 28, 5, 1139-1151, 2017</td> </tr><tr> <td>[21]</td> <td class="space">Hu, J.; Wang, Z.; Liu, G‐P; Zhang, H.; Navaratne, R., A prediction‐based approach to distributed filtering with missing measurements and communication delays through sensor networks, IEEE Trans Syst Man Cybern Syst, 2020 · <a href="https://doi.org/10.1109/TSMC.2020.2966977" class="nowrap">doi:10.1109/TSMC.2020.2966977</a></td> </tr><tr> <td>[22]</td> <td class="space">Shen, B.; Wang, Z.; Tan, H., Guaranteed cost control for uncertain nonlinear systems with mixed time‐delays: the discrete‐time case, Europ J Control, 40, 62-67, 2018 · <a href="/1403.93123" class="nowrap">Zbl 1403.93123</a></td> </tr><tr> <td>[23]</td> <td class="space">Liu, H.; Wang, Z.; Shen, B.; Dong, H., Delay‐distribution‐dependent H_∞ state estimation for discrete‐time memristive neural networks with mixed time‐delays and fading measurements, IEEE Trans Cybern, 50, 2, 440-451, 2020</td> </tr><tr> <td>[24]</td> <td class="space">Ma, L.; Wang, Z.; Han, Q‐L; Liu, Y., Dissipative control for nonlinear Markovian jump systems with actuator failures and mixed time‐delays, Automatica, 98, 358-362, 2018 · <a href="/1406.93372" class="nowrap">Zbl 1406.93372</a></td> </tr><tr> <td>[25]</td> <td class="space">Liu, Y.; Wang, Z.; Ma, L.; Cui, Y.; Alsaadi, FE, Synchronization of directed switched complex networks with stochastic link perturbations and mixed time‐delays, Nonlinear Anal Hybrid Syst, 27, 213-224, 2018 · <a href="/1378.93117" class="nowrap">Zbl 1378.93117</a></td> </tr><tr> <td>[26]</td> <td class="space">Cetinkaya, A.; Ishii, H.; Hayakawa, T., Networked control under random and malicious packet losses, IEEE Trans Automat Control, 62, 5, 2434-2449, 2017 · <a href="/1366.93617" class="nowrap">Zbl 1366.93617</a></td> </tr><tr> <td>[27]</td> <td class="space">Ding, D.; Wang, Z.; Han, Q‐L, Neural‐network‐based output‐feedback control with stochastic communication protocols, Automatica, 106, 221-229, 2019 · <a href="/1429.93112" class="nowrap">Zbl 1429.93112</a></td> </tr><tr> <td>[28]</td> <td class="space">Caballero‐Águila, R.; Hermoso‐Carazo, A.; Linares‐Pérez, J., Distributed fusion filters from uncertain measured outputs in sensor networks with random packet losses, Inf Fusion, 34, 70-79, 2017</td> </tr><tr> <td>[29]</td> <td class="space">Ding, D.; Wang, Z.; Han, Q‐L, A scalable algorithm for event‐triggered state estimation with unknown parameters and switching topologies over sensor networks, IEEE Trans Cybern, 50, 9, 4087-4097, 2019</td> </tr><tr> <td>[30]</td> <td class="space">Han, F.; Wei, G.; Ding, D.; Song, Y., Finite‐horizon bounded H_∞ synchronisation and state estimation for discrete‐time complex networks: local performance analysis, IET Control Theory Appl, 11, 6, 827-837, 2017</td> </tr><tr> <td>[31]</td> <td class="space">Li, W.; Wei, G.; Han, F.; Liu, Y., Weighted average consensus‐based unscented Kalman filtering, IEEE Trans Cybern, 46, 2, 558-567, 2016</td> </tr><tr> <td>[32]</td> <td class="space">Liu, Y.; Shen, B.; Li, Q., State estimation for neural networks with Markov‐based nonuniform sampling: the partly unknown transition probability case, Neurocomputing, 357, 261-270, 2019</td> </tr><tr> <td>[33]</td> <td class="space">Zou, L.; Wang, Z.; Zhou, DH, Moving horizon estimation with non‐uniform sampling under component‐based dynamic event‐triggered transmission, Automatica, 120, 2020 · <a href="/1448.93191" class="nowrap">Zbl 1448.93191</a></td> </tr><tr> <td>[34]</td> <td class="space">Hu, J.; Wang, Z.; Liu, G‐P; Zhang, H., Variance‐constrained recursive state estimation for time‐varying complex networks with quantized measurements and uncertain inner coupling, IEEE Trans Neural Netw Learn Syst, 31, 6, 1955-1967, 2020</td> </tr><tr> <td>[35]</td> <td class="space">Zhang, X‐M; Han, Q‐L; Yu, X., Survey on recent advances in networked control systems, IEEE Trans Ind Inform, 12, 5, 1740-1752, 2016</td> </tr><tr> <td>[36]</td> <td class="space">Wan, X.; Wang, Z.; Han, Q‐L; Wu, M., Finite‐time H_∞ state estimation for discrete time‐delayed genetic regulatory networks under stochastic communication protocols, IEEE Trans Circ Syst Part I, 65, 10, 3481-3491, 2018 · <a href="/1468.93178" class="nowrap">Zbl 1468.93178</a></td> </tr><tr> <td>[37]</td> <td class="space">Zou, L.; Wang, Z.; Han, Q‐L; Zhou, D., Moving horizon estimation for networked time‐delay systems under Round‐Robin protocol, IEEE Trans Automat Control, 64, 12, 5191-5198, 2019 · <a href="/1482.93358" class="nowrap">Zbl 1482.93358</a></td> </tr><tr> <td>[38]</td> <td class="space">Chen, Y.; Wang, Z.; Wang, L.; Sheng, W., Finite‐horizon H_∞ state estimation for stochastic coupled networks with random inner couplings using Round‐Robin protocol, IEEE Trans Cybern, 2020 · <a href="https://doi.org/10.1109/TCYB.2020.3004288" class="nowrap">doi:10.1109/TCYB.2020.3004288</a></td> </tr><tr> <td>[39]</td> <td class="space">Liu, S.; Wang, Z.; Chen, Y.; Wei, G., Protocol‐based unscented Kalman filtering in the presence of stochastic uncertainties, IEEE Trans Automat Control, 65, 3, 1303-1309, 2020 · <a href="/1533.93808" class="nowrap">Zbl 1533.93808</a></td> </tr><tr> <td>[40]</td> <td class="space">Shen, B.; Wang, Z.; Wang, D.; Liu, H., Distributed state‐saturated recursive filtering over sensor networks under Round‐Robin protocol, IEEE Trans Cybern, 50, 8, 3605-3615, 2020</td> </tr><tr> <td>[41]</td> <td class="space">Ding, D.; Wang, Z.; Han, Q‐L; Wei, G., Neural‐network‐based output‐feedback control under round‐robin scheduling protocols, IEEE Trans Cybern, 49, 6, 2372-2384, 2019</td> </tr><tr> <td>[42]</td> <td class="space">Wang, F.; Wang, Z.; Liang, J.; Liu, X., Resilient filtering for linear time‐varying repetitive processes under uniform quantizations and Round‐Robin protocols, IEEE Trans Circ Syst I Reg Pap, 65, 9, 2992-3004, 2018 · <a href="/1468.93176" class="nowrap">Zbl 1468.93176</a></td> </tr><tr> <td>[43]</td> <td class="space">Wan, X.; Wang, Z.; Wu, M.; Liu, X., State estimation for discrete time‐delayed genetic regulatory networks with stochastic noises under the round‐robin protocols, IEEE Trans Nanobiosci, 17, 2, 145-154, 2018</td> </tr><tr> <td>[44]</td> <td class="space">Liu, Y.; Chen, S.; Guan, B.; Xu, P., Layout optimization of large‐scale oil‐gas gathering system based on combined optimization strategy, Neurocomputing, 332, 159-183, 2019</td> </tr><tr> <td>[45]</td> <td class="space">Liu, Y.; Cheng, Q.; Gan, Y.; Wang, Y.; Li, Z.; Zhao, J., Multi‐objective optimization of energy consumption in crude oil pipeline transportation system operation based on exergy loss analysis, Neurocomputing, 332, 100-110, 2019</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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Document Zbl 1525.93135 - zbMATH Open