Document Zbl 1544.93251

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The frequency domain is divided into three frequency domains of low, medium and high, multiple AD filters are designed to work simultaneously. The AD problem is transformed into a multi-objective optimisation schemes for each finite frequency domain. Finally, the experimental results of networked motion control system are given to verify the effectiveness and superiority of proposed method.<br class="zbmathjax-paragraph">© 2021 The Authors. IET Control Theory & Applications published by John Wiley & Sons, Ltd. on behalf of The Institution of Engineering and Technology</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7907100">Cited in <strong>6</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93B70" title="MSC2020">93B70</a> </td> <td class="space"> Networked control </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C83" title="MSC2020">93C83</a> </td> <td class="space"> Control/observation systems involving computers (process control, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C80" title="MSC2020">93C80</a> </td> <td class="space"> Frequency-response methods in control theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Asensors">sensors</a>; <a href="/?q=ut%3Aoptimisation">optimisation</a>; <a href="/?q=ut%3Amotion+control">motion control</a>; <a href="/?q=ut%3Acyber-physical+systems">cyber-physical systems</a>; <a href="/?q=ut%3Afrequency-domain+analysis">frequency-domain analysis</a>; <a href="/?q=ut%3Afilters">filters</a>; <a href="/?q=ut%3Anetworked+control+systems">networked control systems</a>; <a href="/?q=ut%3Acyber-physical+systems">cyber-physical systems</a>; <a href="/?q=ut%3Anetworked+motion+control+system">networked motion control system</a>; <a href="/?q=ut%3Afinite+frequency+domain+partition">finite frequency domain partition</a>; <a href="/?q=ut%3Asensor+attack+detection+problem">sensor attack detection problem</a>; <a href="/?q=ut%3Asensor+AD+problem">sensor AD problem</a>; <a href="/?q=ut%3AAD+filter">AD filter</a>; <a href="/?q=ut%3Amultiobjective+optimisation+schemes">multiobjective optimisation schemes</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 1544.93251" data-ciurl="/ci/07907100" data-biburl="/bibtex/07907100.bib" data-amsurl="/amsrefs/07907100.bib" data-xmlurl="/xml/07907100.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07907100.pdf" title="Zbl 1544.93251 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/10.1049/iet-cta.2019.1140" aria-label="DOI for “Sensor attack detection for cyber-physical systems based on frequency domain partition”" title="10.1049/iet-cta.2019.1140">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">Ding, D.; Han, Q. L.; Xiang, Y., A survey on security control and attack detection for industrial cyber‐physical systems, Neurocomputing, 275, 1674-1683, 2018</td> </tr><tr> <td>[2]</td> <td class="space">Mahmoud, M. S.; Hamdan, M. M.; Baroudi, U. A., Modeling and control of cyber‐physical systems subject to cyber attacks: a survey of recent advances and challenges, Neurocomputing, 338, 101-115, 2019</td> </tr><tr> <td>[3]</td> <td class="space">Dibaji, S. M.; Pirani, M.; Flamholz, D. B., A systems and control perspective of cps security, Annu. Rev. Control, 47, 394-411, 2019</td> </tr><tr> <td>[4]</td> <td class="space">Ashibani, Y.; Mahmoud, Q. H., Cyber physical systems security: analysis, challenges and solutions, Comput. Secur., 68, 81-97, 2017</td> </tr><tr> <td>[5]</td> <td class="space">Ye, D.; Zhang, T. Y., Summation detector for false data‐injection attack in cyber‐physical systems, IEEE Trans. Cybern., PP, 2019 · <a href="https://doi.org/10.1109/TCYB.2019.2915124" class="nowrap">doi:10.1109/TCYB.2019.2915124</a></td> </tr><tr> <td>[6]</td> <td class="space">Yang, H.; Ye, D., Distributed fixed‐time consensus tracking control of uncertain nonlinear multiagent systems: a prioritized strategy, IEEE Trans. Cybern., PP, 1-12, 2019</td> </tr><tr> <td>[7]</td> <td class="space">Wang, X.; Yang, G. H., Fault‐tolerant consensus tracking control for linear multiagent systems under switching directed network, IEEE Trans. Cybern., PP, 1-10, 2019</td> </tr><tr> <td>[8]</td> <td class="space">Zhu, J. W.; Yang, G. H.; Wang, H., Fault estimation for a class of nonlinear systems based on intermediate estimator, IEEE Trans. Autom. Control, 61, 1-1, 2015</td> </tr><tr> <td>[9]</td> <td class="space">Fei, X.; Shah, N.; Verba, N., Cps data streams analytics based on machine learning for cloud and fog computing: a survey, Futur. Gener. Comp. Syst., 90, 435-450, 2019</td> </tr><tr> <td>[10]</td> <td class="space">Abbaspour, A.; Yen, K. K.; Noei, S., Detection of fault data injection attack on uav using adaptive neural network, Procedia Comput. Sci., 95, 193-200, 2016</td> </tr><tr> <td>[11]</td> <td class="space">Yu, J. J.Q.; Hou, Y.; Li, V. O.K., Online false data injection attack detection with wavelet transform and deep neural networks, IEEE Trans. Ind. Inform., 14, 7, 3271-3280, 2018</td> </tr><tr> <td>[12]</td> <td class="space">Ge, H.; Yue, D.; Xie, X., A unified modeling of muti‐sources cyber‐attacks with uncertainties for cps security control, J. Frankl. Inst., 2019 · <a href="/1459.94160" class="nowrap">Zbl 1459.94160</a> · <a href="https://doi.org/10.1016/j.jfranklin.2019.01.006" class="nowrap">doi:10.1016/j.jfranklin.2019.01.006</a></td> </tr><tr> <td>[13]</td> <td class="space">Fritz, R.; Zhang, P., Modeling and detection of cyber attacks on discrete event systems, IFAC‐PapersOnLine, 51, 7, 285-290, 2018</td> </tr><tr> <td>[14]</td> <td class="space">Lu, A. Y.; Yang, G. H., Switched projected gradient descent algorithms for secure state estimation under sparse sensor attacks, Automatica, 103, 503-514, 2019 · <a href="/1415.93253" class="nowrap">Zbl 1415.93253</a></td> </tr><tr> <td>[15]</td> <td class="space">Lu, A. Y.; Yang, G. H., Distributed consensus control for multi‐agent systems under denial‐of‐service, Inf. Sci., 439-440, 95-107, 2018 · <a href="/1448.93297" class="nowrap">Zbl 1448.93297</a></td> </tr><tr> <td>[16]</td> <td class="space">Guan, Y.; Ge, X., Distributed attack detection and secure estimation of networked cyber‐physical systems against false data injection attacks and jamming attacks, IEEE Trans. Signal Inf. Process. Netw., 4, 1, 48-59, 2018</td> </tr><tr> <td>[17]</td> <td class="space">Ge, X.; Han, Q. L.; Zhong, M., Distributed krein space‐based attack detection over sensor networks under deception attacks, Automatica, 109, 108557, 2019 · <a href="/1429.93135" class="nowrap">Zbl 1429.93135</a></td> </tr><tr> <td>[18]</td> <td class="space">Ao, W.; Song, Y.; Wen, C., Finite time attack detection and supervised secure state estimation for cpss with malicious adversaries, Inf. Sci., 451-452, 67-82, 2018 · <a href="/1448.93222" class="nowrap">Zbl 1448.93222</a></td> </tr><tr> <td>[19]</td> <td class="space">Li, Y. G.; Yang, G. H., Optimal stealthy false data injection attacks in cyber‐physical systems, Inf. Sci., 481, 474-490, 2019 · <a href="/1451.94030" class="nowrap">Zbl 1451.94030</a></td> </tr><tr> <td>[20]</td> <td class="space">Hu, L.; Wang, Z.; Han, Q. L., State estimation under false data injection attacks: security analysis and system protection, Automatica, 87, 176-183, 2018 · <a href="/1378.93119" class="nowrap">Zbl 1378.93119</a></td> </tr><tr> <td>[21]</td> <td class="space">Sun, H.; Peng, C.; Zhang, W., Security‐based resilient event‐triggered control of networked control systems under denial of service attacks, J. Frankl. Inst., 356, 17, 10277-10295, 2019 · <a href="/1425.93186" class="nowrap">Zbl 1425.93186</a></td> </tr><tr> <td>[22]</td> <td class="space">Yuan, H.; Xia, Y., Resilient strategy design for cyber‐physical system under dos attack over a multi‐channel framework, Inf. Sci., 454-455, 312-327, 2018 · <a href="/1446.94163" class="nowrap">Zbl 1446.94163</a></td> </tr><tr> <td>[23]</td> <td class="space">Lu, A.; Yang, G., Input‐to‐state stabilizing control for cyber‐physical systems with multiple transmission channels under denial of service, IEEE Trans. Autom. Control., 63, 6, 1813-1820, 2018 · <a href="/1395.93483" class="nowrap">Zbl 1395.93483</a></td> </tr><tr> <td>[24]</td> <td class="space">Chen, X.; Wang, Y.; Hu, S., Event‐based robust stabilization of uncertain networked control systems under quantization and denial‐of‐service attacks, Inf. Sci., 459, 369-386, 2018 · <a href="/1448.93266" class="nowrap">Zbl 1448.93266</a></td> </tr><tr> <td>[25]</td> <td class="space">Sanchez, H. S.; Rotondo, D.; Escobet, T., Detection of replay attacks in cyber‐physical systems using a frequency‐based signature, J. Frankl. Inst., 356, 5, 2798-2824, 2019 · <a href="/1411.93119" class="nowrap">Zbl 1411.93119</a></td> </tr><tr> <td>[26]</td> <td class="space">Zhang, W. A.; Yu, L.; He, D., Sequential fusion estimation for sensor networks with deceptive attacks, IEEE Trans. Aerosp. Electron. Syst., PP, 1-1, 2019</td> </tr><tr> <td>[27]</td> <td class="space">Teixeira, A.; Shames, I.; Sandberg, H., A secure control framework for resource‐limited adversaries, Automatica, 51, 135-148, 2015 · <a href="/1309.93020" class="nowrap">Zbl 1309.93020</a></td> </tr><tr> <td>[28]</td> <td class="space">Li, L.; Ding, S. X., Optimal detection schemes for multiplicative faults in uncertain systems with application to rolling mill processes, IEEE Trans. Control Syst. Technol., 1-13, 2019 · <a href="https://doi.org/10.1109/TCST.2019.2947876" class="nowrap">doi:10.1109/TCST.2019.2947876</a></td> </tr><tr> <td>[29]</td> <td class="space">Ding, S. X.; Li, L.; Kruger, M., Application of randomized algorithms to assessment and design of observer‐based fault detection systems, Automatica, 107, 175-182, 2019 · <a href="/1429.93124" class="nowrap">Zbl 1429.93124</a></td> </tr><tr> <td>[30]</td> <td class="space">Iwasaki, T.; Hara, S., Generalized kyp lemma: unified frequency domain inequalities with design applications, IEEE Trans. Autom. Control., 50, 1, 41-59, 2005 · <a href="/1365.93175" class="nowrap">Zbl 1365.93175</a></td> </tr><tr> <td>[31]</td> <td class="space">Chadli, M.; Abdo, A.; Ding, S. X., $ H_- / H_\infty$ fault detection filter design for discrete‐time Takagi‐Sugeno fuzzy system, Automatica, 49, 7, 1996-2005, 2013 · <a href="/1364.93420" class="nowrap">Zbl 1364.93420</a></td> </tr><tr> <td>[32]</td> <td class="space">Duan, Z.; Ghous, I.; Huang, S., Fault detection observer design for 2‐D continuous nonlinear systems with finite frequency specifications, ISA Trans., 84, 1-11, 2019</td> </tr><tr> <td>[33]</td> <td class="space">Ding, D. W.; Xie, X.; Du, X., Finite‐frequency model reduction of discrete‐time T‐S fuzzy state‐delay systems, Neurocomputing, 203, 121-128, 2016</td> </tr><tr> <td>[34]</td> <td class="space">Chibani, A.; Chadli, M.; Shi, P., Multi‐objective $H_- / H_\infty$ fault detection filter design for T‐S fuzzy systems with finite frequency specifications, IFAC‐PapersOnLine, 51, 24, 1172-1178, 2018</td> </tr><tr> <td>[35]</td> <td class="space">Yang, H.; Xia, Y.; Liu, B., Fault detection for T‐S fuzzy discrete systems in finite‐frequency domain, IEEE Trans. Syst. Man Cybern. Syst., 41, 4, 911-920, 2011</td> </tr><tr> <td>[36]</td> <td class="space">Zhang, K.; Jiang, B.; Shi, P., Analysis and design of robust $H_\infty$ fault estimation observer with finite‐frequency specifications for discrete‐time fuzzy systems, IEEE Trans. Cybern., 45, 7, 1225-1235, 2015</td> </tr><tr> <td>[37]</td> <td class="space">Li, X.; Yang, G., Fault detection in finite frequency domain for Takagi‐Sugeno fuzzy systems with sensor faults, IEEE Trans. Cybern., 44, 8, 1446-1458, 2014</td> </tr><tr> <td>[38]</td> <td class="space">Chibani, A.; Chadli, M.; Ding, S. X., Design of robust fuzzy fault detection filter for polynomial fuzzy systems with new finite frequency specifications, Automatica, 93, 42-54, 2018 · <a href="/1400.93297" class="nowrap">Zbl 1400.93297</a></td> </tr><tr> <td>[39]</td> <td class="space">Long, Y.; Yang, G. H., Fault detection in finite frequency domain for networked control systems with missing measurements, J. Frankl. Inst., 350, 9, 2605-2626, 2013 · <a href="/1287.93083" class="nowrap">Zbl 1287.93083</a></td> </tr><tr> <td>[40]</td> <td class="space">Zhong, G. X.; Yang, G. H., Fault detection for discrete‐time switched systems with sensor stuck faults and servo inputs, ISA Trans., 58, 196-205, 2015</td> </tr><tr> <td>[41]</td> <td class="space">Jing, H.; Wang, R.; Karimi, H. R., Robust output‐feedback based fault‐tolerant control of active suspension with finite‐frequency constraint, IFAC‐PapersOnLine, 48, 21, 1173-1179, 2015</td> </tr><tr> <td>[42]</td> <td class="space">Zhu, X.; Xia, Y.; Fu, M., Fault estimation and active fault‐tolerant control for discrete‐time systems in finite‐frequency domain, ISA Trans., 2019 · <a href="https://doi.org/10.1016/j.isatra.2019.02.014" class="nowrap">doi:10.1016/j.isatra.2019.02.014</a></td> </tr><tr> <td>[43]</td> <td class="space">Duan, Z.; Ghous, I.; Shen, J., Fault detection observer design for discrete‐time 2‐D T‐S fuzzy systems with finite‐frequency specifications, Fuzzy Sets. Syst., 2019 · <a href="https://doi.org/10.1016/i.fss.2019.05.004" class="nowrap">doi:10.1016/i.fss.2019.05.004</a></td> </tr><tr> <td>[44]</td> <td class="space">Zhang, S.; Wang, Z.; Ding, D., On design of robust fault detection filter in finite‐frequency domain with regional pole assignment, IEEE Trans. CIRCUITS‐II, 62, 4, 382-386, 2015</td> </tr><tr> <td>[45]</td> <td class="space">Li, X.; Yang, G., Fault detection for T‐S fuzzy systems with unknown membership functions, IEEE Trans. Fuzzy Syst., 22, 1, 139-152, 2014</td> </tr><tr> <td>[46]</td> <td class="space">Hu, Y.; Li, H.; Luan, T. H., Detecting stealthy attacks on industrial control systems using a permutation entropy‐based method, Future Gener. Comput. Syst., 2018 · <a href="https://doi.org/10.1016/j.future.2018.07.027" class="nowrap">doi:10.1016/j.future.2018.07.027</a></td> </tr><tr> <td>[47]</td> <td class="space">Liu, D. L.; Feng, D. Q., Deceptive attack detection of control system using multi‐scale principal component analysis, Zhejiang Daxue Xuebao (Gongxue Ban)/J. Zhejiang Univ. (Eng. Sci.), 52, 9, 1738-1746, 2018</td> </tr><tr> <td>[48]</td> <td class="space">Frank, P. M., Fault diagnosis in dynamic systems using analytical and knowledge‐based redundancy: a survey and some new results, Automatica, 26, 3, 459-474, 1990 · <a href="/0713.93052" class="nowrap">Zbl 0713.93052</a></td> </tr><tr> <td>[49]</td> <td class="space">Skelton, R.; Iwasaki, T.; Grigoriadis, K., A unified algebraic approach to linear control design, 1997, CRC Press: USA</td> </tr><tr> <td>[50]</td> <td class="space">Tuan, H. D.; Apkarian, P.; Nguyen, T. Q., Robust and reduced‐order filtering: new lmi‐based characterizations and methods, IEEE Trans. Signal Process., 49, 12, 2975-2984, 2001</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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