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(2022)</a>. </div> <div class="abstract">Summary: For attack reconstruction issue of cyber-physical systems (CPSs), it is a challenge to guarantee online computational efficiency and reconstruction accuracy simultaneously, especially for cases with actuator attacks. However, most existing approaches are only focused on sensor attacks. This brief paper studies the multiple attack reconstruction problem for CPSs. A new projected intermediate estimator is proposed, where the computation complexity problem is solved by reducing the number of loop iterations significantly via enhancing the designs of loop iteration, termination condition and parameter selection strategy, without sacrificing the attack reconstruction performance. Compared with traditional intermediate estimators, the dependence of linear matrix inequality technique is completely removed, besides, the intrinsic heavy overshooting problem in transient period is effectively alleviated. The application to a networked motion control system verifies the effectiveness and superiority of the presented method.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7553588">Cited in <strong>5</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%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></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Acyber-physical+systems">cyber-physical systems</a>; <a href="/?q=ut%3Amultiple+attack+reconstruction">multiple attack reconstruction</a>; <a href="/?q=ut%3Aprojected+intermediate+estimator">projected intermediate estimator</a>; <a href="/?q=ut%3Aonline+computation+efficiency">online computation efficiency</a>; <a href="/?q=ut%3Anetworked+motion+control+system">networked motion control system</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">&times;</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 1494.93048" data-ciurl="/ci/07553588" data-biburl="/bibtex/07553588.bib" data-amsurl="/amsrefs/07553588.bib" data-xmlurl="/xml/07553588.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07553588.pdf" title="Zbl 1494.93048 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.2022.110388" aria-label="DOI for “Performance guaranteed attack reconstruction for cyber-physical systems: a new intermediate estimator-based framework”" title="10.1016/j.automatica.2022.110388">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">An, L.; Yang, G. H., State estimation under sparse sensor attacks: A constrained set partitioning approach, IEEE Transactions on Automatic Control, 64, 9, 3861-3868 (2018) &middot; <a href="/1482.93127" class="nowrap">Zbl 1482.93127</a></td> </tr><tr> <td>[2]</td> <td class="space">An, L.; Yang, G. H., Distributed secure state estimation for cyber-physical systems under sensor attacks, Automatica, 107, 526-538 (2019) &middot; <a href="/1429.93237" class="nowrap">Zbl 1429.93237</a></td> </tr><tr> <td>[3]</td> <td class="space">Blumensath, T.; Davies, M. E., Normalized iterative hard thresholding: Guaranteed stability and performance, IEEE Journal of Selected Topics in Signal Processing, 4, 2, 298-309 (2010)</td> </tr><tr> <td>[4]</td> <td class="space">Corradini, M. L.; Cristofaro, A., Robust detection and reconstruction of state and sensor attacks for cyber-physical systems using sliding modes, IET Control Theory &amp; Applications, 11, 11, 1756-1766 (2017)</td> </tr><tr> <td>[5]</td> <td class="space">Krishnan, V.; Pasqualetti, F., Data-driven attack detection for linear systems, IEEE Control Systems Letters, 5, 2, 671-676 (2021)</td> </tr><tr> <td>[6]</td> <td class="space">Lewis, F. L.; Vrabie, D.; Vamvoudakis, K. G., Reinforcement learning and feedback control: using natural decision methods to design optimal adaptive controllers, IEEE Control Systems Magazine, 32, 6, 76-105 (2012) &middot; <a href="/1395.93584" class="nowrap">Zbl 1395.93584</a></td> </tr><tr> <td>[7]</td> <td class="space">Li, Y.; Shi, D.; Chen, T., False data injection attacks on networked control systems: A stackelberg game analysis, IEEE Transactions on Automatic Control, 63, 10, 3503-3509 (2018) &middot; <a href="/1425.91092" class="nowrap">Zbl 1425.91092</a></td> </tr><tr> <td>[8]</td> <td class="space">Lu, A. Y.; Yang, G. H., Secure state estimation for cyber-physical systems under sparse sensor attacks via a switched luenberger observer, Information Sciences, 417, 454-464 (2017) &middot; <a href="/1447.93346" class="nowrap">Zbl 1447.93346</a></td> </tr><tr> <td>[9]</td> <td class="space">Lu, A. Y.; Yang, G. H., Secure switched observers for cyber-physical systems under sparse sensor attacks: a set cover approach, IEEE Transactions on Automatic Control, 64, 9, 3949-3955 (2019) &middot; <a href="/1482.93092" class="nowrap">Zbl 1482.93092</a></td> </tr><tr> <td>[10]</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) &middot; <a href="/1415.93253" class="nowrap">Zbl 1415.93253</a></td> </tr><tr> <td>[11]</td> <td class="space">Mishra, S.; Shoukry, Y.; Karamchandani, N.; Diggavi, S. N.; Tabuada, P., Secure state estimation against sensor attacks in the presence of noise, IEEE Transactions on Control of Network Systems, 4, 1, 49-59 (2016) &middot; <a href="/1370.93259" class="nowrap">Zbl 1370.93259</a></td> </tr><tr> <td>[12]</td> <td class="space">Mo, Y.; Sinopoli, B., Secure estimation in the presence of integrity attacks, IEEE Transactions on Automatic Control, 60, 4, 1145-1151 (2014) &middot; <a href="/1360.68439" class="nowrap">Zbl 1360.68439</a></td> </tr><tr> <td>[13]</td> <td class="space">Pang, Z. H.; Liu, G. P.; Zhou, D.; Hou, F.; Sun, D., Two-channel false data injection attacks against output tracking control of networked systems, IEEE Transactions on Industrial Electronics, 63, 5, 3242-3251 (2016)</td> </tr><tr> <td>[14]</td> <td class="space">Pasqualetti, F.; Dörfler, F.; Bullo, F., Attack detection and identification in cyber-physical systems, IEEE Transactions on Automatic Control, 58, 1, 2715-2729 (2013) &middot; <a href="/1369.93675" class="nowrap">Zbl 1369.93675</a></td> </tr><tr> <td>[15]</td> <td class="space">Sánchez, H. S.; Rotondo, D.; Escobet, T.; Puig, V.; Quevedo, J., Bibliographical review on cyber attacks from a control oriented perspective, Annual Reviews in Control, 48, 103-128 (2019)</td> </tr><tr> <td>[16]</td> <td class="space">Shi, D.; Elliott, R. J.; Chen, T., On finite-state stochastic modeling and secure estimation of cyber-physical systems, IEEE Transactions on Automatic Control, 62, 1, 65-80 (2017) &middot; <a href="/1359.94658" class="nowrap">Zbl 1359.94658</a></td> </tr><tr> <td>[17]</td> <td class="space">Shoukry, Y.; Nuzzo, P.; Puggelli, A.; Sangiovanni-Vincentelli, A. L.; Seshia, S. A.; Tabuada, P., Secure state estimation for cyber-physical systems under sensor attacks: A satisfiability modulo theory approach, IEEE Transactions on Automatic Control, 62, 10, 4917-4932 (2017) &middot; <a href="/1390.93532" class="nowrap">Zbl 1390.93532</a></td> </tr><tr> <td>[18]</td> <td class="space">Shoukry, Y.; Tabuada, P., Event-triggered state observers for sparse sensor noise/attacks, IEEE Transactions on Automatic Control, 61, 8, 2079-2091 (2016) &middot; <a href="/1359.93072" class="nowrap">Zbl 1359.93072</a></td> </tr><tr> <td>[19]</td> <td class="space">Yang, H.; Han, Q. L.; Ge, X.; Ding, L.; Xu, Y.; Jiang, B., Fault tolerant cooperative control of multi-agent systems: A survey of trends and methodologies, IEEE Transactions on Industrial Informatics, 16, 4-17 (2020)</td> </tr><tr> <td>[20]</td> <td class="space">Zhou, J.; Chen, B.; Yu, L., Intermediate-variable-based estimation for FDI attacks in cyber-physical systems, IEEE Transactions on Circuits and Systems II: Express Briefs (2020)</td> </tr><tr> <td>[21]</td> <td class="space">Zhu, J. W.; Yang, G. H.; Wang, H.; Wang, F., Fault estimation for a class of nonlinear systems based on intermediate estimator, IEEE Transactions on Automatic Control, 61, 9, 2518-2524 (2016) &middot; <a href="/1359.93075" class="nowrap">Zbl 1359.93075</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&nbsp;Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> &copy; 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms &amp; Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none">&nbsp;<span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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