Document Zbl 1213.93192

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Control Appl. Methods</a> <a href="/?q=in%3A273608" title="Articles in this Issue">32, No. 1, 47-63 (2011)</a>. </div> <div class="abstract">Summary: We investigate the robust filter design problem for nonlinear systems with multiplicative noises. The aim of the problem is to design a state estimator with a predictor-corrector structure, such that the upper bound on the state estimation error variance is minimized. A Robust Extended Kalman Filter (REKF) is proposed based on a novel method to obtain the upper bound on the variances of the multiplicative noises. Further analysis shows that the proposed filter guarantees a bounded energy gain from the multiplicative noises to the estimation error. The REKF is implemented on the satellite attitude determination system that consists of the gyroscopes and the star sensors. Its performance is illustrated by using the real data obtained from a gyroscope. Simulation results show that the REKF outperforms another robust algorithm.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5866101">Cited in <strong>19</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93E11" title="MSC2020">93E11</a> </td> <td class="space"> Filtering in stochastic control theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C10" title="MSC2020">93C10</a> </td> <td class="space"> Nonlinear systems in control theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93E10" title="MSC2020">93E10</a> </td> <td class="space"> Estimation and detection in stochastic control theory </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%3Arobust+extended+Kalman+filter">robust extended Kalman filter</a>; <a href="/?q=ut%3Amultiplicative+noise">multiplicative noise</a>; <a href="/?q=ut%3Anonlinear+system">nonlinear system</a>; <a href="/?q=ut%3Asatellite+attitude+determination">satellite attitude determination</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" 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class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1002/oca.928" aria-label="DOI for “Robust extended Kalman filtering for nonlinear systems with multiplicative noises”" title="10.1002/oca.928">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">Zhang, Nonlinear observer design for systems with unknown nonlinearity via B-spline network approach, Acta Automatica Sinica 26 (2) pp 199– (2006)</td> </tr><tr> <td>[2]</td> <td class="space">Qiu, Solution of two-stage Kalman filter, IEE Proceedings-Control Theory and Applications 152 (2) pp 152– (2005) · <a href="https://doi.org/10.1049/ip-cta:20045014" class="nowrap">doi:10.1049/ip-cta:20045014</a></td> </tr><tr> <td>[3]</td> <td class="space">Einicke, Optimal and robust noncausal filter formulations, IEEE Transactions on Signal Processing 54 (3) pp 1069– (2006) · <a href="/1373.94572" class="nowrap">Zbl 1373.94572</a> · <a href="https://doi.org/10.1109/TSP.2005.863042" class="nowrap">doi:10.1109/TSP.2005.863042</a></td> </tr><tr> <td>[4]</td> <td class="space">Rawicz, Explicit formulas for two state Kalman, H2 and H target tracking filters, IEEE Transactions of Aerospace and Electronic Systems 39 (1) pp 53– (2003) · <a href="https://doi.org/10.1109/TAES.2003.1188893" class="nowrap">doi:10.1109/TAES.2003.1188893</a></td> </tr><tr> <td>[5]</td> <td class="space">Yu, Nonlinear robust observer design for strapdown INS in-flight alignment, IEEE Transactions on Aerospace and Electronic Systems 40 (3) pp 797– (2004) · <a href="https://doi.org/10.1109/TAES.2004.1337455" class="nowrap">doi:10.1109/TAES.2004.1337455</a></td> </tr><tr> <td>[6]</td> <td class="space">He, Robust H filtering for networked systems with multiple state delays, International Journal of Control 80 (8) pp 1217– (2007) · <a href="/1133.93314" class="nowrap">Zbl 1133.93314</a> · <a href="https://doi.org/10.1080/00207170701196968" class="nowrap">doi:10.1080/00207170701196968</a></td> </tr><tr> <td>[7]</td> <td class="space">Lu, Robust Kalman filtering for discrete-time systems with measurement delay, IEEE Transactions on Circuits and Systems-II: Express Briefs 54 (6) pp 522– (2007) · <a href="https://doi.org/10.1109/TCSII.2007.892223" class="nowrap">doi:10.1109/TCSII.2007.892223</a></td> </tr><tr> <td>[8]</td> <td class="space">Singh, Closed-form minimax time-delay filters for underdamped systems, Optimal Control Applications and Methods 28 (3) pp 157– (2007) · <a href="https://doi.org/10.1002/oca.790" class="nowrap">doi:10.1002/oca.790</a></td> </tr><tr> <td>[9]</td> <td class="space">Haddad, H2 suboptimal estimation control for nonnegative dynamical systems, Optimal Control Applications and Methods 30 (1) pp 27– (2009) · <a href="https://doi.org/10.1002/oca.844" class="nowrap">doi:10.1002/oca.844</a></td> </tr><tr> <td>[10]</td> <td class="space">Xiong, Adaptive robust extended Kalman filter for nonlinear stochastic systems, IET Control Theory and Applications 2 (3) pp 239– (2008) · <a href="https://doi.org/10.1049/iet-cta:20070096" class="nowrap">doi:10.1049/iet-cta:20070096</a></td> </tr><tr> <td>[11]</td> <td class="space">Gao, A robust fault detection filtering for stochastic distribution systems via descriptor estimator and parametric gain design, IET Proceedings-Control Theory Applications 1 (5) pp 1286– (2007) · <a href="https://doi.org/10.1049/iet-cta:20060429" class="nowrap">doi:10.1049/iet-cta:20060429</a></td> </tr><tr> <td>[12]</td> <td class="space">Gao, State/noise estimator for descriptor systems with application to sensor fault diagnosis, IEEE Transactions on Signal Processing 54 (4) pp 1316– (2006) · <a href="/1373.94594" class="nowrap">Zbl 1373.94594</a> · <a href="https://doi.org/10.1109/TSP.2006.870579" class="nowrap">doi:10.1109/TSP.2006.870579</a></td> </tr><tr> <td>[13]</td> <td class="space">Gao, PD observer parameterization design for descriptor systems, Journal of the Franklin Institute 342 (5) pp 551– (2005) · <a href="/1141.93317" class="nowrap">Zbl 1141.93317</a> · <a href="https://doi.org/10.1016/j.jfranklin.2005.03.001" class="nowrap">doi:10.1016/j.jfranklin.2005.03.001</a></td> </tr><tr> <td>[14]</td> <td class="space">Fu, Finite-horizon robust Kalman filter design, IEEE Transactions on Signal Processing 49 (9) pp 2103– (2001) · <a href="/1369.93625" class="nowrap">Zbl 1369.93625</a> · <a href="https://doi.org/10.1109/78.942638" class="nowrap">doi:10.1109/78.942638</a></td> </tr><tr> <td>[15]</td> <td class="space">Ishihara, Robust Kalman filter for descriptor systems, IEEE Transactions on Automatic Control 51 (8) pp 1354– (2006) · <a href="/1366.93657" class="nowrap">Zbl 1366.93657</a> · <a href="https://doi.org/10.1109/TAC.2006.878741" class="nowrap">doi:10.1109/TAC.2006.878741</a></td> </tr><tr> <td>[16]</td> <td class="space">Sun, Robust H2 and H filters for uncertain LFT systems, IEEE Transactions on Automatic Control 50 (5) pp 715– (2005) · <a href="/1365.93515" class="nowrap">Zbl 1365.93515</a> · <a href="https://doi.org/10.1109/TAC.2005.847040" class="nowrap">doi:10.1109/TAC.2005.847040</a></td> </tr><tr> <td>[17]</td> <td class="space">Wang, Robust Kalman filters for linear time-varying systems with stochastic parametric uncertainties, IEEE Transactions on Signal Processing 50 (4) pp 803– (2002) · <a href="/1369.93659" class="nowrap">Zbl 1369.93659</a> · <a href="https://doi.org/10.1109/78.992124" class="nowrap">doi:10.1109/78.992124</a></td> </tr><tr> <td>[18]</td> <td class="space">Wang, Robust steady-state filtering for systems with deterministic and stochastic uncertainties, IEEE Transactions on Signal Processing 51 (10) pp 2550– (2003) · <a href="https://doi.org/10.1109/TSP.2003.816861" class="nowrap">doi:10.1109/TSP.2003.816861</a></td> </tr><tr> <td>[19]</td> <td class="space">Yang, Robust Kalman filtering for discrete time-varying uncertain systems with multiplicative noise, IEEE Transactions on Automatic Control 47 (7) pp 1179– (2002) · <a href="/1364.93817" class="nowrap">Zbl 1364.93817</a> · <a href="https://doi.org/10.1109/TAC.2002.800668" class="nowrap">doi:10.1109/TAC.2002.800668</a></td> </tr><tr> <td>[20]</td> <td class="space">Azimi-Sadjadi, Two-dimensional adaptive block Kalman filtering of SAR Imagery, IEEE Transactions on Geoscience and Remote Sensing 29 (5) pp 742– (1991) · <a href="https://doi.org/10.1109/36.83989" class="nowrap">doi:10.1109/36.83989</a></td> </tr><tr> <td>[21]</td> <td class="space">Wang, Robust filtering for a class of stochastic uncertain nonlinear time-delay systems via exponential state estimation, IEEE Transactions on Signal Processing 49 (4) pp 794– (2001) · <a href="https://doi.org/10.1109/78.912924" class="nowrap">doi:10.1109/78.912924</a></td> </tr><tr> <td>[22]</td> <td class="space">Seo, An extended robust H filter for nonlinear constrained uncertain systems, IEEE Transactions on Signal Processing 54 (11) pp 4471– (2006) · <a href="/1373.93110" class="nowrap">Zbl 1373.93110</a> · <a href="https://doi.org/10.1109/TSP.2006.880308" class="nowrap">doi:10.1109/TSP.2006.880308</a></td> </tr><tr> <td>[23]</td> <td class="space">Calafiore, Reliable localization using set-valued nonlinear filters, IEEE Transactions on Systems, Man and Cybernetics-Part A: Systems and Humans 35 (2) pp 189– (2005) · <a href="https://doi.org/10.1109/TSMCA.2005.843383" class="nowrap">doi:10.1109/TSMCA.2005.843383</a></td> </tr><tr> <td>[24]</td> <td class="space">Xie, Robust Kalman filtering for uncertain discrete-time systems, IEEE Transactions on Automatic Control 39 pp 1310– (1994) · <a href="/0812.93069" class="nowrap">Zbl 0812.93069</a> · <a href="https://doi.org/10.1109/9.293203" class="nowrap">doi:10.1109/9.293203</a></td> </tr><tr> <td>[25]</td> <td class="space">Theodor, Robust discrete-time minimum-variance filtering, IEEE Transactions on Signal Processing 44 pp 181– (1996) · <a href="https://doi.org/10.1109/78.485915" class="nowrap">doi:10.1109/78.485915</a></td> </tr><tr> <td>[26]</td> <td class="space">Xie, Robust Kalman filter for uncertain systems, Systems and Control Letters 22 pp 123– (1994) · <a href="/0792.93118" class="nowrap">Zbl 0792.93118</a> · <a href="https://doi.org/10.1016/0167-6911(94)90106-6" class="nowrap">doi:10.1016/0167-6911(94)90106-6</a></td> </tr><tr> <td>[27]</td> <td class="space">Lefferts, Kalman filtering for spacecraft attitude estimation, Journal of Guidance, Control, and Dynamics 5 pp 417– (1982) · <a href="https://doi.org/10.2514/3.56190" class="nowrap">doi:10.2514/3.56190</a></td> </tr><tr> <td>[28]</td> <td class="space">Crassidis, Unscented filtering for spacecraft attitude estimation, Journal of Guidance, Control, and Dynamics 26 pp 536– (2003) · <a href="https://doi.org/10.2514/2.5102" class="nowrap">doi:10.2514/2.5102</a></td> </tr><tr> <td>[29]</td> <td class="space">Kim YV Difilippo KJ Ng A On the autonomous in orbit calibration of satellite attitude sensors</td> </tr><tr> <td>[30]</td> <td class="space">Markley FL Crassidis JL Cheng Y Nonlinear attitude filtering method</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. 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"> © 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 & 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"> <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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