Document Zbl 1036.93058

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Minimum variance unbiased state estimators are considered, and the solutions are given in the time domain. The general case is investigated where the unknown inputs affect both the model and the outputs. In this case, the major problem is that the state estimation error is correlated with the model and the measurement noises. The unbiasedness of this error leads to the minimization of the variance of the estimation error under algebraic constraints. The authors parametrize all solutions of the algebraic constraints by two parameter matrices. The first matrix plays the role of a gain matrix and the second one is used to guarantee rank conditions. In the general case where the estimation error is correlated with the model and measurement noises, only a suboptimal filter can be obtained. Necessary and sufficient conditions for the existence of this filter are presented. The conditions for the estimation error to be uncorrelated with the system noises are given; these conditions lead to two filters. The first one is an estimator, and the second one is a predictor. All the obtained results are expressed in terms of the initial system matrices. In the time-invariant case, the conditions for the stability of the filters are derived. When the measurements are free of unknown inputs, these conditions become necessary and sufficient.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A3647">Yuliya S. Mishura (Kyïv)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A1963706">Cited in <strong>2</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A1963706">Cited in <strong>47</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><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%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%3A93E11" title="MSC2020">93E11</a> </td> <td class="space"> Filtering in stochastic control theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aminimum+variance+filter">minimum variance filter</a>; <a href="/?q=ut%3Aunknown+inputs">unknown inputs</a>; <a href="/?q=ut%3Astability">stability</a>; <a href="/?q=ut%3Aconvergence">convergence</a>; <a href="/?q=ut%3Alinear+time-varying+systems">linear time-varying systems</a>; <a href="/?q=ut%3Anoise-estimation+error+correlation">noise-estimation error correlation</a>; <a href="/?q=ut%3Aunbiased+estimators">unbiased estimators</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 1036.93058" data-ciurl="/ci/01963706" data-biburl="/bibtex/01963706.bib" data-amsurl="/amsrefs/01963706.bib" data-xmlurl="/xml/01963706.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01963706.pdf" title="Zbl 1036.93058 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/S0005-1098(03)00006-2" aria-label="DOI for “Extension of minimum variance estimation for systems with unknown inputs.”" title="10.1016/S0005-1098(03)00006-2">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://hal.science/hal-00098113" title="Open Access at HAL">HAL </a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Anderson, B.; Moore, J. B., Optimal filtering (1979), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · <a href="/0688.93058" class="nowrap">Zbl 0688.93058</a></td> </tr><tr> <td>[2]</td> <td class="space">Basseville, M.; Nikiforov, I., Detection of abrupt changes: Theory and application (1994), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ</td> </tr><tr> <td>[3]</td> <td class="space">Darouach, M.; Zasadzinski, M., Unbiased minimum variance estimation for systems with unknown exogenous inputs, Automatica, 33, 717-719 (1997) · <a href="/0874.93086" class="nowrap">Zbl 0874.93086</a></td> </tr><tr> <td>[4]</td> <td class="space">Darouach, M.; Zasadzinski, M.; Xu, S. J., Full-order observers for linear systems with unknown inputs, IEEE Transactions on Automatic Control, 39, 606-609 (1994) · <a href="/0813.93015" class="nowrap">Zbl 0813.93015</a></td> </tr><tr> <td>[5]</td> <td class="space">de Souza, C.; Gevers, M. R.; Goodwin, G. C., Riccati equations in optimal filtering of nonstabilizable systems having singular state transition matrices, IEEE Transactions on Automatic Control, 31, 831-838 (1986) · <a href="/0604.93059" class="nowrap">Zbl 0604.93059</a></td> </tr><tr> <td>[6]</td> <td class="space">Hou, M.; Müller, P. C., Design of observers for linear systems with unknown inputs, IEEE Transactions on Automatic Control, 37, 871-875 (1992) · <a href="/0775.93021" class="nowrap">Zbl 0775.93021</a></td> </tr><tr> <td>[7]</td> <td class="space">Hou, M.; Müller, P. C., Disturbance decoupled observer designA unified point of view, IEEE Transactions on Automatic Control, 39, 1338-1341 (1994) · <a href="/0812.93017" class="nowrap">Zbl 0812.93017</a></td> </tr><tr> <td>[8]</td> <td class="space">Hsieh, C., Robust two-stage Kalman filters for systems with unknown inputs, IEEE Transactions on Automatic Control, 45, 2374-2378 (2000) · <a href="/0990.93130" class="nowrap">Zbl 0990.93130</a></td> </tr><tr> <td>[9]</td> <td class="space">Kailath, T.; Sayed, A. H.; Hassibi, B., Linear estimation (2000), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ</td> </tr><tr> <td>[10]</td> <td class="space">Kitanidis, P., Unbiased minimum-variance linear state estimation, Automatica, 23, 775-778 (1987) · <a href="/0627.93065" class="nowrap">Zbl 0627.93065</a></td> </tr><tr> <td>[11]</td> <td class="space">Lancaster, P.; Tismenetsky, M., The theory of matrices (1985), Academic Press: Academic Press Orlando, USA · <a href="/0516.15018" class="nowrap">Zbl 0516.15018</a></td> </tr><tr> <td>[12]</td> <td class="space">Middleton, R.; Goodwin, G. C., Digital control and estimation (1990), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · <a href="/0672.93046" class="nowrap">Zbl 0672.93046</a></td> </tr><tr> <td>[13]</td> <td class="space">Nikoukhah, R., Innovations generation in the presence of unknown inputsApplication to robust failure detection, Automatica, 30, 1851-1867 (1994) · <a href="/0815.93080" class="nowrap">Zbl 0815.93080</a></td> </tr><tr> <td>[14]</td> <td class="space">Park, S.; Kim, P. S.; Kwon, O. H.; Kwon, W. H., Estimation and detection of unknown inputs using optimal FIR filter, Automatica, 36, 1481-1488 (2000) · <a href="/0959.93517" class="nowrap">Zbl 0959.93517</a></td> </tr><tr> <td>[15]</td> <td class="space">Patton, R.; Clark, R. N.; Frank, P. M., Fault diagnosis in dynamic systems (1989), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ</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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