Document Zbl 1238.93108

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Signal Process.</a> <a href="/?q=in%3A289855" title="Articles in this Issue">30, No. 6, 1355-1368 (2011)</a>. </div> <div class="abstract">Summary: In this paper, the robust filter design problem is studied for a class of uncertain dynamical systems with finite-step correlated process noises and missing measurements. The dynamical system under consideration is subject to both deterministic norm-bounded uncertainties in the measurement output and stochastic uncertainties on the system states. The process noises are assumed to be finite-step correlated. The missing measurement phenomenon is modeled as a binary switching sequence. Based on the min-max game theory, a recursive robust filter is designed that is suitable for online application. A particular feature is that, as the proposed robust filters work in a recursive fashion, there is no need to investigate the existence issue of the filters. A simulation example is presented to illustrate the usefulness of the proposed filter.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6021833">Cited in <strong>10</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%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%3A93E10" title="MSC2020">93E10</a> </td> <td class="space"> Estimation and detection in stochastic control theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Arecursive+filters">recursive filters</a>; <a href="/?q=ut%3Arobust+filters">robust filters</a>; <a href="/?q=ut%3Afinite-step+correlated+noises">finite-step correlated noises</a>; <a href="/?q=ut%3Amin-max+game+theory">min-max game theory</a>; <a href="/?q=ut%3Amultiplicative+noises">multiplicative noises</a>; <a href="/?q=ut%3Anorm-bound+uncertainty">norm-bound uncertainty</a>; <a href="/?q=ut%3Amissing+measurements">missing measurements</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 1238.93108" data-ciurl="/ci/06021833" data-biburl="/bibtex/06021833.bib" data-amsurl="/amsrefs/06021833.bib" data-xmlurl="/xml/06021833.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06021833.pdf" title="Zbl 1238.93108 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/10.1007/s00034-011-9289-6" aria-label="DOI for “Recursive robust filtering with finite-step correlated process noises and missing measurements”" title="10.1007/s00034-011-9289-6">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">D.P. Bertsekas, I.B. Rhodes, Recursive state estimate for a membership description of uncertainty. IEEE Trans. Autom. Control 16(2), 117–128 (1971) · <a href="https://doi.org/10.1109/TAC.1971.1099674" class="nowrap">doi:10.1109/TAC.1971.1099674</a></td> </tr><tr> <td>[2]</td> <td class="space">X. Chen, K. Zhou, H Gaussian filter on infinite time horizon. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 49(5), 674–679 (2002) · <a href="/1368.93709" class="nowrap">Zbl 1368.93709</a> · <a href="https://doi.org/10.1109/TCSI.2002.1001957" class="nowrap">doi:10.1109/TCSI.2002.1001957</a></td> </tr><tr> <td>[3]</td> <td class="space">Y.L. Chen, B.S. Chen, Minimax robust deconvolution filters under stochastic parametric and noise uncertainties. IEEE Trans. Signal Process. 42(1), 32–45 (1994) · <a href="https://doi.org/10.1109/78.258119" class="nowrap">doi:10.1109/78.258119</a></td> </tr><tr> <td>[4]</td> <td class="space">J.M.C. Clark, P.A. Kountouriotis, R.B. Vinter, A new algorithm for GMTI tracking problems, subject to a Doppler blind zone constraint, in IET Seminar on Target Tracking and Data Fusion: Algorithms and Applications, vol. 1 (2008), pp. 51–58</td> </tr><tr> <td>[5]</td> <td class="space">S. Cong, L. Hong, J.R. Layne, Iterative robust filtering for ground target tracking. IET Control Theory Appl. 1(1), 372–380 (2007) · <a href="https://doi.org/10.1049/iet-cta:20050378" class="nowrap">doi:10.1049/iet-cta:20050378</a></td> </tr><tr> <td>[6]</td> <td class="space">H. Gao, J. Lam, L. Xie, C. Wang, New approach to mixed H 2/H filtering for polytopic discrete-time systems. IEEE Trans. Signal Process. 53(8), 3183–3192 (2005) · <a href="/1370.93274" class="nowrap">Zbl 1370.93274</a> · <a href="https://doi.org/10.1109/TSP.2005.851116" class="nowrap">doi:10.1109/TSP.2005.851116</a></td> </tr><tr> <td>[7]</td> <td class="space">H. Gao, X. Meng, T. Chen, A parameter-dependent approach to robust H filtering for time-delay systems. IEEE Trans. Autom. Control 53(10), 2420–2425 (2008) · <a href="/1367.93176" class="nowrap">Zbl 1367.93176</a> · <a href="https://doi.org/10.1109/TAC.2008.2007544" class="nowrap">doi:10.1109/TAC.2008.2007544</a></td> </tr><tr> <td>[8]</td> <td class="space">H. Gao, Y. Zhao, J. Lam, K. Chen, H fuzzy filtering of nonlinear systems with intermittent measurements. IEEE Trans. Fuzzy Syst. 17(2), 291–300 (2009) · <a href="https://doi.org/10.1109/TFUZZ.2008.924206" class="nowrap">doi:10.1109/TFUZZ.2008.924206</a></td> </tr><tr> <td>[9]</td> <td class="space">L.E. Ghaoui, State-feedback control of systems with multiplicative noise via linear matrix inequalities. Syst. Control Lett. 24(3), 223–228 (1995) · <a href="/0877.93076" class="nowrap">Zbl 0877.93076</a> · <a href="https://doi.org/10.1016/0167-6911(94)00045-W" class="nowrap">doi:10.1016/0167-6911(94)00045-W</a></td> </tr><tr> <td>[10]</td> <td class="space">J.C. Geromel, J. Bernussou, G. Garcia, M.C. de Oliveira, H 2/H robust filtering for discrete-time linear systems. IEEE Conf. Decis. Control 1, 632–637 (1998)</td> </tr><tr> <td>[11]</td> <td class="space">E. Gershon, U. Shaked, I. Yaesh, H control and filtering of discrete-time stochastic systems with multiplicative noise. Automatica 37(3), 409–417 (2001) · <a href="/0989.93030" class="nowrap">Zbl 0989.93030</a> · <a href="https://doi.org/10.1016/S0005-1098(00)00164-3" class="nowrap">doi:10.1016/S0005-1098(00)00164-3</a></td> </tr><tr> <td>[12]</td> <td class="space">Y.S. Hung, H.T. Ho, A Kalman filter approach to direct depth estimation incorporating surface structure. IEEE Trans. Pattern Anal. Mach. Intell. 21, 570–575 (1999) · <a href="https://doi.org/10.1109/34.771330" class="nowrap">doi:10.1109/34.771330</a></td> </tr><tr> <td>[13]</td> <td class="space">Y.S. Hung, F. Yang, H versus Kalman filtering for depth estimation, in Proc. Int. Conf. Signal Processing Intelligent System, Guangzhou, P.R.C. (1999), pp. 276–281</td> </tr><tr> <td>[14]</td> <td class="space">V.S. Kouikoglou, Y.A. Phillis, Trace bounds on the covariances of continuous-time systems with multiplicative. IEEE Trans. Autom. Control 38(1), 138–142 (1993) · <a href="/0773.93053" class="nowrap">Zbl 0773.93053</a> · <a href="https://doi.org/10.1109/9.186326" class="nowrap">doi:10.1109/9.186326</a></td> </tr><tr> <td>[15]</td> <td class="space">G. Liu, H. Wang, Fault detection and diagnosis for general stochastic systems using B-spline expansions and nonlinear filters. IEEE Trans. Circuits Syst. I, Regul. Pap. 52(8), 1644–1652 (2005) · <a href="/1374.94963" class="nowrap">Zbl 1374.94963</a> · <a href="https://doi.org/10.1109/TCSI.2005.851686" class="nowrap">doi:10.1109/TCSI.2005.851686</a></td> </tr><tr> <td>[16]</td> <td class="space">N. Nhai, Optimal recursive estimation with uncertain observation. IEEE Trans. Inf. Theory 15(4), 457–462 (1969) · <a href="/0174.51102" class="nowrap">Zbl 0174.51102</a> · <a href="https://doi.org/10.1109/TIT.1969.1054329" class="nowrap">doi:10.1109/TIT.1969.1054329</a></td> </tr><tr> <td>[17]</td> <td class="space">Y.A. Phillis, Estimation and control systems with unknown covariance and multiplicative noise. IEEE Trans. Autom. Control 34(10), 1075–1078 (1989) · <a href="/0694.93099" class="nowrap">Zbl 0694.93099</a> · <a href="https://doi.org/10.1109/9.35279" class="nowrap">doi:10.1109/9.35279</a></td> </tr><tr> <td>[18]</td> <td class="space">A.H. Sayed, A framework for state-space estimation with uncertain models. IEEE Trans. Autom. Control 46(7), 998–1013 (2001) · <a href="/1009.93075" class="nowrap">Zbl 1009.93075</a> · <a href="https://doi.org/10.1109/9.935054" class="nowrap">doi:10.1109/9.935054</a></td> </tr><tr> <td>[19]</td> <td class="space">D. Simon, Optimal State Estimation (Wiley/INC Press, New York, 2006)</td> </tr><tr> <td>[20]</td> <td class="space">E. Song, Y. Zhu, Z. You, The Kalman type recursive state estimator with a finite-step correlated process noises, in Proceedings of the IEEE International Conference on Automation and Logistics, Qingdao, China, Sept. 2008</td> </tr><tr> <td>[21]</td> <td class="space">A.M. Tekalp, G. Pavlovic, Image restoration with multiplicative noise: incorporating the sensor nonlinearity. IEEE Trans. Signal Process. 39(9), 2132–2136 (1991) · <a href="https://doi.org/10.1109/78.134455" class="nowrap">doi:10.1109/78.134455</a></td> </tr><tr> <td>[22]</td> <td class="space">J.Y. Tourneret, M. Coulon, M. Doisy, Least-squares estimation of multiple abrupt changes contaminated by multiplicative noise using MCMC, in Proc. IEEE Workshop Higher Order Statistics (1999), pp. 1148–1152</td> </tr><tr> <td>[23]</td> <td class="space">Z. Wang, J. Lam, X. Liu, Robust filtering for discrete-time Markovian jump delay systems. IEEE Signal Process. Lett. 11(8), 659–662 (2004) · <a href="https://doi.org/10.1109/LSP.2004.831729" class="nowrap">doi:10.1109/LSP.2004.831729</a></td> </tr><tr> <td>[24]</td> <td class="space">Z. Wang, Y. Liu, X. Liu, H filtering for uncertain stochastic time-delay systems with sector-bounded nonlinearities. Automatica 44(5), 1268–1277 (2008) · <a href="/1283.93284" class="nowrap">Zbl 1283.93284</a> · <a href="https://doi.org/10.1016/j.automatica.2007.09.016" class="nowrap">doi:10.1016/j.automatica.2007.09.016</a></td> </tr><tr> <td>[25]</td> <td class="space">Z. Wang, D.W.C. Ho, Y. Liu, X. Liu, Robust H control for a class of nonlinear discrete time-delay stochastic systems with missing measurements. Automatica 45(3), 684–691 (2009) · <a href="/1166.93319" class="nowrap">Zbl 1166.93319</a> · <a href="https://doi.org/10.1016/j.automatica.2008.10.025" class="nowrap">doi:10.1016/j.automatica.2008.10.025</a></td> </tr><tr> <td>[26]</td> <td class="space">Z. Wang, D.W.C. Ho, H. Dong, H. Gao, Robust H finite-horizon control for a class of stochastic nonlinear time-varying systems subject to sensor and actuator saturations. IEEE Trans. Autom. Control 55(7), 1716–1722 (2010) · <a href="/1368.93668" class="nowrap">Zbl 1368.93668</a> · <a href="https://doi.org/10.1109/TAC.2010.2047033" class="nowrap">doi:10.1109/TAC.2010.2047033</a></td> </tr><tr> <td>[27]</td> <td class="space">Z. Wang, Y. Liu, X. Liu, Exponential stabilization of a class of stochastic system with Markovian jump parameters and mode-dependent mixed time-delays. IEEE Trans. Autom. Control 55(7), 1656–1662 (2010) · <a href="/1368.93778" class="nowrap">Zbl 1368.93778</a> · <a href="https://doi.org/10.1109/TAC.2010.2046114" class="nowrap">doi:10.1109/TAC.2010.2046114</a></td> </tr><tr> <td>[28]</td> <td class="space">F. Yang, Z. Wang, Y.S. Hung, Robust Kalman filtering for discrete time-varying uncertain systems with multiplicative noise. IEEE Trans. Autom. Control 47(7), 1179–1183 (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>[29]</td> <td class="space">F. Yang, Z. Wang, Y.S. Hung, H. Shu, Mixed H 2/H filtering for uncertain systems with regional pole assignment. IEEE Trans. Aerosp. Electron. Syst. 41(2), 438–448 (2005) · <a href="https://doi.org/10.1109/TAES.2005.1468739" class="nowrap">doi:10.1109/TAES.2005.1468739</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 Open. 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