Document Zbl 1153.93530

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Necessary conditions for reconstruction and stability of system outputs are derived using the Information Transmission theorem and the Shannon lower bound. These conditions are expressed in terms of the Shannon entropy rate and the distortion measure employed to describe reconstruction and stability. The methodology is general, and hence it is applicable to a variety of systems. The results are applied to linear partially observed stochastic Gaussian controlled systems, when the channel is an additive white Gaussian noise channel. For such systems and channels, sufficient conditions are also derived by first showing that the Shannon lower bound is exactly equal to the rate distortion function, and then designing the encoder, decoder and controller which achieve the capacity of the channel. The conditions imposed are the standard detectability and stabilizability of Linear Quadratic Gaussian (LQG) theory, while a separation principle is shown between the design of the control and communication systems, without assuming knowledge of the control sequence at the encoder/decoder.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5495785">Cited in <strong>16</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93E20" title="MSC2020">93E20</a> </td> <td class="space"> Optimal stochastic control </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93E03" title="MSC2020">93E03</a> </td> <td class="space"> Stochastic systems in control theory (general) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A94A40" title="MSC2020">94A40</a> </td> <td class="space"> Channel models (including quantum) in information and communication theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Anetworked">networked</a>; <a href="/?q=ut%3Acapacity">capacity</a>; <a href="/?q=ut%3Arate+distortion">rate distortion</a>; <a href="/?q=ut%3Astability">stability</a>; <a href="/?q=ut%3Aseparation">separation</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 1153.93530" data-ciurl="/ci/05495785" data-biburl="/bibtex/05495785.bib" data-amsurl="/amsrefs/05495785.bib" data-xmlurl="/xml/05495785.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05495785.pdf" title="Zbl 1153.93530 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.2008.05.021" aria-label="DOI for “LQG optimality and separation principle for general discrete time partially observed stochastic systems over finite capacity communication channels”" title="10.1016/j.automatica.2008.05.021">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">Berger, T., Rate distortion theory: A mathematical basis for data compression (1971), Prentice-Hall · <a href="/0200.00001" class="nowrap">Zbl 0200.00001</a></td> </tr><tr> <td>[2]</td> <td class="space">Caines, P. E., Linear stochastic systems (1988), John Wiley and Sons · <a href="/0781.93093" class="nowrap">Zbl 0781.93093</a></td> </tr><tr> <td>[3]</td> <td class="space">Cover, T. M.; Thomas, J. A., Elements of information theory (1991), John Wiley and Sons · <a href="/0762.94001" class="nowrap">Zbl 0762.94001</a></td> </tr><tr> <td>[4]</td> <td class="space">Elia, N., When Bode meets Shannon: Control-oriented feedback communication schemes, IEEE Transactions on Automatic Control, 49, 9, 1477-1488 (2004) · <a href="/1365.94013" class="nowrap">Zbl 1365.94013</a></td> </tr><tr> <td>[5]</td> <td class="space">Li, K.; Baillieul, J., Robust quantization for digital finite communication bandwidth (DFCB) control, IEEE Transactions on Automatic Control, 49, 9, 1573-1584 (2004) · <a href="/1365.93280" class="nowrap">Zbl 1365.93280</a></td> </tr><tr> <td>[6]</td> <td class="space">Liberzon, D.; Hespanha, J. P., Stabilization of nonlinear systems with limited information feedback, IEEE Transactions on Automatic Control, 50, 6, 910-915 (2005) · <a href="/1365.93405" class="nowrap">Zbl 1365.93405</a></td> </tr><tr> <td>[7]</td> <td class="space">Linder, T.; Zamir, R., On the asymptotic tightness of the Shannon lower bound, IEEE Transactions on Information Theory, 40, 6, 2026-2031 (1994) · <a href="/0826.94013" class="nowrap">Zbl 0826.94013</a></td> </tr><tr> <td>[8]</td> <td class="space">Massey, J. (1990). Causality, feedback and directed information. In <span class="zbmathjax-textit">Proceedings of the 1990 symposium on information theory and its applications</span>; Massey, J. (1990). Causality, feedback and directed information. In <span class="zbmathjax-textit">Proceedings of the 1990 symposium on information theory and its applications</span></td> </tr><tr> <td>[9]</td> <td class="space">Malyavej, V.; Savkin, A. V., The problem of optimal robust Kalman state estimation via limited capacity digital communication channels, Systems and Control Letters, 54, 3, 283-292 (2005) · <a href="/1129.93539" class="nowrap">Zbl 1129.93539</a></td> </tr><tr> <td>[10]</td> <td class="space">Nair, G. N., Dey, S., & Evans, R. J. (2003). Infimum data rates for stabilizing Markov jump linear systems. In <span class="zbmathjax-textit">Proceedings of the 42nd IEEE conference on decision and control</span>; Nair, G. N., Dey, S., & Evans, R. J. (2003). Infimum data rates for stabilizing Markov jump linear systems. In <span class="zbmathjax-textit">Proceedings of the 42nd IEEE conference on decision and control</span></td> </tr><tr> <td>[11]</td> <td class="space">Nair, G. N.; Evans, R. J.; Mareels, I. M.Y.; Moran, W., Topological feedback entropy and nonlinear stabilization, IEEE Transations on Automatic Control, 49, 9, 1585-1597 (2004) · <a href="/1365.94158" class="nowrap">Zbl 1365.94158</a></td> </tr><tr> <td>[12]</td> <td class="space">Savkin, A. V.; Petersen, I. R., Set valued state estimation via a limited capacity communication channel, IEEE Transactions on Automatic Control, 48, 4, 676-680 (2003) · <a href="/1364.93784" class="nowrap">Zbl 1364.93784</a></td> </tr><tr> <td>[13]</td> <td class="space">Tatikonda, S.; Mitter, S., Control over noisy channels, IEEE Transactions on Automatic Control, 49, 7, 1196-1201 (2004) · <a href="/1365.94341" class="nowrap">Zbl 1365.94341</a></td> </tr><tr> <td>[14]</td> <td class="space">Tatikonda, S.; Mitter, S., Control under communication constraints, IEEE Transactions on Automatic Control, 49, 7, 1056-1068 (2004) · <a href="/1365.93271" class="nowrap">Zbl 1365.93271</a></td> </tr><tr> <td>[15]</td> <td class="space">Tatikonda, S.; Sahai, A.; Mitter, S., Stochastic linear control over a communication channel, IEEE Transactions on Automatic Control, 49, 9, 1549-1561 (2004) · <a href="/1365.93548" class="nowrap">Zbl 1365.93548</a></td> </tr><tr> <td>[16]</td> <td class="space">Tsumura, K., & Maciejowski, J. (2003). Stabilizability of SISO control systems under constraints of channel capacities. In <span class="zbmathjax-textit">Proceedings of 42nd IEEE conference on decision and control</span>; Tsumura, K., & Maciejowski, J. (2003). Stabilizability of SISO control systems under constraints of channel capacities. In <span class="zbmathjax-textit">Proceedings of 42nd IEEE conference on decision and control</span></td> </tr><tr> <td>[17]</td> <td class="space">Wu, B. F.; Jonckheere, E. A., A simplified approach to Bode’s theorem for continuous-time and discrete-time systems, IEEE Transactions on Automatic Control, 37, 11, 1797-1802 (1992) · <a href="/0778.93076" class="nowrap">Zbl 0778.93076</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. 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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