Document Zbl 1133.93040

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With an introduction to stochastic control theory. 2nd ed.</strong> <i>(English)</i> <a class="label nowrap" href="/1133.93040">Zbl 1133.93040</a> </h2> <div class="source"> <a href="/serials/6303" title="Series Profile">Automation and Control Engineering</a> 26. Boca Raton, FL: CRC Press (ISBN 978-0-8493-9008-1/hbk). xxi, 523 p. (2008). </div> <div class="abstract">In this book, the author discusses optimal and robust estimation with an introduction to stochastic control theory. The authors’ view is that optimal control and estimation are the dual theories that provide the basis for the study of systems in the modern era. In this regard, optimal control can be studied in an exclusively deterministic framework in which we may assume that full information about the system is known. In this case, control problems are solved via a procedure involving a backward progression through time. On the other hand, optimal estimation is related to probabilistic concepts that take into account the casting of information arising from noisy environments into a tractable form. By contrast, optimal estimation problems are solved by means of a forward progression through time. Also, stochastic control theory involves elements from both optimal estimation and deterministic optimal control. This due to the fact that this type of control theory uses information built from noisy measurements to control a system so as to perform in a desirable manner. <br class="zbmathjax-paragraph">The book is arranged in three parts consisting of a total of ten chapters plus an appendix. <br class="zbmathjax-paragraph">Part I of the book is related to optimal estimation. In chapter 1, the authors use classical techniques to estimate an unknown quantity from available data. The techniques used are mean-square estimation and the Wiener filter which includes a discussion of recursive estimation. Chapter 2 of Part I estimates the state of a discrete-time linear system that evolves dynamically through time. In this case, the optimal estimator turns out to be the Kalman filter which is an example of a linear dynamical system. The discrete Kalman filter is extended to the continuous-time case in chapter 3. The fourth chapter of Part I discusses two topics related to practical aspects of Kalman Filter design and application. The first of these topics analyzes the effects of the assumption that a model does not accurately describe the physical system whose state is to be estimated while the second topic involves techniques for designing filters of simplified form. This part of the book concludes with finding time and measurement updates for the entire hyperstate in chapter 5. This involves the conditional probability density function of the unknown when a certain set of data is given. Also, the authors obtain a computationally viable logarithm for general nonlinear systems by engaging the extended Kalman filter (EKF). <br class="zbmathjax-paragraph">In Part II the author describes the concept of robust estimation. This firstly involves a discussion of the design of a filter (in particular, the robust Kalman filter) that possesses specific performance robustness against likely uncertainties in system parameters (Chapter 6). Chapter 7 of Part II investigates estimator design with an $\mathcal{H}_{\infty}$ performance measure. Here system uncertainty is in the form bounded energy noise signals which means that uncertainty in the matrices of of the system-state space model is absent. In this chapter, both the finite-horizon and stationary $\mathcal{H}_{\infty}$ estimation problems are considered. Part II concludes with a study of $\mathcal{H}_{\infty}$ filtering of discrete-time systems. Here both a Riccati equation and polynomial approach is employed. In order to illustrate the design of an $\mathcal{H}_{\infty}$ filter, an application of $\mathcal{H}_{\infty}$ filtering to channel equalization is presented. <br class="zbmathjax-paragraph">Part III of the book is dedicated to optimal stochastic control. In this regard, the author discusses stochastic control for state variable systems (Chapter 9) and stochastic control for polynomial systems (Chapter 10). <br class="zbmathjax-paragraph">The book concludes with an appendix containing a review of matrix algebra.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A10569">Mark A. Petersen (Potchefstroom)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5210013">Cited in <strong>40</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><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%3A93-02" title="MSC2020">93-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to systems and control theory </td> </tr><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%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%3Aoptimal+estimation">optimal estimation</a>; <a href="/?q=ut%3Arobust+estimation">robust estimation</a>; <a href="/?q=ut%3Astochastic+control+theory">stochastic control theory</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 1133.93040" data-ciurl="/ci/05210013" data-biburl="/bibtex/05210013.bib" data-amsurl="/amsrefs/05210013.bib" data-xmlurl="/xml/05210013.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05210013.pdf" title="Zbl 1133.93040 as PDF">Review PDF</a> </div> <div class="fulltexts"> </div> <div class="sfx" style="float: right;"> </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 ? 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