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HTTP/1.1 200 OKServer: nginxDate: Tue, 03 Feb 2026 18:51:13 GMTContent-Type: text/html; charsetUTF-8Transfer-Encoding: chunkedConnection: keep-aliveVary: Accept-EncodingX-Powered-By: PHP/8.2.30Vary: Accept-EncodingX-Powered-By: PleskLin html>head>title>Book: Stability Analysis and Nonlinear Observer Design Using Takagi-Sugeno Fuzzy Models/title>meta http-equivContent-Type contenttext/html; charsetiso-8859-1>meta http-equivDescription contentWeb page for the book Stability Analysis and Nonlinear Observer Design Using Takagi-Sugeno Fuzzy Models, by Zsofia Lendek, Thierry Marie Guerra, Robert Babuska, and Bart De Schutter>meta http-equivKeywords contentTakagi-Sugeno fuzzy systems, fuzzy observers, stability, adaptive observers, distributed observer, Zsofia Lendek, Thierry Marie Guerra, Robert Babuska, Bart De Schutter>style typetext/css>/******************* * MAIN STYLES * *******************/body { margin: 0em .3em 1em .3em; font-family: Arial; font-size: 10pt; color: #0a3080; padding-left: 1em; padding-right: 1.5em;}p { margin-top: .4em; margin-bottom: 1em; text-align: justify;}/** Version of paragraph with tight margins. */p.tight { margin-top: .4em; margin-bottom: .4em;}h1 { margin-top: 1em; margin-bottom: .3em; font-size: 15pt; font-weight: bold; font-style: italic;}h2 { margin-top: .5em; margin-bottom: .4em; font-size: 12pt; font-weight: bold;}h3 { margin-top: .4em; margin-bottom: .3em; font-size: 1em; font-weight: bold;}hr { border: none; height: .2em; background-color: #9098a8;}/* LINK STYLES */a { color: #0a3080; }a:hover { color: #0f70c0; }a img { border: none; }/** Discrete links. **/a.discreet { font-weight: normal; text-decoration: none; border-bottom: 1px #cccccc solid; }a.discreet:hover { text-decoration: underline; border-bottom: none; }/** Strong links. **/a.strong { font-weight: bold; }/* Plain anchor. Remove all styling. */a.plainanchor { color: #0a3080; text-decoration: none; border: none; }a.plainanchor:hover { color: #0a3080; text-decoration: none; border: none; }/* Button link. */a.button { font-weight: bold; color: #214795; text-decoration: none; }a.button:hover { color: #3293C3; text-decoration: underline; }a.hide-button { font-weight: bold; color: #214795; display: block; text-align: right; text-decoration: none;}a.hide-button:hover { color: #3293C3; text-decoration: underline; }a.note-link { margin-left: .2em; margin-right: .2em; font-weight: normal; color: #214795; text-decoration: none;/* border-bottom-style: solid; border-bottom-color: #999999; border-bottom-width: 1px;*/}a.note-link:hover { color: #3293C3; text-decoration: underline; border-bottom: none;}/* GENERIC CLASSES */.note { color: #092051; font-size: .8em;}.footer { margin-top: 1em; color: #092051; font-size: .8em;}ul.toc { list-style-type: none; padding-left: 1.5em; font-size: .9em;}ul.toc ul { list-style-type: none; padding-left: 3em;}li.chapter { font-weight: bold;}td.biophoto { text-align: center;}td.bio { vertical-align: text-top; padding-left: .5em; font-size: 10pt;}/style>script typetext/javascript languagejavascript >/** * Toggle details display for an element. * @param id the ID of the details element. Will also be used as cookie key. * @param mode on or off. */function toggleDetails(id, mode) { switch(mode) { case on: document.getElementById(id).style.display ; document.getElementById(id + -showcontrol).style.display none; document.getElementById(id + -hidecontrol).style.display ; break; case off: document.getElementById(id + ).style.display none; document.getElementById(id + -showcontrol).style.display ; document.getElementById(id + -hidecontrol).style.display none; break; }}function initializePage() { toggleDetails(toc, off); toggleDetails(bio, off);}/script>/head>body onLoadinitializePage();>h1>Stability Analysis and Nonlinear Observer Design Using Takagi-Sugeno Fuzzy Models/h1>p>bya classstrong hrefhttp://lendek.net/home.php>Zsófia Lendek/a>,a classstrong hrefhttp://www.univ-valenciennes.fr/LAMIH-intra/site/commun/membres/page_perso.php?persoguerra_thierry-marie&languelang_fr>Thierry Marie Guerra/a>,a classstrong hrefhttp://www.dcsc.tudelft.nl/~rbabuska>Robert Babuška/a>,a classstrong hrefhttp://deschutter.info/>Bart De Schutter/a>br>Springer Germany, Studies on Fuzziness and Soft Computing, vol. 262, 230 pages, ISBN 978-3-642-16775-1/p>!--*p stylefont-size: .9em>*Navigation: a classnote-link href#features>Features/a>|a classnote-link href#order>Order/a>|a classnote-link href#download>Downloadable material/a>|a classnote-link href#info>Additional information/a>|a classnote-link href#contact>Contact/a>*/p>-->img srcimg/cop_resize.png stylefloat: left; margin-right: 1.5em; margin-bottom: 1em altFront cover height400px>a classplainanchor nameabout>h2>About the book/h2>/a>p>Many problems in decision making, monitoring, fault detection, and control require the knowledge of state variables and time-varying parameters that are not directly measured by sensors. In such situations, observers, or estimators, can be employed that use the measured input and output signals along with a dynamic model of the system in order to estimate the unknown states or parameters. An essential requirement in designing an observer is to guarantee the convergence of the estimates to the true values or at least to a small neighborhood around the true values. However, for nonlinear, large-scale, or time-varying systems, the design and tuning of an observer is generally complicated and involves large computational costs./p>p>This book provides a range of methods and tools to design observers for nonlinear systems represented by a special type of a dynamic nonlinear model - the Takagi-Sugeno (TS) fuzzy model. The TS model is a convex combination of affine linear models, which facilitates its stability analysis and observer design by using effective algorithms based on Lyapunov functions and linear matrix inequalities. Takagi-Sugeno models are known to be universal approximators and, in addition, a broad class of nonlinear systems can be exactly represented as a TS system. Three particular structures of large-scale TS models are considered: cascaded systems, distributed systems, and systems affected by unknown disturbances. The reader will find in-depth theoretic analysis accompanied by illustrative examples and simulations of real-world systems. Stability analysis of TS fuzzy systems is addressed in detail. The intended audience are graduate students and researchers both from academia and industry. For newcomers to the field, the book provides a concise introduction to dynamic TS fuzzy models along with two methods to construct TS models for a given nonlinear system./p>!--a classplainanchor namefeatures>h2 styleclear: left>Features/h2>/a>ul>li>A concise introduction to the basics of RL and DPli>A detailed treatment of RL and DP with function approximators for continuous-variable problems, with theoretical results and illustrative examplesli>A thorough treatment of policy search techniquesli>Extensive experimental studies on a range of control problems, including real-time control resultsli>An extensive, illustrative theoretical analysis of a representative algorithm/ul>-->a classplainanchor nameorder>h2>Order/h2>/a>p>The book can be ordered from a hrefhttp://www.springer.com/engineering/mathematical/book/978-3-642-16775-1>Springer/a> or from a hrefhttp://www.amazon.com/Stability-Analysis-Nonlinear-Fuzziness-Computing/dp/3642167756>Amazon/a>, among other places./p>!--a classplainanchor namedownload>h2>Downloadable material/h2>/a>ul>li>Sample chapter: a hrefhttp://www.dcsc.tudelft.nl/~lbusoniu/files/repository/rlbook_ch3.pdf>Ch. 3 - Dynamic programming and reinforcement learning in large and continuous spaces/a>. The most extensive chapter in the book, it reviews methods and algorithms for approximate dynamic programming and reinforcement learning, with theoretical results, discussion, and illustrative numerical examples. This chapter has been made freely available for download, for a limited time, with the kind permission of Taylor & Francis.li>Code used for the numerical studies in the book: em>ApproxRL, A Matlab toolbox for approximate RL and DP/em>, a hrefhttp://www.dcsc.tudelft.nl/~lbusoniu/files/repository/approxrl.zip>approxrl.zip/a>. See a hrefhttp://www.dcsc.tudelft.nl/~lbusoniu/files/repository/readme_approxrl.html>the readme file of the toolbox/a> for more information.li>a hrefhttp://www.dcsc.tudelft.nl/~lbusoniu/files/repository/sc4081rllectures.pdf>Lecture slides/a> on classical RL and DP (part 1) and on RL and DP with function approximation (part 2)./ul>-->a classplainanchor nameinfo >h2 styleclear:left>Additional information/h2>/a>h3>Table of contents/h3>span idtoc-showcontrol styledisplay: none>a classnote-link hrefjavascript: toggleDetails(toc, on);>show/a>/span>span idtoc-hidecontrol styledisplay: none>a classnote-link hrefjavascript: toggleDetails(toc, off);>hide/a>/span>div idtoc styledisplay: none>ul classtoc>li classchapter>1. Introduction/li> ul> li>1.1 Observer design for TS fuzzy systems li>1.2 Outline /ul>li classchapter>2. Takagi-Sugeno fuzzy models/li> ul> li>2.1 TS fuzzy models li>2.2 Dynamic TS fuzzy models li>2.3 Constructing TS models ul> li>2.3.1 The sector nonlinearity approach li>2.3.2 Linearization /ul> li>2.4 Summary /ul>li classchapter>3. Stability analysis of TS fuzzy systems/li> ul> li>3.1 Introduction li>3.2 Preliminaries ul> li>3.2.1 Notation li>3.2.2 Linear matrix inequalities /ul> li>3.3 Stability analysis of TS systems ul> li>3.3.1 Quadratic stability li>3.3.2 D-stability li>3.3.3 Leaving the quadratic stability framework /ul> li>3.4 State feedback stabilization ul> li>3.4.1 H-infinity attenuation li>3.4.2 Robust control /ul> li>3.5 Output feedback stabilization li>3.6 Input-to-state stability li>3.7 Summary /ul>li classchapter>4. Observers for TS fuzzy systems/li> ul> li>4.1 Observer design for TS systems li>4.2 Observer design: measured scheduling vector li>4.3 Observer design: estimated scheduling vector li>4.4 Observer-based stabilization li>4.5 Summary /ul>li classchapter>5. Cascaded TS systems and observers/li> ul> li>5.1 Introduction li>5.2 Stability of cascaded dynamic systems ul> li>5.2.1 Cascaded dynamic systems li>5.2.2 Partitioning a nonlinear system li>5.2.3 Stability of cascaded systems /ul> li>5.3 Cascaded TS fuzzy systems ul> li>5.3.1 Stability analysis of cascaded TS systems li>5.3.2 Convergence rate of cascaded systems /ul> li>5.4 Cascaded TS fuzzy observers ul> li>5.4.1 Measured scheduling vector li>5.4.2 Estimated scheduling vector /ul> li>5.5 Summary /ul>li classchapter>6. Distributed TS systems and observers/li> ul> li>6.1 Introduction li>6.2 Distributed stability analysis of TS systems ul> li>6.2.1 Parallel stability analysis li>6.2.2 Sequential stability analysis /ul> li>6.3 Distributed observer design for TS systems ul> li>6.3.1 General framework li>6.3.2 Sequential design: measured scheduling vector li>6.3.3 Sequential design: estimated scheduling vector /ul> li>6.4 Summary /ul>li classchapter>7. Adaptive observers for TS systems /li> ul> li>7.1 Introduction li>7.2 Unknown input estimations li>7.3 Estimation of unknown polynomial inputs ul> li>7.3.1 Measured scheduling vector li>7.3.2 Estimated scheduling vector /ul> li>7.4 Estimation of unmodelled dynamics ul> li>7.4.1 Measured scheduling vector li>7.4.2 Estimated scheduling vector /ul> li>7.5 Summary /ul>li classchapter>Glossary/li>li classchapter>References/li>li classchapter>Index/li>/ul>/div>noscript>ul classtoc>li classchapter>1. Introduction/li> ul> li>1.1 Observer design for TS fuzzy systems li>1.2 Outline /ul>li classchapter>2. Takagi-Sugeno fuzzy models/li> ul> li>2.1 TS fuzzy models li>2.2 Dynamic TS fuzzy models li>2.3 Constructing TS models ul> li>2.3.1 The sector nonlinearity approach li>2.3.2 Linearization /ul> li>2.4 Summary /ul>li classchapter>3. Stability analysis of TS fuzzy systems/li> ul> li>3.1 Introduction li>3.2 Preliminaries ul> li>3.2.1 Notation li>3.2.2 Linear matrix inequalities /ul> li>3.3 Stability analysis of TS systems ul> li>3.3.1 Quadratic stability li>3.3.2 D-stability li>3.3.3 Leaving the quadratic stability framework /ul> li>3.4 State feedback stabilization ul> li>3.4.1 H-infinity attenuation li>3.4.2 Robust control /ul> li>3.5 Output feedback stabilization li>3.6 Input-to-state stability li>3.7 Summary /ul>li classchapter>4. Observers for TS fuzzy systems/li> ul> li>4.1 Observer design for TS systems li>4.2 Observer design: measured scheduling vector li>4.3 Observer design: estimated scheduling vector li>4.4 Observer-based stabilization li>4.5 Summary /ul>li classchapter>5. Cascaded TS systems and observers/li> ul> li>5.1 Introduction li>5.2 Stability of cascaded dynamic systems ul> li>5.2.1 Cascaded dynamic systems li>5.2.2 Partitioning a nonlinear system li>5.2.3 Stability of cascaded systems /ul> li>5.3 Cascaded TS fuzzy systems ul> li>5.3.1 Stability analysis of cascaded TS systems li>5.3.2 Convergence rate of cascaded systems /ul> li>5.4 Cascaded TS fuzzy observers ul> li>5.4.1 Measured scheduling vector li>5.4.2 Estimated scheduling vector /ul> li>5.5 Summary /ul>li classchapter>6. Distributed TS systems and observers/li> ul> li>6.1 Introduction li>6.2 Distributed stability analysis of TS systems ul> li>6.2.1 Parallel stability analysis li>6.2.2 Sequential stability analysis /ul> li>6.3 Distributed observer design for TS systems ul> li>6.3.1 General framework li>6.3.2 Sequential design: measured scheduling vector li>6.3.3 Sequential design: estimated scheduling vector /ul> li>6.4 Summary /ul>li classchapter>7. Adaptive observers for TS systems /li> ul> li>7.1 Introduction li>7.2 Unknown input estimations li>7.3 Estimation of unknown polynomial inputs ul> li>7.3.1 Measured scheduling vector li>7.3.2 Estimated scheduling vector /ul> li>7.4 Estimation of unmodelled dynamics ul> li>7.4.1 Measured scheduling vector li>7.4.2 Estimated scheduling vector /ul> li>7.5 Summary /ul>li classchapter>Glossary/li>li classchapter>References/li>li classchapter>Index/li>/ul>/noscript>!--h3>About the authors/h3>span idbio-showcontrol styledisplay: none>a classnote-link hrefjavascript: toggleDetails(bio, on);>show/a>/span>span idbio-hidecontrol styledisplay: none>a classnote-link hrefjavascript: toggleDetails(bio, off);>hide/a>/span>div idbio styledisplay: none>table>tr>td classbiophoto>img srcimg/biophoto_lb.jpg altPhoto Lucian Babuska height150px>/td>td classbio>strong>Lucian Busoniu/strong> is a postdoctoral fellow at the Delft Center for Systems and Control of Delft University of Technology, in the Netherlands. He received his PhD degree (cum laude) in 2009 from the Delft University of Technology, and his MSc degree in 2003 from the Technical University of Cluj-Napoca, Romania. His current research interests include reinforcement learning and dynamic programming with function approximation, intelligent and learning techniques for control problems, and multi-agent learning./td>/tr>tr>td classbiophoto>img srcimg/biophoto_rb.jpg altPhoto Robert Babuska height150px>/td>td classbio>strong>Robert Babuska/strong> is a full professor at the Delft Center for Systems and Control of Delft University of Technology in the Netherlands. He received his PhD degree (cum laude) in Control in 1997 from the Delft University of Technology, and his MSc degree (with honors) in Electrical Engineering in 1990 from Czech Technical University, Prague. His research interests include fuzzy systems modeling and identification, data-driven construction and adaptation of neuro-fuzzy systems, model-based fuzzy control and learning control. He is active in applying these techniques in robotics, mechatronics, and aerospace./td>/tr>tr>td classbiophoto>img srcimg/biophoto_bds.jpg altPhoto Bart De Schutter height150px>/td>td classbio>strong>Bart De Schutter/strong> is a full professor at the Delft Center for Systems and Control and at the Marine & Transport Technology department of Delft University of Technology in the Netherlands. He received the PhD degree in Applied Sciences (summa cum laude with congratulations of the examination jury) in 1996 from K.U. Leuven, Belgium. His current research interests include multi-agent systems, hybrid systems control, discrete-event systems, and control of intelligent transportation systems./td>/tr>tr>td classbiophoto>img srcimg/biophoto_de.jpg altPhoto Damien Ernst height150px>/td>td classbio>strong>Damien Ernst/strong> received the MSc and PhD degrees from the University of Liège in 1998 and 2003, respectively. He is currently a Research Associate of the Belgian FRS-FNRS and he is affiliated with the Systems and Modeling Research Unit of the University of Liège. Damien Ernst spent the period 2003--2006 with the University of Liège as a Postdoctoral Researcher of the FRS-FNRS and held during this period positions as visiting researcher at CMU, MIT and ETH. He spent the academic year 2006--2007 working at Supélec (France) as professor. His main research interests are in the fields of power system dynamics, optimal control, reinforcement learning, and design of dynamic treatment regimes./td>/tr>/table>/div>noscript>table>tr>td classbiophoto>img srcimg/biophoto_lb.jpg altPhoto Lucian Babuska height150px>/td>td classbio>strong>Lucian Busoniu/strong> is a postdoctoral fellow at the Delft Center for Systems and Control of Delft University of Technology, in the Netherlands. He received his PhD degree (cum laude) in 2009 from the Delft University of Technology, and his MSc degree in 2003 from the Technical University of Cluj-Napoca, Romania. His current research interests include reinforcement learning and dynamic programming with function approximation, intelligent and learning techniques for control problems, and multi-agent learning./td>/tr>tr>td classbiophoto>img srcimg/biophoto_rb.jpg altPhoto Robert Babuska height150px>/td>td classbio>strong>Robert Babuska/strong> is a full professor at the Delft Center for Systems and Control of Delft University of Technology in the Netherlands. He received his PhD degree (cum laude) in Control in 1997 from the Delft University of Technology, and his MSc degree (with honors) in Electrical Engineering in 1990 from Czech Technical University, Prague. His research interests include fuzzy systems modeling and identification, data-driven construction and adaptation of neuro-fuzzy systems, model-based fuzzy control and learning control. He is active in applying these techniques in robotics, mechatronics, and aerospace./td>/tr>tr>td classbiophoto>img srcimg/biophoto_bds.jpg altPhoto Bart De Schutter height150px>/td>td classbio>strong>Bart De Schutter/strong> is a full professor at the Delft Center for Systems and Control and at the Marine & Transport Technology department of Delft University of Technology in the Netherlands. He received the PhD degree in Applied Sciences (summa cum laude with congratulations of the examination jury) in 1996 from K.U. Leuven, Belgium. His current research interests include multi-agent systems, hybrid systems control, discrete-event systems, and control of intelligent transportation systems./td>/tr>tr>td classbiophoto>img srcimg/biophoto_de.jpg altPhoto Damien Ernst height150px>/td>td classbio>strong>Damien Ernst/strong> received the MSc and PhD degrees from the University of Liège in 1998 and 2003, respectively. He is currently a Research Associate of the Belgian FRS-FNRS and he is affiliated with the Systems and Modeling Research Unit of the University of Liège. Damien Ernst spent the period 2003--2006 with the University of Liège as a Postdoctoral Researcher of the FRS-FNRS and held during this period positions as visiting researcher at CMU, MIT and ETH. He spent the academic year 2006--2007 working at Supélec (France) as professor. His main research interests are in the fields of power system dynamics, optimal control, reinforcement learning, and design of dynamic treatment regimes./td>/tr>/table>/noscript>-->a classplainanchor namecontact>h2>Contact/h2>/a>Comments, suggestions, and questions concerning the book or the Web site are welcome. Please contact preferably the first author, a hrefmailto: zsofia@lendek.net>Zsófia Lendek/a>, or otherwise any of the other authors (see their respective websites for contact information).p classnote>The template for this webpage has been used with the kind permission of a hrefhttp://www.dcsc.tudelft.nl/~lbusoniu/home.php>Lucian Busoniu/a>./body>
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