Document Zbl 1010.93522

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We show that for Mamdani-type fuzzy systems with correlation-product inference, centroid defuzzification, and triangular membership functions, optimizing the membership functions can be viewed as an identification problem for a nonlinear dynamic system. This identification problem can be solved with an extended Kalman filter. We describe the algorithm and compare it with gradient descent and with adaptive neuro-fuzzy inference system (ANFIS) based optimization of fuzzy membership functions. The methods discussed in this paper are illustrated on a fuzzy filter for motor winding current estimation and are compared with Butterworth filtering. We demonstrate that the Kalman filter can be an effective tool for improving the performance of a fuzzy system.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A1891675">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%3A93C42" title="MSC2020">93C42</a> </td> <td class="space"> Fuzzy 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%3Alearning">learning</a>; <a href="/?q=ut%3Afiltering">filtering</a>; <a href="/?q=ut%3Aestimation">estimation</a>; <a href="/?q=ut%3Atraining">training</a>; <a href="/?q=ut%3Aoptimization">optimization</a>; <a href="/?q=ut%3Agradient+descent">gradient descent</a>; <a href="/?q=ut%3AKalman+filtering">Kalman filtering</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 1010.93522" data-ciurl="/ci/01891675" data-biburl="/bibtex/01891675.bib" data-amsurl="/amsrefs/01891675.bib" data-xmlurl="/xml/01891675.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01891675.pdf" title="Zbl 1010.93522 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/S0165-0114(01)00241-X" aria-label="DOI for “Training fuzzy systems with the extended Kalman filter”" title="10.1016/S0165-0114(01)00241-X">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://engagedscholarship.csuohio.edu/cgi/viewcontent.cgi?article=1026&context=enece_facpub" title="Full Text Link">Link</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">G. Abdelnour, S. Chand, S. Chiu, Applying fuzzy logic to the Kalman filter divergence problem, IEEE Internat. Conf. on Systems, Man and Cybernetics, Le Touquet, France, 1993, pp. 630-634.; G. Abdelnour, S. Chand, S. Chiu, Applying fuzzy logic to the Kalman filter divergence problem, IEEE Internat. Conf. on Systems, Man and Cybernetics, Le Touquet, France, 1993, pp. 630-634.</td> </tr><tr> <td>[2]</td> <td class="space">S. Akbar, R. Parra-Loera, Optimal self-tuning fuzzy controller, SPIE Conf. on Machine Tool, In-Line, and Robot Sensors and Controls, Philadelphia, 1995, pp. 130-137.; S. Akbar, R. Parra-Loera, Optimal self-tuning fuzzy controller, SPIE Conf. on Machine Tool, In-Line, and Robot Sensors and Controls, Philadelphia, 1995, pp. 130-137.</td> </tr><tr> <td>[3]</td> <td class="space">Anderson, B.; Moore, J., Optimal Filtering (1979), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · <a href="/0688.93058" class="nowrap">Zbl 0688.93058</a></td> </tr><tr> <td>[4]</td> <td class="space">Atkinson, K., An Introduction to Numerical Analysis (1989), Wiley: Wiley New York · <a href="/0718.65001" class="nowrap">Zbl 0718.65001</a></td> </tr><tr> <td>[5]</td> <td class="space">Barada, W.; Singh, H., Generating optimal adaptive fuzzy-neural models of dynamical systems with applications to control, IEEE Trans. Systems Man Cybernet.—Part C, 28, 371-391 (1998)</td> </tr><tr> <td>[6]</td> <td class="space">J. Deskur, R. Muszynski, D. Sarnowski, Tuning and investigation of combined fuzzy controller, Internat. Conf. on Power Electronics and Variable Speed Drives, London, 1998, pp. 353-357.; J. Deskur, R. Muszynski, D. Sarnowski, Tuning and investigation of combined fuzzy controller, Internat. Conf. on Power Electronics and Variable Speed Drives, London, 1998, pp. 353-357.</td> </tr><tr> <td>[7]</td> <td class="space">B. Demaya, R. Palm, S. Boverie, A. Titli, Multilevel qualitative and numerical optimization of fuzzy controller, IEEE Internat. Conf. on Fuzzy Systems, Yokohama, Japan, 1995, pp. 1149-1154.; B. Demaya, R. Palm, S. Boverie, A. Titli, Multilevel qualitative and numerical optimization of fuzzy controller, IEEE Internat. Conf. on Fuzzy Systems, Yokohama, Japan, 1995, pp. 1149-1154.</td> </tr><tr> <td>[8]</td> <td class="space">W. Eradus, H. Scholten, A. Udink ten Cate, Optimized fuzzy inference system for oestrus detection in dairy cattle, Internat. Symp. on Neuro-Fuzzy Systems, Lausanne, Switzerland, 1996, pp. 171-176.; W. Eradus, H. Scholten, A. Udink ten Cate, Optimized fuzzy inference system for oestrus detection in dairy cattle, Internat. Symp. on Neuro-Fuzzy Systems, Lausanne, Switzerland, 1996, pp. 171-176.</td> </tr><tr> <td>[9]</td> <td class="space">Feucht, D., Sensorless start-up positioning of brushless DC motors, PCIM Mag., 19, 24-27 (1993)</td> </tr><tr> <td>[10]</td> <td class="space">Figueiredo, M.; Gomide, F., Design of fuzzy systems using neurofuzzy networks, IEEE Trans. Neural Networks, 10, 815-827 (1999)</td> </tr><tr> <td>[11]</td> <td class="space">Gelb, A., Applied Optimal Estimation (1974), MIT Press: MIT Press Cambridge, MA</td> </tr><tr> <td>[12]</td> <td class="space">J. Goddard, R. Parrazales, I. Lopez, A. de Luca, Rule learning in fuzzy systems using evolutionary programs, IEEE Midwest Symp. on Circuits and Systems, Ames, Iowa, 1996, pp. 703-709.; J. Goddard, R. Parrazales, I. Lopez, A. de Luca, Rule learning in fuzzy systems using evolutionary programs, IEEE Midwest Symp. on Circuits and Systems, Ames, Iowa, 1996, pp. 703-709.</td> </tr><tr> <td>[13]</td> <td class="space">F. Guély, P. Siarry, Gradient descent method for optimizing various fuzzy rule bases, IEEE Internat. Conf. on Fuzzy Systems, San Francisco, 1993, pp. 1241-1246.; F. Guély, P. Siarry, Gradient descent method for optimizing various fuzzy rule bases, IEEE Internat. Conf. on Fuzzy Systems, San Francisco, 1993, pp. 1241-1246.</td> </tr><tr> <td>[14]</td> <td class="space">Hsiao, C., Modified gain fuzzy Kalman filtering algorithm, JSME Internat. J. Ser. C, 42, 363-368 (1999)</td> </tr><tr> <td>[15]</td> <td class="space">Jacomet, M.; Stahel, A.; Walti, R., On-line optimization of fuzzy systems, Inform. Sci., 98, 301-313 (1997)</td> </tr><tr> <td>[16]</td> <td class="space">Jang, J.; Sun, C.; Mizutani, E., Neuro-Fuzzy and Soft Computing (1997), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ</td> </tr><tr> <td>[17]</td> <td class="space">Jiang, T.; Li, Y., Generalized defuzzification strategies and their parameter learning procedures, IEEE Trans. Fuzzy Systems, 4, 64-71 (1996)</td> </tr><tr> <td>[18]</td> <td class="space">K. Kobayashi, K. Cheok, K. Watanabe, Estimation of the absolute vehicle speed using fuzzy logic rule-based Kalman filter, American Control Conf., Seattle, 1995, pp. 3086-3090.; K. Kobayashi, K. Cheok, K. Watanabe, Estimation of the absolute vehicle speed using fuzzy logic rule-based Kalman filter, American Control Conf., Seattle, 1995, pp. 3086-3090.</td> </tr><tr> <td>[19]</td> <td class="space">Kosko, B., Neural Networks and Fuzzy Systems (1992), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · <a href="/0755.94024" class="nowrap">Zbl 0755.94024</a></td> </tr><tr> <td>[20]</td> <td class="space">Kovacevic, R.; Zhang, Y., Neurofuzzy model-based weld fusion state estimation, IEEE Control Systems Mag., 17, 30-42 (1997)</td> </tr><tr> <td>[21]</td> <td class="space">Krause, P.; Wasynczuk, O., Electromechanical Motion Devices (1989), McGraw-Hill: McGraw-Hill New York</td> </tr><tr> <td>[22]</td> <td class="space">J. Lalk, Intelligent Adaptation of Kalman filters using fuzzy logic, IEEE Conf. Fuzzy Systems (1994) 744-749.; J. Lalk, Intelligent Adaptation of Kalman filters using fuzzy logic, IEEE Conf. Fuzzy Systems (1994) 744-749.</td> </tr><tr> <td>[23]</td> <td class="space">Magdalena, L.; Monasterio-Huelin, F., Fuzzy logic controller with learning through the evolution of its knowledge base, Internat. J. Approx. Reasoning, 16, 335-358 (1997) · <a href="/0947.93513" class="nowrap">Zbl 0947.93513</a></td> </tr><tr> <td>[24]</td> <td class="space">K. Mostov, A. Soloviev, Fuzzy adaptive stabilization of higher order Kalman filters in application to precision kinematic GPS, Institute of Navigation Meeting of the Satellite Division, Vol. 2, 1996, pp. 1451-1456.; K. Mostov, A. Soloviev, Fuzzy adaptive stabilization of higher order Kalman filters in application to precision kinematic GPS, Institute of Navigation Meeting of the Satellite Division, Vol. 2, 1996, pp. 1451-1456.</td> </tr><tr> <td>[25]</td> <td class="space">E. Nakamura, N. Kehtarnavez, Optimization of fuzzy membership function parameters, IEEE International Conference on Systems Man and Cybernetics, Vancouver, Canada, 1995, pp. 1-6.; E. Nakamura, N. Kehtarnavez, Optimization of fuzzy membership function parameters, IEEE International Conference on Systems Man and Cybernetics, Vancouver, Canada, 1995, pp. 1-6.</td> </tr><tr> <td>[26]</td> <td class="space">Oppenheim, A.; Willsky, A.; Young, I., Signals and Systems (1983), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · <a href="/0539.93001" class="nowrap">Zbl 0539.93001</a></td> </tr><tr> <td>[27]</td> <td class="space">Puskorius, G.; Feldkamp, L., Neurocontrol of nonlinear dynamical systems with Kalman filter trained recurrent networks, IEEE Trans. Neural Networks, 5, 279-297 (1994)</td> </tr><tr> <td>[28]</td> <td class="space">P. Ramaswamy, Mr. Riese, R. Edwards, K. Lee, Two approaches for automating the tuning process of fuzzy logic controllers, IEEE Conf. on Decision and Control, San Antonio, TX, 1993, pp. 1753-1758.; P. Ramaswamy, Mr. Riese, R. Edwards, K. Lee, Two approaches for automating the tuning process of fuzzy logic controllers, IEEE Conf. on Decision and Control, San Antonio, TX, 1993, pp. 1753-1758.</td> </tr><tr> <td>[29]</td> <td class="space">Shah, S.; Palmieri, F.; Datum, M., Optimal filtering algorithms for fast learning in feedforward neural Networks, Neural Networks, 5, 779-787 (1992)</td> </tr><tr> <td>[30]</td> <td class="space">D. Simon, Fuzzy estimation of DC motor winding currents, North American Fuzzy Information Processing Society Conf., New York, 1999, pp. 859-863.; D. Simon, Fuzzy estimation of DC motor winding currents, North American Fuzzy Information Processing Society Conf., New York, 1999, pp. 859-863.</td> </tr><tr> <td>[31]</td> <td class="space">Simon, D.; El-Sherief, H., Fuzzy logic for digital phase-locked loop filter design, IEEE Trans. Fuzzy Systems, 3, 211-218 (1995)</td> </tr><tr> <td>[32]</td> <td class="space">Simon, D., Design and rule base reduction of a fuzzy filter for the estimation of motor currents, Internat. J. Approx. Reasoning, 25, 145-167 (2000) · <a href="/0969.68615" class="nowrap">Zbl 0969.68615</a></td> </tr><tr> <td>[33]</td> <td class="space">D. Simon, Sum normal optimization of fuzzy membership functions, submitted for publication.; D. Simon, Sum normal optimization of fuzzy membership functions, submitted for publication.</td> </tr><tr> <td>[34]</td> <td class="space">Singhal, S.; Wu, L., Training multilayer perceptrons with the extended Kalman algorithm, (Touretzky, D., Advances in Neural Information Processing Systems, Vol. 1 (1989), Morgan Kaufmann: Morgan Kaufmann San Mateo, CA), 133-140</td> </tr><tr> <td>[35]</td> <td class="space">Smith, S.; Comer, D., Automated calibration of a fuzzy logic controller using a cell state space algorithm, IEEE Control Systems Mag., 11, 18-28 (1991)</td> </tr><tr> <td>[36]</td> <td class="space">Sugeno, M.; Tanaka, K., Successive identification of a fuzzy model and its applications to prediction of a complex system, Fuzzy Sets and Systems, 42, 315-334 (1991) · <a href="/0741.93052" class="nowrap">Zbl 0741.93052</a></td> </tr><tr> <td>[37]</td> <td class="space">Sum, J.; Leung, C.; Young, G.; Kan, W., On the Kalman filtering method in neural-network training and pruning, IEEE Trans. Neural Networks, 10, 161-166 (1999)</td> </tr><tr> <td>[38]</td> <td class="space">Surmann, H., Genetic optimization of a fuzzy system for charging batteries, IEEE Trans. Ind. Electron., 43, 541-548 (1996)</td> </tr><tr> <td>[39]</td> <td class="space">Tao, C.; Taur, J., Design of fuzzy controllers with adaptive rule insertion, IEEE Trans. Systems Man Cybernetics—Part B, 29, 389-397 (1999)</td> </tr><tr> <td>[40]</td> <td class="space">L. Wang, J. Mendel, Back-propagation of fuzzy systems as nonlinear dynamic system identifiers, IEEE Internat. Conf. on Fuzzy Systems, San Diego, CA, 1992, pp. 1409-1418.; L. Wang, J. Mendel, Back-propagation of fuzzy systems as nonlinear dynamic system identifiers, IEEE Internat. Conf. on Fuzzy Systems, San Diego, CA, 1992, pp. 1409-1418.</td> </tr><tr> <td>[41]</td> <td class="space">Wang, L., Adaptive Fuzzy Systems and Control: Design and Stability Analysis (1994), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ</td> </tr><tr> <td>[42]</td> <td class="space">Wang, L.; Yen, J., Extracting fuzzy rules for system modeling using a hybrid of genetic algorithms and Kalman filter, Fuzzy Sets and Systems, 101, 353-362 (1998)</td> </tr><tr> <td>[43]</td> <td class="space">Wu, R.; Chen, S., A new method for constructing membership functions and fuzzy rules from training examples, IEEE Trans. Systems Man Cybernet—Part B, 29, 25-40 (1999)</td> </tr><tr> <td>[44]</td> <td class="space">Yam, Y.; Baranyi, P.; Yang, C., Reduction of fuzzy rule base via singular value decomposition, IEEE Trans. Fuzzy Systems, 7, 120-132 (1999)</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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