Fuzzy Modeling and Control: Theory and Applications

Fernando MatíaG. Nicolás MarichalEmilio Jiménez Editors

Atlantis Computational Intelligence Systems Series Editors: J. Lu · J. Montero

Atlantis Computational Intelligence Systems

Volume 9

Series editors

Jie Lu, University of Technology, Sydney, AustraliaJavier Montero, Complutense University of Madrid, Spain

Aims and scope of the series

The series ‘Atlantis Computational Intelligence Systems’ aims at covering state-of-the-artresearch and development in all fields where computational intelligence is investigatedand applied. The series seeks to publish monographs and edited volumes on foundations andnew developments in the field of computational intelligence, including fundamentaland applied research as well as work describing new, emerging technologies originatingfrom computational intelligence research. Applied CI research may range from CIapplications in the industry to research projects in the life sciences, including research inbiology, physics, chemistry and the neurosciences.

All books in this series are co-published with Springer.

For more information on this series and our other book series, please visit our website at:www.atlantis-press.com/publications/books.Atlantis Press8, Square des Bouleaux75019 Paris, France

Fernando Matía • G. Nicolás MarichalEmilio JiménezEditors

Fuzzy Modeling and Control:Theory and Applications

EditorsFernando MatíaCenter for Automation and RoboticsUniversidad Politécnica de Madrid—

Consejo Superior de InvestigacionesCientíficas (UPM—CSIC)

MadridSpain

G. Nicolás MarichalPolytechnic School of EngineeringUniversidad de La LagunaSan Cristóbal de La Laguna, Santa Cruz de

TenerifeSpain

Emilio JiménezDepartamento de Ingeniería EléctricaUniversidad de La RiojaLa Rioja, NavarraSpain

ISSN 1875-7650ISBN 978-94-6239-081-2 ISBN 978-94-6239-082-9 (eBook)DOI 10.2991/978-94-6239-082-9

Library of Congress Control Number: 2014945966

� Atlantis Press and the authors 2014This book is published under the Creative Commons Attribution-Non-commercial license, meaning thatcopying, distribution, transmitting and adapting the book is permitted, provided that this is done fornon-commercial purposes and that the book is attributed.This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by anymeans, electronic or mechanical, including photocopying, recording or any information storage andretrieval system known or to be invented, without prior permission from the Publisher.

Printed on acid-free paper

Foreword

Almost half a century has passed since the publication of the seminal paper byProf. Lotfi A. Zadeh. Fifty years that have witnessed an impressive growth of fuzzysystems from both the theoretical and applied points of view. Impressive growthand at the same time following an unforeseen way, since as Prof. Zadeh has saidmany times, his expectations when writing his first paper on fuzzy sets in 1965were that fuzzy sets would find its main applications in the realm of humanisticsystems (economics, linguistics, psychology, etc.). Now the reality is that most ofthe success of fuzzy systems research (at least in its initial decades) was related toits application to mechanistic instead of humanistic systems. This book concen-trates on modeling and control of mechanistic systems, as can be seen whenreading the Table of Contents, first considering important theoretical aspects andthen showing its application to quite interesting real-world problems.

Any book on fuzzy modeling and control is somehow a tribute to those whopaved the way for the topic. In this case, we need to consider three key persons andthree related milestones in the history of fuzzy sets. First, the already mentionedProf. Zadeh and his 1973 paper ‘‘Outline of a New Approach to the Analysis ofComplex Systems and Decision Processes,’’ IEEE Transactions on Systems, Man,and Cybernetics 3: 28–44 (1973), the paper where concepts as linguistic variableand fuzzy if-then rules took form, and at the same time, the paper that triggered thesecond hint in the road to build up the theory of fuzzy modeling and control. AbeMamdani was responsible for this second step, the birth of fuzzy control, with thepublication of his papers ‘‘Applications of Fuzzy Algorithm for Control a SimpleDynamic Plant,’’ Proceedings of the IEEE 121(12): 1585–1588 (1974), and ‘‘AnExperiment in Linguistic Synthesis with a Fuzzy Logic Controller,’’ InternationalJournal of Man-Machine Studies 7: 1–13 (1975), in this case jointly with hisstudent S. Assilian. Ten years later we find the third key person, Michio Sugeno,who jointly with T. Takagi closed this collection of seminal works with the paper‘‘Fuzzy Identification of Systems and Its Application to Modeling and Control,’’IEEE Transactions on Systems, Man, and Cybernetics 15(1): 116–132 (1985).

v

Three persons and 20 years needed to establish the foundations of the theory thatmotivates the present book.

And once considered the history, it is time to pay attention to the significance ofthe topic. What is the weight of modeling and control in the field of fuzzy systems?To answer this question we can simply analyze the ratio of publications onmodeling and control inside the overall set of publications in fuzzy sets andsystems. Two queries on Scopus will give as the answer, the first one for paperscontaining the term ‘‘fuzzy’’ in its title, abstract, or keywords, and the second forpapers containing ‘‘fuzzy’’ and either ‘‘control’’ or ‘‘model’’ (and their derivates:controller, controlling, modeling, etc.) in its title, abstract, or keywords. The resultis 185,274 papers containing ‘‘fuzzy,’’ from which 117,589 also contain ‘‘con-trol*’’ or ‘‘model*’’. This means approximately a 63 %, so, almost two-thirds ofthe publications on fuzzy sets and systems are somehow related to modeling andcontrol. Moreover, how has this ratio evolved along the 50 years of research infuzzy sets? The answer is that the ratio was obviously lower in the initial period,under 40 % in the 1970s and 1980s, reaching 60 % in 1993, and being quiteconstant in the last 20 years (in the range from 60 to 70 %). So again we can saythat fuzzy modeling and control represents two-thirds of the present research infuzzy systems.

Finally, once established the significance of the topic, why is the present bookimportant? Even considering the present success and recognition of fuzzy mod-eling and control, its development has not been free of controversy. In the earlystages of fuzzy control, the control systems community did not easily accept thesenew ideas, particularly due to the absence of a sound theoretical analysis ofquestions as system stability. In recent years fuzzy control and modeling have beenanalyzed from a ‘‘classical’’ point of view, offering us a clearer and more objectiveperception of its capabilities, goodness, and, obviously, some weak points. For thatreason, it is quite important that the two communities (fuzzy systems and controlsystems communities) met together in an initiative where the theoretical andapplied concepts related to fuzzy modeling and control are scrutinized from boththe fuzzy and control perspectives. In that sense, this joint initiative of CEA-IFAC(a control society) and EUSFLAT (a fuzzy society) deserves a loud applause andall our support for future continuation. The result of this encounter is a magnificentbook that every theoretician or practitioner of fuzzy modeling and control shouldconsider.

I have to congratulate Profs. Matía, Marichal, and Jiménez for conceiving thisproject, and giving us the opportunity to enjoy this book.

Luis Magdalena

vi Foreword

Preface

Our motivation when enroling this project was to cover the lack of a book wheremost of the fuzzy modeling and control techniques, frequently used by Europeanresearch groups, could be put together. Moreover, some chapters have been addedin order to illustrate the application of these techniques in several scenarios. Thebook was elaborated by members of the CEA-IFAC Intelligent Control Group andthe EUSFLAT Working Group on Fuzzy Control. Both associations includeamong their members some of the most well-recognized experts in the fuzzycontrol field in Spain and Europe, respectively.

The book covers the range from systems modeling to controllers design and aninteresting set of applications, being divided into the following three parts. Part I isdedicated to describe fuzzy modeling techniques. It is important to note that inmost of the chapters Takagi-Sugeno (TS) model is used instead of Mamdani one,which is a clear demonstration of how this modeling technique has been spread forthe last decades in the fuzzy control area. Three chapters deal with this topic.Chapter 1 describes new approaches to improve the local and global approxima-tion and modeling capability of TS fuzzy model, solving traditional identificationproblems when membership functions are overlapped by pairs, something verypopular in industrial control applications. Chapter 2 shows the application ofextended Kalman filter for the parametric adaptation of a TS fuzzy model, whichcan be implemented online based on input/output data with high accuracy and in acomputationally efficient manner. Chapter 3 presents a systematic technique toreduce the number of rules and so the complexity of TS fuzzy systems, by usingFunctional Principal Components Analysis. It is shown that the main problemwhich arises, the lack of rules interpretability, is not a problem in predictivecontrol applications.

Part II is dedicated to the parameters adjustment of fuzzy systems. Chapter 4presents a formal design methodology for the synthesis of fuzzy control systemswhich guarantees the stability of the closed-loop system, without requiring tosearch for Lyapunov functions. Chapter 5 extends the well-known sector nonlin-earity of TS fuzzy modeling methodology to the polynomial framework. While themajor part of existent results in the literature do not pay attention to the validity

vii

region of the obtained solution, here the problem is addressed presenting amethodology to estimate the domain of attraction. Chapter 6 is concerned with theintroduction of a Fault Tolerant Control framework using uncertain TS fuzzymodels. The design is performed using a Linear Matrix Inequality-based synthesisthat directly takes into account the TS description of the system and its uncer-tainties. Chapter 7 is focused on the use of learning and gradient-free optimizationtechniques for optimal tuning of fuzzy and neuro-fuzzy systems. Single feedbackcontrol and internal model control schemes are considered in the two proposedstrategies. Chapter 8 presents aspects concerning the tuning of simple TS PI-likefuzzy controllers by adaptive evolutionary optimization algorithms, which arebuilt by using Gravitational Search Algorithm and Charged System Search algo-rithm structures.

Finally, Part III is dedicated to show a variety of fuzzy control applications.Chapter 9 deals with vibrations as a very common problem which appears in manyengineering applications. In order to reduce (or ultimately eliminate) theirpotential negative impact, special attention is paid to semi-active dampers. Theirmain advantage is the ability to modify its structural parameters requiring lessamount of power. Chapter 10 presents a general neuro-fuzzy model for the fastferry vertical motion. Intelligent controllers using fuzzy logic are proposed tostabilize the vertical motion of the craft and therefore to improve the comfort andsafety of sailing. Chapter 11 shows the application of soft computing techniques tocontrol road vehicles. AUTOPÍA project combines different artificial intelligenttechniques: fuzzy logic allows modeling the behavior of the planning, cooperation,and decision agents according to the human way of doing; evolutionary algorithmsallow to adjust the agent behavior to optimize the operation or cost criteria; andswarm algorithms allow modeling the emergent behavior of the urban traffic flow.Chapter 12 proposes the use of a fuzzy logic controller to drive an electrolyzer inan experimental hybrid renewable energy system. The controller is able to dealwith the interactions of the various system components and the energy flow,providing a stable and reliable source of energy. Chapter 13 describes a newdesign methodology for automatic synthesis of fuzzy logic-based inference sys-tems on programmable logic devices and Application-Specific Integrated Circuits(ASIC). It describes both an efficient architecture for hardware implementation anda set of design tools that allow accelerating the exploration of the design space offuzzy inference modules. Chapter 14 presents a TS fuzzy control system devel-oped for the throttle and brake pedals of a car, whose objective is to follow anotherone which precedes it, while maintaining a safe distance depending on the speed.A genetic algorithm and a mathematical model are used for the tuning of theparameters in the consequents of the rules.

We hope you enjoy this book as much as we did coordinating its elaboration.

Fernando MatíaG. Nicolás Marichal

Emilio Jiménez

viii Preface

Acknowledgments

The book editors want to thank the Spanish Committee on Automation, division ofthe International Federation of Automatic Control (CEA-IFAC) and the EuropeanSociety for Fuzzy Logic and Technology (EUSFLAT) for their support as well asfor the coordinated work of the CEA-IFAC Intelligent Control Group and theEUSFLAT Working Group on Fuzzy Control on the elaboration of this book.

We also want to acknowledge John Wiley & Sons for allowing to reuse materialfrom Optimal Control Applications and Methods journal, as well as Elsevier forallowing to reuse material from Engineering Applications of Artificial Intelligenceand Control Engineering Practice journals.

ix

Contents

Part I Fuzzy Systems and Fuzzy Models

1 New Concepts for the Estimation of Takagi-SugenoModel Based on Extended Kalman Filter . . . . . . . . . . . . . . . . . . 3Basil Mohammed Al-Hadithi, Agustín Jiménez, Fernando Matía,José Manuel Andújar and Antonio Javier Barragán

2 Suboptimal Recursive Methodology for Takagi-SugenoFuzzy Models Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25José Manuel Andújar, Antonio Javier Barragán,Basil Mohammed Al-Hadithi, Fernando Matía and Agustín Jiménez

3 Complexity Reduction in Fuzzy Systems Using FunctionalPrincipal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 49Juan Manuel Escaño and Carlos Bordons

Part II Fuzzy Systems Adjustment

4 Stable Fuzzy Control System by Design. . . . . . . . . . . . . . . . . . . . 69José Manuel Andújar, Antonio Javier Barragán,Basil Mohammed Al-Hadithi, Fernando Matía and Agustín Jiménez

5 Polynomial Fuzzy Systems: Stability and Control . . . . . . . . . . . . 95José Luis Pitarch, Antonio Sala and Carlos Vicente Ariño

6 Robust Fault Tolerant Control Framework UsingUncertain Takagi-Sugeno Fuzzy Models . . . . . . . . . . . . . . . . . . . 117Damiano Rotondo, Fatiha Nejjari and Vicenç Puig

xi

7 Intelligent Tuning of Fuzzy Controllers by Learningand Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Rodolfo Haber, Raúl Mario del Toro, Jorge Godoyand Agustín Gajate

8 Adaptive Evolutionary Optimization Algorithmsfor Simple Fuzzy Controller Tuning Dedicatedto Servo Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159Radu-Codrut� David, Radu-Emil Precup, Emil M. Petriu,Stefan Preitl, Mircea-Bogdan Radac and Lucian-Ovidiu Fedorovici

Part III Fuzzy Control Applications

9 Neuro-Fuzzy Control for Semi-active Dampers . . . . . . . . . . . . . . 177G. Nicolás Marichal, Ángela Hernández, Isidro Padrón,Alfonso Poncela, María Tomás-Rodríguez and Evelio González

10 Neuro-Fuzzy Modeling and Fuzzy Controlof a Fast Ferry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191Matilde Santos

11 AUTOPÍA Program for Intelligent Control of Vehicles . . . . . . . . 205Teresa de Pedro

12 Management of a PEM Electrolyzer in HybridRenewable Energy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217Antonio José Calderón, Isaías González and Manuel Calderón

13 Hardware Implementation of Embedded Fuzzy Controllerson FPGAs and ASICs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235Santiago Sánchez-Solano and María Brox

14 Genetic Optimization of Fuzzy Adaptive Cruise Controlfor Urban Traffic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255Luciano Alonso and Juan Pérez-Oria

Appendix A: Jacobian Matrix of a Fuzzy Control System. . . . . . . . . . 273

Appendix B: Code of Examples of Chapter 5 . . . . . . . . . . . . . . . . . . . 285

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

xii Contents

Contributors

Basil Mohammed Al-Hadithi Center for Automation and Robotics, UniversidadPolitécnica de Madrid—Consejo Superior de Investigaciones Científicas (UPM—CSIC), Madrid, Spain

Luciano Alonso Department of Electronics Technology, Systems and Automa-tion Engineering, University of Cantabria, Cantabria, Spain

José Manuel Andújar Departamento de Ingeniería Electrónica, de SistemasInformáticos y Automática, Universidad de Huelva, Huelva, Spain

Carlos Vicente Ariño Department of Systems Engineering and Design,Universitat Jaume I de Castellón, Castellón, Spain

Antonio Javier Barragán Departamento de Ingeniería Electrónica, de SistemasInformáticos y Automática, Universidad de Huelva, Huelva, Spain

Carlos Bordons Department of Systems Engineering and Automatic Control,University of Seville, Seville, Spain

María Brox Department of Computer Architecture, Instituto de Microelectrónicade Sevilla, CNM-CSIC, Universidad de Córdoba, Córdoba, Sapin

Antonio José Calderón Departamento de Ingenierìa Eléctrica, Electrónica yAutomática, Universidad de Extremadura, Badajoz, Spain

Manuel Calderón Departamento de Ingenierìa Eléctrica, Electrónica yAutomática, Universidad de Extremadura, Badajoz, Spain

Radu-Codrut� David Department of Automation and Applied Informatics,Politehnica University of Timisoara, Timisoara, Romania

Teresa de Pedro Center for Automation and Robotics, Consejo Superior de In-vestigaciones Científicas—Universidad Politécnica de Madrid (CSIC—UPM),Madrid, Spain

xiii

Raúl Mario del Toro Center for Automation and Robotics, Consejo Superior deInvestigaciones Científicas—Universidad Politécnica de Madrid (CSIC—UPM),Madrid, Spain

Juan Manuel Escaño Nimbus Centre, Cork Institute of Technology, Cork,Ireland

Lucian-Ovidiu Fedorovici Department of Automation and Applied Informatics,Politehnica University of Timisoara, Timisoara, Romania

Agustín Gajate Center for Automation and Robotics, Consejo Superior deInvestigaciones Científicas—Universidad Politécnica de Madrid (CSIC—UPM),Madrid, Spain

Jorge Godoy Center for Automation and Robotics, Consejo Superior de Inves-tigaciones Científicas—Universidad Politécnica de Madrid (CSIC—UPM),Madrid, Spain

Evelio González Departamento de Informática, Universidad de La Laguna, SantaCruz de Tenerife, Spain

Isaías González Departamento de Ingenierìa Eléctrica, Electrónica y Automática,Universidad de Extremadura, Badajoz, Spain

Rodolfo Haber Center for Automation and Robotics, Consejo Superior deInvestigaciones Científicas—Universidad Politécnica de Madrid (CSIC—UPM),Madrid, Spain

Ángela Hernández Polytechnic School of Engineering, Universidad de LaLaguna, Santa Cruz de Tenerife, Spain

Agustín Jiménez Center for Automation and Robotics, Universidad Politécnicade Madrid—Consejo Superior de Investigaciones Científicas (UPM—CSIC),Madrid, Spain

G. Nicolás Marichal Polytechnic School of Engineering, Universidad de LaLaguna, Santa Cruz de Tenerife, Spain

Fernando Matía Center for Automation and Robotics, Universidad Politécnicade Madrid—Consejo Superior de Investigaciones Científicas (UPM—CSIC),Madrid, Spain

Fatiha Nejjari Advanced Control Systems Group, Universitat Politècnica deCatalunya, Barcelona, Spain

Isidro Padrón Polytechnic School of Engineering, Universidad de La Laguna,Santa Cruz de Tenerife, Spain

Juan Pérez-Oria Department of Electronics Technology, Systems and Auto-mation Engineering, University of Cantabria, Cantabria, Spain

xiv Contributors

Emil M. Petriu School of Electrical Engineering and Computer Science,University of Ottawa, Ottawa, Canada

José Luis Pitarch Department of Systems Engineering and Automatic Control,Universidad de Valladolid, Valladolid, Spain

Alfonso Poncela Instituto de las Tecnologías Avanzadas de la Producción,Universidad de Valladolid, Valladolid, Spain

Radu-Emil Precup Department of Automation and Applied Informatics,Politehnica University of Timisoara, Timisoara, Romania

Stefan Preitl Department of Automation and Applied Informatics, PolitehnicaUniversity of Timisoara, Timisoara, Romania

Vicenç Puig Advanced Control Systems Group, Universitat Politècnica deCatalunya, Barcelona, Spain

Mircea-Bogdan Radac Department of Automation and Applied Informatics,Politehnica University of Timisoara, Timisoara, Romania

Damiano Rotondo Advanced Control Systems Group, Universitat Politècnica deCatalunya, Barcelona, Spain

Antonio Sala Systems Engineering and Control Department, I. U. de Automáticae Informática Industrial (ai2), Universitat Politècnica de València, Valencia, Spain

Santiago Sánchez-Solano Department of Computer Architecture, Instituto deMicroelectrónica de Sevilla, CNM-CSIC, Universidad de Córdoba, Seville, Sapin

Matilde Santos Department of Computer Architecture and Automatic Control,Universidad Complutense de Madrid, Madrid, Spain

María Tomás-Rodríguez Department of Engineering and Mathematics, CityUniversity London, London, UK

Contributors xv

Part IFuzzy Systems and Fuzzy Models

Chapter 1New Concepts for the Estimation ofTakagi-Sugeno Model Based on ExtendedKalman Filter

Basil Mohammed Al-Hadithi, Agustín Jiménez, Fernando Matía,José Manuel Andújar and Antonio Javier Barragán

1.1 Introduction

Nonlinear control systems based on the Takagi-Sugeno (TS) fuzzy model (Takagiand Sugeno 1985) have attracted lots of attention during the last 20 years (Chenet al. 2007; Gang 2006; Guerra and Vermeiren 2004; Hou et al. 2007; Hseng et al.2007; Jae-Hun et al. 2007; Jiang and Han 2007; Lian et al. 2006; Tanaka et al.2003), in opposition to nonlinear control systems design methods based on Mam-dani fuzzy model (Matía et al. 1992). It provides a powerful solution for developmentof function approximation, systematic techniques to stability analysis and controllerdesign of fuzzy control systems in view of fruitful conventional control theory andtechniques. They also allow relatively easy application of powerful learning tech-niques for their identification from data (Cordon et al. 2001). This model (Takagiand Sugeno 1985) is formed by using a set of fuzzy rules to represent a nonlinearsystem as a set of local affine models which are connected by fuzzy membership func-tions. The authors divide the identification process in three steps; premise variables,membership functions and consequent parameters. With respect to membership func-tions, they apply nonlinear programming technique using the complex method forthe minimization of the performance index. This fuzzy modeling method presentsan alternative technique to represent complex nonlinear systems (Tanaka and Wang2001) and reduces the number of rules in modeling higher order nonlinear systems(Gang 2006; Takagi and Sugeno 1985). TS fuzzy models are proved to be universalfunction approximators as they are able to approximate any smooth nonlinear func-tions to any degree of accuracy in any convex compact region (Johansen et al. 2000;

B.M. Al-Hadithi (B) · A. Jiménez · F. MatíaCenter for Automation and Robotics, Universidad Politécnica de Madrid—Consejo Superior deInvestigaciones Científicas (UPM—CSIC), Madrid, Spaine-mail: basil.alhadithi@upm.es: basil@etsii.upm.es

J.M. Andújar · A.J. BarragánDepartamento de Ingeniería Electrónica, de Sistemas Informáticos y Automática,Universidad de Huelva, Huelva, Spain

© Atlantis Press and the authors 2014F. Matía et al. (eds.), Fuzzy Modeling and Control: Theory and Applications,Atlantis Computational Intelligence Systems 9, DOI 10.2991/978-94-6239-082-9_1

3

4 B.M. Al-Hadithi et al.

Lian et al. 2006; Tanaka and Wang 2001). This result provides a theoretical foundationfor applying TS fuzzy models to represent complex nonlinear systems approximately.

Several results have been obtained about the identification of TS fuzzy models(Cao et al. 1997; Johansen et al. 2000; Mollov et al. 2004). They are based upontwo kinds of approaches, one is to linearize the original nonlinear system in variousoperating points when the model of the system is known, and the other is based onthe input-output data collected from the original nonlinear system when its modelis unknown. The authors in Cao et al. (1997) use a fuzzy clustering method toidentify TS fuzzy models, including identification of the number of fuzzy rules andparameters of fuzzy membership functions, and identification of parameters of locallinear models by using a least squares method (Skrjanc et al. 2005).

In Hong and Lee (1996), have analyzed that the disadvantages of most fuzzysystems are that the membership functions and fuzzy rules should be predefined tomap numerical data into linguistic terms and to make fuzzy reasoning work. Theysuggested a method based on the fuzzy clustering technique and the decision tables toderive membership functions and fuzzy rules from numerical data. However, Hongand Lee’s algorithm presented needs to predefine the membership functions of theinput linguistic variables and it simplifies fuzzy rules by a series of merge operations.As the number of variables becomes larger, the decision table will grow tremendouslyand the process of the rule simplification based on the decision tables becomes morecomplicated. The authors in Johansen et al. (2000) suggest a method to identify TSfuzzy models. Their method aims at improving the local and global approximationof TS model. However, this complicates the approximation in order to obtain bothtargets. It has been shown that constrained and regularized identification methodsmay improve interpretability of constituent local models as local linearizations, andlocally weighted least squares method may explicitly address the trade-off betweenthe local and global accuracy of TS fuzzy models.

In Matía et al. (2011), the authors proposed to obtain the best features ofMamdani and TS models by using an affine global model with function approx-imation capabilities which maintains local interpretation. The suggested model iscomposed of variant coefficients which are independently governed by a zerothorder fuzzy inference system. This model may be interpreted as a generalization ofTS model in which dynamics coefficients have been decoupled. They have shownthat Mamdani and TS models can be combined so that local and global interpretationsare preserved.

In Al-Hadithi et al. (2011), Jiménez et al. (2012), Al-Hadithi et al. (2012) newand efficient approaches are presented to improve the local and global estimationof TS model. The aim is obtaining high function approximation accuracy and fastconvergence. The main problem is that TS identification method can not be appliedwhen the Membership Functions (MFs) are overlapped by pairs. The first approachuses the minimum norm method to search for an exact optimum solution at theexpense of increasing complexity and computational cost. The second one is a simpleand less computational method, based on weighting of parameters. This restricts theuse of the TS method because this type of membership function has been widelyused during the last 2 decades.

1 New Concepts for the Estimation of Takagi-Sugeno Model. . . 5

In Skrjanc et al. (2005) a method of interval fuzzy model identification was devel-oped. The method combines a fuzzy identification methodology with some ideas fromlinear programming theory. The idea is then extended to modeling the optimal lowerand upper bound functions that define the band which contains all the measurementvalues. This results in lower and upper fuzzy models or a fuzzy model with a set oflower and upper parameters. This approach can also be used to compress informationin the case of large amount of data and in the case of robust system identification.The method can be efficiently used in the case of the approximation of the nonlinearfunctions family.

In Kumar et al. (2006), a study has outlined a new min-max approach to thefuzzy clustering, estimation, and identification with uncertain data. The proposedapproach minimizes the worst-case effect of data uncertainties and modeling errorson estimation performance without making any statistical assumption and requiring apriori knowledge of uncertainties. Simulation studies have been provided to show thebetter performance of the proposed method in comparison to the standard techniques.The developed fuzzy estimation theory was applied to a real world application ofphysical fitness classification and modeling.

A fuzzy system containing a dynamic rule base is proposed in Chen and Saif(2005). The characteristic of the proposed system is in the dynamic nature of its rulebase which has a fixed number of rules and allows the fuzzy sets to dynamicallychange or move with the inputs. The number of the rules in the proposed system canbe small, and chosen by the designer. The proposed system is capable of approxi-mating any continuous function on an arbitrarily large compact domain. Moreover, itcan even approximate any uniformly continuous function on infinite domains. Thispaper addresses existence conditions, and as well provides constructive sufficientconditions so that the new fuzzy system can approximate any continuous functionwith bounded partial derivatives.

Several methods are used to deal with the problem of optimizing membershipfunctions, which are either derivative-based or derivative-free methods. The deriva-tive free approaches are desirable because they are more robust than derivative-basedmethods with respect to finding global minimum and with respect to a wide rangeof objective function and MFs types. The main drawback is that they converge moreslowly than derivative-based techniques (Tao and Taur 1999). On the other hand,derivative-based methods tend to converge to local minimums. In addition, they arelimited to specific objective functions and types of inference and MFs. The mostcommon approaches are: gradient descent (Simon 2000a), least squares (Skrjancet al. 2005), back propagation (Wangand and Mendel 1992) and Kalman filtering(Simon 2002b, c).

The relation of Kalman filter techniques with fuzzy models have been widelyshown in different applications. In Matía et al. (2006) a fuzzy Kalman filter wasintroduced to use a possibilistic instead of probabilistic representation of uncertainty.But the use of the probabilistic Kalman filter training to optimize the MFs of a fuzzysystem was introduced by Simon (2002c) for motor winding current estimation.The used MFs were assumed as symmetric triangular forms. The Kalman filter train-ing was extended to asymmetric triangles in Simon (2002b), and a matrix was defined

6 B.M. Al-Hadithi et al.

relating the parameters of the MFs together based on the sum-normality conditions,then projecting this matrix in each iteration of optimization to constrain the MFs tosum normal types.

Since the derivatives of the functions are used in Kalman filtering, it is limitedto special type of MFs because of complicated and time consuming calculations. Sofar only triangular types are optimized for both inputs and outputs of a Fuzzy LogicController (FLC) (Simon 2002b, c).

The rest of the chapter is organized as follows. Section 1.2 recalls the estimationof TS fuzzy model. Section 1.3 introduces restrictions of TS identification Method.In Sect. 1.4 the proposed non iterative approach is explained. The proposed iterativeapproach by applying Extended Kalman filter is explained in Sect. 1.5. Section 1.6entails an example of an inverted pendulum to demonstrate the validity of the pro-posed approach. The results show that the proposed approach is less conservativethan those based on (standard) TS model and illustrate the utility of the proposedapproach in comparison with TS model. The conclusions of the effectiveness andvalidity of the proposed approach are explained in Sect. 1.7.

1.2 Estimation of Fuzzy TS Model’s Parameters

An interesting method of identification is presented in Takagi and Sugeno (1985). Theidea is based on estimating the nonlinear system parameters minimizing a quadraticperformance index. The method is based on the identification of functions of thefollowing form:

f : �n −→ � (1.1)

y = f (x1, x2, . . . , xn) (1.2)

Each IF-THEN rule Ri1...in , for an nth order system can be written as follows:

S(i1...in) : I f x1 is Mi11 and . . . xn is Min

n then (1.3)

y = p(i1...in)0 + p(i1...in)

1 x1 + p(i1...in)2 x2 + · · · + p(i1...in)

n xn (1.4)

where the fuzzy estimation of the output is:

y =∑r1

i1=1 . . .∑rn

in=1 w(i1...in )(x)[

p(i1...in )0 + p(i1...in )

1 x1 + p(i1...in )2 x2 + · · · + p(i1...in )

n xn

]

∑r1i1=1 . . .

∑rnin=1 w(i1...in)(x)

(1.5)

1 New Concepts for the Estimation of Takagi-Sugeno Model. . . 7

where,w(i1...in)(x) = μ1i1(x1)μ2i2(x2) . . . μnin (xn) (1.6)

being μ j i j (x j ) the membership function that corresponds to the fuzzy set Mi jj .

Let m be a set of input/output system samples {x1k, x2k, . . . , xnk, yk}. The para-meters of the fuzzy system can be calculated as a result of minimizing a quadraticperformance index:

J =m∑

k=1

(yk − yk)2 = ‖Y − X P‖2 (1.7)

where

Y = [y1 y2 . . . ym

]T (1.8)

P =[

p(1...1)0 p(1...1)1 p(1...1)2 . . . p(1...1)n . . . p(r1...rn)0 . . . p(r1...rn)

n

]T(1.9)

X =[β(1...1)1 β

(1...1)1 x11 . . . β

(1...1)1 xn1 . . . β

(r1...rn)1 . . . β

(r1...rn)1 xn1

β1...1m β

(1...1)m x1m . . . β

(1...1)m xnm . . . β

(r1...rn)m . . . β

(r1...rn)m xnm

]

(1.10)

and

β(i1...in)k = w(i1...in)(xk)

∑r1i1=1 . . .

∑rnin=1 w(i1...in)(xk)

(1.11)

If X is a matrix of full rank, the solution is obtained as follows:

J = ‖Y − X P‖2 = (Y − X P)T (Y − X P) (1.12)

∇ J = X T (Y − X P) = X T Y − X T X P = 0 (1.13)

P = (X T X)−1 X T Y (1.14)

1.3 Restrictions of TS Identification Method

The method proposed in Takagi and Sugeno (1985) arises serious problems as it cannot be applied in the most common case where the membership functions are thoseshown in Fig. 1.1.

The membership functions μi1(xi ) = bi −xibi −ai

and μi2(xi ) = xi −aibi −ai

are defined inan interval [ai , bi ] which should verify:

μi1(ai ) = 1 μi1(bi ) = 0 (1.15)

μi2(ai ) = 0 μi2(bi ) = 1 (1.16)

8 B.M. Al-Hadithi et al.

Fig. 1.1 Membershipfunctions of the fuzzysystem

μi1(xi )+ μi2(xi ) = 1 (1.17)

For this case which is widely used, it can be easily demonstrated (Jiménez et al.2012) that the matrix X is not of full rank and therefore X T X is not invertible,which makes the mentioned method of TS invalid. This result can be easily provenas follows:

Supposing that it exists:

f : �n −→ � (1.18)

y = f (x) (1.19)

in which each row of the matrix X is of the form:

Xk = [μ1(xk) μ1(xk)xk μ2(xk) μ2(xk)xk

]

=[

b−xkb−a

b−xkb−a xk

xk−ab−a

xk−ab−a xk

](1.20)

verifying that:

[b−xkb−a

b−xkb−a xk

xk−ab−a

xk−ab−a xk

]

⎡

⎢⎢⎣

−a1−b1

⎤

⎥⎥⎦ = 0 (1.21)

The rank of X in this case is 3 in other words, the columns of X are linearly depen-dent which in turn makes impossible the use of the above mentioned identificationmethod proposed in Takagi and Sugeno (1985). Analyzing another example of twovariables:

f : �2 −→ � (1.22)

y = f (x1, x2) (1.23)

1 New Concepts for the Estimation of Takagi-Sugeno Model. . . 9

Each row of the matrix X is of the form:

Xk = [μ11μ21 μ11μ21x1k μ11μ21x2k μ11μ22 μ11μ22x1k μ11μ22x2k

μ12μ21 μ12μ21x1k μ12μ21x2k μ12μ22 μ12μ22x1k μ12μ22x2k] (1.24)

It can be noticed that the columns 1, 3, 4 and 6 have the same form as in the previousexample multiplied by a constant μ11 and therefore they are linearly dependent aswell. The same thing happens with the columns 6, 9, 10 and 12, etc. In fact, the rankof the matrix in this case is 8.

The solution proposed in Takagi and Sugeno (1985) avoids the occurrence of thissituation. In order to identify a function in the interval [ai , bi ] using TS method,certain intermediate points are chosen of the form:

a∗i ∈ [ai , bi ] and b∗

i ∈ [ai , bi ] (1.25)

and they use membership functions which verify:

μi1(x) ={

xi −b∗i

ai −b∗i

ai ≤ x ≤ b∗i

0 b∗i ≤ x ≤ bi

(1.26)

μi2(x) ={

0 ai ≤ x ≤ a∗i

xi −a∗i

bi −a∗i

a∗i ≤ x ≤ bi

(1.27)

and thus:

μi1(ai ) = 1 μi1(b∗i ) = 0 (1.28)

μi2(a∗i ) = 0 μi2(bi ) = 1 (1.29)

which impedes that the domains of these functions being overlapped and thereforeit can be observed that, except for some isolated points,

μi1(xi )+ μi2(xi ) = 1 (1.30)

and thus, in general, the matrix X will be of full rank and the method is applicable.This solution can be clearly seen in Takagi and Sugeno (1985) where the authors findthe optimum membership functions minimizing the performance index and reducingthe problem to a nonlinear programming one. For this reason, they use the well-known complex method for the minimization. This can obviously be observed in theillustrative examples selected by the authors in Takagi and Sugeno (1985) where allthe identified memberships are non overlapping ones.

10 B.M. Al-Hadithi et al.

1.4 Non Iterative Approach

The restriction of TS identification method for the case presented in the previoussection does not mean the non-existence of solutions rather than an incentive fortheir search. As it has been seen, the problem comes from the fact that the solutionshould fulfil:

∇ J = Xt Y − Xt X P = 0 (1.31)

But as it was shown above, the columns of the matrix X are linearly dependent andconsequently Xt X is not an invertible matrix, therefore it is impossible to calculateP through:

P = (Xt X)−1 Xt Y (1.32)

Nevertheless, as the rows of Xt are linearly dependent, the independent term inequation (1.31) Xt Y will have the same dependence among its rows and thereuponthe rank of the system matrix will be the same as the rank of the extended matrixwith the independent term.

rank(Xt X) = rank(Xt X |Xt Y ) (1.33)

And so the system has solution. In other words, the system is a compatible inde-terminate one, that is, if P is a solution of (1.31) and K belongs to the Kernel ofXt X

K ∈ K er(Xt X) (1.34)

then P∗ = P + K will also be a solution. Therefore, the problem is not the lack ofa solution rather the existence of infinite solutions and the key idea is the ability tofind one of them. Several proposals can be made to select a solution. In our case, theaim is to find solutions with lower norm.

1.4.1 Parameters’ Weighting Method

An effective approach with few computational effort, based on the well known para-meters’ weighting method, is proposed. The main target is to improve the choice ofthe performance index and minimize it. It is characterized by extending the objectivefunction by including a weighting γ of the norm of P vector.

J =m∑

k=1

(yk − yk)2 + γ 2

∑

j

p2j = ‖Y − X P‖2 + γ 2 ‖P‖2 (1.35)

1 New Concepts for the Estimation of Takagi-Sugeno Model. . . 11

This can be rewritten as follows:

J = ‖Y − X P‖2 + γ 2 ‖P‖2 =∥∥∥∥

[Y0

]

−[

Xγ I

]

P

∥∥∥∥

2

= ‖Ya − Xa P‖2

(1.36)

Now the extended matrix Xa is of full rank, and the vector P can be computed as:

P = (Xta Xa)

−1 XtaYa (1.37)

Obviously the solution that minimizes this index is not optimum. However, for asmall value of γ , it will be close to the optimum one and it will be unique as well.

The weighting of γ does not need to have a unique value, rather than some valuescan be weighted more than others to choose the most suitable one.

J =m∑

k=1

(yk − yk)2 +

∑

j

γ 2i p2

j = ‖Y − X P‖2 + ‖�P‖2 =∥∥∥∥

[Y0

]

−[

X�

]

P

∥∥∥∥

2

(1.38)

where � is a diagonal matrix formed by the values γ 2 which should necessarily havea value other than zero to guarantee that the extended matrix is invertible.

1.4.2 Parameters Tuning Using the Parameters’Weighting Method

The parameters’ weighting method can also be used for parameters tuning of TSmodel from local parameters obtained through the identification of a system in anoperating region or from any physical input/output data.

We suppose that in this case we have a first estimation

P0 = [p00 p0

1 p02 . . . p0

n]T (1.39)

of the TS model parameters. In order to obtain such an estimation, the classical leastsquare method can be used around the equilibrium point. The objective is to obtaina global approximation of the system.

y = p00 + p0

1 x1 + p02 x2 + · · · + p0

n xn (1.40)

Let us analyze a set of input/output system samples {x1k, x2k, . . . , xnk, yk}. Theparameters of the global approximation can be calculated by minimizing the follow-ing quadratic performance index:

12 B.M. Al-Hadithi et al.

J =m∑

k=1

(yk − yk)2 = ∥

∥Y − Xg P0∥∥2 (1.41)

where

Y = [y1 y2 y3 . . . ym

]T (1.42)

P0 = [p0

0 p01 p0

2 . . . p0n

]T(1.43)

Xg =

⎡

⎢⎢⎢⎣

1 x11 x21 . . . xn11 x12 x22 . . . xn2......

.... . ....

1 x1m x2m . . . xnm

⎤

⎥⎥⎥⎦

(1.44)

In this case, if we select a sufficient number of points distributed in the regionwhere it is required to obtain the approximation, then the matrix Xg will be of a fullrank and therefore, the solution becomes unique, which can be calculated as follows:

P0 = (X Tg Xg)

−1 X Tg Y (1.45)

This first approximation can be utilized as reference parameters for all the subsys-tems. Then, the parameters’ vector of the fuzzy model can be obtained minimizing:

J =m∑

k=1

(yk − yk)2 + γ 2

r1∑

i1=1

. . .

rn∑

in=1

n∑

j=0

(p0j − p(i1...in)

j )2 = ‖Y − X p‖2 + γ 2 ‖p0 − p‖2

=∥∥∥∥

[Yγ p0

]

−[

Xγ I

]

p

∥∥∥∥

2

= ‖Ya − Xa p‖2 (1.46)

wherep0 = [P0 P0 . . . P0]T

︸ ︷︷ ︸r1.r2...rn

(1.47)

In this case, the factor γ represents the degree of confidence of the parametersinitially estimated. In a similar way to the previous case, different weight factorsof γ (i1...in)

j can be used to each one of the parameters p(i1...in)j depending on the

reliability of the initial parameter p0j in the specific rule.

J =m∑

k=1

(yk − yk)2 +

r1∑

i1=1

. . .

rn∑

in=1

n∑

j=0

γ(i1...in)2j (p0

j − p(i1...in)j )2

= ‖Y − X p‖2 + ‖�(p0 − p)‖2

=∥∥∥∥

[Y�p0

]

−[

X�

]

p

∥∥∥∥

2

= ‖Ya − Xa p‖2 (1.48)

1 New Concepts for the Estimation of Takagi-Sugeno Model. . . 13

where � is a diagonal matrix with the weight factor γ (i1...in)j . It is not necessary to

apply this process for all the parameters. If the values of some of them are known,they can be fixed beforehand or we can assign them a weighting factor γ (i1...in)

jcomparatively high.

1.5 Iterative Parameters’ Identification

The inconvenient feature of the above mentioned method is the amplification of thematrix X throughout the time, so that it becomes inappropriate to be used in real timeapplications as adaptive control for example. The solution is find an iterative methodso that the dimension of the calculation will not be augmented for each sample. Asexplained in detail in Sect. 1.3 that the solution developed in Takagi and Sugeno(1985) to find the optimum membership functions is invalid when memberships areoverlapping ones. In this section, we use an iterative method based on the extendedKalman filter to solve with this problem.

1.5.1 The Kalman Filter

Kalman filter is widely used for stochastic estimation. It is developed by RudolphE. Kalman, through a recursive method for the discrete data linear filtering (Kalman1960). Kalman filter is known to be optimum for linear systems with white processand measurement noises. It is assumed that the system is described by the followingsampled model:

x(k + 1) = �x(k)+ �u(k)+ v(k) (1.49)

y(k) = Cx(k)+ e(k) (1.50)

with

x(k), x(k + 1), v(k) ∈ �n (1.51)

u(k) ∈ �m (1.52)

y(k), e(k) ∈ �p (1.53)

where x(k) represents the state of the dynamic system, u(k) is the input vector andy(k) is the output vector. The vector v(k) represents the Gaussian-white noise of thesystem and e(k) is the measured Gaussian-white noise. Both of them are independentfrom each other with zero mean. The objective of the Kalman filter is to obtain anoptimum estimation x(k) of the state x(k) from measurements of the input/outputvectors. The covariance matrices are supposed to be known and are given as:

14 B.M. Al-Hadithi et al.

R1 = E(v(k), vt (k)) (1.54)

R12 = E(v(k), et (k)) (1.55)

R2 = E(e(k), et (k)) (1.56)

It is also assumed that the initial condition x(0) is Gaussian distributed with

m0 = E(x(0)) (1.57)

R(0) = E((x(0)− m0) (x(0)− m0)

t) (1.58)

where E(.) is the expectation operator. It is supposed that x(k/k − 1), u(k) and y(k)are known and the objective is to estimate x(k +1/k). The prediction problem can beimproved by introducing the difference between the measured and estimated outputs,(y(k)− Cx(k/k − 1)

)as a feedback gain:

x(k + 1/k) = �x(k/k − 1)+ �u(k)+ K (k)(y(k)− Cx(k/k − 1)) (1.59)

The resultant prediction error is the difference between the state of the real systemand the estimated one which can be stated as the following:

ε(k + 1) = x(k + 1)− x(k + 1/k) (1.60)

It should be observed that as above mentioned Gaussian errors v(k) and e(k) arewith zero mean, it can be verified that:

ε(k + 1) = (�− K (k)C)ε(k) (1.61)

Thus,ε(0)

(⇒ x(0) = m0) ⇒ ∀k > 0 ε(k) = 0 (x(k) = mk) (1.62)

And if the dynamics of (1.61) is stable, then:

∀x(0) limk→∞ e(k) = 0 ⇒ lim

k→∞ x(k) = mk (1.63)

The secondary objective is to minimize the covariance matrix which is denotedas P(k),

P(k) = E((ε − ε).(ε − ε)t ) (1.64)

in the sense that it approaches its minimum for:

min(αt P(k)α)∀α ∈ �n (1.65)

The algorithm of Kalman filter can be summarized by the following iterativeprocess:

1 New Concepts for the Estimation of Takagi-Sugeno Model. . . 15

K (k) = (�P(k/k − 1)Ct + R12

) (C P(k/k − 1)Ct + R2

)−1 (1.66)

x(k + 1/k) = �x(k/k − 1)+ �u(k)+ K (k)(y(k)− Cx(k/k − 1)) (1.67)

P(k + 1/k) = (�P(k/k − 1)�t + R1 − K (k)

) (C P(k/k − 1)�t + Rt

12)

(1.68)

This process is initialized with x(0) = m0 and P(0) = R0 which have beeninitially estimated. The classic formulation of Kalman filter can be complementedwith an additional useful filtering process for certain applications.

1.5.2 Extended Kalman Filter

Kalman filter can also be used for state estimation of nonlinear systems. It is basedon the same idea of parameters’ weighting method. For nonlinear systems, e.g. fuzzysystems, the Kalman filter can not be applied directly; but if the nonlinearity of thesystem be sufficiently smooth, then we can linearize it about the current mean andcovariance of the state estimation. This is called Extended Kalman Filter (EKF)with white process and measurement noises. The extended Kalman filter can beconsidered as a predictor-corrector type linear estimator obtained by the linearizationof a nonlinear model at each time step. It is used to estimate the states and parametersof a nonlinear system through the measurements using a function of the linearizedmodel with additive Gaussian white noise. The EKF procedure consists of two steps:time update step and measurement update step. The time update step projects forwardthe current state and error covariance estimates to obtain the a priori estimates for thenext time step. The measurement update step incorporates a new measurement intothe a priori estimate to get an improved a posteriori estimate. In other words, the timeupdate step is a model prediction and the measurement update step is a measurementcorrection. In this section, we briefly outline the algorithm and show how it can beapplied to fuzzy system optimization. Consider a nonlinear discrete time system ofthe form:

x(k + 1) = f (x(k), u(k))+ v(k) (1.69)

y(k) = g(x(k))+ e(k) (1.70)

In this case, Jacobian matrices are those which represent the nonlinear systems:

�(x(k), u(k)) = ∂ f

∂x|x=x(k),u=u(k) (1.71)

�(x(k), u(k)) = ∂ f

∂u|x=x(k),u=u(k) (1.72)

C(x(k)) = ∂ f

∂x|x=x(k) (1.73)

16 B.M. Al-Hadithi et al.

Moreover, the prediction formula for the nonlinear case is the following:

x(k + 1) = �x(k/k − 1)+ �(k)+ K (k)(y(k)− g(x(k/k − 1), u(k))) (1.74)

It must be noted that in this case, the system matrices in this depend on boththe state and input of the system in each instant. Thus, it becomes necessary thecalculation of these matrices in each iteration of the algorithm.

1.5.3 Kalman Filter for Parameters’ Estimation

One of the applications of Kalman filter is the identification of parameters. Let ussuppose that a function depends on q parameters p1, p2 . . . pq

f : �n → �m (1.75)

y = f (x1, x2, . . . , xn, p1, p2 . . . pq) = f (x, p) (1.76)

The problem of identification of parameters can be explained as a problem ofestimation of systems’ states.

p(k + 1) = p(k) (1.77)

y(k + 1) = f (p(k))+ e(k) (1.78)

Then, if we have a set of m samples {x1k, x2k, . . . , xnk, yk} of the function to beidentified, Kalman filter can be used with the following particularities. The matrix�will be an identity matrix in this case. It is assumed a free system without an externalinput so the matrix � is null and the matrix C can be calculated as follows:

C(p(k)) = ∂ f

∂p|p= p(k) (1.79)

The matrices R1 and R12 become null, while R2 is selected based on trial and error.If y ∈ � and we suppose that R2 = I , which would correspond to Gaussians errorfunctions N(0,1), and the function is a linear one, the algorithm becomes equivalentto the recursive minimum square one. The initial covariance state matrix is supposedto be P(0) = cI where C is a number relatively large with respect to the data of theproblem. The formulation of the algorithm becomes:

K (k) = (P(k/k − 1)Ct ) (

C P(k/k − 1)Ct + R2)−1 (1.80)

p(k + 1/k) = p(k/k − 1)+ K (k)(yk − f (x1k , x2k , . . . , xnk , p(k/k − 1))) (1.81)

P(k + 1/k) = P(k/k − 1)− K (k)C P(k/k − 1) (1.82)