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THESIS ON INFORMATICS AND SYSTEM ENGINEERING Transparent Fuzzy Systems: Modeling and Control Andri Riid TALLINN TECHNICAL UNIVERSITY FACULTY OF INFORMATION TECHNOLOGY DEPARTMENT OF COMPUTER CONTROL Thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Engineering in Tallinn Technical University © Andri Riid, 2002 ii Abstract During the last twenty years, fuzzy logic has been successfully applied to many modeling and control problems. One of the reasons of success is that fuzzy logic provides human-friendly and understandable knowledge representation that can be utilized in expert knowledge extraction and implementation. It is observed, however, that transparency, which is vital for undistorted information transfer, is not a default property of fuzzy systems, moreover, application of algorithms that identify fuzzy systems from data will most likely destroy any semantics a fuzzy system ever had after initialization. This thesis thoroughly investigates the issues related to transparency. Fuzzy systems are generally divided into two classes. It is shown here that for these classes different definitions of transparency apply. For standard fuzzy systems that use fuzzy propositions in IF-THEN rules, explicit transparency constraints have been derived. Based on these constraints, exploitation/modification schemes of existing identification algorithms are suggested, moreover, a new algorithm for training standard fuzzy systems has been proposed, with a considerable potential to reduce the gap between accuracy and transparency in fuzzy modeling. For 1st order Takagi-Sugeno systems that are interpreted in terms of local linear models, such conditions cannot be derived due to system architecture and its undesirable interpolation properties of 1st order TS systems. It is, however, possible to solve the transparency preservation problem in the context of modeling with another proposed method that benefits from rule activation degree exponents. 1st order TS systems that admit valid interpretation of local models as linearizations of the modeled system are useful, for example, in gain-scheduled control. Transparent standard fuzzy systems, on the other hand, are vital to this branch of intelligent control that seeks solutions by emulating the mechanisms of reasoning and decision processes of human beings not limited to knowledge- based fuzzy control. Performing the local inversion of the modeled system it is possible to extract relevant control information, which is demonstrated with the application of fed-batch fermentation. The more a fuzzy controller resembles the expert’s role in a control task, the higher will be the implementation benefit of the fuzzy engine. For example, a hierarchy of fuzzy (and non-fuzzy) controllers simulates an existing hierarchy in the human decision process and leads to improved control performance. Another benefit from hierarchy is that it assumes problem decomposition. This is especially important with fuzzy logic where large number of system variables leads to exponential explosion of rules (curse of dimensionality) that makes controller design extremely difficult or even impossible. The advantages of hierarchical control are illustrated with truck backer-upper applications. iii Kokkuvõte Viimaste aastakümnete vältel on hägus loogika leidnud edukat rakendust mitmesuguste juhtimis- ja modelleerimisprobleemide lahendamisel. Edu üheks pandiks on olnud asjaolu, et informatsiooni esitus hägus loogika kaudu on lähedane informatsiooni esitusele neis otsustusmehhanismides, mida inimene oma igapäevaelus kasutab. Seejuures tuleb arvestada, et läbipaistvus, hägusate süsteemide omadus, mis on paljude antud rakenduste edukuse oluliseks eelduseks, ei ole vaikimisi tagatud, samuti puudub algoritmide kasutamisel, mis on suutelised andmekogumi põhjal hägusaid süsteeme identifitseerima, igasugune garantii, et tulemuseks on läbipaistev hägus mudel. Käesolevas töös kontsentreerutakse hägusate süsteemide läbipaistvusega seonduvale. Kui tavakäsitluses jagatakse hägusad süsteemid kahte eri klassi, siis töös on näidatud, et nende klasside puhul kehtivad erinevad läbipaistvuse definitsioonid. Klassikaliste hägusate süsteemide puhul, kus KUI-SIIS reeglid seostavad hägusaid määratlusi, on võimalik esitada läbipaistvuse tingimused ilmutatud kujul. Esitatud tingimuste alusel on hinnatud olemasolevate identifitseerimisalgoritmide omadusi ja kasutusvõimalusi. Lisaks on väljatöötatud uudne algoritm, millega on võimalik vähendada eksisteerivat lõhet täpsuse ja läbipaistvuse vahel hägusas modelleerimises. Esimest järku Takagi- Sugeno hägusate süsteemide jaoks ilmutatud läbipaistvuse tingimuste andmise võimalus puudub, kuid probleemile on võimalik leida lahendus modelleerimise kontekstis, seda teise töös väljatöötatud meetodiga. Esimest järku Takagi-Sugeno süsteemide läbipaistvus on kasulik näiteks metoodikas, tuntud termini gain-scheduling all. Läbipaistvate hägusate klassikaliste süsteemide kasutusvaldkond on veelgi suurem, laiendudes nendele juhtimismeetoditele, mis otsivad lahendusi inimese otsustus ja mõtlemisprotsesside emuleerimise läbi ja ei piirdu vaid teadmuspõhise juhtimisega. Protsessi lingvistilise mudeli piiratud pööramise kaudu on võimalik omandada olulist juhtimisinformatsiooni, mille näiteks on töös esitatud fermentatsiooniprotsessi juhtimise rakendus. Hägusa loogika kasutegur on seda suurem, mida enam regulaatori ülesanne meenutab eksperdi rolli. Regulaatorite hierarhia kopeerib tegelikku hierarhiat inimese otsustusprotsessis ja tagab juhtimiskvaliteedi paranemise. Kuivõrd hierarhilise juhtimissüsteemi konstrueerimise eelduseks on probleemi dekompositsioon, on kasu hägusa loogika valdkonnas veelgi suurem, sest hägus juhtimine on eriti tundlik juhtimisparameetrite paljususe suhtes. Hierarhilise juhtimissüsteemi eeliseid on demonstreeritud auto tagurdamissüsteemi näitel. iv Acknowledgements First I would like to thank my supervisor, prof. Ennu Rüstern, for introducing me to the subject, providing excellent working conditions and continuous support throughout the studies. Special thanks go to ex-colleague Mati Pirn for many fruitful discussions in the early stadium of the work. I am even more grateful to Raul Isotamm who did a lot of work on the implementation of algorithms described in the thesis and other students I supervised during those years who all contributed to my work in one way or another. Andres Rähni and colleagues in the Department of Computer Control also deserve a mention here. I would also like to mention gratefully other researchers all over the world who have made their papers available online or sent their papers at my modest request, as well as people who stand behind www.researchindex.org. In this corner of the world it is sometimes difficult to obtain relevant scientific information and cooperation of all such people has been of great help. Many thanks to prof. em. Hanno Sillamaa for proofreading the first draft of the manuscript and pointing out numerous mistakes and how the work could be improved. I am indebted to my family. What one may accomplish in terms of professional career is quite meaningless compared to the importance of having children and not ruining their lives. At least, this is what I think. Andri Riid Tallinn, April-September 2001, December 2001, February-March 2002 v vi Contents 1 Introduction ……………………………………………………. 1 1.1 General Background ………………………………………. 1 1.2 Problem statement ………………………………………… 6 1.3 Original contribution ……………………………………… 6 1.4 Outline of the thesis ……………………………………….. 7 2 Fuzzy systems ………………………………………………….. 9 2.1 Fuzzy sets …………………………………………………. 9 2.2 Basic properties of fuzzy sets ..……………………………. 10 2.3 Fuzzy partition ..…………………………………………… 11 2.4 Operations on fuzzy sets and fuzzy logic …………………. 13 2.5 Fuzzy systems ..……………………………………………. 15 2.6 Rule base properties ………………………...……………... 19 2.7 Inference examples ..………………………………………. 21 2.8 Takagi-Sugeno fuzzy systems…...………………………… 25 2.9 Design of fuzzy systems ..…………………………………. 27 2.10 Summary ..……………………………………………….. 28 vii 3 Interpolation and transparency in fuzzy systems …………… 31 3.1 Transparency and interpretability …………………………. 31 3.2. Transparency of standard fuzzy systems …………………. 33 3.3 Interpolation in standard systems …………………………. 37 3.3.1 Role of defuzzification……………………………… 37 3.3.2 Role of MF type ………..…………………………... 38 3.3.3 Role of inference parameters ………………………. 40 3.3.4 Interpolation in multidimensional space ………….... 40 3.4. Interpolation in 1st order TS systems …..…………………. 41 3.5 Transparency of 1st order TS systems ………...…………… 45 3.6 Relationship between 0th and 1st order TS systems ……….. 47 3.7 Summary…………………………………………………… 49 4 Fuzzy modeling ………………..………………………………. 51 4.1 Introduction ……………………..………………………… 51 4.2 Fuzzy systems as universal approximators ………………... 54 4.3 Selection of input-output data ...…………………………… 54 4.4 Rule-based approaches ..…………………………………... 56 4.4.1 Fuzzy template modeling …………..………………. 56 4.4.2. Yager-Filev fuzzy template modeling algorithm ….. 58 4.4.3 Rule weights in modeling ……….…………………. 59 4.4.4 Wang-Mendel rule extraction ….…………………... 62 4.5 Least squares method …………...…………………………. 64 4.6 Gradient descent ..…………………………………………. 68 4.6.1 Gradient descent for fuzzy systems …..……………. 68 4.6.2 The learning process...……………………………… 72 4.6.3 Convergence issues and higher order methods …….. 73 4.6.4 Overfitting…...……………………………………… 76 4.7 Clustering algorithms…...…………………………………. 78 4.7.1 Extraction of fuzzy rules and membership functions. 81 4.7.2. Clustering example ……………………………..…. 83 viii 4.8 Genetic Algorithms …...…………………………………… 86 4.9 Transparency protection ..…………………………………. 89 4.9.1 Transparency protection of 0th order TS systems and standard fuzzy systems ...………………………………… 90 4.9.2 Transparency protection of 1st order TS systems …... 92 4.10 Comparison of gradient-based methods …………………. 94 4.10.1 Modeling of a SISO system ……………………..... 94 4.10.2 Modeling of a TISO system ………………………. 100 4.11 Modeling of large systems ……………………..………… 101 4.12 Summary and conclusions. ………………………………. 103 5 Fuzzy control …...………………………….…………………... 105 5.1 Introduction ………………………..……………………... 105 5.2 Fuzzy setpoint controllers ………………………………... 107 5.3. Fusion of fuzzy and PID control ………………………….. 113 5.4. Inversion of fuzzy systems ………………………………. 116 5.4.1 Numerical inversion of fuzzy systems……………… 116 5.4.2 Non-numerical inversion of fuzzy systems………..... 118 5.4.3 Control by inverting a fuzzy model ………………... 124 5.5. Control example…………..….…………………………… 128 5.6. Stability issues …..………………………….…………….. 134 5.7. Summary and conclusions..……………………………….. 136 6 Applications ……………………...…………………………….. 139 6.1 Introduction…………………...…………………………… 139 6.2 Backing up the truck and truck-and-trailer ………………... 140 6.2.1 Truck backer-upper …..…………………………….. 140 6.2.2 Backing up the truck and trailer ……………………. 149 ix 6.3 Control of a fed-batch fermentation ………………………. 153 6.3.1 Control system for fed-batch fermentation process with a single substrate feed ………………………………. 154 6.3.2 Fed-batch fermentation control (two substrate process)…………………………………………………… 160 6.4 Conclusions and comments ……………..………………… 175 7 Conclusions …………………………………………...………... 179 7.1 Transparency conditions ……..……………………………. 179 7.2 Transparent modeling algorithms ..………………………... 180 7.3 Transparent fuzzy control ..………………………………... 181 7.4 Suggestions for further research ..…………………………. 182 References ………………………………………………...……… 183 Symbols and abbreviations..……………………………………. 193 List of publications ………………………………………………. 195 Appendix A ………………………………………………………. 197 Appendix B ………………………………………………………. 201 Appendix C ………………………………………………………. 205 Appendix D …...………………………………………………….. 213 x