Symbols and abbreviations..…………………………………….

227 Pages·2002·3.23 MB·English

by  Andri Riid

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

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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.
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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.
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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 academically is
absolutely meaningless compared to joy of having children and seeing them
developing. At least, this is what I think.

Andri Riid

Tallinn, April-September 2001, December 2001, February-March 2002

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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

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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 ………...…………… 44
3.6 Relationship between 0th and 1st order TS systems ……….. 46
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
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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

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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

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