ISTANBUL TECHNICAL UNIVERSITY GRADUATE SCHOOL OF SCIENCE ENGINEERING AND TECHNOLOGY
M.Sc. THESIS
JULY 2014
LEARNING OF INTERVAL TYPE-2 FUZZY LOGIC SYSTEMS USING BIG BANG – BIG CRUNCH OPTIMIZATION
Cihan ÖZTÜRK
Department of Control Engineering
Control and Automation Engineering Programme
Anabilim Dalı : Herhangi Mühendislik, Bilim Programı : Herhangi Program
JULY 2014
ISTANBUL TECHNICAL UNIVERSITY GRADUATE SCHOOL OF SCIENCE ENGINEERING AND TECHNOLOGY
LEARNING OF INTERVAL TYPE-2 FUZZY LOGIC SYSTEMS USING BIG BANG – BIG CRUNCH OPTIMIZATION
M.Sc. THESIS Cihan ÖZTÜRK
(504121110)
Department of Control Engineering
Control and Automation Engineering Programme
Anabilim Dalı : Herhangi Mühendislik, Bilim Programı : Herhangi Program
TEMMUZ 2014
İSTANBUL TEKNİK ÜNİVERSİTESİ FEN BİLİMLERİ ENSTİTÜSÜ
ARALIK DEĞERLİ TİP-2 BULANIK SİSTEMLERİN BÜYÜK PATLAMA – BÜYÜK ÇÖKÜŞ OPTİMİZASYONUYLA EĞİTİLMESİ
YÜKSEK LİSANS TEZİ Cihan ÖZTÜRK
(504121110)
Kontrol Mühendisliği Anabilim Dalı Kontrol ve Otomasyon Mühendisliği Programı
Anabilim Dalı : Herhangi Mühendislik, Bilim Programı : Herhangi Program
Thesis Advisor : Asst. Prof. Dr. Engin YEŞİL
Istanbul Technical University ...
Jury Members : Asst. Prof. Dr. Tufan KUMBASAR
Istanbul Technical University ...
Asst. Prof. Dr. Özgür Turay KAYMAKÇI
Yıldız Technical University ... Cihan Öztürk, a M.Sc. student of ITU Graduate School of Science Engineering and Technology student ID 504121110, successfully defended the thesis entitled “LEARNING OF INTERVAL TYPE-2 FUZZY LOGIC SYSTEMS USING BIG BANG – BIG CRUNCH OPTIMIZATION” which he prepared after fulfilling the requirements specified in the associated legislations, before the jury whose signatures are below.
FOREWORD
I would like to express my deep appreciation and thanks to my advisor Asst. Prof. Dr. Engin Yeşil and Asst. Prof. Dr. Tufan Kumbasar for their tolerance and always positive approach during my whole education period. I would like to present my appreciation to my family and friends since they always believed in me and never gave up supporting me. At last, I would like to thank specially Beste Özsoy for her encouragement and accompany.
I would like to thank the Scientific and Technological Research Council of Turkey (TUBITAK) for their support and graduate student scholarship under the research project 113E206 (1001, the Support Program for Scientific and Technological Research Project). I would also like to thank project members M. Furkan Dodurka and Ahmet Sakallı for their cooperation.
I would like to thank Istanbul Technical University Scientific Research Projects Unit (ITU BAP) for their support to my master thesis under the Support Program for Graduate Thesis Research Project.
TABLE OF CONTENTS Page FOREWORD ... ix TABLE OF CONTENTS ... xi ABBREVIATIONS ... xiii LIST OF TABLES ... xv
LIST OF FIGURES ... xvii
SUMMARY ...xix
ÖZET ...xxi
1. INTRODUCTION ...1
1.1 Purpose of Thesis ...1
1.2 Literature Review ...2
2. TYPE-1 FUZZY LOGIC SYSTEMS ...7
2.1 Introduction ...7
2.2 Type-1 Fuzzy Sets and Memberships...7
2.2.1 Triangular membership function ...8
2.2.2 Trapezoidal membership function ...8
2.2.3 Gaussian membership function ...9
2.3 Fuzzy Inference System (FIS)... 10
2.3.1 Fuzzification ... 11
2.3.2 Rule base ... 11
2.3.3 Inference mechanism ... 11
2.3.4 Defuzzification ... 11
3. INTERVAL TYPE-2 FUZZY LOGIC SYSTEMS ... 13
3.1 Introduction ... 13
3.2 Type-2 Fuzzy Logic Systems ... 14
3.3 Interval Type-2 Fuzzy Logic Systems ... 17
3.3.1 IF-THEN rules, inference and defuzzification ... 17
3.3.2 Type reduction methods ... 19
3.3.2.1 Karnik-Mendel type reduction method ... 19
3.3.2.2 Wu-Mendel uncertainty bound method ... 21
4. FUZZY MODELING AND LEARNING ... 23
4.1 Introduction ... 23
4.2 White-Box, Black-Box and Gray-Box Modeling ... 25
4.3 Fuzzy Model Identification ... 27
4.4 Classes of Fuzzy Models ... 28
4.5 Constructing Fuzzy Models from Input - Output Data ... 29
4.5.1 Mosaic or table lookup scheme ... 30
4.5.2 Gradient descent method ... 31
4.5.3 Clustering and gradient descent method ... 32
5. PROPOSED LEARNING METHODOLOGY OF INTERVAL TYPE-2
FUZZY LOGIC SYSTEMS ... 35
5.1 Introduction ... 35
5.2 Choosing The Regressors ... 36
5.2.1 Exhaustive search (Brute force search) ... 36
5.2.2 Heuristic search ... 37
5.2.3 Pseudo-random methods ... 39
5.3 Choosing The Structure ... 40
5.4 Proposed Method ... 41
6. BIG BANG – BIG CRUNCH OPTIMIZATION ALGORITHM ... 45
7. SIMULATION STUDY ... 49
7.1 Heat Transfer Process Trainer (PT-326) ... 49
7.1.1 PT-326 - type-1 fuzzy modeling ... 52
7.1.1.1 Data pre-processing ... 52
7.1.1.2 Model structure and regressor selection ... 53
7.1.1.3 Type-1 fuzzy models ... 57
7.1.2 Generating interval type-2 fuzzy models ... 63
7.1.2.1 Pure data - interval type-2 fuzzy models ... 65
7.1.2.2 Noisy data - interval type-2 fuzzy models ... 68
7.2 El Nino Southern Oscillation (ENSO) Index Data Set ... 71
7.2.1 Type-1 fuzzy modeling... 71
7.2.1.1 Data pre-processing ... 71
7.2.1.2 Model structure and regressor selection ... 72
7.2.1.3 Type-1 fuzzy models ... 74
7.2.2 Generating interval type-2 fuzzy models ... 75
8. CONCLUSION ... 79
REFERENCES ... 81
ABBREVIATIONS
ANFIS : Adaptive Neuro-Fuzzy Inference System AR : Auto-Regressive
BB-BC : Big Bang – Big Crunch COA : Center of Area
DC : Direct Current
ENSO : El Nino Southern Oscillation FIS : Fuzzy Inference System FLC : Fuzzy Logic Controller FLS : Fuzzy Logic System FM : Fuzzy Modeling
FOU : Footprint of Uncertainty
FS : Fuzzy Set
GK : Gustafson-Kessel IT2 : Interval Type-2
KM : Karnik-Mendel
LMF : Lower Membership Function MF : Membership Function
MOM : Mean of Maximum
NAR : Nonlinear Auto-Regressive
NARMAX : Nonlinear Auto-Regressive Moving Average with eXogenous input NARX : Nonlinear Auto-Regressive eXogenous input
NBJ : Nonlinear Box-Jenkins
NFIR : Nonlinear Finite Impulse Response NOE : Nonlinear Output Error
RLS : Recursive Least Squares RMSE : Root Mean Square Error
PID : Proportional Integral Derivative T1FS : Type-1 Fuzzy Set
T1FLC : Type-1 Fuzzy Logic Controller T1FLS : Type-1 Fuzzy Logic System T2FS : Type-2 Fuzzy Set
T2FLC : Type-2 Fuzzy Logic Controller T2FLS : Type-2 Fuzzy Logic System TR : Type Reduction
TS : Takagi-Sugeno
UB : Uncertainty Bound
LIST OF TABLES
Page
Table 3.1 : Classification of type-2 FLSs………..17
Table 4.1 : Parameters adjusted by the different fuzzy modeling methods……...30
Table 6.1 : BB-BC optimization algorithm ………...47
Table 7.1 : The parameters of type-1 MFs for PT326 pure data (2 regressors)...57
Table 7.2 : The fuzzy rules of type-1 MFs (Pure data - 2 regressors)...58
Table 7.3 : The parameters of type-1 MFs for PT326 pure data (3 regressors)...59
Table 7.4 : The fuzzy rules of type-1 MFs (Pure data - 3 regressors)...60
Table 7.5 : The parameters of type-1 MFs for PT326 noisy data (2 regressors)...60
Table 7.6 : The fuzzy rules of type-1 MFs (Noisy data - 2 regressors)...61
Table 7.7 : The parameters of type-1 MFs for PT326 noisy data (3 regressors)...62
Table 7.8 : The fuzzy rules of type-1 MFs (Noisy data - 3 regressors)...63
Table 7.9 : New tuning parameter details...64
Table 7.10 : New parameter distrubution for 2-component regressors...64
Table 7.11 : BB-BC training parameters ...64
Table 7.12 : New parameter distrubution for 3-component regressors...65
Table 7.13 : RMSE comparison of T1 and IT2 fuzzy models (PT-326–Pure)...67
Table 7.14 : RMSE comparison of T1 and IT2 fuzzy models (PT-326–Noisy)...70
Table 7.15 : The parameters of type-1 MFs for ENSO...74
Table 7.16 : The fuzzy rules of type-1 MFs (ENSO)...75
LIST OF FIGURES
Page
Figure 2.1 : Triangle membership function. ...8
Figure 2.2 : Trapezoidal membership function. ...9
Figure 2.3 : Gaussian membership function. ... 10
Figure 2.4 : Type-1 fuzzy logic system structure. ... 10
Figure 3.1 : a) Primary membership function of type-2 fuzzy set, b) Interval secondary membership function, c) Triangular secondary membership function………...16
Figure 3.2 : Type-2 fuzzy logic system structure...17
Figure 5.1 : Sequence for regressors selection...37
Figure 5.2 : Heuristic search for regressors selection...38
Figure 5.3 : Regressors selection using genetic algorithms...39
Figure 5.4 : Type-1 and interval type-2 fuzzy modeling procedure...43
Figure 5.5 : Creation of triangle shaped interval type-2 MFs from type-1 MFs...44
Figure 5.6 : Creation of triangle shaped interval type-2 MFs from type-1 MFs...44
Figure 7.1 : Schematic working principle of the Heat Transfer Process Trainer...50
Figure 7.2 : Block diagram of Heat Transfer Process Trainer...50
Figure 7.3 : View of the Heat Transfer Process Trainer...50
Figure 7.4 : Input voltage - output temperature data of Heat Transfer Process...51
Figure 7.5 : 1000 data points of the output y(k)...51
Figure 7.6 : Training and checking data seperation of pure data...52
Figure 7.7 : Training and checking data seperation of noisy data...53
Figure 7.8 : Seq. search training and checking errors of 2-components regressors...55
Figure 7.9 : Seq. search training and checking errors of 3-components regressors...55
Figure 7.10 : Seq. search training and checking errors of 2-components regressors (Noisy)...56
Figure 7.11 : Seq. search training and checking errors of 3-components regressors (Noisy)...56
Figure 7.12 : PT326 - Type-1 fuzzy model MFs (Pure data, 2 regressors)...58
Figure 7.13 : PT326 - Type-1 fuzzy model MFs (Pure data, 3 regressors)...59
Figure 7.14 : PT326 - Type-1 fuzzy model MFs (Noisy data, 2 regressors)...61
Figure 7.15 : PT326 - Type-1 fuzzy model MFs (Noisy data, 3 regressors)...62
Figure 7.16 : PT326 – IT2 fuzzy model MFs (Pure data, 2 regressors)...66
Figure 7.17 : PT326 – IT2 fuzzy model MFs (Pure data, 3 regressors)...66
Figure 7.18 : Output comparison of T1 and IT2 fuzzy models (PT-326–Pure- 2reg)...67
Figure 7.19 : Output comparison of T1 and IT2 fuzzy models (PT-326–Pure- 3reg)...68
Figure 7.21 : PT326 – IT2 fuzzy model MFs (Noisy data, 3 regressors)...69
Figure 7.22 : Output comparison of T1 and IT2 fuzzy models (PT326–Noisy- 2reg)...70
Figure 7.23 : Output comparison of T1 and IT2 fuzzy models (PT326–Noisy- 3reg)...71
Figure 7.24 : ENSO index data set...72
Figure 7.25 : Seq. search training and checking errors of 2-components regressors...73
Figure 7.26 : Seq. search training and checking errors of 3-components regressors...74
Figure 7.27 : ENSO - Type-1 fuzzy model membership functions...75
Figure 7.28 : IT2 fuzzy model MFs (2 regressors - ENSO)...76
LEARNING OF INTERVAL TYPE-2 FUZZY LOGIC SYSTEMS USING BIG BANG – BIG CRUNCH OPTIMIZATION
SUMMARY
History of the fuzzy logic began with the first article in 1965 written by Prof. Lotfi A. Zadeh. Fuzzy logic is flexible application of classical logic rules, fuzzy sets is the extension of the classical set notation. In classical logic approach, an entity is a member of a cluster in a precise manner or not. However, in fuzzy logic approach, an entity may be a cluster member in a partial manner.
In 1965, Zadeh proposed the fuzzy membership functions containing type-1 fuzzy systems. These systems are widely used in engineering problems, modeling and control of the nonlinear systems whose the mathematical model is not exactly obtained. However, in recent years it has been shown that type-1 fuzzy sets can define their members only a number in [0,1] interval. That’s why type-1 fuzzy models obtained with type-1 fuzzy sets remain incapable of containing the uncertainties in the actual system dynamics.
The concept of 2 fuzzy sets are also proposed by Zadeh as an extension of type-1 fuzzy sets (classical fuzzy sets). Type-2 fuzzy systems include at least one type-2 fuzzy set. The most important property of type-2 fuzzy systems is their ability to express the uncertainties in system dynamics with another fuzzy set at membership functions. In other words, type-2 fuzzy sets are fuzzy-fuzzy sets. As shown in recent studies, type-2 fuzzy models outperforms type-1 fuzzy models in identification and modeling of the systems with uncertainties and/or nonlinearities.
Type-2 fuzzy sets are also used in fuzzy controller design and it has been shown that type-2 fuzzy sets are good at expressing the relationships between input and outputs of fuzzy controllers. On the other hand, working with type-2 fuzzy systems leads to more computational complexity according to the study with type-1 fuzzy systems. Therefore, a special case of type-2 fuzzy sets called interval type-2 fuzzy sets was proposed. The efficiency and advantages of interval type-2 fuzzy systems are also shown in control and modeling applications.
In literature, there are many systematic methodology of designing and modeling of type-2 fuzzy systems. In this thesis, primarily type-1 fuzzy sets and systems are explained, after type-2 fuzzy systems that use type-2 fuzzy sets are have been introduced, and the advantages of interval type-2 fuzzy sets are highlighted. Early proposed type reduction and defuzzification structures for type-2 fuzzy systems have been mentioned. After this stage, type-2 fuzzy modeling of the systems has started. Primarily, the systems are modeled with type-1 fuzzy systems and secondly better interval type-2 fuzzy models are investigated from type-1 fuzzy models point of view. Type-1 fuzzy modeling has been performed by MATLAB Adaptive Neuro-Fuzzy Inference System (ANFIS) toolbox. Then, Big Bang-Big Crunch (BB-BC) algorithm is employed using root mean square error (RMSE) cost function in order to obtain better interval type-2 fuzzy models than type-1 fuzzy models by minimizing
membership (locations) functions are searched. As output membership functions, Takagi-Sugeno linear type is selected and they remain the same as in type-1 fuzzy models. Simulations are carried on with the input-output data of PT-326 Heat Transfer Process Trainer dynamic system and the real world data of El Nino Southern Oscillation (ENSO) index. Simulation studies show that better interval type-2 fuzzy models can be found as compared with the optimal type-1 fuzzy models which leads to state that interval type-2 fuzzy models can express the nonlinearity and uncertainties better than type-1 fuzzy models. The superiority of interval type-2 fuzzy models are more obvious to see when the measurement noise is added to the input-output data.
ARALIK DEĞERLİ TİP-2 BULANIK SİSTEMLERİN BÜYÜK PATLAMA – BÜYÜK ÇÖKÜŞ OPTİMİZASYONUYLA EĞİTİLMESİ
ÖZET
Bulanık mantık kavramının tarihi 1965 yılında Prof. Lotfi A. Zadeh’in ilk makalesiyle başlamıştır.. Bulanık mantık, klasik mantık kurallarının esnek bir biçimde uygulanması, bulanık kümeler ise klasik küme gösteriminin genişletilmiş halidir. Klasik mantık yaklaşımda bir varlık bir kümenin ya kesin bir biçimde elemanıdır ya da değildir. Ancak bulanık mantık yaklaşımında, bir varlık bir kümenin kısmi biçimde elemanı olabilir.
1965 yılında Zadeh’in önerdiği şekilde bulanık üyelik fonksiyonları içeren tip-1 bulanık sistemler, mühendisliği problemlerinde, matematiksel modeli tam olarak elde edilemeyen ve/veya doğrusal olmayan sistemlerin modellenmesinde ve kontrol edilmesinde etkin bir araç olarak kullanılmaktadır. Ancak son yıllarda gösterilmiştir ki tip-1 bulanık kümeler kendilerine ait elemanları sadece [0, 1] aralığında bir sayıyla ifade ettikleri için bu kümelerle elde edilen tip-1 bulanık sistem modelleri gerçek sistem dinamiklerinde mevcut olan belirsizlikleri içermekte bazen yetersiz kalmaktadır. Bu yetersizliğin temelde tip-1 bulanık kümelerin keskin üyelik değerlerinden kaynaklandığı gösterilmiştir.
Tip-2 bulanık küme kavramı da yine Zadeh tarafından tip-1 bulanık kümelerin (klasik bulanık kümeler) bir genişlemesi olarak sunulmuştur. Bünyesinde en az bir tane tip-2 bulanık küme bulunduran bulanık sistemlere tip-2 bulanık sistem denmektedir. Tip-2 bulanık sistemlerin en önemli özelliği sistem dinamiğindeki belirsizlikleri üyelik fonksiyonlarında bir başka bulanık küme ile ifade edebilme yetenekleridir. Bir diğer deyişle tip-2 bulanık kümeler bulanık-bulanık kümelerdir. Ancak tip-2 bulanık kümelerin karmaşık iç yapıları ve tip-2 bulanık kümelerin hesaplama yükü sebebiyle tip-2 bulanık sistemler günümüzde hala yeterince geliştirilememiştir. Çünkü tip-1 bulanık kümeler yerine tip-2 bulanık kümeler ile uğraşmak bulanık çıkarım mekanizmasına tip indirgeme adında yeni bir operasyon eklemiştir. Tip indirgeme işlemi, bulanık kuralların öncül kısmındaki tip-2 bulanık kümelerini durulama işlemi öncesinde tip-1 bulanık kümelere indirgemek için kullanılır. Ancak tip indirgemenin genellikle iteratif algoritmalar ile gerçeklenmesi tip-2 bulanık sistemlerin hesaplama yükünü arttırmış ve iç yapısının incelenmesini engellemiştir. Bu sebeple tip-2 bulanık sistemlerin zorluklarını yok etmek için, tip-2 bulanık sistemlerin özel bir hali olan aralık değerli tip-2 bulanık sistemler önerilmiştir. Aralık değerli tip–2 bulanık sistemler, EĞER-O HALDE şeklindeki bulanık kurallardan oluşmaktadır. Tip–2 bulanık mantık sistemlerin kural yapısındaki öncül ve/veya sonuç önermeleri aralık değerli tip–2 bulanık kümeleriyle ifade edilmektedir. Sistemin çıkışını hesaplayabilmek için, ilk önce bulanıklaştırma bloğunda keskin girişler tip–2 bulanık kümelere dönüştürülürler. Daha sonra, çıkarım mekanizması tanımlanmış kuralları kullanarak giriş değerlerini tip–2 bulanık değerlerine dönüştürür. Elde edilen tip–2 bulanık küme çıkışları, tip indirgeme mekanizması ile tip–1 bulanık kümelere dönüşürler. Tip indirgeme işlemi ile elde
edilen kümeler durulaştırıcı mekanizması ile keskin çıkışlara dönüştürülür. Günümüzde de aralık değerli tip-2 bulanık sistem açık bir şekilde bulanık mantık alanında yapılan çalışmalara yön vermektedir.
Aralık değerli tip-2 bulanık mantık sistemler, sıvı seviye kontrolü, otonom mobil robot kontrolü, süreç kontrolü, pH kontrolü, biyoreaktör kontrolü ve modelleme gibi birçok farklı kontrol alanındaki uygulamalarda gerçeklenmiştir. Yapılan çalışmalarla gösterilmiştir ki aralık değerli tip-2 bulanık mantık sistemleri öncül üyelik fonksiyonlarındaki belirsizlik izdüşümü tarafından sağlanan fazladan serbestlik derecesi sayesinde tip-1 eşdeğerlerine kıyasla daha iyi modelleme ve kontrol performansları sağlamaktadır. Ayrıca aralık değerli tip-2 bulanık mantık tabanlı sistemler, tip-1 eşdeğerlerine kıyasla daha yumuşak çıkışlar üretmektedir ve bu durum belirsizlikler karşısında daha iyi performans göstermesini sağlamaktadır. Yapılan bazı çalışmalarda değinilmiş olsa da, aralık değerli tip-2 bulanık mantık sistemlerinin tasarımı günümüzde hala geliştirilmeye açık bir konu olarak değerlendirilmektedir. Bunların yanında, hesaplama yükünü azaltmak için orijinal Karnik-Mendel algoritmasına alternatif bir çok tip indirgeme algoritması önerilmiştir. Ancak tip indirgemenin temel problemlerinin üstesinden hala gelinememiştir çünkü alternatif tip indirgeme yöntemleri Karnik-Mendel algoritmasının yenilikçi ve uyarlamalı olmak üzere iki temel özelliğini taşımamaktadır.
Literatür incelendiğinde araştırmacıların son yıllarda ilgisi aralık değerli tip-2 bulanık mantık sistemleri daha iyi anlamak, kararlılık analizlerini gerçekleştirmek, farklı amaçlarla tasarım yöntemleri geliştirebilmek için devam etmektedir.
Bu tez içerisinde, öncelikle tip-1 bulanık kümeler ve sistemlerden bahsedilmiş, bir bulanık mantık sistemin parçaları olan giriş-çıkış üyelik fonksiyonları, bulanıklaştırma, çıkarım mekanizması ve durulamanın üzerinde durulmuştur. Daha sonra tip-2 bulanık kümelerin kullanıldığı tip-2 bulanık mantık sistemler tanıtılmış, buradan hareketle aralık değerli tip-2 bulanık kümelere geçilip aralık değerli tip-2 bulanık sistemlerin avantajları öne çıkarılmıştır. Araştırmacılar tarafından tip-2 bulanık sistemler için önerilmiş olan tip indirgeme algoritmalarından söz edilmiştir. Bu başlangıç kısımlarından sonra, önerilen Büyük Patlama - Büyük Çöküş (BP-BÇ) global optimizasyon algoritması ile aralık değerli tip-2 bulanık mantık sistemlerin eğitilmesi yöntemi detaylı olarak anlatılmıştır.
BP-BÇ global optimizasyon algoritması, genetik algoritmada olduğu gibi doğadan esinlenmiştir. Evrenin oluşum teorilerinden biri olan Büyük Patlama – Büyük Çöküş evrim teorisine dayanmaktadır. Büyük Patlama aşamasında, arama uzayında rastgele çözümler üretilir, daha sonra Büyük Çöküş aşamasında bir daraltma operatörü yardımıyla bir sonraki nesil için bir başlangıç vektörü üretilir. Klasik genetik arama yöntemlerinde olduğu gibi, evrimsel operatörlere ihtiyaç duymadığı için az bir hesaplama zamanına ve yüksek yakınsama hızına sahiptir. Bu özelliğinden dolayı çevrimiçi kontrol uygulamalarında kullanılma uygundur. BP-BÇ Optimizasyonu 2 ana aşamadan oluşmaktadır. Birinci aşamada Büyük Patlama gerçekleşir. Bu bölümde arama uzayında rastgele dağıtılmış problemin çözümü olabilecek bireyler oluşturulur. Büyük Çöküşün gerçekleştiği ikinci aşamada ise bir daraltma prosedürü gerçekleştirilerek popülasyonun ağırlık merkezi bulunur. İlk Büyük Patlama aşamasında, diğer evrimsel arama algoritmalarında olduğu gibi bireyler bütün arama uzayını kapsayacak şekilde rastlantısal olarak oluşturulur. Bunu takıp eden Büyük Patlama aşamalarında ise bireyler ağırlık merkezinin ya da en iyi birey etrafında rastlantısal olarak dağıtılmış olarak oluşturulur. Kısaca, bu yeni evrimsel arama
algoritmasının çalışma prensibi yakınsamış bir çözümü kaotik bir duruma dönüştürerek yeni çözüm kümeleri oluşturmaktadır.
Önerilen yöntemin temeli belli koşullar altında elde edilmiş olan en iyi tip-1 bulanık mantık sistem modelinden yola çıkarak daha iyi bir aralık değerli tip-2 bulanık mantık sistem modeli elde etmeye dayanmaktadır. Buradaki amaç, aralık değerli tip-2 bulanık mantık sistemlerin farklı koşullar altındaki (saf data, gürültülü data) modelleme performanslarının elde edilebilen en iyi tip-1 eşdeğerlerinden daha iyi olduğunu göstermektir.
Bu aşamadan sonra farklı özelliklere sahip olan sistemleri tip-2 bulanık modelleme aşamasına geçilmiştir. Benzetim çalışmalarında kullanılan veriler, Isı İletimi Süreci dinamik sistemine ait saf giriş-çıkış dataları, aynı sisteme ait gürültülü giriş-çıkış dataları ve gerçek verilerden oluşan El Nino Southern Oscillation (ENSO) indeksi zaman serisi datalarıdır. Benzetimlerde öncelikle sistemleri tip-1 bulanık modelleyip sonrasında ilgili tip-1 bulanık modellerden daha iyi aralık değerli tip-2 bulanık modeller elde edilmeye çalışılmıştır. MATLAB’ın Adaptive Neuro-Fuzzy Inference System (ANFIS) programı ile tip-1 modellemeler yapılmıştır. Tip-1 modellemelerde ANFIS’in kullandığı ortalama hata kareleri kökü amaç fonksiyonu kullanılmış, her girişe 2 adet üyelik fonksiyonu atanmış ve giriş üyelik fonksiyonlarının tipi Gauss olarak seçilmiştir. Giriş üyelik fonksiyonu tipi olarak Gauss seçilmesinin nedeni sadece 2 (merkez, sigma) parametre ile üyelik fonksiyonun tanımlanabilmesidir. Isı İletim Süreci dinamik sistemine ait giriş-çıkış datalarının modellenmesinde seçilen regresyon yapısı hem geçmiş giriş, hem de geçmiş çıkış değerlerine dayanmaktadır. ENSO indeksi zaman serisi datalarının modellenmesinde seçilen regresyon yapısı ise sadece geçmiş çıkış değerlerine dayanmaktadır. Isı İletimi Süreci verileri için 2 ve 3 elemanlı (2 ve 3 girişli) regresyon vektörleri kullanılırken, ENSO indeksi zaman serisi modellemesinde sadece 2 elemanlı (2 girişli) regresyon vektörü kullanılmıştır. Regresyon vektörlerinin elemanlarının seçiminde ortalama hata kareleri kökü amaç fonksiyonuna dayanarak bir seçim yapılmıştır. Öncelikle aday regresyon elemanları tanımlanmıştır. Isı İletimi Süreci modellemesi için aday regresyon elemanları olarak girişin önceki son 6 datası, çıkışın önceki son 4 datası olmak üzere 10 adettir. ENSO indeksi zaman serisinin modellenmesinde ise aday regresyon elemanları olarak çıkışın önceki son 12 datası ve çıkışın önceki 26. datası olmak üzere 13 aday kullanılmıştır. Bu aday regresyon elemanlarından sezgisel bir yöntemle ortalama hata kareleri kökü amaç fonksiyonunu minimize edecek şekilde en iyi regresyon ikilileri ve üçlüleri elde edilmiştir. Kullanılan sezgisel yöntem, öncelikle her bir regresyon elemanı adayının ortalama hata kareleri kökü değerine bakıp bunlardan en küçük olanını seçip, bu seçilen regresyon elemanının yanına kalanları teker teker ekleyip elde edilen ortalama hata kareleri kökü değerlerine göre en iyi ikiliyi üretip, gerekirse aynı prosedürle en iyi üçlüleri elde etmektedir. Daha sonra BP-BÇ global optimizasyon algoritması kullanılarak ortalama hata kareleri kökü amaç fonksiyonu değerleri tip-1 bulanık modellere göre daha da minimize edilmeye çalışılarak aralık değerli tip-2 bulanık modellere ulaşılmıştır. Optimizasyon sırasında giriş Gauss üyelik fonksiyonlarının merkez değerleri tip-1 modellerdeki gibi bırakılmış, sadece sigma (σ) değerleri optimizasyon parametreleri olarak alınmıştır. Çıkış üyelik fonksiyonları ise Takagi-Sugeno lineer tip olarak seçilmiş olup, tip-1 modellerdeki gibi bırakılmıştır. Benzetim çalışmalarında elde edilen performanslara bakılarak belli koşullar altında elde edilmiş olan en iyi tip-1 bulanık modellerden daha iyi aralık değerli tip-2 bulanık modeller elde edilebildiği, aralık değerli tip-2 bulanık modellerin sisteme ait nonlineerlik ve belirsizlikleri tip-1 eşdeğerlerine nazaran daha iyi ifade ettikleri görülmüştür.
1. INTRODUCTION
Generally, conventional control techniques, for instance proportional-integral-derivative (PID), adaptive control, robust control, nonlinear control, need the mathematical model of the systems which the controller design is based on. In other words; if the parameters of a system can be obtained precisely, then its control would be a relatively straightforward problem and these model-based approaches could be used. However, in real-time industrial systems, it is common that there exist considerable difficulties in obtaining an accurate model. Even if the model is sufficiently accurate, there may be many other uncertainties caused by the sensitivity of the sensors, noise of the sensors and nonlinear characteristics of the actuators. Then, the performance of model-based approaches may decrease where the complexity of the controller design may increase, too. In such cases, model-free approaches are usually favored for both modeling and control purposes. There are numerous types of model-free approaches but the most common approach are fuzzy logic systems (FLSs) which are based on fuzzy logic and sets.
Zadeh introduced the concept of fuzzy sets in 1965 which are sets with uncertain amplitudes (Zadeh, 1965). The concept of fuzzy sets is based on the degree of memberships rather than true or false. Zadeh makes the statement that ”fuzzy logic is a precise conceptual system of reasoning, deduction and computation in which the objects of discourse and analysis are, or are allowed to be, associated with imperfect information. Imperfect information is information which in one or more respects is imprecise, uncertain, incomplete, unreliable, vague or partially true” (Zadeh, 2009).
1.1 Purpose of Thesis
In literature, there are many systematic or heuristic methodology of designing and modeling of type-2 fuzzy systems. In this thesis, primarily type-1 fuzzy sets and systems are explained, after type-2 fuzzy systems that use type-2 fuzzy sets are have been introduced, and the advantages of interval type-2 fuzzy sets are highlighted
Early proposed type reduction and defuzzifiction structures for type-2 fuzzy systems have been mentioned.
After this stage, type-2 fuzzy modeling of the systems has started. Primarily, the systems are modeled with type-1 fuzzy systems and secondly better interval type-2 fuzzy models are investigated from type-1 fuzzy models point of view. Type-1 fuzzy modeling has been performed by MATLAB Adaptive Neuro-Fuzzy Inference System (ANFIS) toolbox. Then, optimization algorithms are emplyed using different cost functions (root mean square error (RMSE), etc.) in order to obtain better interval type-2 fuzzy models than type-1 fuzzy models by minimizing the cost function. During the optimization process, only the parameters of input membership (locations) functions are searched. As output membership functions, Takagi-Sugeno linear type is selected and they remain the same as in type-1 fuzzy models. Simulations are carried on with static functions and the input-output data of some dynamic systems. Simulation studies show that better interval type-2 fuzzy models can be found as compared with the optimal type-1 fuzzy models which leads to state that interval type-2 fuzzy models can express the nonlinearity and uncertainties better than type-1 fuzzy models.
1.2 Literature Review
After Zadeh published his first paper on fuzzy sets (Zadeh, 1965), he opened the door for other researchers to study type-1 fuzzy logic (T1FL) and he quickly amassed a large number of followers. On his second paper, he introduced FLSs, which are well known for their ability to model linguistics and system uncertainties (Zadeh, 1975). After this initial break-through start, fuzzy logic began to gain popularity as a method of control. Another critical point for fuzzy logic came about in the year 1974, when Ebrahim Mamdani and S. Assilian used fuzzy logic to control a steam engine for the first time (Mamdani, 1974).
While the fuzzy control literature was being progressed, FLS is also used for modeling since it is a widely known fact that fuzzy models can easily have integration with a priori knowledge of information obtained from measured process data (Babuska, 1998; Abonyi 2003). Hence, FLSs are strong tools in representing nonlinear and/or uncertain systems. The nonlinearities / uncertainties of the process
Türksen 2009). Because of this capability, fuzzy systems are very satisfactory in modelling and control of complex nonlinear systems whose complete mathematical models are not simply present. Since most of the systems and processes are characterized by uncertainties and nonlinearities, various fuzzy modeling and control strategies have been successfully implemented in many problems during the last few decades. T1FL has been applied in many industrial areas such as elevator drive systems, robotics, DC-DC converters. Since the proportional (P) and integral (I) values speed controller cannot generally be set to a large value due to its mechanical resonance, instead of conventional speed controller, a fuzzy logic controller (FLC) was used in elevator drive systems (Yu, 2007). Experimental results show that for high-performance elevator drive systems, the proposed FLC is better than the conventional PI controller in speed control. In order to accomplish a real-time and robust control performance in reactive manners, FLC is used to encode the behaviors for the quadruped walking robots which learn and execute soccer-playing behaviors (Dongbing, 2007). These experimental studies show the effectiveness of the controller in representing behavior of the robots. In order to smoothen the output power fluctuation of a variable-speed wind farm, a FLC is used as a reference adjuster to control the DC DC converter (Muyeen, 2009). Simulation studies demonstrate that using FLC enhances the control ability of the overall system, and reduces the cost of the energy capacitor system, yet keeps the size of the energy storage system small.
On the other hand, when a system has a large amount of uncertainties, type-1 fuzzy logic systems (T1FLSs) may not be capable of achieving the desired level of performance with a reasonable complexity of structure (Mendel, 1998). Researchers demonstrated that there might be limitations in the ability of ordinary fuzzy logic systems in the ability of to model the uncertainties and nonlinearities (Hagras, 2004). This limitation is mainly caused because of the fact that, the membership grade for each input value is a crisp value. In such cases, the use of type-2 fuzzy sets were introduced as a preferable approach in the literature by Zadeh in 1975 as an extension of type-1 fuzzy sets. Mendel, (2000) stated that fuzzy logic systems that are described with at least one type-2 fuzzy set are named as type-2 fuzzy logic systems. The major contribution of type-2 fuzzy logic systems is the footprint of uncertainty (FOU) that models the uncertainties/nonlinearities in a secondary
membership function as the second degree of freedom. Nevertheless, the main obscure with type-2 fuzzy logic system is the computational complexity in the type reduction operations (Liang and Mendel, 2000; Wu and Mendel, 2002; Mendel et al., 2006). Herewith, to cope with this obscure, Liang et al. (2000) and Mendel (2000) proposed a special type of type-2 fuzzy sets called interval type-2 fuzzy sets (IT2-FSs). IT2-FSs have attracted much research interest in recent years due to their ability to cope with uncertainty and robustness in comparison with ordinary T1FLSs. Several control and engineering applications such as liquid-level process control (Wu and Wantan, 2006); autonomous mobile robots (Castillo et al., 2009); prediction of air pollutant (Zarandi et al., 2012); pH control (Kumbasar, 2011); control and the identification of a real-time servo system (Kayacan et al., 2011) and face recognition (Castillo et al, 2009) illustrate the advantages of interval type-2 fuzzy sets (IT2FS). Studies in the literature show that the interval type-2 fuzzy logic controllers (IT2-FLCs) are generally more robust than type-1 fuzzy logic controllers (T1-(IT2-FLCs) (Hagras, 2004; Wu, 2012).
Fuzzy inference systems are universal approximators. This property means that FISs are capable of approximating any continuous function into a compact domain with a certain level of accuracy. The universal approximation property of the fuzzy models is not the only remarkable property. Fuzzy models add a new dimension to the information that can be extracted from the model. The new dimension is the linguistic dimension, which provides intuitive (linguistic) descriptions over the behavior of the modeled system.
Fuzzy models can be dynamic or static. Different types of fuzzy models have been proposed in the literature. Perhaps the most used are the rulebased fuzzy system (Zadeh, 1973; Driankov et al., 1993). These models are characterized by having fuzzy propositions as antecedents and consequences of the rules (Mamdani models). Another important type of models are the Takagi–Sugeno fuzzy models (Takagi and Sugeno, 1985), where the consequences of the rules are crisp functions of the antecedents.
After the model structures were proposed, many models were developed based on “pure” empirical knowledge, although in many applications this proved to be insufficient and not very efficient because most of the quantitative information was not used. Several data-driven techniques are mentioned in the literature. Some of
them attempt to tune the parameters of the fuzzy systems once the structure was selected (Wang, 1995; Jang and Sun, 1995; Jager, 1995; Sugeno and Yasukawa, 1993); others try to use the data to tune not only the parameters but also the structure (Yager and Filev, 1994; Lori and Branco, 1995; Jang, 1994; Tan and Vandewalle, 1995).
2. TYPE-1 FUZZY LOGIC SYSTEMS
2.1 Introduction
The fuzzy theory was first introduced into the scientific literature in 1965 by Professor Lotfi A. Zadeh at the University of California at Berkeley who proposed a set theory that operated over the range [0,1]. He published a paper titled “Fuzzy Sets” in the journal Information and Control (Zadeh, 1965). While Boolean logic results are restricted to 0 and 1, fuzzy logic results are between 0 and 1. In other words, fuzzy logic defines some intermediate values between sharp evaluations like absolute true and absolute false. This means that fuzzy sets can handle some concepts that we commonly meet in daily life, like “very old”, “old”, “young”, “very young”. Fuzzy logic is more like human thinking because it is based on degrees of truth and uses linguistic variables.
2.2 Type-1 Fuzzy Sets and Memberships
Fuzzy logic begins with the concept of a fuzzy set. A fuzzy set is a set that does not have a crisp boundary. A fuzzy set can contain components with only a partial degree of membership. Let X be the universe of discourse, whose elements are denoted as x. A fuzzy set A in X may be defined as follows:
{( ( )) | (2.1)
where ( ) is the MF of x in , which represents the degree of the x belongs to . The MF ( ( )) maps each element in X to a continuous unit interval [0,1]. If ( ) , it means x is surely not an element of fuzzy set and if ( ) , it indicates that x falls in the fuzzy boundary of fuzzy set . It is clear that a fuzzy set is an extension of a classical crisp set by generalizing the range of the characteristic function from the crisp numbers 0, 1 to the unit interval [0,1].
MF states that values assigned to the elements of the universal set, X, fall within a specified range. At the same time, it also represents the membership grade of these
elements in fuzzy set . In literature, there are three prevailingly used membership functions that are triangular, trapezoidal and Gaussian MFs.
2.2.1 Triangular membership function
A type-1 triangular membership function is specified by three parameters {a, b, c} where a is the left endpoint, b is the central point, and c is the right endpoint as follows: ( ) ( ) { (2.2)
The parameters a,b,c (where a < b < c) determine the x coordinates of the three corners of the underlying triangular membership function as it can be seen in Figure 2.1.
Figure 2.1: Triangle membership function.
2.2.2 Trapezoidal membership function
A type-1 trapezoidal membership function is specified by four parameters {a,b,c,d} as follows:
( ) ( ) { (2.3)
The parameters a,b,c,d (where a < b < c < d) determine the x coordinates of the four corners of the underlying trapezoidal membership function as it can be seen in Figure 2.2.
Figure 2.2: Trapezoidal membership function.
2.2.3 Gaussian membership function
A type-1 Gaussian membership function is specified by two parameters { }: ( ) ( ) ( )
(2.4) A Gaussian membership function is determined completely by c and ; c represents the membership functions center and determines membership functions width as can be seen in Figure 2.3.
Figure 2.3: Gaussian membership function. 2.3 Fuzzy Inference System (FIS)
In this section, fuzzy inference system (FIS) which was first introduced by Zadeh is depicted. The main goal of a FIS is to incorporate human’s knowledge into a set of fuzzy IF-THEN rules. A classic structure of a FIS consists of four components:
The "fuzzification" block converts the crisp inputs to fuzzy sets,
the "rule base" contains a selection of fuzzy rules,
the "inference mechanism" uses the fuzzy rules in the rule base to produce fuzzy conclusions
the "defuzzification" block converts the fuzzy conclusions into the crisp outputs. The basic structure of a T1FLS can be seen in Figure 2.4.
2.3.1 Fuzzification
The fuzzification block is responsible for converting crisp values into grades of membership for linguistic terms of fuzzy sets. Each linguistic term is associated with a MF by a membership grade.
2.3.2 Rule base
The fuzzy rule base is built by a gathering of fuzzy IF-THEN rules. The rule structure of T1FLS with p inputs ( ) and one output is as follows:
(2.5)
where n = 1, ..., M, M is the number of rules and A1, ..., Ap are the values for each
input linguistic variables in the universes of discourse. This rule represents a relation between the input space , and the output space, Y, of the fuzzy logic system.
2.3.3 Inference mechanism
Once inputs are fuzzified, it is known that the degree to which each part of the antecedent is satisfied for each rule. If the antecedent of a given rule has more than one part, the fuzzy operator is applied to obtain one number that represents the result of the antecedent for that rule. The input for the implication process is a single number given by the antecedent, and the output is a fuzzy set. Implication is implemented for each rule. Since decisions are based on the testing of all of the rules in a FIS, the rules must be combined in some manner in order to make a decision. Aggregation is the process by which the fuzzy sets that represent the outputs of each rule are combined into a single fuzzy set. Aggregation only occurs once for each output variable, just prior to the final step, defuzzification.
2.3.4 Defuzzification
Defuzzification, is a process which is performed by defuzzifier, refers to the way a crisp value is extracted from a fuzzy set as a representative value, which is a
necessary step as in often case a crisp number is required for real application. There are three most commonly used methods for defuzzifying a fuzzy set that are the center of area (COA), the mean of maximum (MOM) and height in generating the crisp system output. A brief explanation of these defuzzification methods are as following:
Center of area: The center of the fuzzy output area is obtained by this method. The crisp system output for discrete universe of discourse is calculated as
∑∑ ( ) ( )
(2.6)
where n is the number of the discrete elements in the universe of discourse, xi is
the value of the discrete element and ( ) represents the corresponding MF value at xi. For continuous universe of discourse, the crisp output can be
calculated via using integral instead of sum operator.
Mean of maximum: Let m be the number of the output points, whose MF values reach the maximum value within the universe of discourse. The mean value of all the m output points is obtaned by using MOM defuzzification method. In the case of a discrete universe, the crisp output is expressed as
∑ (2.7)
where xi is the support value at these points, whose MF reaches the maximum
value ( ). The MOM method does not consider the shape of the fuzzy output, but the defuzzification calculation is relatively simple.
Height: This method takes the peak value of each consequence and makes a weighted sum of these peak values, where the weights are the degree of membership of the fired rule. The method is equal to the center of area (COA) when the consequence membership functions are singletons.
3. INTERVAL TYPE-2 FUZZY LOGIC SYSTEMS
3.1 Introduction
The existence of uncertainties and the lack of information in many real world problems makes it difficult to model some real-world problems. Uncertainty in fuzzy set theory has been recognised and divided into three types (Klir and Wierman, 1998) which are:
Fuzziness (or vagueness): This results from the imprecise boundaries of fuzzy sets.
Non-specificity (or imprecision): It is linked to relevant sets of alternatives when some alternative belongs to a specific set of alternatives but we do not know which one in the set it is.
Strife (or discord): which expresses conflicts among the various sets of alternatives.
Type-1 fuzzy logic has been used successfully in a wide range of problems such as control system design, decision making, classification, system modeling and information retrieval (Ross, 2004; Hoffmann, 2001). However, type-1 approach is not able to directly model uncertainties and minimise its effects (Mendel and John, 2002). These uncertainties exist in a large number of real world applications. It can be a result of uncertainty in inputs, uncertainty in outputs, uncertainty that is related to the linguistic differences, uncertainty caused by the change of conditions in the operation and uncertainty associated with the noisy data when training the fuzzy logic system (Mendel, 2001). All these uncertainties translate into uncertainties about fuzzy sets membership functions (Mendel and John, 2002). Therefore, existence of uncertainties in the majority of real world applications makes the use of type-1 fuzzy logic inappropriate in many cases especially with problems related to inefficiency of performance in fuzzy logic control (Hagras, 2007). Also, it is arguable weather human brain uses crisp images of membership functions (Zimmermann, 2001). Problems related to modelling uncertainty using crisp membership functions of
type-of fuzzy sets called type-n fuzzy sets including type-2 fuzzy sets (Mendel, 2003; Zimmermann, 2001). Type-2 fuzzy logic systems might have many advantages compared with type-1 fuzzy logic systems.
These advantages include (Hagras, 2007):
Type-2 fuzzy set can handle numerical and linguistic uncertainties because its membership function is fuzzy and has a footprint of uncertainty (FOU) while type-1 fuzzy sets membership function is precise.
Using type-2 fuzzy sets to represent inputs and outputs results in using less rules compared with using type-1 fuzzy sets as a result of the wider coverage obtained by footprint of uncertainty (FOU).
Type-2 fuzzy sets embed a large number of type-1 fuzzy sets to describe variables with a detailed description adding extra levels of smooth control surface and response.
The extra dimension provided by the FOU enables a type-2 fuzzy logic system to produce outputs that cannot be achieved by type-1 fuzzy logic system using the same number of membership functions (Wu and Tan, 2005a).
Two factors should be considered regarding the widespread perception that general type-2 fuzzy logic system should outperform the interval form which also should outperform type-1 fuzzy logic system (Wagner and Hagras, 2010). These two factors are the dependence of performance on the choice of the model parameters as well as on the variability of uncertainty within the application (Wagner and Hagras, 2010). Therefore, a good choice of the model's parameters using automated methods is desired to get more clearer conclusions regarding this comparison. Type 2 fuzzy logic is a growing research area with much evidence of successful applications (John and Coupland, 2007; Mendel, 2007).
3.2 Type-2 Fuzzy Logic Systems
Type-2 fuzzy sets were introduced by Zadeh in 1975 as an extension of type-1 fuzzy sets. Mendel and Karnik have developed the theory of type-2 fuzzy sets further in (Mendel et al., 2006). The theoretical background of interval type-2 fuzzy system and its design principles are described in (Liang et al, 2000). T2FLSs appear to be a more promising method than their type-1 counterparts for handling uncertainties
such as noisy data and changing environments (Wu and Wantan, 2006; Castillo, 2009). The effects of the measurement noise in type and type-2 FLCs (T2FLCs) and identifiers are simulated to perform a comparative analysis (Zarandi et al., 2012; Kumbasar, 2011). It is concluded that the use of T2FLCs in real world applications which exhibit measurement noise and modeling uncertainties can be a better option than type-1 FLCs (T1FLCs).
There are numerous methods by which to represent T2 fuzzy sets and perform the type-reduction phase for a T2FLSs. Such methods include approaches like the Karnik-Mendel (KM) algorithms, enhanced KM algorithms, z-slice, α-plane method, wavy slice, point-valued, horizontal slice, and centroid type-reduction (Wu and Mendel, 2009; Wagner and Hagras, 2008; Mendel, Liu, and Zhai, 2009; Mendel and Liu, 2008; Liu, 2007; Nie and Wan Tan, 2008; Wu and Wan Tan, 2005). Despite the improvements to speed achieved by using one of the above mentioned type-reducers, T2FLSs still possess a great deal of mathematical, theoretical, and computational complexity. When discussing the complexities of T2FL, Mendel and John (2002) make the following observations stating that, Type-2 fuzzy sets are difficult to understand and use because:
The three-dimensional nature of type-2 fuzzy sets makes them very difficult to draw and visualize.
There is no simple collection of well-defined terms that let us effectively communicate about type-2 fuzzy sets, and to then be mathematically precise about them (terms do exist but have not been precisely defined).
Derivations for (set theory) formulas for the union, intersection, and complement of type-2 fuzzy sets all rely on using Zadeh’s Extension Principle (Zadeh, 1975).
Using 2 fuzzy sets is computationally more complicated than using type-1 fuzzy sets.
Therefore, T2FS has two membership functions with the names of ‘primary membership function’ and ‘secondary membership function’. For T2FS, secondary membership function can be different types such as crisp, triangle and so on. There is a special type of T2FS called interval type-2 fuzzy set (IT2FS) which is proposed in (Liang and Mendel, 2000). The MF of an interval type-2 fuzzy set has the secondary membership function which is an interval set to 1 as shown in Figure 3.1.b. If Figure
3.1.c is the secondary membership function, then the fuzzy set is generalized type-2 fuzzy set. An "upper membership function (UMF)" and a "lower membership function (LMF)" are two type-1 membership functions that are the bounds for the FOU of a type-2 fuzzy set.
Figure 3.1 : a) Primary membership function of type-2 fuzzy set,
b) Interval secondary membership function, c) Triangular secondary membership function.
3.3 Interval Type-2 Fuzzy Logic Systems
A fuzzy logic system described with at least one type-2 fuzzy set called a type-2 fuzzy logic system (T2FLS). A block diagram of a T2FLS that is a special fuzzy logic system is given in Figure 3.2. Similar to a type-1 FLS, a type-2 FLS includes type-2 fuzzifier, rule-base, inference engine, and substitutes the defuzzifier by the output processor.
Figure 3.2 : Type-2 fuzzy logic system structure. 3.3.1 IF-THEN rules, inference and defuzzification
The difference between type-1 and type-2 comes from the nature of MFs. The structure of the rules remains the same in type-2 case, but now some or all of the MFs are type-2. According to the rule-base formation, there are three possible structures for building such FLCs as classified in Table 3.1. The most common case called Model I and this structure has type-2 fuzzy sets in antecedents, and type-1 fuzzy sets in consequents. There are two special cases of Model I: Model II with crisp numbers (singletons) in the consequents; Model III type-1 fuzzy sets in antecedents and in consequents (Mendel, 2000).
Table 3.1 : Classification of type-2 FLSs. Consequent MF Antecedent MF
Type-1 FS Type-2 FS T0-FS (Crisp) T1-FLS Model II T1-FS (Interval) Model III Model I
The rule structure of type-2 fuzzy logic system with n inputs ( ) and one output can be defined as following:
̃ ̃ (3.1)
where N is the number of rules. This rule defines a type-2 relation between the input space , and the output space, , of the type-2 fuzzy logic system. is the consequent interval sets of the jth IF-THEN rule.
̃ [ ] ̃ [ ]
̃
(3.2)
where and are the firing sets of the lower membership function and upper membership function, respectively. The firing sets inherit the uncertainties of the antecedents and affect the interval consequent sets. The total firing set for each rule is defined as:
[ ] (3.3)
where
( ) ( ) ( )
( ) ( ) ( ) (3.4)
The consequent of the rule is also an interval set, defined as:
(3.5)
is the crisp output of type-2 fuzzy logic system (T2FLS). It inherits the uncertainty of the output of a type-2 FLS due to antecedent or consequent parameter uncertainties. and can be calculated using a type reduction method (Mendel, 2000, Mendel et al., 2006). Then, the type reduced fuzzy set for an interval type-2
( ) ( ) (3.6)
where is an interval type-1 fuzzy set determined by its two end points and . The output of the T2FLS can be obtained by using the average value of and . Therefore, the crisp output of T2FLS is calculated using
( ) ( ) ( ) (3.7)
3.3.2 Type reduction methods
There are more than ten different type reduction methods proposed for IT2-FLS in literature. In this subsection, the most used and efficient proposed type-reduction methods for IT2-FLS are explained. The most commonly used type reduction and defuzzification method is the iterative Karnik and Mendel algorithm (Karnik and Mendel, 1999) Furthermore, there are five approximations to the Karnik-Mendel method in order to decrease the computational cost and to represent the output in a closed loop form. Wu (2011) remarked two properties of the Karnik-Mendel type reduction method that are adaptiveness and inconsistency. Adaptiveness means that the bounds of the type-reduced interval set change as the inputs change. Inconsistency has the meaning that the upper and lower membership functions of the same type-2 fuzzy set may be used at the same time for computing the each bound of the type-reduced interval. These two properties are should be simultaneously captured. However, it has been stated that the alternative five approximations (Begian et al., 2008; Du and Ying, 2010; Nie and Tan, 2010; Wu and Tan, 2005b; Wu and Mendel, 2002) to the KM algorithms cannot simultaneously capture adaptiveness and inconsistency properties (Wu, 2011). These two properties provide the model to handle the uncertainties that may appear within systems much better. Because of this reason, the Karnik-Mendel type reduction method is preferred in this thesis. Two most common type reduction methods, The Karnik Mendel Algorithm and Wu Mendel Uncertainty Bound Method, are explained in details below.
3.3.2.1 Karnik-Mendel type reduction method
1. Sort ( ) in increasing order such that . Match the corresponding weights . (with their index corresponds to the renumbered ) 2. Initialize by setting: (3.8) Compute ∑ ∑ (3.9)
3. Find the switch point ( ) such that 4. Set { (3.10) and compare ∑∑ (3.11)
5. Check if . If not go to step 3 and set . If yes stop and set .
KM algorithm for computing :
1. Sort ( ) in increasing order such that . Match the corresponding weights . (with their index corresponds to the renumbered )
2. Initialize by setting:
(3.12)
∑
∑ (3.13)
3. Find the switch point ( ) such that 4. Set { (3.14) and compare ∑ ∑ (3.15)
5. Check if . If not go to step 3 and set . If yes stop and set .
3.3.2.2 Wu-Mendel uncertainty bound method
In this type-reduction mechanism proposed by (Wu and Mendel, 2002), the inner and outer bound sets for the type-reduced set of an IT2-FLS is derived. The bound sets estimates the uncertainty contained in the output of the IT2-FLS.
The UB type-reducer computes the type reduced set as follows: (3.16) (3.17) where ( ) ( ) (3.18) ( ) ( ) (3.19) ∑ ( ) ∑ ∑ ∑ ( ) ∑ ( ) ∑ ( ) ∑ ( ) (3.20)
∑ ( ) ∑ ∑ ∑ ( ) ∑ ( ) ∑ ( ) ∑ ( ) (3.21) ( ) ∑ ∑ ( ) ∑ ∑ ( ) ∑ ∑ ( ) ∑ ∑ (3.22)
Unlike the KM algorithms, the uncertainty bound method does not require { } and { } to be sorted, though it still needs to identify the minimum and maximum of { } and { }.
4. FUZZY MODELING AND LEARNING
4.1 Introduction
In this chapter, fuzzy modeling of complex, nonlinear and partially uncertain systems is emphasized. During the chapter, primarily fuzzy modeling concept and later on fuzzy modeling methods is explained.
Many disciplines in science and engineering in the creation of mathematical models of real systems pose an important issue. Since mathematical models are used in control systems in simulations, analysis and control of the dynamic model of the system, it is essential to coincide with nearly all of the dynamics of the system. If the model is not sufficiently correct, the analysis, estimation and control procedures are impossible to succeed. There is an inverse relationship between the accuracy of the model and the complexity of the system. Hence, when the system becomes more complex, less accurate model is obtained. Models must be free from unnecessary details of the system and the system should be able to be expressed as much as possible by the model. The system model can not be expressed exactly and can not fulfill the purpose of the model if the model of the system is too simple. On the other hand, if the model is too complicated, it loses the usability.
In the industry, production techniques are being developed everyday and modeling of processes has gained importance. Most systems in the industry can not be modeled well, since they can not be analyzed by conventional techniques. This is because of the nonlinear characteristics of the actual system and the time dependencies as well as their high uncertainties.
Before starting to explain details about fuzzy modeling and regression, that is nice to give a literature review about fuzzy modeling and learning subject.
Many approaches of fuzzy systems have been proposed and researched in the framework of soft computing. These approaches have been proposed because of the lack of learning capabilities of the fuzzy systems (Alcala et al., 1999). Fuzzy systems
are good at explaining how they reached a decision but can not automatically acquire the rules or membership functions to make a decision (Goonatilake and Khebbal, 1995). On the other hand, learning methods such as neural networks can not explain how a decision was reached but have a good learning capability while hybridisation overcomes the limitations of each method in one approach such as neurofuzzy systems or genetic fuzzy systems (Goonatilake and Khebbal, 1995). Two of the most known approaches for adding learning capability to fuzzy systems are genetic algorithms and neural networks. Genetic algorithms techniques are called genetic fuzzy systems when hybridised with fuzzy systems (Hoffmann, 2001; Herrera, 2005; Kim and Kim, 2002) while neural networks techniques are known as neuro-fuzzy systems (Jang and Sun, 1995; Horikawa et al., 1992). In fact, genetic algorithms are one of the most common search algorithms used with fuzzy systems that has been applied to a wide range of problems such as control system design, decision making, optimisation classification, modelling and information retrieval (Hoffmann, 2001). There are a wide range of other methods used to design fuzzy systems from other optimisation techniques to ad-hoc methods where the learning of fuzzy rules are guided by covering criteria of the data in the training set (Alcala et al., 1999). In 1992 and 1993, three articles appeared in the literature about designing fuzzy logic systems parameters by using the numerical training data rather than using fixed parameters chosen by the designer arbitrarily (Mendel, 2001). These works are Wang-Mendel algorithm (Wang and Mendel, 1992), a neuro-fuzzy system called ANFIS (Jang, 1993) and fuzzy neural networks with the back-propagation algorithm (Horikawa et al., 1992). All these works reported above were based on using type-1 fuzzy sets to build type-1 fuzzy logic systems. Many other methods used to learn fuzzy logic systems including local search algorithms such as gradient decent (Musikasuwan et al., 2004) and classical learning methods such as least-squares method (Mendel, 2001). The next section will present some attempts to design type-2 fuzzy logic systems.
The advancement on research in type-2 fuzzy sets and systems encouraged many researchers to apply some learning methods to type-2 fuzzy logic systems. For example:
Type-2 fuzzy logic controllers for autonomous robots were evolved using genetic algorithms (Wagner and Hagras, 2007).
Type-2 fuzzy logic controllers used for a coupled-tank liquid-level control systemand evolved using genetic algorithms (Wu and Wan Tan, 2006).
A type-2 fuzzy system was trained by particle swarm algorithm to estimate the blood pressure mean (Al-Jaafreh and Al-Jumaily, 2007).
Type-2 fuzzy logic systems were designed using orthogonal least-squares and backpropagation (Mendez and de los Angeles Hernandez, 2009).
Type-2 fuzzy logic systems were tuned using gradient descent to predict the Mackey-Glass time series with various levels of added noise (Musikasuwan et al., 2004).
Type-2 fuzzy logic systems were designed using back-propagation method to predict the Mackey-Glass time series with added noise (Mendel, 2001).
Type-2 fuzzy logic systems were tuned using genetic algorithms to design a trajectory tracking controller (Martinez et al., 2009).
There are many other research using some kind of learning or tuning to type-2 fuzzy logic systems reported in the literature. In some other experiments, some manual tuning and settings of interval or general type-2 fuzzy logic systems were used as reported by (Coupland et al., 2006; Sepulveda et al., 2007). It is indicated from above works and many others that interval type-2 fuzzy logic system can add more abilities to handle the uncertainties than type-1 fuzzy logic system.
4.2 White-Box, Black-Box and Gray-Box Modeling
Traditionally, modeling is thought to understand and express the nature and mathematical behavior of the system. This approach (physical, mechanical) is called white-box modeling. However, to create a model for a system which is complex, difficult to understand and has uncertainties is difficult in practice. On the other hand, even if such a model is achieved, this model can be much more difficult to understand. In addition, to understand the inner structure of the systems are in practice can be time consuming or very costly. Even sometimes, such a model of the system may be impossible to acquire.
For some systems, even the inner structure of the system is determined, determining the parameter exactly is another problem to overcome. At this point, the data obtained from the system through the estimation of system parameters which can be
