Table of Content Abstract Ⅰ Table of Content Ⅱ List of Figures Ⅳ List of Tables Ⅶ ChapterPage Chapter 1 Introduction 1 1.1 Background 1 1.2 Motivation 2 1.3 Organization 3 Chapter 2 Notions of the Fuzzy Logic Controller4 2.1 The Basic Notion of the Ordinary Fuzzy Controller4 2.2 The Basic Structure of the PID-type Controller6 2.3 The Simulations of Comparison9 Chapter 3 A Single-input Fuzzy Logic Controller via the Tuning of Scaling Factors Based on Genetic Algorithms12 3.1 Introduction 12 3.2 Design of the Single-input FLC13 3.2-1 Problem Formulation 13 3.2-2 Design of S-FLC 15 3.3 The Performance-oriented Objective Function for Genetic Algorithms18 3.3-1 The Basic Concept of Genetic Algorithms18 3.3-2 The Scaling Factor Based on Genetic Algorithms 19 3.4 Simulation Results20 3.4-1 Example 1 20 3.4-2 Example 2 23 3.5 Conclusions 28 Chapter 4 Comparison of Adaptive Controllers Using Different Basis Functions29 4.1 Introduction 29 4.2 Basis Functions 30 4.2-1 Fuzzy Basis Functions 30 4.2-2 Gaussian Radial Basis Functions 31 4.2-3 Orthogonal Basis Functions 32 4.2-3-A Legendre Polynomials 33 4.2-3-B Laguerre Polynomials 33 4.2-3-C Hermitte Polynomials 34 4.2-3-D Tchebycheff Polynomials of the First Kind 35 4.2-3-E Tchebycheff Polynomials of the Second Kind 35 4.3 The Problem Formulation and the Adaptive Controller Design Using Different Basis Functions36 4.4 Simulation and Discussion39 4.5 Conclusions 51 Chapter 5 Conclusions and Recommendations 52 References 54 List of Figures Page Fig. 2.1 The basic structure of fuzzy controller4 Fig. 2.2 Four types of the fuzzy membership functions 5 Fig. 2.3 The fuzzy PD-type control system 6 Fig. 2.4 The fuzzy PI-type control system6 Fig. 2.5 Time domain7 Fig. 2.6 Phase plane8 Fig. 2.7 The fuzzy PID-type control system9 Fig. 2.8 The membership function of antecedent and consequent 9 Fig. 2.9 Comparison of the fuzzy PD-type and PID-type controller system 10 Fig. 2.10 Comparison of the fuzzy PI-type and PID-type controller system 10 Fig. 3.1 Rule table with infinitesimal quantization levels 15 Fig. 3.2 Derivation of a signed distance 16 Fig. 3.3 The algorithm structure of SGA 18 Fig. 3.4 The scaling factor on S-FLC used genetic algorithms 19 Fig. 3.5 The membership function of antecedent and consequent 20 Fig. 3.6 The response of of S-FLC21 Fig. 3.7 The response of of S-FLC22 Fig. 3.8 The response of of S-FLC based on genetic algorithms22 Fig. 3.9 The response of of S-FLC based on genetic algorithms23 Fig. 3.10 Structure of an inverted pendulum system 24 Fig. 3.11 Time response of of S-FLC25 Fig. 3.12 Time response of of S-FLC25 Fig. 3.13 Time response of of S-FLC based on genetic algorithms26 Fig. 3.14 Time response of of S-FLC based on genetic algorithms26 Fig. 3.15 The control input of S-FLC 27 Fig. 3.16 The control input of S-FLC based on genetic algorithms27 Fig. 4.1 Basic configuration of fuzzy systems 30 Fig. 4.2 Six fuzzy membership functions defined over the state space41 Fig. 4.3(a) The response of the different positive initial states using the fuzzy basis functions41 Fig. 4.3(b) The response of the different negative initial states using the fuzzy basis functions42 Fig. 4.4(a) The control inputs of different positive initial states using the fuzzy basis functions 42 Fig. 4.4(b) The control inputs of different negative initial states using the fuzzy basis functions 42 Fig. 4.5(a) The response of the different positive initial states using the radial basis functions43 Fig. 4.5(b) The response of the different negative initial states using the radial basis functions43 Fig. 4.6(a) The control inputs of different positive initial states using the radial basis functions 43 Fig. 4.6(b) The control inputs of different negative initial states using the radial basis functions 44 Fig. 4.7(a) The response of the different positive initial states using the Legendre polynomial functions44 Fig. 4.7(b) The response of the different negative initial states using the Legendre polynomial functions44 Fig. 4.8(a) The control inputs of the different positive initial states using the Legendre polynomial functions45 Fig. 4.8(b) The control inputs of the different negative initial states using the Legendre polynomial functions45 Fig. 4.9 The response of the different initial states using the Laguerre polynomial functions45 Fig. 4.10 The control inputs of the different initial states using the Laguerre polynomial functions46 Fig. 4.11(a) The response of the different positive initial states using the Hermite polynomial functions46 Fig. 4.11(b) The response of the different negative initial states using the Hermite polynomial functions46 Fig. 4.12(a) The control inputs of the different positive initial states using the Hermite polynomial functions47 Fig. 4.12(b) The control inputs of the different negative initial states using the Hermite polynomial functions47 Fig. 4.13(a) The response of the different positive initial states using the Tchebycheff polynomial of the first kind functions47 Fig. 4.13(b) The response of the different negative initial states using the Tchebycheff polynomial of the first kind functions48 Fig. 4.14(a) The response of the different positive initial states using the Tchebycheff polynomial of the second kind functions48 Fig. 4.14(b) The response of the different positive initial states using the Tchebycheff polynomial of the second kind functions48 Fig. 4.15 The response of the positive initial state using the Walsh functions 50 Fig. 4.16 The response of the positive initial using the Walsh functions 50 Fig. 4.17 The response of the negative initial state using the Walsh functions 50 Fig. 4.18 The response of the negative initial using the Walsh functions 51