The Development of Neural Network Based System Identification ...

More documents

Recommendations

Info

104 CHAPTER 4 NEURAL NETWORK BASED SYSTEM IDENTIFICATION minimum of the error criterion by iteratively solving the following update equation: [ −1 θ (i+1) = θ (i) − R (i) (θ) + λ I] (i) G (i) (θ) (4.34) where I is the identity matrix with a size equal to Hessian R (θ) matrix and the λ (i) constant is a damping factor used for deciding the step size. The value of λ (i) is selected to be λ (i) ≥ 0. The usage of the constant λ (i) can also be viewed as a blending factor between Gauss-Newton (GN) and Steepest Descent (SD) update, which provides LM algorithm with fast convergence speed of the GN algorithm and the stability of SD method [Yu and Wilamowski, 2011]. The largest update can be possibly achieved by choosing λ (i) = 0 which gives a full GN step. Shorter step length as in SD algorithm is achieved by taking λ (i) → ∞ which will cause the diagonal elements of R (i) (θ) to dominate [Ngia and Sjoberg, 2000]. In order to determine λ, the indirect method used in Norgaard [2000] and Fletcher [1981] is adopted by calculating the following ratio to determine the accuracy of approximation: r (i) = 2 [ ( V N θ (i) ) ( , Z N − VN θ (i) + f (i) )] , Z N (f (i)) T ( G θ (i)) ( + f (i)) T f (i) λ (i) } {{ } reduction approximation (4.35) The main purpose of introducing the ratio calculation is to measure how well the reduction of the criterion V N (θ, Z N ) matches the reduction predicted by approximation terms in denominator of ratio calculation in Equation (4.35). The damping factor λ is adjusted accordingly to the ratio r (i) by some factor [Norgaard, 2000]. The procedure for the LM algorithm using indirect method to determine λ is given in Figure 4.11. The reduction approximation (denominator term in Equation (4.35)) is most likely a close approximation to error criterion V N (θ, Z N ), if the ratio r (i) value is close to one and parameter λ should be reduced by some factor. However, if the ratio r (i) is small or a negative value, parameter λ should be increased. Additional stopping criteria are normally introduced to this algorithm to prevent minimisation problems or to force early stopping such as stopping criteria based on maximum number of iterations, sum of square error that drops below a certain threshold, upper bound for gradient and
4.3 SYSTEM IDENTIFICATION WITH NEURAL NETWORK 105 maximum weight change, maximum value of parameter λ or early stopping criterion due to training time constraint. Initialize Parameters Vector, Input matrix, N and 0 Damping factor, 0 Calculate Jacobian Matrix t | i1 i 2 Determine the search direction. i i i i R I f G i 1 i 2 i1 i Update Parameter Vector i1 i i f r i 0.25 Calculate Mean Square Error Criterion and i ratio, r r i 0.75 V k 1 N V MAX END Figure 4.11 The Levenberg-Marquardt (LM) algorithm with step involving λ determination. 4.3.4.1 Jacobian Matrix Calculation The calculation of Jacobian matrix ψ(t|θ) is an important step in Gradient or Newton based training algorithms. For the SISO case, the Jacobian matrix ψ(t|θ) is a d × N matrix where N denotes the number of samples in the training data set. The dimension
Page 1:
The Development of Neural Network B
Page 5 and 6:
ABSTRACT This thesis presents the d
Page 7:
vii the MLP network, the prediction
Page 11:
PUBLICATIONS The following is a lis
Page 14 and 15:
xiv TABLE OF CONTENTS CHAPTER 4 CHA
Page 17 and 18:
LIST OF FIGURES 1.1 Examples of typ
Page 19 and 20:
LIST OF FIGURES xix 3.13 Overview o
Page 21 and 22:
LIST OF FIGURES xxi 5.2 The predict
Page 23 and 24:
LIST OF FIGURES xxiii 5.13 The pred
Page 25 and 26:
LIST OF FIGURES xxv 7.1 The locatio
Page 27 and 28:
LIST OF TABLES 3.1 The specificatio
Page 29 and 30:
NOMENCLATURE AFCS Automatic Flight
Page 31:
LIST OF TABLES xxxi NNARX QP RBF RC
Page 34 and 35:
2 CHAPTER 1 INTRODUCTION Surveillan
Page 36 and 37:
4 CHAPTER 1 INTRODUCTION is then de
Page 38 and 39:
6 CHAPTER 1 INTRODUCTION the dynami
Page 40 and 41:
8 CHAPTER 1 INTRODUCTION models [Sa
Page 42 and 43:
10 CHAPTER 1 INTRODUCTION improved
Page 44 and 45:
12 CHAPTER 1 INTRODUCTION Chapter 3
Page 47 and 48:
Chapter 2 LITERATURE REVIEW The pur
Page 49 and 50:
2.1 INTRODUCTION 17 • Inefficient
Page 51 and 52:
2.1 INTRODUCTION 19 Figure 2.2 The
Page 53 and 54:
2.2 HELICOPTER DYNAMICS MODELLING A
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
2.3 NEURAL NETWORK BASED SYSTEM IDE
Page 61 and 62:
Page 63 and 64:
Page 65 and 66:
Page 67 and 68:
Page 69 and 70:
2.4 AUTOMATIC FLIGHT CONTROL SYSTEM
Page 71 and 72:
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
Page 81 and 82:
Page 83 and 84:
Page 85: 2.5 SUMMARY 53 design based on the
Page 88 and 89: 56 CHAPTER 3 AERIAL PLATFORM AND CU
Page 109 and 110: Chapter 4 NEURAL NETWORK BASED SYST
Page 111 and 112: 4.2 THE ARTIFICIAL NEURAL NETWORKS
Page 123 and 124: 4.3 SYSTEM IDENTIFICATION WITH NEUR
Page 135: 4.3 SYSTEM IDENTIFICATION WITH NEUR
Page 149 and 150: 4.4 SUMMARY 117 Segment 1 Segment 2
Page 151 and 152: Chapter 5 NN BASED SYSTEM IDENTIFIC
Page 153 and 154: 5.2 OFF-LINE BASED SYSTEM IDENTIFIC
Page 177 and 178: 5.6 ON-LINE SYSTEM IDENTIFICATION 1
Page 179 and 180: 5.6 ON-LINE SYSTEM IDENTIFICATION 1
Page 181 and 182: 5.7 MODEL PERFORMANCE COMPARISON US
Page 183 and 184: 5.8 SUMMARY 151 methods such as wei
Page 185: 5.8 SUMMARY 153 would enable the im
Page 188 and 189:
156 CHAPTER 6 NEURAL NETWORK BASED
Page 190 and 191:
Page 192 and 193:
Page 194 and 195:
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
Page 204 and 205:
Page 206 and 207:
Page 208 and 209:
Page 210 and 211:
Page 212 and 213:
Page 215 and 216:
Chapter 7 FLIGHT CONTROL SYSTEM DES
Page 217 and 218:
7.2 FLIGHT TESTS IMPLEMENTATION 185
Page 219 and 220:
7.3 EXPERIMENTAL RESULTS 187 1 0.8
Page 221 and 222:
7.3 EXPERIMENTAL RESULTS 189 Longit
Page 223 and 224:
7.3 EXPERIMENTAL RESULTS 191 Pitch
Page 225 and 226:
7.3 EXPERIMENTAL RESULTS 193 Flight
Page 227 and 228:
7.3 EXPERIMENTAL RESULTS 195 140 Ya
Page 229 and 230:
7.3 EXPERIMENTAL RESULTS 197 indica
Page 231 and 232:
7.3 EXPERIMENTAL RESULTS 199 20 100
Page 233 and 234:
7.3 EXPERIMENTAL RESULTS 201 Yaw Ra
Page 235 and 236:
7.3 EXPERIMENTAL RESULTS 203 mainta
Page 237 and 238:
7.3 EXPERIMENTAL RESULTS 205 Yaw Ra
Page 239:
7.4 SUMMARY 207 data measured from
Page 242 and 243:
210 CHAPTER 8 CONCLUSIONS AND FUTUR
Page 244 and 245:
212 CHAPTER 8 CONCLUSIONS AND FUTUR
Page 246 and 247:
214 REFERENCES B. W. Bequette. Non-
Page 248 and 249:
216 REFERENCES Konstantinos Dalamag
Page 250 and 251:
218 REFERENCES Wayne Johnson. Helic
Page 252 and 253:
220 REFERENCES B. Mettler, Mark B.
Page 254 and 255:
222 REFERENCES V. R. Puttige and S.
Page 256 and 257:
224 REFERENCES D.H. Shim. Hierarchi
Page 258 and 259:
226 REFERENCES Nikos Vitzilaios and
show all

The Development of Neural Network Based System Identification ...

Create successful ePaper yourself

Delete template?

Save as template?