The Development of Neural Network Based System Identification ...

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124 CHAPTER 5 NN BASED SYSTEM IDENTIFICATION: RESULTS AND DISCUSSION in Figure 5.3(c) shows that the weights gradually increase and remain constant beyond iteration point i = 185. The network training ends much quicker compared with the network training without regularisation term since the weights do not change after i = 185. The MSE calculation on the validation data set also shows that the optimum weights are found at i = 185, thus demonstrating the effectiveness of the regularisation method to prevent the over-fitting problem. The effect of using the regularisation term basically introduces a smoothing effect on the error criterion V N (θ, Z N ) in such a way that weights that have less important influence on error are forced to decay towards zero [Samarasinghe, 2007]. In this process, only the important weights that minimise the error are allowed to grow and stabilise at their optimum values. The MSE result for network training with large regularisation parameter α = 0.8 is shown in Figure 5.3(f). The figure indicates that the network training stops much earlier than the network training with α = 0 and α = 0.0001. However, the network model has higher MSE values in both training and validation data. Further increase in regularisation parameter would make the weights less adaptable as the weights only grow in limited small magnitude (±1.5). This would make the network less flexible and would result in poor prediction performance due to severe bias. 5.2.2 Model Structure Selection Results The optimum model structure can be found using the Lipschitz coefficient, and it is possible to determine the proper lag space via experimental data [Norgaard, 2000]. The result of the Lipschitz coefficient calculation for a pair of input and output data (δ lat and p) is shown in Figure 5.4(a). It is shown that the Lipschitz coefficient curve decreases and stabilises at n y = 3 and n u = 1 for that particular pair of input and output data selection. For this input-output pair, the reasonable network structure to describe the attitude dynamic (p/δlat) is by selecting a number of past outputs n y = 3 and number of past inputs n u = 1. By summing all the stabilising points for each data pair, a total number of 8 time regressors are fed to the neural network. Finally, the selected neural network based on the ARX model (NNARX) structure to identify the non-linear relationship of helicopter’s attitude dynamics is shown in Figure 5.4(b).
5.2 OFF-LINE BASED SYSTEM IDENTIFICATION FOR MLP NETWORK 125 Lipschitz Coefficient for Lateral Cyclic, δ lat and Roll Rate, p 10 1 10 0 1 10 2 Number of past inputs, n u Order index 2 3 Number of past outputs, n y 4 5 6 7 7 6 5 4 3 2 1 (a) Past Outputs pt ( 3) pt ( 2) pt ( 1) qt ( 3) qt ( 2) qt ( 1) Past Inputs ( t lon 1) ( t 1) lat REGRESSION VECTOR NNARX Predicted Outputs ˆp t ˆq t (b) Figure 5.4 Preliminary NN model structure selection from experimental input-output data set using the Lipschitz coefficient (a) The Lipschitz coefficient plot obtained for a pair of input and output data; and (b) The NNARX model structure with preselected regression vectors obtained after determining each individual Lipschitz coefficient from respective input-output pair. The regression vector size can be selected using a much higher number of past outputs and inputs. However, the choice of higher number of past outputs and inputs will result in a larger network architecture that may lower the mean square error (MSE) but with poor generalisation ability [Billings et al., 1992]. This means that the network model would predict the training data set with great accuracy but fail to represent a new data that has not been used in training process. The method to determine the model order such as the Lipschitz coefficient is known to be sensitive to noise [Sragner and Horvath, 2003]. Normally, if the Lipschitz
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The Development of Neural Network B
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ABSTRACT This thesis presents the d
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vii the MLP network, the prediction
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PUBLICATIONS The following is a lis
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xiv TABLE OF CONTENTS CHAPTER 4 CHA
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LIST OF FIGURES 1.1 Examples of typ
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LIST OF FIGURES xix 3.13 Overview o
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LIST OF FIGURES xxi 5.2 The predict
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LIST OF FIGURES xxiii 5.13 The pred
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LIST OF FIGURES xxv 7.1 The locatio
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LIST OF TABLES 3.1 The specificatio
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NOMENCLATURE AFCS Automatic Flight
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LIST OF TABLES xxxi NNARX QP RBF RC
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2 CHAPTER 1 INTRODUCTION Surveillan
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4 CHAPTER 1 INTRODUCTION is then de
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6 CHAPTER 1 INTRODUCTION the dynami
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8 CHAPTER 1 INTRODUCTION models [Sa
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10 CHAPTER 1 INTRODUCTION improved
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12 CHAPTER 1 INTRODUCTION Chapter 3
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Chapter 2 LITERATURE REVIEW The pur
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2.1 INTRODUCTION 17 • Inefficient
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2.1 INTRODUCTION 19 Figure 2.2 The
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2.2 HELICOPTER DYNAMICS MODELLING A
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2.3 NEURAL NETWORK BASED SYSTEM IDE
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2.4 AUTOMATIC FLIGHT CONTROL SYSTEM
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2.5 SUMMARY 53 design based on the
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56 CHAPTER 3 AERIAL PLATFORM AND CU
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174 CHAPTER 6 NEURAL NETWORK BASED
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Chapter 7 FLIGHT CONTROL SYSTEM DES
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7.2 FLIGHT TESTS IMPLEMENTATION 185
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7.3 EXPERIMENTAL RESULTS 187 1 0.8
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7.3 EXPERIMENTAL RESULTS 189 Longit
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7.3 EXPERIMENTAL RESULTS 191 Pitch
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7.3 EXPERIMENTAL RESULTS 193 Flight
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7.3 EXPERIMENTAL RESULTS 195 140 Ya
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7.3 EXPERIMENTAL RESULTS 197 indica
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7.3 EXPERIMENTAL RESULTS 199 20 100
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7.3 EXPERIMENTAL RESULTS 201 Yaw Ra
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7.3 EXPERIMENTAL RESULTS 203 mainta
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7.3 EXPERIMENTAL RESULTS 205 Yaw Ra
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7.4 SUMMARY 207 data measured from
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210 CHAPTER 8 CONCLUSIONS AND FUTUR
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212 CHAPTER 8 CONCLUSIONS AND FUTUR
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214 REFERENCES B. W. Bequette. Non-
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216 REFERENCES Konstantinos Dalamag
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218 REFERENCES Wayne Johnson. Helic
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220 REFERENCES B. Mettler, Mark B.
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222 REFERENCES V. R. Puttige and S.
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224 REFERENCES D.H. Shim. Hierarchi
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226 REFERENCES Nikos Vitzilaios and
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