The Development of Neural Network Based System Identification ...

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98 CHAPTER 4 NEURAL NETWORK BASED SYSTEM IDENTIFICATION and outputs of the neural network are presented by m and n respectively. Similarly, the NNARX model can also be represented using HMLP network by introducing the lagged time input variables into the neural network model. The Elman network with modification in the network’s internal dynamics is found capable of identifying an n th order discrete dynamic system in Equation (4.23), through empirical findings in Pham and Liu [1993]. Furthermore, Pham and Liu [1996] suggested that in order to model the dynamic system in Equation (4.23) from experimental data using MLP network, a n y + n u input (regressor) nodes are needed to be included in the network. However, if an Elman network is used to model a NNARX model, only current measurement data are needed to be fed into the input nodes as shown in Figure 4.9(c). Therefore, the modified Elman network is significantly smaller in size compared with MLP or HMLP network when large time lags (model order) are used. 4.3.2.1 Lag Space Selection for Feed-Forward MLP or HMLP Network After deciding the input parameters to be used in the model, the number of past inputs and outputs fed into the MLP or HMLP neural networks were decided based on the calculation of the Lipschitz coefficient given in Norgaard [2000]. Using this coefficient, it is possible to determine the proper lag space via experimental data. The sizes of output and input time regression vectors depend on the degree of non-linearity of the Lipschitz coefficients where an insufficient number of regressors will result in high Lipschitz coefficients and small numerical values for extra regressors. The Lipschitz coefficient is calculated using the following formula for each input u i (t) and output y j (t) pairs: q ij = y (t i ) − y (t j ) ∣ϕ (t i ) − ϕ (t j ) ∣ i ≠ j (4.27) The outline of Lipschitz coefficient approach is given as follows: 1. Determine the Lipschitz quotients for all combination of input-output pairs using (4.27) for a given choice of number of past outputs and inputs.
4.3 SYSTEM IDENTIFICATION WITH NEURAL NETWORK 99 2. Select the p largest quotients with p selected as p = 0.01N ∼ 0.02N. 3. Calculate criterion according to: ( p∏ ¯q (n) ij = k=1 √ (n) nq ij (k) ) 1 p where n = n y + n u 4. Repeat step (1)-(3) for different lag structures. 5. Plot the calculated criterion as a function of number of past outputs and past inputs (lag space). 4.3.3 Off-line and Recursive Methods The system identification for inferring the helicopter dynamic model can be conducted using off-line (batch) and recursive based system identification methods. The estimation of a dynamic model in off-line neural network identification method involves the training process being carried out over some finite data gathered beforehand. Over the whole length of the data record, we determine the best weights (parameters vector θ) that give the best fit for the measurement data over repetitive iterations. Obviously, the off-line methods have a disadvantage such as their unsuitability for tracking time varying dynamics, as the amount of computation time for the training phase in each iteration might exceed the available processing time [Norgaard, 2000]. The adaptive control is an example where a model needs to be identified at the same time as a control law is calculated to compensate the time varying control variables. Even though the off-line methods are not suitable for real-time implementation, several researchers have used a mini-batch data size for off-line method implementation in real-time as proposed in Samal [2009], Puttige and Anavatti [2006], Puttige [2009]. However, in these examples, the NN estimation and control were restricted only for SISO case and smaller network due to limited computation capabilities to invert big Hessian matrix at each iteration. To overcome the disadvantages of off-line methods, the recursive based system identification methods can be used in tracking time varying dynamics. The recursive
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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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5.7 MODEL PERFORMANCE COMPARISON US
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5.8 SUMMARY 151 methods such as wei
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5.8 SUMMARY 153 would enable the im
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156 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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