The Development of Neural Network Based System Identification ...

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114 CHAPTER 4 NEURAL NETWORK BASED SYSTEM IDENTIFICATION Initialize Parameters Vector, Forgetting factor, 0 and matrix P 0 Q 0 I ˆ 0 Update Training Data at time, Calculate Jacobian Matrix t | Calculate step (1) – (5) in Potter’s Square Root Algorithm Calculate prediction error e(t) and compute, tˆ t Update t ˆ t Qt End Condition END Figure 4.13 The recursive Gauss-Newton (rGN) algorithm with Potter’s square root factorisation. nearly singular if the model set contains too many parameters or if the input signal is not general enough [Ljung and Soderstrom, 1983]. This problem can be overcome by introducing a lower and upper bounds on the eigenvalues of P (t). Several variations of recursive Gauss-Newton algorithms such as Constant Trace (CT) and Exponential Forgetting and Resetting Algorithm (EFRA) have been proposed in various examples to overcome the unstable numerical P (t) recursion [Norgaard, 2000, Salgado et al., 1988]. By using the CT method, Step 6 in Table 4.1 is introduced to bound the eigenvalues of P (t). The ρ max and ρ min are the maximum and minimum eigenvalues respectively, and the values are selected so that ρ max /ρ min ≃ 10 5 . The initial Q(0) should be selected as
4.3 SYSTEM IDENTIFICATION WITH NEURAL NETWORK 115 a diagonal matrix, ρ min I ≤ Q(0) ≤ ρ max I. 4.3.6 Model Validation The model validation process is performed using the second data set Z V = [y V (t), u V (t)] that is different from the training data set. The validation results are based on three analyses: one-step ahead predictions, k-step ahead predictions and k-folds cross validation of the predictions. One-step ahead predictions are a simple plot that compares the actual measurement data with the model prediction over a test data set. The k-step ahead predictions are normally carried out to detect further deficiency in the fitted model since under high frequency sampling, a one-step ahead visual inspection usually gives a very small prediction error. The calculation example of k-step ahead prediction is shown in Figure 4.14. The k-step ahead prediction is calculated starting from the first step prediction using past output and input data. The predicted outputs from the first step are used as substitutes for the measured output data for the second step prediction since the actual system observations are not available in future predictions. This process continues until the final k-step ahead prediction is obtained. The k-step ahead prediction in a mathematical compact form is given in Norgaard [2000]. The overall accuracy of the prediction error can be represented in scalar quantity according to mean square error (MSE) criterion or root mean square error (RMSE), percentage RMSE and the R 2 criterion as in the following formulation [Suresh et al., 2003, Norgaard, 2000, Shamsudin and Chen, 2012a]: RMSE = √ 1 N N∑ (ŷ i (t) − y i (t)) 2 (4.59) t=1 %RMSE = √ ∑N t=1 (ŷ i(t) − y i (t)) 2 ∑ N t=1 (y i(t) − y i (t)) 2 (4.60) R 2 = 1 − ∑ N t=1 (ŷ i(t) − y i (t)) 2 ∑ N t=1 (y i(t) − y i (t)) 2 (4.61) Cross validation is a statistical method that is normally used in data mining problems to determine the model structure selection and to compare generalisation
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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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164 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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