The Development of Neural Network Based System Identification ...

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86 CHAPTER 4 NEURAL NETWORK BASED SYSTEM IDENTIFICATION ( H ) ∑ ŷ i (t) = g i W 2 ih V h (t) + B2 i h=1 + g i ⎛ ⎝ ⎞ m∑ W 3 ij X j (t) ⎠ for i = 1, 2, 3 · · · n (4.7) j=1 where m, n and H denote the number of inputs, outputs and hidden nodes of the HMLP network respectively, B1 h and B2 i is the bias elements of input and output layer and X j (t) denotes the inputs data fed into the HMLP network. The weight matrix that connects the input layer to the hidden layer is given by W 1 hj and W 2 ih indicates weight matrix that connect the hidden layer to the output layer. The linear connections of HMLP that connect input and output layer are represented by W 3 ij . The function g h and g i are the activation function used in the hidden and output layer similar to MLP architecture. In our work, the hyperbolic tangent and linear activation function are used in the hidden and output layer respectively. The reason to choose hyperbolic tangent as activation function in neural network model training is because of its superior learning speed and accuracy compared with other activation functions such as Bipolar Sigmoid, Unipolar Sigmoid, Conic Section and Radial Basis Function [Karlik and Olgac, 2010]. Both MLP and HMLP networks can be used for prediction or forecasting time series by presenting inputs to the network with lags of variables to be predicted. This would create a set of input variables with delays being fed to the network input layer, which in turn serve as a short term memory built into network’s weights. This modelling method using feed-forward network with time lagged inputs is also known as Neural Network based Autoregressive structure with extra inputs (NNARX) model and its formulation and representation are given in Section 4.3.2. 4.2.3 Elman Network The Elman network is a simple dynamically driven recurrent network which attempts to capture long term history in measurement data [Samarasinghe, 2007]. Elman network was first introduced by Elman [1990] and it consists of a feed-forward network where hidden neuron signals are copied to a context/memory units and fed back into the network in the next time step. The Elman network realised the long term history of data through its internal memory without relying on externally provided memory as in NNARX network. Using this recurrence loop concept, the network can develop
4.2 THE ARTIFICIAL NEURAL NETWORKS 87 memories of the past activation of the hidden units. The Elman network offers benefits over NNARX network in such a way that the regression structure and memory dynamics are determined by the network itself [Samarasinghe, 2007]. Thus, it is not necessary to provide the time lagged inputs to the network as in NNARX network. Figure 4.6(a) shows the basic Elman network which has similarity to typical feedforward MLP network in term of input, hidden and output layers. The external inputs, X j from the measurements are fed to the input layer and propagated forward to the hidden neurons. The outputs of from the hidden neurons, v h are then sent forward to the output layers to produce the network output predictions, y i at the next time step t + 1. The Elman network also consists of extra processing units called context units. The context units receive and store the output signals from the hidden neurons and with a one step delay [Pham and Liu, 1993]. At the next time step, the outputs of context units, x k would send the delayed hidden neuron outputs to the hidden neurons. Note that solid lines indicate weights that are allows to adapt according to an optimisation routine, while dashed lines denotes recurrent connections from hidden neurons to context units which are not modifiable and usually their strengths are fixed as 1. Findings from Pham and Liu [1993] have suggested that the basic Elman network was found unable to identify higher order linear or non-linear dynamic system due to insufficient memory in the network. Pham and Liu [1996] proposed a modified version of Elman network to increase the memory capabilities of the basic Elman network though the use of self-connections in the context units. Figure 4.6(b) shows that the modified version of basic Elman network with self-connections in the context units indicate by scalar value α. The gradient calculation of the modified Elman network has similarity in structure to gradient calculation from dynamic back-propagation (DBP) or back-propagation through time (BPTT) algorithm [Pham and Liu, 1996]. This algorithm provides a dynamic trace of gradients in parameter space and would enable the network to model dynamics system of higher order. In this work, the modified version of the basic Elman network is used for system identification of helicopter dynamics system. Further modification has been made to the modified Elman network proposed in Pham and Liu [1993] by reducing the number of
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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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5.3 OFF-LINE BASED SYSTEM IDENTIFIC
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5.6 ON-LINE SYSTEM IDENTIFICATION 1
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5.6 ON-LINE SYSTEM IDENTIFICATION 1
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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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