The Development of Neural Network Based System Identification ...

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164 CHAPTER 6 NEURAL NETWORK BASED PREDICTIVE CONTROL SYSTEM 6.4 NON-MINIMAL STATE SPACE MODEL REALISATION There are generally three types of models used in MPC design such as finite impulse response (FIR)/step response models, transfer function models and state space models [Rossiter, 2003]. The state space model is preferred over other model types in MPC controller design due to its simplicity and effective handling of multi-variable problems. The transfer function models obtained from instantaneous linearisation of the NN model can also be represented in terms of state space model realisation. This can be done by reformulating the state variables of the state space model to be identical to the feedback variables that have been used in the ARX model [Wang, 2009b, Ordys and Clarke, 1993]. Consider the general discrete time ARX model that describes the input and output relationship as: A(q −1 )(1 − q −1 )y(k) = B(q −1 )∆u(k) + q −1 ɛ(t) (6.16) where ɛ(t) is the input disturbance which is assumed to be a sequence of integrated white noise, A(q −1 ) and B(q −1 ) are polynomials in the time shift operator given by the following forms: A(q −1 ) = 1 + a 1 q −1 + a 2 q −2 · · · + a n q −n B(q −1 ) = b 1 q −1 + b 2 q −2 · · · + b n q −n (6.17) Let the polynomial A(q −1 )(1 − q −1 ) be referred as: A(q −1 )(1 − q −1 ) = 1 + ā 1 q −1 + ā 2 q −2 + · · · + ā n+1 q −(n+1) (6.18) Next, by choosing the state variable vector as: x(k) = [y(k) y(k − 1) · · · y(k − n − 1) ∆u(k − 1) ∆u(k − 2) · · · ∆u(k − n)] T (6.19)
6.4 NON-MINIMAL STATE SPACE MODEL REALISATION 165 the corresponding state space model with non-minimal realisation is formulated as: x m (k + 1) = A m x m (k) + B m ∆u(k) + B d ɛ(k) (6.20) y(k) = C m x m (k) (6.21) where A m denotes a square matrix of (2n + 1) × (2n + 1) size, B m is a matrix of size (2n + 1) × 1 and C m is a matrix of 1 × (2n + 1) size. The detailed formulation of matrix A m , B m and C m are given as follows: ⎡ ⎤ ⎡ ⎤ −ā 1 −ā 2 · · · −ā n−1 −ā n b 2 · · · b n−1 b n b 1 1 0 · · · 0 0 0 · · · 0 0 0 0 1 · · · 0 0 0 · · · 0 0 0 . . . .. . . · · · . . . . A m = 0 0 · · · 1 0 0 · · · 0 0 ; B m = 0 0 0 · · · 0 0 0 · · · 0 0 1 0 0 · · · 0 0 1 · · · 0 0 0 . ⎢ . . .. . . . .. . .. . . ⎥ ⎢ . ⎥ ⎣ ⎦ ⎣ ⎦ 0 0 · · · 0 0 0 · · · 1 0 0 [ ] [ C m = 1 0 0 · · · 0 0 0 0 ; B d = 1 0 0 · · · 0 0 0 0 ] T
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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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Chapter 4 NEURAL NETWORK BASED SYST
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4.2 THE ARTIFICIAL NEURAL NETWORKS
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4.3 SYSTEM IDENTIFICATION WITH NEUR
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Page 149 and 150: 4.4 SUMMARY 117 Segment 1 Segment 2
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Page 183 and 184: 5.8 SUMMARY 151 methods such as wei
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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
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Page 229 and 230: 7.3 EXPERIMENTAL RESULTS 197 indica
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Page 233 and 234: 7.3 EXPERIMENTAL RESULTS 201 Yaw Ra
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Page 242 and 243: 210 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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