The Development of Neural Network Based System Identification ...

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174 CHAPTER 6 NEURAL NETWORK BASED PREDICTIVE CONTROL SYSTEM input variable constraints, matrices C 1 and C 2 are defined as follows: ⎡ ⎤ ⎡ ⎤ I I 0 0 · · · 0 I I I 0 · · · 0 C 1 = I ; C 2 = I I I · · · 0 ⎢. ⎥ ⎢. ⎥ ⎣ ⎦ ⎣ ⎦ I I I I · · · 0 In the cost function (6.31), the matrix Φ T Φ + ¯R is also known as the Hessian matrix and is assumed to be positive definite. Since the cost function in Equation (6.31) is considered quadratic with linear inequality constraints, the problem of minimising the cost function becomes the equivalent problem for finding the optimal solution in Quadratic Programming (QP) study. The constraint equation in Equation (6.35) can be expressed in a compact form as: M∆U ≤ γ (6.36) where M is a matrix representing the constraints with its number of rows equal to the number of constraints and the number of column equal to the dimension of vector ∆U (if considering SISO case). When the system is fully imposed with all three types of constraints, the number of constraints are equal to D = (4 × m × N c ) + (2 × q × N p ), where the constant m is the number of inputs and the constant q is the number of outputs in the system. 6.8.1 The Hildreth’s Quadratic Programming Procedure In the formulation of MPC problem, constraints imposed in the problem formulation represent the desired range of operation for the plant. The Active Set method, Primal Dual Inferior Point method, Hildreth’s QP procedure and Shor’s r-Algorithm are some examples of methods that handle optimisation solutions involving constraints [Wang, 2009d, Truong, 2007, Luenberger and Ye, 2008]. In the Active Set method, the active constraints (constraints that meet the condition M∆U = γ) need to be identified along
6.8 MODEL PREDICTIVE CONTROL WITH CONSTRAINTS 175 with optimal control decision variable ∆U. The Active Set method requires an iterative updating procedure to test the constraint conditions (active or inactive) before solving for optimal decision variable. The main drawback of the Active Set method is that it produces quite a high computation load if many constraints are imposed on the optimisation problem [Wang, 2009d, Truong, 2007]. The Primal-Dual method such as Hildreth’s QP procedure can be used to overcome the computation burden of the Active Set method. Generally, the Primal-Dual method systematically identifies the inactive constraints and eliminated them directly in the solution [Wang, 2009d]. This would lead to a much simpler programming implementation for finding the optimal solution for the constrained control problem. To be consistent with QP literature notation, the cost function in Equation (6.31) and the associate constraints are reformulated as: J = 1 2 xT Ex + x T F subject to Mx ≤ γ (6.37) where matrix E and F are compatible matrices and vectors in the quadratic programming problem in Equation (6.31). Variable x denotes the decision variable or the control decision variable ∆U. Matrix E is also known as the Hessian matrix with symmetric mN c × mN c dimension and F is a column vector with mN c elements. In order to minimise the objective function (6.31) subject to inequality constraints Mx ≤ γ, the following Lagrange expression is considered: J = 1 2 xT Ex + x T F + λ T (Mx − γ) (6.38) where λ denotes the Lagrange Multiplier vector with a dimension equivalent to D number of constraints. The Equation (6.38) is also known as the primal problem in literature. The Lagrange Multiplier λ indicates whether a constraint is either active or inactive. By definition, the elements in Lagrange Multiplier vector are non-negative and if the element is λ i = 0 then the i th constraint is inactive, which indicates that a
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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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4.4 SUMMARY 117 Segment 1 Segment 2
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Chapter 5 NN BASED SYSTEM IDENTIFIC
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5.2 OFF-LINE BASED SYSTEM IDENTIFIC
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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 239: 7.4 SUMMARY 207 data measured from
Page 242 and 243: 210 CHAPTER 8 CONCLUSIONS AND FUTUR
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Page 246 and 247: 214 REFERENCES B. W. Bequette. Non-
Page 248 and 249: 216 REFERENCES Konstantinos Dalamag
Page 250 and 251: 218 REFERENCES Wayne Johnson. Helic
Page 252 and 253: 220 REFERENCES B. Mettler, Mark B.
Page 254 and 255: 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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