Approximation of Hessian Matrix for Second-order SPSA Algorithm ...

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List of Figures Fig. 1.1 Example of stochastic optimization algorithm minimizing loss function L θ 1 θ ) 3 ( , 2 Fig. 1.2 Performance of SPSA algorithm (two measurements). 9 Fig. 2.1 The two-recursions in 2nd-SPSA Algorithm 21 Fig. 2.2 Diagram of method for forming estimate F ( ) 39 M , N θ Fig. 2.3 Split uniform distribution 56 Fig. 2.4 Inverse split uniform distribution 57 Fig. 2.5 Symmetric double triangular distribution 57 Fig. 2.6 Identification with an unknown observation system 65 Fig. 2.7 Identification results (with bias compensation) 75 Fig. 2.8 Identification results (without bias compensation) 76 Fig. 3.1 One-link flexible arm 82 Fig. 3.2 Sliding mode surface 88 Fig. 3.3 Block diagram of the sliding mode control system incorporating the non-linear observer 91 Fig. 3.4 Motor angle. Without M2-SPSA and MR-SMC (dotted-line (-.-)).With RM-SA algorithm and MR-SMC (dashed-line (- -)). With LS algorithm and MR-SMC (dash-dot-line (-.)).With M2-SPSA and MR-SMC (solid-line (-)) 94 Fig. 3.5 Tip position. Without M2-SPSA and MR-SMC (dotted-line (-.-)).With RM-SA algorithm and MR-SMC (dashed-line (- -)). With LS algorithm and MR-SMC (dash-dot-line (-.)).With M2-SPSA and MR-SMC (solid-line (-)) 95 Fig. 3.6 Tip Velocity. Without M2-SPSA and MR-SMC (dotted-line (-.-)).With RM-SA algorithm and MR-SMC (dashed-line (- -)). With LS algorithm and MR-SMC (dash-dot-line (-.)).With M2-SPSA and MR-SMC (solid-line (-)) 95 Fig. 3.7 Control torque. Without M2-SPSA and MR-SMC (dotted-line (-.-)).With RM-SA algorithm and MR-SMC (dashed-line (- -)). With LS algorithm and MR-SMC (dash-dot-line (-.)).With M2-SPSA and MR-SMC (solid-line (-)) 96 Fig. 3.8 Motor angle. Simulation using x 1 with M2-SPSA and MR-SMC (solid-line). Simulation using x m with M2-SPSA and MR-SMC (dashed-line) 96 Fig. 3.9 Tip position. Simulation using x 3 with M2-SPSA and MR-SMC (solid-line). Simulation using ˆx 3 with M2-SPSA and MR-SMC (dashed-line) 96 Fig. 3.10 Tip velocity. Simulation using x 4 with M2-SPSA and MR-SMC (solid-line). Tip velocity. Simulation using ˆx 4 with M2-SPSA and MR-SMC (dashed-line) 97 xi
LIST OF FIGURES Fig. 4.1 Block diagram of the SHARF lattice algorithm 107 Fig. 4.2 Block diagram of the SM lattice algorithm 109 Fig. 4.3 Convergence of the proposed SHARF algorithm and M2-SPSA 111 Fig. 4.4 Instability of the existing SHARF algorithm 111 Fig. 4.5 Instability of the existing SM algorithm 112 Fig. 4.6 Convergence of the proposed SM algorithm and M2-SPSA 112 T Fig. 5.1 ML Parameter estimateθ = θ , θ , for the bi-modal non-linear model using k [ 1, k 2, k θ3 , k] M2-SPSA. The true parameters in the model are defined by * =[0.5, 25, 8] θ T . 122 Fig. 5.2 Parameter estimation using 2nd-SPSA and FDSA 123 xii
Page 1 and 2: Approximation of Hessian Matrix for
Page 3 and 4: Copyright 2009 by Jorge Ivan Medina
Page 5 and 6: ここで提案するアルゴ
Page 7 and 8: ABSTRACT shown that for the same as
Page 9 and 10: Contents 1. Introduction 1 1.1 Moti
Page 11: CONTENTS 5.3 Parameter Estimation b
Page 15 and 16: List of Abbreviations SPSA 1st-SPSA
Page 17 and 18: CHAPTER 1. INTRODUCTION the converg
Page 19 and 20: CHAPTER 1.INTRODUCTION approximatio
Page 21 and 22: CHAPTER 1. INTRODUCTION and simulta
Page 23 and 24: CHAPTER 1. INTRODUCTION Typical app
Page 25 and 26: CHAPTER 1. INTRODUCTION 1.4--Featur
Page 27 and 28: CHAPTER 1. INTRODUCTION Some of the
Page 29 and 30: CHAPTER 1. INTRODUCTION M − k ( k
Page 31 and 32: CHAPTER 1. INTRODUCTION usually, a
Page 33 and 34: CHAPTER 1. INTRODUCTION Main Disadv
Page 35 and 36: CHAPTER 2. PROPOSED SPSA ALGORITHM
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CHAPTER 2. PROPOSED SPSA ALGORITHM
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CHAPTER 3. APPLICATION USING M2-SPS
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CHAPTER 6. CONCLUSIONS AND FUTURE W
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REFERENCE [10] S. A. Billings, G. N
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REFERENCE [29] M. Metivier and P. P
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REFERENCE [51] D. Parikh, N. Ahmed
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REFERENCE [72] N. J. Gordon, D. J S
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APPENDIX A that this random vector
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APPENDIX A Part 2: To show that ~
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APPENDIX A Proof of Theorem 2a (M2-
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APPENDIX A 1 ⎡~ ~ ~ ~ ( ( ) ( ))
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APPENDIX A ˆ θ * −α * k+ 1 −
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APPENDIX A results. Here, zk+n+ 1 i
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APPENDIX A Because the second eleme
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154 APPENDIX A
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APPENDIX B The Wei [48] approach is
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158 APPENDIX B
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LIST OF THE PUBLICATIONS AND INTERN
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LIST OF THE PUBLICATIONS AND INTERN
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Author Biography Jorge Ivan Medina
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Approximation of Hessian Matrix for Second-order SPSA Algorithm ...

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