Multi-Resolution Methods for Modeling and Control of Dynamical Systems (Chapman & Hall CRC Applied Mathematics & Nonlinear Science)

c7699fm......Page 1
Multi-Resolution Methods for Modeling and Control of Dynamical Systems......Page 3
Contents......Page 6
Preface......Page 10
Acknowledgments......Page 13
Appendix......Page 0
1.1 Introduction......Page 15
1.2 The Least Squares Algorithm......Page 16
1.3.1 Batch Least Squares Method......Page 17
1.3.2 Sequential Least Squares Algorithm......Page 19
1.4 Non-Linear Least Squares Algorithm......Page 22
1.5 Properties of Least Squares Algorithms......Page 24
1.6.1 Smooth Function Approximation......Page 25
1.6.2 Star Camera Calibration......Page 26
1.7 Summary......Page 33
2.1 Introduction......Page 34
2.2 Gram-Schmidt Procedure of Orthogonalization......Page 35
2.2.1 Three-Term Recurrence Relation to Generate Orthogonal Polynomials......Page 37
2.2.2 Uniqueness of Orthogonal Polynomials......Page 38
2.3 Hypergeometric Function Approach to Generate Orthogonal Polynomials......Page 43
2.3.1 Derivation of Rodrigues’s Formula for Continuous Variable Polynomials......Page 47
2.3.2 Leading Coefficients for Three-Term Recurrence Formula......Page 49
2.4 Discrete Variable Orthogonal Polynomials......Page 51
2.4.1 Hypergeometric Type Difference Equation......Page 52
2.4.2 Derivation of Rodrigues’s Formula for Discrete Variable Orthogonal Polynomials......Page 55
2.4.3 Leading Coefficients for Three-Term Recurrence Formula for Discrete Variable Orthogonal Polynomials......Page 57
2.5 Approximation Properties of Orthogonal Polynomials......Page 58
2.6 Summary......Page 61
3.1 Introduction......Page 62
3.1.1 Radial Basis Function Networks......Page 63
3.2 Direction-Dependent Approach......Page 68
3.3 Directed Connectivity Graph......Page 73
3.3.1 Estimation Algorithm......Page 75
3.3.2 Spectral Decomposition of the Covariance Matrix......Page 77
3.3.4 Cholesky Decomposition of the Covariance Matrix......Page 79
3.4 Modified Minimal Resource Allocating Algorithm (MMRAN)......Page 82
3.5 Numerical Simulation Examples......Page 85
3.5.1 Test Example 1: Function Approximation......Page 86
3.5.2 Test Example 2: 3-Input 1-Output Continuous Function Approximation......Page 93
3.5.3 Test Example 3: Dynamical System Identification......Page 95
3.5.4 Test Example 4: Chaotic Time Series Prediction......Page 99
3.5.5 Test Example 5: Benchmark Against the On-Line Structural Adaptive Hybrid Learning (ONSAHL) Algorithm......Page 103
3.6 Summary......Page 106
4.1 Introduction......Page 108
4.2 Wavelets......Page 110
Example 4.1......Page 113
Example 4.2......Page 114
4.3 Bèzier Spline......Page 118
Example 4.4......Page 120
4.4 Moving Least Squares Method......Page 123
4.5 Adaptive Multi-Resolution Algorithm......Page 125
4.6.1 Calibration of Vision Sensors......Page 129
4.6.2 Simulation and Results......Page 130
4.6.3 DCG Approximation Result......Page 132
4.7 Summary......Page 134
5.1 Introduction......Page 136
5.2 Basic Ideas......Page 138
5.3 Approximation in 1, 2 and N Dimensions Using Weighting Functions......Page 141
5.4 Global-Local Orthogonal Approximation in 1-, 2- and N-Dimensional Spaces......Page 149
5.4.1 1-Dimensional Case......Page 152
5.4.2 2-Dimensional Case......Page 153
5.4.3 N-Dimensional Case......Page 155
5.5 Algorithm Implementation......Page 157
5.5.1 Sequential Version of the GLO-MAP Algorithm......Page 159
5.6.1 Approximation Error......Page 162
5.6.2 Bounds on Approximation Error......Page 163
5.6.3 Probabilistic Analysis of the GLO-MAP Algorithm......Page 165
5.7.1 Function Approximation......Page 168
5.7.2 Synthetic Jet Actuator Modeling......Page 173
5.7.2.1 Experimental Setup......Page 176
5.7.3 Space-Based Radar (SBR) Antenna Shape Approximation......Page 179
5.7.4 Porkchop Plot Approximations for Mission to Near-Earth Objects (NEOs)......Page 183
5.8 Summary......Page 187
6.1 Introduction......Page 191
6.2 Problem Statement and Background......Page 192
6.3 Novel System Identification Algorithm......Page 194
6.3.1 Linear System Identification......Page 197
6.3.2 State Variable Estimation......Page 201
6.4.1 Learning Algorithm for State Model Perturbation Approach (SysID 1)......Page 202
6.4.1.1 Adaption Law Derivation Using the GLO-MAP Model......Page 205
6.4.2 Learning Algorithm for Output Model Perturbation Approach (SysID 2)......Page 209
6.5.1 Dynamic System Identification of Large Space Antenna......Page 211
6.6 Summary......Page 217
7.1 Introduction......Page 218
7.2 MLPG-Moving Least Squares Approach......Page 221
7.2.1 Poisson Equation......Page 225
7.2.2 Comments on the MLPG Algorithm......Page 231
7.3 Partition of Unity Finite Element Method......Page 233
7.3.1 Poisson Equation......Page 239
7.3.2 Fokker-Planck-Kolmogorov Equation......Page 246
7.4 Summary......Page 251
8.1 Introduction......Page 253
8.2 Problem Statement and Background......Page 255
8.3 Control Distribution Functions......Page 261
8.3.1 Radial Basis Functions......Page 264
8.3.2 Global Local Orthogonal Basis Functions......Page 265
8.4 Hierarchical Control Distribution Algorithm......Page 269
8.5.1 Control Allocation for a Morphing Wing......Page 274
8.6 Summary......Page 283
Appendix......Page 285
References......Page 294