Multi-resolution methods for modeling and control of dynamical systems / Puneet Singla, John L. Junkins.

Format:

Book

Author/Creator:

Singla, Puneet.

Contributor:

Junkins, John L.

Louis A. Duhring Fund.

Series:

Chapman & Hall/CRC applied mathematics and nonlinear science series

Language:

English

Subjects (All):

Systems engineering--Mathematical models.

Systems engineering.

Physical Description:

xvi, 299 pages : illustrations (some color), maps (some color) ; 25 cm.

Place of Publication:

Boca Raton : CRC Press, [2009]

Summary:

Unifying the most important methodology in this field, Multi-Resolution Methods for Modeling and Control of Dynamical Systems explores existing approximation methods as well as develops new ones for the approximate solution of large-scale dynamical system problems. It brings together a wide set of material from classical orthogonal function approximation, neural network input-output approximation, finite element methods for distributed parameter systems, and various approximation methods employed in adaptive control and learning theory.

With sufficient rigor and generality, the book promotes a qualitative understanding of the development of key ideas. It facilitates a deep appreciation of the important nuances and restrictions implicit in the algorithms that affect the validity of the results produced. The text features benchmark problems throughout to offer insights and illustrate some of the computational implications. The authors provide a framework for understanding the advantages, drawbacks, and application areas of existing and new algorithms for input-output approximation. They also present novel adaptive learning algorithms that can be adjusted in real time to the various parameters of unknown mathematical models.

Contents:

1 Least Squares Methods 1

1.2 The Least Squares Algorithm 2

1.3 Linear Least Squares Methods 3

1.3.1 Batch Least Squares Method 3

1.3.2 Sequential Least Squares Algorithm 5

1.4 Non-Linear Least Squares Algorithm 8

1.5 Properties of Least Squares Algorithms 10

1.6.1 Smooth Function Approximation 11

1.6.2 Star Camera Calibration 12

2 Polynomial Approximation 21

2.2 Gram-Schmidt Procedure of Orthogonalization 22

2.2.1 Three-Term Recurrence Relation to Generate Orthogonal Polynomials 24

2.2.2 Uniqueness of Orthogonal Polynomials 25

2.3 Hypergeometric Function Approach to Generate Orthogonal Polynomials 30

2.3.1 Derivation of Rodrigues's Formula for Continuous Variable Polynomials 34

2.3.2 Leading Coefficients for Three-Term Recurrence Formula 36

2.4 Discrete Variable Orthogonal Polynomials 38

2.4.1 Hypergeometric Type Difference Equation 39

2.4.2 Derivation of Rodrigues's Formula for Discrete Variable Orthogonal Polynomials 42

2.4.3 Leading Coefficients for Three-Term Recurrence Formula for Discrete Variable Orthogonal Polynomials 44

2.5 Approximation Properties of Orthogonal Polynomials 45

3 Artificial Neural Networks for Input-Output Approximation 49

3.1.1 Radial Basis Function Networks 50

3.2 Direction-Dependent Approach 55

3.3 Directed Connectivity Graph 60

3.3.1 Estimation Algorithm 62

3.3.2 Spectral Decomposition of the Covariance Matrix 64

3.3.3 Additive Decomposition of the Covariance Matrix 66

3.3.4 Cholesky Decomposition of the Covariance Matrix 66

3.4 Modified Minimal Resource Allocating Algorithm (MMRAN) 69

3.5 Numerical Simulation Examples 72

3.5.1 Test Example 1: Function Approximation 73

3.5.2 Test Example 2: 3-Input 1-Output Continuous Function Approximation 80

3.5.3 Test Example 3: Dynamical System Identification 82

3.5.4 Test Example 4: Chaotic Time Series Prediction 86

3.5.5 Test Example 5: Benchmark Against the On-Line Structural Adaptive Hybrid Learning (ONSAHL) Algorithm 90

4 Multi-Resolution Approximation Methods 95

4.2 Wavelets 97

4.3 Bezier Spline 105

4.4 Moving Least Squares Method 110

4.5 Adaptive Multi-Resolution Algorithm 112

4.6 Numerical Results 116

4.6.1 Calibration of Vision Sensors 116

4.6.2 Simulation and Results 117

4.6.3 DCG Approximation Result 119

4.6.4 Local Approximation Results 121

5 Global Local Orthogonal Polynomial MAPping (GLO-MAP) in N Dimensions 123

5.3 Approximation in 1, 2 and N Dimensions Using Weighting Functions 128

5.4 Global-Local Orthogonal Approximation in 1-, 2- and N-Dimensional Spaces 136

5.4.1 1-Dimensional Case 139

5.4.2 2-Dimensional Case 140

5.4.3 N-Dimensional Case 142

5.5 Algorithm Implementation 144

5.5.1 Sequential Version of the GLO-MAP Algorithm 146

5.6 Properties of GLO-MAP Approximation 149

5.6.1 Approximation Error 149

5.6.2 Bounds on Approximation Error 150

5.6.3 Probabilistic Analysis of the GLO-MAP Algorithm 152

5.7 Illustrative Engineering Applications 155

5.7.1 Function Approximation 155

5.7.2 Synthetic Jet Actuator Modeling 160

5.7.3 Space-Based Radar (SBR) Antenna Shape Approximation 166

5.7.4 Porkchop Plot Approximations for Mission to Near-Earth Objects (NEOs) 170

6 Nonlinear System Identification 179

6.2 Problem Statement and Background 180

6.3 Novel System Identification Algorithm 182

6.3.1 Linear System Identification 185

6.3.2 State Variable Estimation 189

6.4 Nonlinear System Identification Algorithm 190

6.4.1 Learning Algorithm for State Model Perturbation Approach (SysID 1) 190

6.4.2 Learning Algorithm for Output Model Perturbation Approach (SysID 2) 197

6.5 Numerical Simulation 199

6.5.1 Dynamic System Identification of Large Space Antenna 199

7 Distributed Parameter Systems 207

7.2 MLPG-Moving Least Squares Approach 210

7.2.1 Poisson Equation 214

7.2.2 Comments on the MLPG Algorithm 220

7.3 Partition of Unity Finite Element Method 222

7.3.1 Poisson Equation 228

7.3.2 Fokker-Planck-Kolmogorov Equation 235

8 Control Distribution for Over-Actuated Systems 243

8.2 Problem Statement and Background 245

8.3 Control Distribution Functions 251

8.3.1 Radial Basis Functions 254

8.3.2 Global Local Orthogonal Basis Functions 255

8.4 Hierarchical Control Distribution Algorithm 259

8.5 Numerical Results 264

8.5.1 Control Allocation for a Morphing Wing 264.

Notes:

"A Chapman & Hall book."

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Louis A. Duhring Fund.

ISBN:

9781584887690

1584887699

OCLC:

144520994