 | Zeroing dynamics, gradient dynamics, and Newton iterations / Y.Zhang et al. - Boca Raton: CRC/Taylor & Francis, 2016. - xxix, 309 p.: ill., tab. - Bibliogr.: p.297-305. - Ind.: p.307-309.
- ISBN 978-1-4987-5376-0
Шифр: (И/В19-Z57) 02
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List of Figures ................................................ xi
List of Tables ................................................ xix
Preface ....................................................... xxi
Author Biographies .......................................... xxvii
Acknowledgments .............................................. xxix
I Time-Varying Root Finding .................................... 1
1 Time-Varying Square Root Finding ............................. 3
1.1 Introduction ............................................ 3
1.2 Problem Formulation and Continuous-Time (CT) Models ..... 4
1.3 S-DTZD Model and Newton Iteration ....................... 7
1.4 Illustrative Examples ................................... 8
1.5 Summary ................................................ 12
2 Time-Varying Cube Root Finding .............................. 13
2.1 Introduction ........................................... 13
2.2 ZD Models for Time-Varying Case ........................ 14
2.3 Simplified ZD Models for Constant Case and Newton
Iteration .............................................. 16
2.4 Illustrative Examples .................................. 18
2.5 Summary ................................................ 26
3 Time-Varying 4th Root Finding ............................... 27
3.1 Introduction ........................................... 27
3.2 Problem Formulation and ZD Models ...................... 28
3.3 GD Model ............................................... 32
3.4 Illustrative Examples .................................. 32
3.5 Summary ................................................ 37
4 Time-Varying 5th Root Finding ............................... 39
4.1 Introduction ........................................... 39
4.2 ZD Models for Time-Varying Case ........................ 40
4.3 Simplified ZD Models for Constant Case and Newton
Iteration .............................................. 43
4.4 Illustrative Examples .................................. 44
4.5 Summary ................................................ 49
Appendix: Extension to Time-Varying pth Root Finding ........ 50
II Nonlinear Equation Solving ................................. 53
5 Time-Varying Nonlinear Equation Solving ..................... 55
5.1 Introduction ........................................... 55
5.2 Problem Formulation and Solution Models ................ 56
5.3 Convergence Analysis ................................... 57
5.4 Illustrative Example ................................... 60
5.5 Summary ................................................ 63
6 Static Nonlinear Equation Solving ........................... 65
6.1 Problem Formulation and Continuous-Time Models ......... 66
6.2 DTZD Models ............................................ 71
6.3 Comparison between CTZD Model and Newton Iteration ..... 76
6.4 Further Discussion to Avoid Local Minimum .............. 81
6.5 Summary ................................................ 89
7 System of Nonlinear Equations Solving ....................... 91
7.1 Problem Formulation and CTZD Model ..................... 91
7.2 Discrete-Time Models ................................... 96
7.3 Summary ............................................... 105
III Matrix Inversion ......................................... 107
8 ZD Models and Newton Iteration ............................. 109
8.1 Introduction .......................................... 109
8.2 ZD Models ............................................. 110
8.3 Choices of Initial State X0 ........................... 112
8.4 Choices of Step Size h ................................ 116
8.5 Illustrative Examples ................................. 121
8.6 New DTZD Models Aided with Line-Search Algorithm ...... 124
8.7 Summary ............................................... 130
9 Moore-Penrose Inversion .................................... 131
9.1 Introduction .......................................... 131
9.2 Preliminaries ......................................... 132
9.3 ZD Models for Moore-Penrose Inverse ................... 133
9.4 Comparison between ZD and GD Models ................... 137
9.5 Simulation and Verification ........................... 139
9.6 Application to Robot Arm .............................. 141
9.7 Summary ............................................... 148
IV Matrix Square Root Finding ................................ 149
10 ZD Models and Newton Iteration ............................. 151
10.1 Introduction .......................................... 151
10.2 Problem Formulation and ZD Models ..................... 152
10.3 Link and Explanation to Newton Iteration .............. 156
10.4 Line-Search Algorithm ................................. 157
10.5 Illustrative Examples ................................. 158
10.6 Summary ............................................... 160
11 ZD Model Using Hyperbolic Sine Activation Functions ........ 161
11.1 Model and Activation Functions ........................ 161
11.2 Convergence Analysis .................................. 162
11.3 Robustness Analysis ................................... 166
11.4 Illustrative Examples ................................. 167
11.5 Summary ............................................... 175
V Time-Varying Quadratic Optimization ........................ 177
12 ZD Models for Quadratic Minimization ....................... 179
12.1 Introduction .......................................... 179
12.2 Problem Formulation and CTZD Model .................... 180
12.3 DTZD Models ........................................... 181
12.4 GD Models ............................................. 182
12.5 Illustrative Example .................................. 182
12.6 Summary ............................................... 189
13 ZD Models for Quadratic Programming ........................ 191
13.1 Introduction .......................................... 191
13.2 CTZD Model ............................................ 192
13.3 DTZD Models ........................................... 194
13.4 Illustrative Examples ................................. 196
13.5 Summary ............................................... 201
14 Simulative and Experimental Application to Robot Arms ...... 203
14.1 Problem Formulation and Reformulation ................. 203
14.2 Solution Models ....................................... 205
14.3 Computer Simulations .................................. 206
14.4 Hardware Experiments .................................. 210
14.5 Summary ............................................... 215
VI Time-Varying Inequality Solving ............................ 217
15 Linear Inequality Solving .................................. 219
15.1 Introduction .......................................... 220
15.2 Time-Varying Linear Inequality ........................ 221
15.3 Constant Linear Inequality ............................ 224
15.4 Illustrative Examples ................................. 225
15.5 System of Time-Varying Linear Inequalities ............ 229
15.6 Illustrative Examples ................................. 239
15.7 Summary ............................................... 253
16 System of Time-Varying Nonlinear Inequalities Solving ...... 255
16.1 Introduction .......................................... 255
16.2 Problem Formulation ................................... 257
16.3 CZD Model and Convergence Analysis .................... 257
16.4 MZD Model and Convergence Analysis .................... 259
16.5 Illustrative Example .................................. 264
16.6 Summary ............................................... 272
VII Application to Fractals .................................. 273
17 Fractals Yielded via Static Nonlinear Equation ............. 275
17.1 Introduction .......................................... 275
17.2 Complex-Valued ZD Models .............................. 276
17.3 Illustrative Examples ................................. 278
17.4 Summary ............................................... 286
18 Fractals Yielded via Time-Varying Nonlinear Equation ....... 287
18.1 Introduction .......................................... 287
18.2 Complex-Valued ZD Models .............................. 288
18.3 Illustrative Examples ................................. 291
18.4 Summary ............................................... 294
Glossary ...................................................... 295
Bibliography .................................................. 297
Index ......................................................... 307
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