Zeroing Neural Networks Finite-time Convergence Design, Analysis and Applications

by Xiao, Lin; Jia, Lei

• ISBN13: 9781119985990
• ISBN10: 1119985994
• Edition: 1st
• Format: Hardcover
• Copyright: 2022-12-01
• Publisher: Wiley-IEEE Press

Summary

Zeroing Neural Networks

Describes the theoretical and practical aspects of finite-time ZNN methods for solving an array of computational problems

Zeroing Neural Networks (ZNN) have become essential tools for solving discretized sensor-driven time-varying matrix problems in engineering, control theory, and on-chip applications for robots. Building on the original ZNN model, finite-time zeroing neural networks (FTZNN) enable efficient, accurate, and predictive real-time computations. Setting up discretized FTZNN algorithms for different time-varying matrix problems requires distinct steps.

Zeroing Neural Networks provides in-depth information on the finite-time convergence of ZNN models in solving computational problems. Divided into eight parts, this comprehensive resource covers modeling methods, theoretical analysis, computer simulations, nonlinear activation functions, and more. Each part focuses on a specific type of time-varying computational problem, such as the application of FTZNN to the Lyapunov equation, linear matrix equation, and matrix inversion. Throughout the book, tables explain the performance of different models, while numerous illustrative examples clarify the advantages of each FTZNN method. In addition, the book:

• Describes how to design, analyze, and apply FTZNN models for solving computational problems
• Presents multiple FTZNN models for solving time-varying computational problems
• Details the noise-tolerance of FTZNN models to maximize the adaptability of FTZNN models to complex environments
• Includes an introduction, problem description, design scheme, theoretical analysis, illustrative verification, application, and summary in every chapter

Zeroing Neural Networks: Finite-time Convergence Design, Analysis and Applications is an essential resource for scientists, researchers, academic lecturers, and postgraduates in the field, as well as a valuable reference for engineers and other practitioners working in neurocomputing and intelligent control.

Author Biography

LIN XIAO, PhD, is a Professor in the College of Information Science and Engineering at Hunan Normal University, Changsha, China. He has authored more than 100 papers in international conferences and journals, including IEEE-TCYB, IEEE-TII, IEEE-TSMCS. Professor Xiao is Associate Editor of IEEE-TNNLS.

LEI JIA is a PhD degree candidate in Operations Research and Control in the College of Mathematics and Statistics at Hunan Normal University, Changsha, China. She has authored or co-authored more than 20 scientific articles, including 13 IEEE-transaction papers.

Table of Contents

List of Figures xv

List of Tables xxxi

Author Biographies xxxiii

Preface xxxv

Acknowledgments xlv

Part I Application to Matrix Square Root 1

1 FTZNN for Time-varying Matrix Square Root 3

1.1 Introduction 3

1.2 Problem Formulation and ZNN Model 4

1.3 FTZNN Model 4

1.3.1 Model Design 5

1.3.2 Theoretical Analysis 7

1.4 Illustrative Verification 8

1.5 Chapter Summary 11

References 11

2 FTZNN for Static Matrix Square Root 13

2.1 Introduction 13

2.2 Solution Models 14

2.2.1 OZNN Model 14

2.2.2 FTZNN Model 15

2.3 Illustrative Verification 17

2.3.1 Example 1 18

2.3.2 Example 2 20

2.4 Chapter Summary 21

References 21

Part II Application to Matrix Inversion 23

3 Design Scheme I of FTZNN 25

3.1 Introduction 25

3.2 Problem Formulation and Preliminaries 25

3.3 FTZNN Model 26

3.3.1 Model Design 26

3.3.2 Theoretical Analysis 29

3.4 Illustrative Verification 30

3.4.1 Example 1: Nonrandom Time-varying Coefficients 30

3.4.2 Example 2: Random Time-varying Coefficients 34

3.5 Chapter Summary 35

References 36

4 Design Scheme II of FT ZNN 39

4.1 Introduction 39

4.2 Preliminaries 40

4.2.1 Mathematical Preparation 40

4.2.2 Problem Formulation 41

4.3 NT-FTZNN Model 41

4.4 Theoretical Analysis 43

4.4.1 NT-FTZNN in the Absence of Noises 43

4.4.2 NT-FTZNN in the Presence of Noises 44

4.5 Illustrative Verification 46

4.5.1 Example 1: Two-dimensional Coefficient 47

4.5.2 Example 2: Six-dimensional Coefficient 52

4.5.3 Example 3: Application to Mobile Manipulator 54

4.5.4 Example 4: Physical Comparative Experiments 54

4.6 Chapter Summary 57

References 57

5 Design Scheme III of FTZNN 61

5.1 Introduction 61

5.2 Problem Formulation and Neural Solver 61

5.2.1 FPZNN Model 62

5.2.2 IVP-FTZNN Model 63

5.3 Theoretical Analysis 64

5.4 Illustrative Verification 70

5.4.1 Example 1: Two-Dimensional Coefficient 70

5.4.2 Example 2: Three-Dimensional Coefficient 73

5.5 Chapter Summary 78

References 78

Part III Application to Linear Matrix Equation 81

6 Design Scheme I of FTZNN 83

6.1 Introduction 83

6.2 Convergence Speed and Robustness Co-design 84

6.3 R-FTZNN Model 90

6.3.1 Design of R-FTZNN 90

6.3.2 Analysis of R-FTZNN 91

6.4 Illustrative Verification 93

6.4.1 Numerical Example 93

6.4.2 Applications: Robotic Motion Tracking 98

6.5 Chapter Summary 101

References 102

7 Design Scheme II of FTZNN 105

7.1 Introduction 105

7.2 Problem Formulation 106

7.3 FTZNN Model 106

7.4 Theoretical Analysis 108

7.4.1 Convergence 108

7.4.2 Robustness 112

7.5 Illustrative Verification 118

7.5.1 Convergence 118

7.5.2 Robustness 121

7.6 Chapter Summary 122

References 122

Part IV Application to Optimization 125

8 FTZNN for Constrained Quadratic Programming 127

8.1 Introduction 127

8.2 Preliminaries 128

8.2.1 Problem Formulation 128

8.2.2 Optimization Theory 128

8.3 U-FTZNN Model 130

8.4 Convergence Analysis 131

8.5 Robustness Analysis 134

8.6 Illustrative Verification 136

8.6.1 Qualitative Experiments 136

8.6.2 Quantitative Experiments 139

8.7 Application to Image Fusion 143

8.8 Application to Robot Control 146

8.9 Chapter Summary 149

References 149

9 FTZNN for Nonlinear Minimization 151

9.1 Introduction 151

9.2 Problem Formulation and ZNN Models 151

9.2.1 Problem Formulation 152

9.2.2 ZNN Model 152

9.2.3 RZNN Model 154

9.3 Design and Analysis of R-FTZNN 154

9.3.1 Second-Order Nonlinear Formula 155

9.3.2 R-FTZNN Model 159

9.4 Illustrative Verification 161

9.4.1 Constant Noise 161

9.4.2 Dynamic Noise 163

9.5 Chapter Summary 165

References 166

10 FTZNN for Quadratic Optimization 169

10.1 Introduction 169

10.2 Problem Formulation 170

10.3 Related Work: GNN and ZNN Models 172

10.3.1 GNN Model 172

10.3.2 ZNN Model 173

10.4 N-FTZNN Model 174

10.4.1 Models Comparison 175

10.4.2 Finite-Time Convergence 176

10.5 Illustrative Verification 178

10.6 Chapter Summary 181

References 181

Part V Application to the Lyapunov Equation 183

11 Design Scheme I of FTZNN 185

11.1 Introduction 185

11.2 Problem Formulation and Related Work 186

11.2.1 GNN Model 186

11.2.2 ZNN Model 187

11.3 FTZNN Model 187

11.4 Illustrative Verification 190

11.5 Chapter Summary 193

References 193

12 Design Scheme II of FTZNN 197

12.1 Introduction 197

12.2 Problem Formulation and Preliminaries 197

12.3 FTZNN Model 198

12.3.1 Design of FTZNN 199

12.3.2 Analysis of FTZNN 200

12.4 Illustrative Verification 202

12.5 Application to Tracking Control 205

12.6 Chapter Summary 207

References 207

13 Design Scheme III of FTZNN 209

13.1 Introduction 209

13.2 N-FTZNN Model 210

13.2.1 Design of N-FTZNN 210

13.2.2 Re-Interpretation from Nonlinear PID Perspective 211

13.3 Theoretical Analysis 212

13.4 Illustrative Verification 219

13.4.1 Numerical Comparison 219

13.4.2 Application Comparison 224

13.4.3 Experimental Verification 228

13.5 Chapter Summary 229

References 229

Part VI Application to the Sylvester Equation 231

14 Design Scheme I of FTZNN 233

14.1 Introduction 233

14.2 Problem Formulation and ZNN Model 233

14.3 N-FTZNN Model 235

14.3.1 Design of N-FTZNN 235

14.3.2 Theoretical Analysis 237

14.4 Illustrative Verification 243

14.5 Robotic Application 248

14.6 Chapter Summary 251

References 251

15 Design Scheme II of FTZNN 255

15.1 Introduction 255

15.2 ZNN Model and Activation Functions 256

15.2.1 ZNN Model 256

15.2.2 Commonly Used AFs 257

15.2.3 Two Novel Nonlinear AFs 257

15.3 NT-PTZNN Models and Theoretical Analysis 258

15.3.1 NT-PTZNN1 Model 258

15.3.2 NT-PTZNN2 Model 262

15.4 Illustrative Verification 266

15.4.1 Example 1 266

15.4.2 Example 2 269

15.4.3 Example 3 273

15.5 Chapter Summary 274

References 274

16 Design Scheme III of FTZNN 277

16.1 Introduction 277

16.2 ZNN Model and Activation Function 278

16.2.1 ZNN Model 278

16.2.2 Sign-bi-power Activation Function 279

16.3 FTZNN Models with Adaptive Coefficients 282

16.3.1 SA-FTZNN Model 282

16.3.2 PA-FTZNN Model 284

16.3.3 EA-FTZNN Model 286

16.4 Illustrative Verification 289

16.5 Chapter Summary 294

References 294

Part VII Application to Inequality 297

17 Design Scheme I of FTZNN 299

17.1 Introduction 299

17.2 FTZNN Models Design 299

17.2.1 Problem Formulation 300

17.2.2 ZNN Model 300

17.2.3 Vectorization 300

17.2.4 Activation Functions 301

17.2.5 FTZNN Models 302

17.3 Theoretical Analysis 303

17.3.1 Global Convergence 303

17.3.2 Finite-Time Convergence 304

17.4 Illustrative Verification 309

17.5 Chapter Summary 314

References 314

18 Design Scheme II of FTZNN 317

18.1 Introduction 317

18.2 NT-FTZNN Model Deisgn 318

18.2.1 Problem Formulation 318

18.2.2 ZNN Model 318

18.2.3 NT-FTZNN Model 319

18.2.4 Activation Functions 319

18.3 Theoretical Analysis 320

18.3.1 Global Convergence 320

18.3.2 Finite-Time Convergence 321

18.3.3 Noise-Tolerant Convergence 326

18.4 Illustrative Verification 327

18.5 Chapter Summary 334

References 335

Part VIII Application to Nonlinear Equation 337

19 Design Scheme I of FTZNN 339

19.1 Introduction 339

19.2 Model Formulation 339

19.2.1 OZNN Model 340

19.2.2 FTZNN Model 340

19.2.3 Models Comparison 341

19.3 Convergence Analysis 341

19.4 Illustrative Verification 343

19.4.1 Nonlinear Equation f (u) with Simple Root 343

19.4.2 Nonlinear Equation f (u) with Multiple Root 346

19.5 Chapter Summary 347

References 347

20 Design Scheme II of FTZNN 349

20.1 Introduction 349

20.2 Problem and Model Formulation 349

20.2.1 GNN Model 350

20.2.2 OZNN Model 350

20.3 FTZNN Model and Finite-Time Convergence 351

20.4 Illustrative Verification 354

20.5 Chapter Summary 356

References 356

21 Design Scheme III of FTZNN 359

21.1 Introduction 359

21.2 Problem Formulation and ZNN Models 359

21.2.1 Problem Formulation 360

21.2.2 ZNN Model 360

21.3 Robust and Fixed-Time ZNN Model 361

21.4 Theoretical Analysis 362

21.4.1 Case 1: No Noise 362

21.4.2 Case 2: Under External Noises 363

21.5 Illustrative Verification 367

21.6 Chapter Summary 370

References 371

Index 375