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

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.

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