Zeroing Neural Networks

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.

• List of Figures xvList of Tables xxxiAuthor Biographies xxxiiiPreface xxxvAcknowledgments xlvPart I Application to Matrix Square Root 11 FTZNN for Time-varying Matrix Square Root 31.1 Introduction 31.2 Problem Formulation and ZNN Model 41.3 FTZNN Model 41.3.1 Model Design 51.3.2 Theoretical Analysis 71.4 Illustrative Verification 81.5 Chapter Summary 11References 112 FTZNN for Static Matrix Square Root 132.1 Introduction 132.2 Solution Models 142.2.1 OZNN Model 142.2.2 FTZNN Model 152.3 Illustrative Verification 172.3.1 Example 1 182.3.2 Example 2 202.4 Chapter Summary 21References 21Part II Application to Matrix Inversion 233 Design Scheme I of FTZNN 253.1 Introduction 253.2 Problem Formulation and Preliminaries 253.3 FTZNN Model 263.3.1 Model Design 263.3.2 Theoretical Analysis 293.4 Illustrative Verification 303.4.1 Example 1: Nonrandom Time-varying Coefficients 303.4.2 Example 2: Random Time-varying Coefficients 343.5 Chapter Summary 35References 364 Design Scheme II of FT ZNN 394.1 Introduction 394.2 Preliminaries 404.2.1 Mathematical Preparation 404.2.2 Problem Formulation 414.3 NT-FTZNN Model 414.4 Theoretical Analysis 434.4.1 NT-FTZNN in the Absence of Noises 434.4.2 NT-FTZNN in the Presence of Noises 444.5 Illustrative Verification 464.5.1 Example 1: Two-dimensional Coefficient 474.5.2 Example 2: Six-dimensional Coefficient 524.5.3 Example 3: Application to Mobile Manipulator 544.5.4 Example 4: Physical Comparative Experiments 544.6 Chapter Summary 57References 575 Design Scheme III of FTZNN 615.1 Introduction 615.2 Problem Formulation and Neural Solver 615.2.1 FPZNN Model 625.2.2 IVP-FTZNN Model 635.3 Theoretical Analysis 645.4 Illustrative Verification 705.4.1 Example 1: Two-Dimensional Coefficient 705.4.2 Example 2: Three-Dimensional Coefficient 735.5 Chapter Summary 78References 78Part III Application to Linear Matrix Equation 816 Design Scheme I of FTZNN 836.1 Introduction 836.2 Convergence Speed and Robustness Co-design 846.3 R-FTZNN Model 906.3.1 Design of R-FTZNN 906.3.2 Analysis of R-FTZNN 916.4 Illustrative Verification 936.4.1 Numerical Example 936.4.2 Applications: Robotic Motion Tracking 986.5 Chapter Summary 101References 1027 Design Scheme II of FTZNN 1057.1 Introduction 1057.2 Problem Formulation 1067.3 FTZNN Model 1067.4 Theoretical Analysis 1087.4.1 Convergence 1087.4.2 Robustness 1127.5 Illustrative Verification 1187.5.1 Convergence 1187.5.2 Robustness 1217.6 Chapter Summary 122References 122Part IV Application to Optimization 1258 FTZNN for Constrained Quadratic Programming 1278.1 Introduction 1278.2 Preliminaries 1288.2.1 Problem Formulation 1288.2.2 Optimization Theory 1288.3 U-FTZNN Model 1308.4 Convergence Analysis 1318.5 Robustness Analysis 1348.6 Illustrative Verification 1368.6.1 Qualitative Experiments 1368.6.2 Quantitative Experiments 1398.7 Application to Image Fusion 1438.8 Application to Robot Control 1468.9 Chapter Summary 149References 1499 FTZNN for Nonlinear Minimization 1519.1 Introduction 1519.2 Problem Formulation and ZNN Models 1519.2.1 Problem Formulation 1529.2.2 ZNN Model 1529.2.3 RZNN Model 1549.3 Design and Analysis of R-FTZNN 1549.3.1 Second-Order Nonlinear Formula 1559.3.2 R-FTZNN Model 1599.4 Illustrative Verification 1619.4.1 Constant Noise 1619.4.2 Dynamic Noise 1639.5 Chapter Summary 165References 16610 FTZNN for Quadratic Optimization 16910.1 Introduction 16910.2 Problem Formulation 17010.3 Related Work: GNN and ZNN Models 17210.3.1 GNN Model 17210.3.2 ZNN Model 17310.4 N-FTZNN Model 17410.4.1 Models Comparison 17510.4.2 Finite-Time Convergence 17610.5 Illustrative Verification 17810.6 Chapter Summary 181References 181Part V Application to the Lyapunov Equation 18311 Design Scheme I of FTZNN 18511.1 Introduction 18511.2 Problem Formulation and Related Work 18611.2.1 GNN Model 18611.2.2 ZNN Model 18711.3 FTZNN Model 18711.4 Illustrative Verification 19011.5 Chapter Summary 193References 19312 Design Scheme II of FTZNN 19712.1 Introduction 19712.2 Problem Formulation and Preliminaries 19712.3 FTZNN Model 19812.3.1 Design of FTZNN 19912.3.2 Analysis of FTZNN 20012.4 Illustrative Verification 20212.5 Application to Tracking Control 20512.6 Chapter Summary 207References 20713 Design Scheme III of FTZNN 20913.1 Introduction 20913.2 N-FTZNN Model 21013.2.1 Design of N-FTZNN 21013.2.2 Re-Interpretation from Nonlinear PID Perspective 21113.3 Theoretical Analysis 21213.4 Illustrative Verification 21913.4.1 Numerical Comparison 21913.4.2 Application Comparison 22413.4.3 Experimental Verification 22813.5 Chapter Summary 229References 229Part VI Application to the Sylvester Equation 23114 Design Scheme I of FTZNN 23314.1 Introduction 23314.2 Problem Formulation and ZNN Model 23314.3 N-FTZNN Model 23514.3.1 Design of N-FTZNN 23514.3.2 Theoretical Analysis 23714.4 Illustrative Verification 24314.5 Robotic Application 24814.6 Chapter Summary 251References 25115 Design Scheme II of FTZNN 25515.1 Introduction 25515.2 ZNN Model and Activation Functions 25615.2.1 ZNN Model 25615.2.2 Commonly Used AFs 25715.2.3 Two Novel Nonlinear AFs 25715.3 NT-PTZNN Models and Theoretical Analysis 25815.3.1 NT-PTZNN1 Model 25815.3.2 NT-PTZNN2 Model 26215.4 Illustrative Verification 26615.4.1 Example 1 26615.4.2 Example 2 26915.4.3 Example 3 27315.5 Chapter Summary 274References 27416 Design Scheme III of FTZNN 27716.1 Introduction 27716.2 ZNN Model and Activation Function 27816.2.1 ZNN Model 27816.2.2 Sign-bi-power Activation Function 27916.3 FTZNN Models with Adaptive Coefficients 28216.3.1 SA-FTZNN Model 28216.3.2 PA-FTZNN Model 28416.3.3 EA-FTZNN Model 28616.4 Illustrative Verification 28916.5 Chapter Summary 294References 294Part VII Application to Inequality 29717 Design Scheme I of FTZNN 29917.1 Introduction 29917.2 FTZNN Models Design 29917.2.1 Problem Formulation 30017.2.2 ZNN Model 30017.2.3 Vectorization 30017.2.4 Activation Functions 30117.2.5 FTZNN Models 30217.3 Theoretical Analysis 30317.3.1 Global Convergence 30317.3.2 Finite-Time Convergence 30417.4 Illustrative Verification 30917.5 Chapter Summary 314References 31418 Design Scheme II of FTZNN 31718.1 Introduction 31718.2 NT-FTZNN Model Deisgn 31818.2.1 Problem Formulation 31818.2.2 ZNN Model 31818.2.3 NT-FTZNN Model 31918.2.4 Activation Functions 31918.3 Theoretical Analysis 32018.3.1 Global Convergence 32018.3.2 Finite-Time Convergence 32118.3.3 Noise-Tolerant Convergence 32618.4 Illustrative Verification 32718.5 Chapter Summary 334References 335Part VIII Application to Nonlinear Equation 33719 Design Scheme I of FTZNN 33919.1 Introduction 33919.2 Model Formulation 33919.2.1 OZNN Model 34019.2.2 FTZNN Model 34019.2.3 Models Comparison 34119.3 Convergence Analysis 34119.4 Illustrative Verification 34319.4.1 Nonlinear Equation f (u) with Simple Root 34319.4.2 Nonlinear Equation f (u) with Multiple Root 34619.5 Chapter Summary 347References 34720 Design Scheme II of FTZNN 34920.1 Introduction 34920.2 Problem and Model Formulation 34920.2.1 GNN Model 35020.2.2 OZNN Model 35020.3 FTZNN Model and Finite-Time Convergence 35120.4 Illustrative Verification 35420.5 Chapter Summary 356References 35621 Design Scheme III of FTZNN 35921.1 Introduction 35921.2 Problem Formulation and ZNN Models 35921.2.1 Problem Formulation 36021.2.2 ZNN Model 36021.3 Robust and Fixed-Time ZNN Model 36121.4 Theoretical Analysis 36221.4.1 Case 1: No Noise 36221.4.2 Case 2: Under External Noises 36321.5 Illustrative Verification 36721.6 Chapter Summary 370References 371Index 375