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

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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(s): Lin Xiao, Lei Jia
Publisher: Wiley-IEEE Press
Year: 2022

Language: English
Pages: 433
City: Piscataway

Cover
Title Page
Copyright
Contents
List of Figures
List of Tables
Author Biographies
Preface
Acknowledgments
Part I Application to Matrix Square Root
Chapter 1 FTZNN for Time‐varying Matrix Square Root
1.1 Introduction
1.2 Problem Formulation and ZNN Model
1.3 FTZNN Model
1.3.1 Model Design
1.3.2 Theoretical Analysis
1.4 Illustrative Verification
1.5 Chapter Summary
References
Chapter 2 FTZNN for Static Matrix Square Root
2.1 Introduction
2.2 Solution Models
2.2.1 OZNN Model
2.2.2 FTZNN Model
2.3 Illustrative Verification
2.3.1 Example 1
2.3.2 Example 2
2.4 Chapter Summary
References
Part II Application to Matrix Inversion
Chapter 3 Design Scheme I of FTZNN
3.1 Introduction
3.2 Problem Formulation and Preliminaries
3.3 FTZNN Model
3.3.1 Model Design
3.3.2 Theoretical Analysis
3.4 Illustrative Verification
3.4.1 Example 1: Nonrandom Time‐varying Coefficients
3.4.2 Example 2: Random Time‐varying Coefficients
3.5 Chapter Summary
References
Chapter 4 Design Scheme II of FTZNN
4.1 Introduction
4.2 Preliminaries
4.2.1 Mathematical Preparation
4.2.2 Problem Formulation
4.3 NT‐FTZNN Model
4.4 Theoretical Analysis
4.4.1 NT‐FTZNN in the Absence of Noises
4.4.2 NT‐FTZNN in the Presence of Noises
4.4.2.1 Dynamic Bounded Gradually Disappearing Noise
4.4.2.2 Dynamic Bounded Non‐disappearing Noise
4.5 Illustrative Verification
4.5.1 Example 1: Two‐dimensional Coefficient
4.5.2 Example 2: Six‐dimensional Coefficient
4.5.3 Example 3: Application to Mobile Manipulator
4.5.4 Example 4: Physical Comparative Experiments
4.6 Chapter Summary
References
Chapter 5 Design Scheme III of FTZNN
5.1 Introduction
5.2 Problem Formulation and Neural Solver
5.2.1 FPZNN Model
5.2.2 IVP‐FTZNN Model
5.3 Theoretical Analysis
5.4 Illustrative Verification
5.4.1 Example 1: Two‐Dimensional Coefficient
5.4.2 Example 2: Three‐Dimensional Coefficient
5.5 Chapter Summary
References
Part III Application to Linear Matrix Equation
Chapter 6 Design Scheme I of FTZNN
6.1 Introduction
6.2 Convergence Speed and Robustness Co‐design
6.3 R‐FTZNN Model
6.3.1 Design of R‐FTZNN
6.3.2 Analysis of R‐FTZNN
6.4 Illustrative Verification
6.4.1 Numerical Example
6.4.1.1 No Noise Considered
6.4.1.2 With Noises Considered
6.4.2 Applications: Robotic Motion Tracking
6.5 Chapter Summary
References
Chapter 7 Design Scheme II of FTZNN
7.1 Introduction
7.2 Problem Formulation
7.3 FTZNN Model
7.4 Theoretical Analysis
7.4.1 Convergence
7.4.2 Robustness
7.5 Illustrative Verification
7.5.1 Convergence
7.5.2 Robustness
7.6 Chapter Summary
References
Part IV Application to Optimization
Chapter 8 FTZNN for Constrained Quadratic Programming
8.1 Introduction
8.2 Preliminaries
8.2.1 Problem Formulation
8.2.2 Optimization Theory
8.3 U‐FTZNN Model
8.4 Convergence Analysis
8.5 Robustness Analysis
8.6 Illustrative Verification
8.6.1 Qualitative Experiments
8.6.2 Quantitative Experiments
8.7 Application to Image Fusion
8.8 Application to Robot Control
8.9 Chapter Summary
References
Chapter 9 FTZNN for Nonlinear Minimization
9.1 Introduction
9.2 Problem Formulation and ZNN Models
9.2.1 Problem Formulation
9.2.2 ZNN Model
9.2.3 RZNN Model
9.3 Design and Analysis of R‐FTZNN
9.3.1 Second‐Order Nonlinear Formula
9.3.2 R‐FTZNN Model
9.4 Illustrative Verification
9.4.1 Constant Noise
9.4.2 Dynamic Noise
9.5 Chapter Summary
References
Chapter 10 FTZNN for Quadratic Optimization
10.1 Introduction
10.2 Problem Formulation
10.3 Related Work: GNN and ZNN Models
10.3.1 GNN Model
10.3.2 ZNN Model
10.4 N‐FTZNN Model
10.4.1 Models Comparison
10.4.2 Finite‐Time Convergence
10.5 Illustrative Verification
10.6 Chapter Summary
References
Part V Application to the Lyapunov Equation
Chapter 11 Design Scheme I of FTZNN
11.1 Introduction
11.2 Problem Formulation and Related Work
11.2.1 GNN Model
11.2.2 ZNN Model
11.3 FTZNN Model
11.4 Illustrative Verification
11.5 Chapter Summary
References
Chapter 12 Design Scheme II of FTZNN
12.1 Introduction
12.2 Problem Formulation and Preliminaries
12.3 FTZNN Model
12.3.1 Design of FTZNN
12.3.2 Analysis of FTZNN
12.4 Illustrative Verification
12.5 Application to Tracking Control
12.6 Chapter Summary
References
Chapter 13 Design Scheme III of FTZNN
13.1 Introduction
13.2 N‐FTZNN Model
13.2.1 Design of N‐FTZNN
13.2.2 Re‐Interpretation from Nonlinear PID Perspective
13.3 Theoretical Analysis
13.4 Illustrative Verification
13.4.1 Numerical Comparison
13.4.2 Application Comparison
13.4.3 Experimental Verification
13.5 Chapter Summary
References
Part VI Application to the Sylvester Equation
Chapter 14 Design Scheme I of FTZNN
14.1 Introduction
14.2 Problem Formulation and ZNN Model
14.3 N‐FTZNN Model
14.3.1 Design of N‐FTZNN
14.3.2 Theoretical Analysis
14.4 Illustrative Verification
14.5 Robotic Application
14.6 Chapter Summary
References
Chapter 15 Design Scheme II of FTZNN
15.1 Introduction
15.2 ZNN Model and Activation Functions
15.2.1 ZNN Model
15.2.2 Commonly Used AFs
15.2.3 Two Novel Nonlinear AFs
15.3 NT‐PTZNN Models and Theoretical Analysis
15.3.1 NT‐PTZNN1 Model
15.3.1.1 Case 1
15.3.1.2 Case 2
15.3.2 NT‐PTZNN2 Model
15.3.2.1 Case 1
15.3.2.2 Case 2
15.4 Illustrative Verification
15.4.1 Example 1
15.4.2 Example 2
15.4.3 Example 3
15.5 Chapter Summary
References
Chapter 16 Design Scheme III of FTZNN
16.1 Introduction
16.2 ZNN Model and Activation Function
16.2.1 ZNN Model
16.2.2 Sign‐bi‐power Activation Function
16.3 FTZNN Models with Adaptive Coefficients
16.3.1 SA‐FTZNN Model
16.3.2 PA‐FTZNN Model
16.3.3 EA‐FTZNN Model
16.4 Illustrative Verification
16.5 Chapter Summary
References
Part VII Application to Inequality
Chapter 17 Design Scheme I of FTZNN
17.1 Introduction
17.2 FTZNN Models Design
17.2.1 Problem Formulation
17.2.2 ZNN Model
17.2.3 Vectorization
17.2.4 Activation Functions
17.2.5 FTZNN Models
17.3 Theoretical Analysis
17.3.1 Global Convergence
17.3.2 Finite‐Time Convergence
17.4 Illustrative Verification
17.5 Chapter Summary
References
Chapter 18 Design Scheme II of FTZNN
18.1 Introduction
18.2 NT‐FTZNN Model Deisgn
18.2.1 Problem Formulation
18.2.2 ZNN Model
18.2.3 NT‐FTZNN Model
18.2.4 Activation Functions
18.3 Theoretical Analysis
18.3.1 Global Convergence
18.3.2 Finite‐Time Convergence
18.3.3 Noise‐Tolerant Convergence
18.4 Illustrative Verification
18.5 Chapter Summary
References
Part VIII Application to Nonlinear Equation
Chapter 19 Design Scheme I of FTZNN
19.1 Introduction
19.2 Model Formulation
19.2.1 OZNN Model
19.2.2 FTZNN Model
19.2.3 Models Comparison
19.3 Convergence Analysis
19.4 Illustrative Verification
19.4.1 Nonlinear Equation f(u) with Simple Root
19.4.2 Nonlinear Equation f(u) with Multiple Root
19.5 Chapter Summary
References
Chapter 20 Design Scheme II of FTZNN
20.1 Introduction
20.2 Problem and Model Formulation
20.2.1 GNN Model
20.2.2 OZNN Model
20.3 FTZNN Model and Finite‐Time Convergence
20.4 Illustrative Verification
20.5 Chapter Summary
References
Chapter 21 Design Scheme III of FTZNN
21.1 Introduction
21.2 Problem Formulation and ZNN Models
21.2.1 Problem Formulation
21.2.2 ZNN Model
21.3 Robust and Fixed‐Time ZNN Model
21.4 Theoretical Analysis
21.4.1 Case 1: No Noise
21.4.2 Case 2: Under External Noises
21.5 Illustrative Verification
21.6 Chapter Summary
References
Index
EULA