**107,99 €**

- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
- Veröffentlichungsjahr: 2023

**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.

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Cover

Title Page

Copyright

List of Figures

List of Tables

Author Biographies

Preface

Acknowledgments

Part I: Application to Matrix Square Root

1 FTZNN for Time‐varying Matrix Square Root

1.1 Introduction

1.2 Problem Formulation and ZNN Model

1.3 FTZNN Model

1.4 Illustrative Verification

1.5 Chapter Summary

References

2 FTZNN for Static Matrix Square Root

2.1 Introduction

2.2 Solution Models

2.3 Illustrative Verification

2.4 Chapter Summary

References

Part II: Application to Matrix Inversion

3 Design Scheme I of FTZNN

3.1 Introduction

3.2 Problem Formulation and Preliminaries

3.3 FTZNN Model

3.4 Illustrative Verification

3.5 Chapter Summary

References

4 Design Scheme II of FTZNN

4.1 Introduction

4.2 Preliminaries

4.3 NT‐FTZNN Model

4.4 Theoretical Analysis

4.5 Illustrative Verification

4.6 Chapter Summary

References

5 Design Scheme III of FTZNN

5.1 Introduction

5.2 Problem Formulation and Neural Solver

5.3 Theoretical Analysis

5.4 Illustrative Verification

5.5 Chapter Summary

References

Part III: Application to Linear Matrix Equation

6 Design Scheme I of FTZNN

6.1 Introduction

6.2 Convergence Speed and Robustness Co‐design

6.3 R‐FTZNN Model

6.4 Illustrative Verification

6.5 Chapter Summary

References

7 Design Scheme II of FTZNN

7.1 Introduction

7.2 Problem Formulation

7.3 FTZNN Model

7.4 Theoretical Analysis

7.5 Illustrative Verification

7.6 Chapter Summary

References

Part IV: Application to Optimization

8 FTZNN for Constrained Quadratic Programming

8.1 Introduction

8.2 Preliminaries

8.3 U‐FTZNN Model

8.4 Convergence Analysis

8.5 Robustness Analysis

8.6 Illustrative Verification

8.7 Application to Image Fusion

8.8 Application to Robot Control

8.9 Chapter Summary

References

9 FTZNN for Nonlinear Minimization

9.1 Introduction

9.2 Problem Formulation and ZNN Models

9.3 Design and Analysis of R‐FTZNN

9.4 Illustrative Verification

9.5 Chapter Summary

References

10 FTZNN for Quadratic Optimization

10.1 Introduction

10.2 Problem Formulation

10.3 Related Work: GNN and ZNN Models

10.4 N‐FTZNN Model

10.5 Illustrative Verification

10.6 Chapter Summary

References

Part V: Application to the Lyapunov Equation

11 Design Scheme I of FTZNN

11.1 Introduction

11.2 Problem Formulation and Related Work

11.3 FTZNN Model

11.4 Illustrative Verification

11.5 Chapter Summary

References

12 Design Scheme II of FTZNN

12.1 Introduction

12.2 Problem Formulation and Preliminaries

12.3 FTZNN Model

12.4 Illustrative Verification

12.5 Application to Tracking Control

12.6 Chapter Summary

References

13 Design Scheme III of FTZNN

13.1 Introduction

13.2 N‐FTZNN Model

13.3 Theoretical Analysis

13.4 Illustrative Verification

13.5 Chapter Summary

References

Part VI: Application to the Sylvester Equation

14 Design Scheme I of FTZNN

14.1 Introduction

14.2 Problem Formulation and ZNN Model

14.3 N‐FTZNN Model

14.4 Illustrative Verification

14.5 Robotic Application

14.6 Chapter Summary

References

15 Design Scheme II of FTZNN

15.1 Introduction

15.2 ZNN Model and Activation Functions

15.3 NT‐PTZNN Models and Theoretical Analysis

15.4 Illustrative Verification

15.5 Chapter Summary

References

16 Design Scheme III of FTZNN

16.1 Introduction

16.2 ZNN Model and Activation Function

16.3 FTZNN Models with Adaptive Coefficients

16.4 Illustrative Verification

16.5 Chapter Summary

References

Part VII: Application to Inequality

17 Design Scheme I of FTZNN

17.1 Introduction

17.2 FTZNN Models Design

17.3 Theoretical Analysis

17.4 Illustrative Verification

17.5 Chapter Summary

References

18 Design Scheme II of FTZNN

18.1 Introduction

18.2 NT‐FTZNN Model Deisgn

18.3 Theoretical Analysis

18.4 Illustrative Verification

18.5 Chapter Summary

References

Part VIII: Application to Nonlinear Equation

19 Design Scheme I of FTZNN

19.1 Introduction

19.2 Model Formulation

19.3 Convergence Analysis

19.4 Illustrative Verification

19.5 Chapter Summary

References

20 Design Scheme II of FTZNN

20.1 Introduction

20.2 Problem and Model Formulation

20.3 FTZNN Model and Finite‐Time Convergence

20.4 Illustrative Verification

20.5 Chapter Summary

References

21 Design Scheme III of FTZNN

21.1 Introduction

21.2 Problem Formulation and ZNN Models

21.3 Robust and Fixed‐Time ZNN Model

21.4 Theoretical Analysis

21.5 Illustrative Verification

21.6 Chapter Summary

References

Index

End User License Agreement

Chapter 2

Table 2.1 The main novelties and differences of the FTZNN model from the GN...

Chapter 4

Table 4.1 The main differences of the NT‐FTZNN model from other models (i.e...

Table 4.2 Comparisons and differences of commonly used activation functions...

Chapter 6

Table 6.1 The main differences of the R‐FTZNN model from the GNN model ([16...

Chapter 10

Table 10.1 The main differences between this chapter and Ref. [12].

Chapter 13

Table 13.1 The main differences of the N‐FTZNN model from existing neural n...

Chapter 14

Table 14.1 The main novelties and differences of the N‐FTZNN model from the...

Chapter 16

Table 16.1 The convergence time of different ZNN models.

Chapter 1

Figure 1.1 Transient behavior of synthesized by the OZNN model (1.5) start...

Figure 1.2 Transient behavior of synthesized by FTZNN model (1.13) startin...

Figure 1.3 Transient behavior of the residual error corresponding to syn...

Chapter 2

Figure 2.1 Simulative results of OZNN model (2.3) using linear activation fu...

Figure 2.2 Simulative results of OZNN model (2.3) using power‐sigmoid activa...

Figure 2.3 Simulative results of FTZNN model (2.5) under the condition of ,...

Figure 2.4 Transient behavior of residual error synthesized by FTZNN model...

Chapter 3

Figure 3.1 Transient behavior of synthesized by GNN model 3.2 starting wit...

Figure 3.2 Transient behavior of synthesized by OZNN model 3.3 starting wi...

Figure 3.3 Transient behavior of the residual error corresponding to syn...

Figure 3.4 Transient behavior of synthesized by FTZNN model 3.11 starting ...

Figure 3.5 Transient behavior of the residual error synthesized by FTZNN m...

Figure 3.6 Transient behavior of the residual error synthesized by FTZNN m...

Figure 3.7 Transient behavior of the residual error synthesized by FTZNN m...

Chapter 4

Figure 4.1 Transient behavior of NT‐FTZNN model 4.6 activated by VAF for sol...

Figure 4.2 Transient behavior of the ZNN model activated by LAF for solving ...

Figure 4.3 Transient behavior of the ZNN model activated by power‐sum activa...

Figure 4.4 Transient behavior of the ZNN model activated by SBPAF for solvin...

Figure 4.5 Transient behavior of NT‐FTZNN model 4.6 activated by VAF for sol...

Figure 4.6 Transient behavior of residual errors synthesized by NT‐FTZNN m...

Figure 4.7 Transient behavior of residual errors synthesized by NT‐FTZNN m...

Figure 4.8 Transient behavior of residual errors synthesized by NT‐FTZNN m...

Figure 4.9 Circular task tracking synthesized by the original ZNN model acti...

Figure 4.10 Circular task tracking synthesized by NT‐FTZNN model 4.6 in the ...

Figure 4.11 Physical comparative experiments of a butterfly‐path tracking ta...

Chapter 5

Figure 5.1 Simulative results using FPZNN model (5.4) with SBPAF when solvin...

Figure 5.2 Simulative results using EVPZNN model (5.7) with SBPAF when solvi...

Figure 5.3 Simulative results using VPZNN model (5.9) with SBPAF when solvin...

Figure 5.4 Simulative results using IVP‐FTZNN model (5.11) with SBPAF when s...

Figure 5.5 Residual errors of FPZNN 5.4, EVPZNN 5.7, VPZNN 5.9, and IVP‐FT...

Figure 5.6 Residual errors of IVP‐FTZNN model (5.11) with different activa...

Figure 5.7 Simulative results using IVP‐FTZNN model (5.11) with SBPAF when s...

Figure 5.8 Simulative residual errors using FPZNN 5.4, EVPZNN 5.7, VPZNN 5.9...

Figure 5.9 Simulative residual errors using FPZNN 5.4, EVPZNN 5.7, VPZNN 5.9...

Figure 5.10 Simulative residual errors using FPZNN 5.4, EVPZNN 5.7, VPZNN 5....

Chapter 6

Figure 6.1 Simulative results generated by the R‐FTZNN model (6.26) for solv...

Figure 6.2 Simulative results generated by ZNN model (6.27) for solving the ...

Figure 6.3 Simulative results generated by the R‐FTZNN model (6.26) for solv...

Figure 6.4 Simulative results generated by ZNN model (6.27) for solving the ...

Figure 6.5 Residual errors generated by the R‐FTZNN model (6.26) and ZNN m...

Figure 6.6 Actual ellipse‐tracking results synthesized by ZNN model (6.27) i...

Figure 6.7 Actual ellipse‐tracking results synthesized by R‐FTZNN model (6.2...

Figure 6.8 Actual circle‐tracking results of the thee‐dimensional (3D) manip...

Chapter 7

Figure 7.1 Solved by the FTZNN1 model with and , where dash curves repres...

Figure 7.2 Solved by the FTZNN2 model with and , where dash curves repres...

Figure 7.3 Steady‐state error produced by proposed FTZNN models with and...

Figure 7.4 Steady‐state error . (a) By ZNN model (7.5) activated by differe...

Figure 7.5 Steady‐state error produced by perturbed ZNN model (7.11) activ...

Chapter 8

Figure 8.1 The state solutions computed by U‐FTZNN model (8.9) on solving di...

Figure 8.2 The residual error generated by U‐FTZNN model (8.9) on solving ...

Figure 8.3 Comparisons of residual error produced by different models with...

Figure 8.4 The convergence time of the U‐FTZNN model (8.10) solving the TVQP...

Figure 8.5 The maximum steady‐state residual error (MSSRE) of noise pertur...

Figure 8.6 Results of image fusion on a gray scale picture using U‐FTZNN m...

Figure 8.7 Computation error synthesized by U‐FTZNN model (8.9) during ima...

Figure 8.8 Results of image fusion on T1‐MR images using U‐FTZNN model (8.10...

Figure 8.9 Residual error synthesized by U‐FTZNN model (8.10) during image...

Figure 8.10 Simulation results of using U‐FTZNN model (8.10) to control PUM5...

Chapter 9

Figure 9.1 Computing nonlinear minimization problem by ZNN model (9.6) using...

Figure 9.2 Computing nonlinear minimization problem by RZNN model (9.8) with...

Figure 9.3 Computing nonlinear minimization problem by R‐FTZNN model (9.27) ...

Figure 9.4 Computing nonlinear minimization problem by ZNN model (9.6) using...

Figure 9.5 Computing nonlinear minimization problem by RZNN model (9.8) with...

Figure 9.6 Computing nonlinear minimization problem by R‐FTZNN model (9.27) ...

Figure 9.7 Residual error generated by R‐FTZNN model (9.8), ZNN model (9.6...

Chapter 10

Figure 10.1 “Moving” nonlinear constraint, “Moving” objective function, and ...

Figure 10.2 Simulative results of real‐time solution to nonstationary quadra...

Figure 10.3 Simulative results of real‐time solution to nonstationary quadra...

Figure 10.4 Simulative results of real‐time solution to nonstationary quadra...

Chapter 11

Figure 11.1 Transient behavior of synthesized by GNN model (11.3) and ZNN ...

Figure 11.2 Transient behavior of synthesized by GNN model (11.3) and ZNN ...

Figure 11.3 Simulative results synthesized by FTZNN model (11.8) starting fr...

Figure 11.4 Transient behavior of synthesized by FTZNN model (11.8) starti...

Chapter 12

Figure 12.1 Transient behavior of neural state synthesized by original ZNN...

Figure 12.2 Transient behavior of neural state synthesized by FTZNN model ...

Figure 12.3 Transient behavior of residual error corresponding to the neur...

Figure 12.4 Tracking ellipse‐path results of the mobile manipulator synthesi...

Figure 12.5 Tracking ellipse‐path results of the mobile manipulator synthesi...

Chapter 13

Figure 13.1 Block diagram of the control architecture of N‐FTZNN model (13.4...

Figure 13.2 Convergence property of each element of state output correspon...

Figure 13.3 Convergence property of each element of state output correspon...

Figure 13.4 Convergence of residual error produced by two different models...

Figure 13.5 Convergence of residual error produced by N‐FTZNN model (13.5)...

Figure 13.6 Convergence of residual error produced by N‐FTZNN model (13.5)...

Figure 13.7 Circular tracking results of the planar six‐link manipulator syn...

Figure 13.8 Circular tracking results of the planar six‐link manipulator syn...

Figure 13.9 Circular tracking experiment results synthesized by the N‐FTZNN‐...

Figure 13.10 Data profiles during the circular tracking experiment of the 3‐...

Chapter 14

Figure 14.1 The circuit topology of design formula (14.6) for hardware imple...

Figure 14.2 Trajectories of theoretical solution and state output genera...

Figure 14.3 Trajectories of residual error generated by N‐FTZNN model (14....

Figure 14.4 Trajectories of theoretical solution and state output genera...

Figure 14.5 Trajectories of residual error generated by two different mode...

Figure 14.6 Trajectories of residual error generated by two different mode...

Figure 14.7 Trajectories of residual error generated by two different mode...

Figure 14.8 Motion trajectories of a two‐link planar manipulator synthesized...

Figure 14.9 Motion trajectories of a two‐link planar manipulator synthesized...

Chapter 15

Figure 15.1 Transient behavior of state solutions generated by NT‐PTZNN1 m...

Figure 15.2 Transient behavior of residual errors generated by NT‐PTZNN1 m...

Figure 15.3 Transient behavior of residual errors synthesized by NT‐PTZNN1...

Figure 15.4 Transient behavior of state solution generated by NT‐PTZNN1 mo...

Figure 15.5 Transient behavior of residual errors synthesized by NT‐PTZNN1...

Figure 15.6 Transient behavior of residual errors synthesized by NT‐PTZNN1...

Figure 15.7 Transient behavior of residual errors synthesized by NT‐PTZNN1...

Chapter 16

Figure 16.1 Trajectories of theoretical solutions (dotted lines) of the time...

Figure 16.2 Comparisons of the computational errors generated by SA‐FTZNN mo...

Figure 16.3 Comparisons of the SA‐FTZNN model (16.22) with different values ...

Figure 16.4 Comparisons of SA‐FTZNN model (16.22), PA‐FTZNN model (16.29), E...

Chapter 17

Figure 17.1 Trajectories of state vector by applying FTZNN‐1 model (17.17)...

Figure 17.2 Comparisons of three FTZNN models with the conventional ZNN mode...

Figure 17.3 Comparisons of three FTZNN models with the conventional ZNN mode...

Figure 17.4 Comparisons among three cases of FTZNN‐3 model (17.21) with diff...

Figure 17.5 Transient behaviors of the error function synthesized by FTZNN...

Chapter 18

Figure 18.1 Trajectories of state matrix by applying NT‐FTZNN model (18.6)...

Figure 18.2 Trajectories of five random state matrix by applying NT‐FTZNN ...

Figure 18.3 Comparisons of the error function generated by NT‐FTZNN model (1...

Figure 18.4 Comparisons of the error functions generated by NT‐FTZNN model (...

Figure 18.5 Trajectories of the error function generated by NT‐FTZNN model (...

Figure 18.6 Comparisons among three cases of NT‐FTZNN model (18.6) with diff...

Figure 18.7 Comparisons among three cases of NT‐FTZNN model (18.6) with an...

Figure 18.8 Trajectories of theoretical solutions of the higher‐order time‐v...

Figure 18.9 Comparisons of the error functions generated by NT‐FTZNN model (...

Chapter 19

Figure 19.1 Transient behavior of neural states synthesized by different d...

Figure 19.2 Transient behavior of residual functions synthesized by differ...

Figure 19.3 Transient behavior of residual functions synthesized by FTZNN ...

Figure 19.4 Transient behavior of residual functions synthesized by differ...

Chapter 20

Figure 20.1 Transient behavior of neural states solved by GNN model (20.2)...

Figure 20.2 Transient behavior of residual errors synthesized by GNN model...

Figure 20.3 Transient behavior of simulation results synthesized by FTZNN mo...

Chapter 21

Figure 21.1 Simulative results of solving nonlinear equation (21.9) by ZNN m...

Figure 21.2 Simulative results of solving nonlinear equation (21.9) by ZNN m...

Figure 21.3 Simulative results of solving nonlinear equation (21.9) by R‐FTZ...

Figure 21.4 Transient behavior of residual errors by different ZNN models ...

Figure 21.5 Residual errors synthesized by different ZNN models under diff...

Cover Page

Table of Contents

Series Page

Title Page

Copyright

Dedication

List of Figures

List of Tables

Author Biographies

Preface

Acknowledgments

Begin Reading

Index

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IEEE Press

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IEEE Press Editorial Board

Sarah Spurgeon, Editor in Chief

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Peter (Yong) Lian

Andreas Molisch

Saeid Nahavandi

Jeffrey Reed

Thomas Robertazzi

Diomidis Spinellis

Ahmet Murat Tekalp

Lin Xiao

Hunan Normal University

Lei Jia

Hunan Normal University

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To our parents and ancestors, as always

Figure 1.1

Transient behavior of synthesized by the OZNN model (

1.5

) starting with 10 randomly generated initial states, where solid curves correspond to neural state , and dash curves correspond to theoretical matrix square root .

Figure 1.2

Transient behavior of synthesized by FTZNN model (

1.13

) starting with 10 randomly generated initial states, where solid curves correspond to neural state , and dash curves correspond to theoretical time‐varying matrix square root .

Figure 1.3

Transient behavior of the residual error corresponding to synthesized by the OZNN model (

1.5

) and FTZNN model (

1.13

). (a) By the OZNN model (

1.5

) and (b) by FTZNN model (

1.13

).

Figure 2.1

Simulative results of OZNN model (

2.3

) using linear activation functions under the condition of and a randomly generated close to . (a) Transient behavior of state matrix and (b) transient behavior of residual error .

Figure 2.2

Simulative results of OZNN model (

2.3

) using power‐sigmoid activation functions under the condition of and a randomly generated close to . (a) Transient behavior of state matrix and (b) transient behavior of residual error .

Figure 2.3

Simulative results of FTZNN model (

2.5

) under the condition of , and a randomly generated close to . (a) Transient behavior of state matrix and (b) transient behavior of residual error .

Figure 2.4

Transient behavior of residual error synthesized by FTZNN model (

2.5

) under the condition of and a randomly generated close to . (a) With and (b) with .

Figure 3.1

Transient behavior of synthesized by GNN model (

3.2

) starting with 8 randomly generated initial states under the condition of .

Figure 3.2

Transient behavior of synthesized by OZNN model (

3.3

) starting with 8 randomly generated initial states under the condition of .

Figure 3.3

Transient behavior of the residual error corresponding to synthesized by GNN model (

3.2

) and OZNN model (

3.3

). (a) By GNN model (

3.2

) and (b) by OZNN model (

3.3

).

Figure 3.4

Transient behavior of synthesized by FTZNN model (

3.11

) starting with 8 randomly generated initial states under the conditions of .

Figure 3.5

Transient behavior of the residual error synthesized by FTZNN model (

3.11

) under the conditions of . (a) and (b) .

Figure 3.6

Transient behavior of the residual error synthesized by FTZNN model (

3.11

) under the conditions of . (a) and (b) .

Figure 3.7

Transient behavior of the residual error synthesized by FTZNN model (

3.11

) using random time‐varying coefficients under the conditions of . (a) and (b) .

Figure 4.1

Transient behavior of NT‐FTZNN model (

4.6

) activated by VAF for solving time‐dependent matrix inversion (

4.9

) without noise. (a) State solutions and (b) residual error.

Figure 4.2

Transient behavior of the ZNN model activated by LAF for solving time‐dependent matrix inversion (

4.9

) with noise . (a) State solutions and (b) residual error.

Figure 4.3

Transient behavior of the ZNN model activated by power‐sum activation function (

PSAF

) for solving time‐dependent matrix inversion (

4.9

) with noise . (a) State solutions and (b) residual error.

Figure 4.4

Transient behavior of the ZNN model activated by SBPAF for solving time‐dependent matrix inversion (

4.9

) with noise . (a) State solutions and (b) residual error.

Figure 4.5

Transient behavior of NT‐FTZNN model (

4.6

) activated by VAF for solving time‐dependent matrix inversion (

4.9

) with noise . (a) State solutions and (b) residual error.

Figure 4.6

Transient behavior of residual errors synthesized by NT‐FTZNN model (

4.6

) activated by VAF and other ZNN models activated by different activation functions in different noise environments for solving time‐dependent matrix inversion (

4.9

). (a) Noise with , (b) noise with , (c) noise with , and (d) noise with .

Figure 4.7

Transient behavior of residual errors synthesized by NT‐FTZNN model (

4.6

) activated by VAF and other ZNN models activated by different activation functions in different noise environments for solving time‐dependent matrix inversion (

4.9

) with different values of parameter . (a) Noise with and (b) noise with .

Figure 4.8

Transient behavior of residual errors synthesized by NT‐FTZNN model (

4.6

) activated by VAF, GNN model, IEZNN model, and other ZNN models activated by LAF, PSAF, and SBPAF under different noise environments for solving time‐dependent matrix inversion (

4.11

). (a) , (b) , and (c) .

Figure 4.9

Circular task tracking synthesized by the original ZNN model activated by the SBP activation function in the presence of additive noise . (a) Whole tracking process, (b) task comparison, and (c) position error.

Figure 4.10

Circular task tracking synthesized by NT‐FTZNN model (

4.6

) in the presence of additive noise . (a) Whole tracking process, (b) task comparison, and (c) position error.

Figure 4.11

Physical comparative experiments of a butterfly‐path tracking task generated by different ZNN models and performed on the Kinova robot manipulator when disturbed by external noise. (a) Failure by SBPAF activated ZNN and (b) success by NT‐FTZNN.

Figure 5.1

Simulative results using FPZNN model (

5.4

) with SBPAF when solving TVMI (

5.1

) of Example 1 with . (a) State solution and (b) residual error .

Figure 5.2

Simulative results using EVPZNN model (

5.7

) with SBPAF when solving TVMI (

5.1

) of Example 1 with . (a) State solution and (b) residual error .

Figure 5.3

Simulative results using VPZNN model (

5.9

) with SBPAF when solving TVMI (

5.1

) of Example 1 with . (a) State solution and (b) residual error .

Figure 5.4

Simulative results using IVP‐FTZNN model (

5.11

) with SBPAF when solving TVMI (

5.1

) of Example 1 with . (a) State solution and (b) residual error .

Figure 5.5

Residual errors of FPZNN (

5.4

), EVPZNN (

5.7

), VPZNN (

5.9

), and IVP‐FTZNN (

5.11

) with SBPAF when solving TVMI (

5.1

) of Example 1 with .

Figure 5.6

Residual errors of IVP‐FTZNN model (

5.11

) with different activation functions when solving TVMI (

5.1

) of Example 1 with .

Figure 5.7

Simulative results using IVP‐FTZNN model (

5.11

) with SBPAF when solving TVMI (

5.1

) of Example 2 with . (a) State solution and (b) residual error .

Figure 5.8

Simulative residual errors using FPZNN (

5.4

), EVPZNN (

5.7

), VPZNN (

5.9

), and IVP‐FTZNN (

5.11

) with SBPAF when solving TVMI (

5.1

) of Example 2 with different and . (a) With , (b) with , (c) with , and (d) with .

Figure 5.9

Simulative residual errors using FPZNN (

5.4

), EVPZNN (

5.7

), VPZNN (

5.9

), and IVP‐FTZNN (

5.11

) with SBPAF when solving TVMI (

5.1

) of Example 2 with different , and noises . (a) With and , (b) with and , (c) with and , and (d) With and .

Figure 5.10

Simulative residual errors using FPZNN (

5.4

), EVPZNN (

5.7

), VPZNN (

5.9

), and IVP‐FTZNN (

5.11

) with SBPAF when solving TVMI (

5.1

) of Example 2 with and different noises . (a) With and with .

Figure 6.1

Simulative results generated by the R‐FTZNN model (

6.26

) for solving the time‐varying linear equation system with no noise. (a) The first element of the neural state and the theoretical solution . (b) The second element of neural state and the theoretical solution . (c) The residual error corresponding to the neural state .

Figure 6.2

Simulative results generated by ZNN model (

6.27

) for solving the time‐varying linear equation system with no noise. (a) The first element of the neural state and the theoretical solution . (b) The second element of neural state and the theoretical solution . (c) The residual error corresponding to the neural state .

Figure 6.3

Simulative results generated by the R‐FTZNN model (

6.26

) for solving the time‐varying linear equation system with the constant noise . (a) The first element of the neural state and the theoretical solution . (b) The second element of neural state and the theoretical solution . (c) The residual error corresponding to the neural state .

Figure 6.4

Simulative results generated by ZNN model (

6.27

) for solving the time‐varying linear equation system with the constant noise . (a) The first element of the neural state and the theoretical solution . (b) The second element of neural state and the theoretical solution . (c) The residual error corresponding to the neural state .

Figure 6.5

Residual errors generated by the R‐FTZNN model (

6.26

) and ZNN model (

6.27

) for solving the time‐varying linear equation system with the different noises and parameters. (a) The bounded noise with . (b) The time‐varying noise with . (c) The bounded noise with .

Figure 6.6

Actual ellipse‐tracking results synthesized by ZNN model (

6.27

) in the presence of the additive noise . (a) Actual ellipse‐tracking process. (b) Desired ellipse path and actual ellipse‐tracking trajectory. (c) Position error between desired ellipse path and actual ellipse‐tracking trajectory.

Figure 6.7

Actual ellipse‐tracking results synthesized by R‐FTZNN model (

6.26

) in the presence of the additive noise . (a) Actual ellipse‐tracking process. (b) Desired ellipse path and actual ellipse‐tracking trajectory. (c) Position error between desired ellipse path and actual ellipse‐tracking trajectory.

Figure 6.8

Actual circle‐tracking results of the thee‐dimensional (3D) manipulator synthesized by R‐FTZNN model (

6.26

) in the presence of the additive noise . (a) Desired circle path and actual circle‐tracking trajectory. (b) Position error between desired circle path and actual circle‐tracking trajectory. (c) Dynamic behavior of joint angle.

Figure 7.1

Solved by the FTZNN1 model with and , where dash curves represent theoretical solution of (

7.1

) and solid curves represent state solution of (

7.5

).

Figure 7.2

Solved by the FTZNN2 model with and , where dash curves represent theoretical solution of (

7.1

) and solid curves represent state solution of (

7.5

).

Figure 7.3

Steady‐state error produced by proposed FTZNN models with and . (a) By FTZNN1. (b) By FTZNN2.

Figure 7.4

Steady‐state error . (a) By ZNN model (

7.5

) activated by different activation functions with and . (b) By the perturbed ZNN model (

7.11

) activated by different activation functions with and .

Figure 7.5

Steady‐state error produced by perturbed ZNN model (

7.11

) activated by NFTAF activation functions with . (a) By NFTAF1 activation function with different . (b) By NFTAF2 activation function with different .

Figure 8.1

The state solutions computed by U‐FTZNN model (

8.9

) on solving different TVQP problems. (a) TVQPEI (

8.27

)–(

8.29

). (b) TVQPE (

8.30

)–(

8.31

).

Figure 8.2

The residual error generated by U‐FTZNN model (

8.9

) on solving different TVQP problems. (a) TVQPEI (

8.27

)–(

8.29

). (b) TVQPE (

8.30

)–(

8.31

).

Figure 8.3

Comparisons of residual error produced by different models with different noises. (a) by perturbed ZNN (

8.21

) and perturbed U‐FTZNN (

8.19

) with . (b) by perturbed U‐FTZNN with and .

Figure 8.4

The convergence time of the U‐FTZNN model (

8.10

) solving the TVQPEI‐C2 problem starting from different initial condition with different model variable settings. is the predefined convergence time bound by

Eq. (8.18)

. (a) while changing . (b) while changing . (c) while changing and .

Figure 8.5

The maximum steady‐state residual error (MSSRE) of noise perturbed U‐FTZNN model (

8.21

) solving the TVQPEI‐C2 problem with different model variable settings, where is the predicted MSSRE bound by inequality (

8.26

) when . (a) while changing and . (b) while changing and . (c) while changing and .

Figure 8.6

Results of image fusion on a gray scale picture using U‐FTZNN model (

8.10

). (a) Original image. (b) Noisy image with SNR = 1 dB. (c) Fused image when . (d) Fused image when .

Figure 8.7

Computation error synthesized by U‐FTZNN model (

8.9

) during image fusion with different . (a) . (b) .

Figure 8.8

Results of image fusion on T1‐MR images using U‐FTZNN model (

8.10

). (a) Slice 50 of T1‐MR. (b) Slice 50 polluted by Gaussian noise. (c) Fused image of noise‐polluted slice 50 when . (d) Slice 80 of T1‐MR. (e) Slice 80 polluted by the Rician noise. (f) Fused image of noise‐polluted slice 80 when .

Figure 8.9

Residual error synthesized by U‐FTZNN model (

8.10

) during image fusion when using different T1‐MR images. (a) Using Gaussian noise polluted Slice 50. (b) Using Rician noise polluted Slice 80.

Figure 8.10

Simulation results of using U‐FTZNN model (

8.10

) to control PUM560 robot manipulator to track the Lissajous curve path. (a) Relationship between and . (b) Trajectory of end effector and desired path. (c) Full view of path tracking with robot arm in Cartesian space. (d) Traces of joint angles. (e) Traces of joint velocities. (f) Position errors of the end effector in three dimensions.

Figure 9.1

Computing nonlinear minimization problem by ZNN model (

9.6

) using the sign‐bi‐power activation function with in front of constant noise . (a) Neural output . (b) Residual error .

Figure 9.2

Computing nonlinear minimization problem by RZNN model (

9.8

) with in front of constant noise . (a) Neural output . (b) Residual error .

Figure 9.3

Computing nonlinear minimization problem by R‐FTZNN model (

9.27

) using the sign‐bi‐power activation function with in front of constant noise . (a) Neural output . (b) Residual error .

Figure 9.4

Computing nonlinear minimization problem by ZNN model (

9.6

) using the sign‐bi‐power activation function with under constant noise . (a) Neural output . (b) Residual error .

Figure 9.5

Computing nonlinear minimization problem by RZNN model (

9.8

) with under dynamic noise . (a) Neural output . (b) Residual error .

Figure 9.6

Computing nonlinear minimization problem by R‐FTZNN model (

9.27

) using the sign‐bi‐power activation function with in front of dynamic noise . (a) Neural output . (b) Residual error .

Figure 9.7

Residual error generated by R‐FTZNN model (

9.8

), ZNN model (

9.6

), and RZNN model (

9.8

) with in front of different types of noises. (a) Bounded additive noise . (b) Linearly increasing noise .

Figure 10.1

“Moving” nonlinear constraint, “Moving” objective function, and “Moving” optimal solution of nonstationary quadratic optimization (

10.1

) and (

10.2

). (a) Snapshot at second. (b) Snapshot at seconds. (c) Snapshot at seconds. (d) Snapshot at seconds.

Figure 10.2

Simulative results of real‐time solution to nonstationary quadratic optimization (

10.4

) and (

10.5

) synthesized by GNN model (

10.12

) with . (a) Transient behavior of neural state . (b) Transient behavior of .

Figure 10.3

Simulative results of real‐time solution to nonstationary quadratic optimization (

10.4

) and (

10.5

) synthesized by ZNN model (

10.15

) with . (a) Transient behavior of neural state . (b) Transient behavior of .

Figure 10.4

Simulative results of real‐time solution to nonstationary quadratic optimization (

10.4

) and (

10.5

) synthesized by N‐FTZNN model (

10.18

) with . (a) Transient behavior of neural state . (b) Transient behavior of .

Figure 11.1

Transient behavior of synthesized by GNN model (

11.3

) and ZNN model (

11.5

) starting from randomly generated initial state matrices . (a) By GNN model (

11.3

) and (b) by ZNN model (

11.5

).

Figure 11.2

Transient behavior of synthesized by GNN model (

11.3

) and ZNN model (

11.5

) starting from randomly generated initial state matrices . (a) By GNN model (

11.3

) and (b) by ZNN model (

11.5

).

Figure 11.3

Simulative results synthesized by FTZNN model (

11.8

) starting from randomly generated initial state matrices . (a) Transient behavior of state matrices and (b) transient behavior of residual errors.

Figure 11.4

Transient behavior of synthesized by FTZNN model (

11.8

) starting with a randomly generated initial state. (a) With and (b) With .

Figure 12.1

Transient behavior of neural state synthesized by original ZNN model (

12.4

).

Figure 12.2

Transient behavior of neural state synthesized by FTZNN model (

12.11

).

Figure 12.3

Transient behavior of residual error corresponding to the neural state synthesized by original ZNN model (

12.4

) and FTZNN model (

12.11

). (a) By the original ZNN model (

12.4

) and (b) by FTZNN model (

12.11

).

Figure 12.4

Tracking ellipse‐path results of the mobile manipulator synthesized by the proposed model (

12.15

). (a) Whole tracking motion trajectories and (b) top graph of tracking motion trajectories.

Figure 12.5

Tracking ellipse‐path results of the mobile manipulator synthesized by the proposed model (

12.15

). (a) Desired path and actual trajectory and (b) tracking errors at the joint position level.

Figure 13.1

Block diagram of the control architecture of N‐FTZNN model (

13.4

) for solving dynamic Lyapunov from the nonlinear PID perspective.

Figure 13.2

Convergence property of each element of state output corresponding to the one of theoretical solution synthesized by N‐FTZNN model (

13.5

) with in the presence of additive constant noise .

Figure 13.3

Convergence property of each element of state output corresponding to the one of theoretical solution synthesized by ZNN model (

13.6

) with in the presence of additive constant noise .

Figure 13.4

Convergence of residual error produced by two different models with in the presence of additive constant noise . (a) By N‐FTZNN model (

13.5

). (b) By ZNN model (

13.6

).

Figure 13.5

Convergence of residual error produced by N‐FTZNN model (

13.5

) with different values of parameters and in the presence of additive constant noise . (a) . (b) .

Figure 13.6

Convergence of residual error produced by N‐FTZNN model (

13.5

) and ZNN model (

13.6

) with in the presence of different kinds of additive noises. (a) Bounded random additive . (b) Time‐varying additive .

Figure 13.7

Circular tracking results of the planar six‐link manipulator synthesized by the N‐FTZNN‐based control law with in the presence of additive constant noise . (a) Joint motion trajectories of planar six‐link manipulator. (b) Comparison between the desired path and the actual trajectory. (c) Position error between the desired path and the actual trajectory. (d) Dynamic behavior of joint angle.

Figure 13.8

Circular tracking results of the planar six‐link manipulator synthesized by the ZNN‐based control law with in the presence of additive constant noise . (a) Joint motion trajectories of planar six‐link manipulator. (b) Comparison between the desired path and the actual trajectory. (c) Position error between the desired path and the actual trajectory. (d) Dynamic behavior of joint angle.

Figure 13.9

Circular tracking experiment results synthesized by the N‐FTZNN‐based control law with in the presence of additive constant noise .

Figure 13.10

Data profiles during the circular tracking experiment of the 3‐axis physical manipulator. (a) Comparison between the desired path and the actual trajectory. (b) Position error between the desired path and the actual trajectory. (c) Dynamic behavior of joint angle. (d) Dynamic behavior of joint velocity.

Figure 14.1

The circuit topology of design formula (

14.6

) for hardware implementation.

Figure 14.2

Trajectories of theoretical solution and state output generated by N‐FTZNN model (

14.9

) with in front of no noise, where solid curves denote the elements of state output , and dash curves denotes the elements of .

Figure 14.3

Trajectories of residual error generated by N‐FTZNN model (

14.9

) with different values of and in front of no noise. (a) With . (b) With .

Figure 14.4

Trajectories of theoretical solution and state output generated by ZNN model (

14.4

) with in front of the additive constant noise , where solid curves denote the elements of state output , and dash curves denotes the elements of .

Figure 14.5

Trajectories of residual error generated by two different models with in front of additive constant noise . (a) By ZNN model (

14.5

). (b) By N‐FTZNN model (

14.9

).

Figure 14.6

Trajectories of residual error generated by two different models with in front of additive constant noise . (a) By ZNN model (

14.5

). (b) By N‐FTZNN model (

14.9

).

Figure 14.7

Trajectories of residual error generated by two different models with in front of additive dynamic noise . (a) By ZNN model (

14.5

). (b) By N‐FTZNN model (

14.9

).

Figure 14.8

Motion trajectories of a two‐link planar manipulator synthesized by the N‐FTZNN‐based model (

14.30

) with in front of the additive constant noise . (a) The actual circle trajectory and the desired path. (b) The whole motion process. (c) The control law . (d) The position tracking error .

Figure 14.9

Motion trajectories of a two‐link planar manipulator synthesized by ZNN‐based model (

14.31

) with in front of the additive constant noise . (a) The actual circle trajectory and the desired path. (b) The whole motion process. (c) The control law . (d) The position tracking error .

Figure 15.1

Transient behavior of state solutions generated by NT‐PTZNN1 model (

15.9

) and NT‐PTZNN2 model (

15.13

) when solving time‐variant Sylvester equation of Example 1 with noise . (a) By NT‐PTZNN1 model (

15.9

). (b) By NT‐PTZNN2 model (

15.13

).

Figure 15.2

Transient behavior of residual errors generated by NT‐PTZNN1 model (

15.9

) and NT‐PTZNN2 model (

15.13

) when solving time‐variant Sylvester equation of Example 1 with noise . (a) By NT‐PTZNN1 model (

15.9

). (b) By NT‐PTZNN2 model (

15.13

).

Figure 15.3

Transient behavior of residual errors synthesized by NT‐PTZNN1 model (

15.9

) activated by AF (

15.5

), NT‐PTZNN2 model (

15.13

) activated by AF (

15.6

) and ZNN model (

15.4

) activated by LAF, PSAF, and SBPAF under different kinds of noises . (a) With noise . (b) With noise . (c) With noise . (d) With noise .

Figure 15.4

Transient behavior of state solution generated by NT‐PTZNN1 model (

15.9

) and NT‐PTZNN2 model (

15.13

) when solving time‐variant Sylvester equation of Example 2 with noise . (a) By NT‐PTZNN1 model (

15.9

). (b) By NT‐PTZNN2 model (

15.13

).

Figure 15.5

Transient behavior of residual errors synthesized by NT‐PTZNN1 model (

15.9

), NT‐PTZNN2 model (

15.13

), and ZNN model (

15.4

) activated by LAF, PSAF, and SBPAF under different kinds of noises with . (a) With noise . (b) With noise . (c) With noise . (d) With noise .

Figure 15.6

Transient behavior of residual errors synthesized by NT‐PTZNN1 model (

15.9

), NT‐PTZNN2 model (

15.13

), and ZNN model (

15.4

) activated by LAF, PSAF, and SBPAF under different kinds of noises with and . (a) With noise and . (b) With noise and .

Figure 15.7

Transient behavior of residual errors synthesized by NT‐PTZNN1 model (

15.9

), NT‐PTZNN2 model (

15.13

), and ZNN model (

15.4

) activated by LAF, PSAF, and SBPAF under different kinds of noises with . (a) With noise . (b) With noise . (c) With noise . (d) With noise .

Figure 16.1

Trajectories of theoretical solutions (dotted lines) of the time‐varying Sylvester equation and state solutions (dash‐dotted, dashed, and solid curves) generated by SA‐FTZNN model (

16.22

), PA‐FTZNN model (

16.29

), and EA‐FTZNN model (

16.37

).

Figure 16.2

Comparisons of the computational errors generated by SA‐FTZNN model (

16.22

), PA‐FTZNN model (

16.29

), EA‐FTZNN model (

16.37

), and the ZNN activated by sign‐bi‐power function (

16.5

).

Figure 16.3

Comparisons of the SA‐FTZNN model (

16.22

) with different values of .

Figure 16.4

Comparisons of SA‐FTZNN model (

16.22

), PA‐FTZNN model (

16.29

), EA‐FTZNN model (

16.37

) and the ZNN activated by SP‐1 function (

16.7

) and SP‐2 function (

16.8

) while setting a large and small initial state. (a) Large initial state. (b) Small initial state.

Figure 17.1

Trajectories of state vector by applying FTZNN‐1 model (

17.17

) to solve LMI (

17.1

) when is outside with and . (a) , (b) , (c) , (d) , (e) , and (f) .

Figure 17.2

Comparisons of three FTZNN models with the conventional ZNN models activated by other AFs with and when . (a) By FTZNN‐1 model (

17.17

) and FTZNN‐2 model (

17.19

). (b) By FTZNN‐1 model (

17.17

) and FTZNN‐3 model (

17.21

).

Figure 17.3

Comparisons of three FTZNN models with the conventional ZNN models activated by other AFs with and when . (a) By FTZNN‐1 model (

17.17

) and FTZNN‐2 model (

17.19

). (b) By FTZNN‐1 model (

17.17

) and FTZNN‐3 model (

17.21

).

Figure 17.4

Comparisons among three cases of FTZNN‐3 model (

17.21

) with different tunable parameters. (a) . (b) .

Figure 17.5

Transient behaviors of the error function synthesized by FTZNN‐3 model (

17.21

) with , and different values of .

Figure 18.1

Trajectories of state matrix by applying NT‐FTZNN model (

18.6

) to solve linear inequalities (

18.1

) when is outside with and . (a) , (b) , (c) , and (d) .

Figure 18.2

Trajectories of five random state matrix by applying NT‐FTZNN model (

18.6

) with and to solve linear inequalities (

18.1

) when is outside .

Figure 18.3

Comparisons of the error function generated by NT‐FTZNN model (

18.6

) with and and the error function created by the CZNN model with which are both activated by linear AF.

Figure 18.4

Comparisons of the error functions generated by NT‐FTZNN model (

18.6

) with and which are activated by several of AFs under the same conditions.

Figure 18.5

Trajectories of the error function generated by NT‐FTZNN model (

18.6

) activated by two sign‐bi‐power AFs with , and to solve linear inequalities (

18.1

). (a) When the initial state . (b) When the initial state .

Figure 18.6

Comparisons among three cases of NT‐FTZNN model (

18.6

) with different to solve linear time‐varying inequalities (

18.1

).

Figure 18.7

Comparisons among three cases of NT‐FTZNN model (

18.6

) with and different to solve linear time‐varying inequalities (

18.1

).

Figure 18.8

Trajectories of theoretical solutions of the higher‐order time‐varying inequalities and actual solutions starting with random initial state using NT‐FTZNN model (

18.6

) with , , and .

Figure 18.9

Comparisons of the error functions generated by NT‐FTZNN model (

18.6

) with and which are activated by different AFs for solving higher‐order time‐varying inequalities.

Figure 19.1

Transient behavior of neural states synthesized by different dynamical models with and 50 randomly generated initial states within . (a) By OZNN model (

19.4

). (b) By FTZNN model (

19.5

).

Figure 19.2

Transient behavior of residual functions synthesized by different dynamical models with and 50 randomly generated initial states within . (a) By OZNN model (

19.4

). (b) By FTZNN model (

19.5

).

Figure 19.3

Transient behavior of residual functions synthesized by FTZNN model (

19.5

) with different values of and 50 randomly generated initial states within . (a) . (b) .

Figure 19.4

Transient behavior of residual functions synthesized by different dynamical models with and 5 randomly generated initial states within . (a) By OZNN model (

19.4

). (b) By FTZNN model (

19.5

).

Figure 20.1

Transient behavior of neural states solved by GNN model (

20.2

) and OZNN model (

20.4

) starting with 20 randomly generated initial states and with , where solid curves correspond to neural state , and dash curves correspond to time‐varying theoretical solutions . (a) By GNN model (

20.2

). (b) By OZNN model (

20.4

).

Figure 20.2

Transient behavior of residual errors synthesized by GNN model (

20.2

) and OZNN model (

20.4

) starting with 20 randomly generated initial states and with . (a) By GNN model (

20.2

). (b) By OZNN model (

20.4

).

Figure 20.3

Transient behavior of simulation results synthesized by FTZNN model (

20.7

) starting with 20 randomly generated initial states and with . (a) Neural state . (b) Residual error .

Figure 21.1

Simulative results of solving nonlinear equation (

21.9

) by ZNN model (

21.4

) with the power‐sigmoid activation function from 18 different initial states. (a) State trajectories. (b) Error trajectories.

Figure 21.2

Simulative results of solving nonlinear equation (

21.9

) by ZNN model (

21.4

) with the SBP activation function from 18 different initial states. (a) State trajectories. (b) Error trajectories.

Figure 21.3

Simulative results of solving nonlinear equation (

21.9

) by R‐FTZNN model (

21.6

) from 18 different initial states and with . (a) State trajectories. (b) Error trajectories.

Figure 21.4

Transient behavior of residual errors by different ZNN models from 18 different initial states and with and constant noise . (a) By R‐FTZNN model (

21.6

). (b) By ZNN model (

21.4

) with SBP function.

Figure 21.5

Residual errors synthesized by different ZNN models under different types of external noises with . (a) By the vanishing noise . (b) By the vanishing noise . (c) By the periodic noise . (d) By the non‐vanishing noise .

Table 2.1

The main novelties and differences of the FTZNN model from the GNN model and the OZNN model for matrix square root finding.

Table 4.1

The main differences of the NT‐FTZNN model from other models (i.e. GNN model,

ZNN

model, and EIZNN model) for time‐independent matrix inversion.

Table 4.2

Comparisons and differences of commonly used activation functions.

Table 6.1

The main differences of the R‐FTZNN model from the GNN model

[16]

, the ZNN model

[17]

and the FTZNN model

[18]

for systems of linear equations.

Table 10.1

The main differences between this chapter and Ref.

[12]

.

Table 13.1

The main differences of the N‐FTZNN model from existing neural network models for Lyapunov equations.

Table 14.1

The main novelties and differences of the N‐FTZNN model from the other models (i.e. GRNN model

[11]

, ZNN model

[12]

, and FTZNN model

[13]

) for Sylvester equation.

Table 16.1

The convergence time of different ZNN models.

Lin Xiao received the B.S. degree in Electronic Information Science and Technology from Hengyang Normal University, Hengyang, China, in 2009, and the Ph.D. degree in Communication and Information Systems from Sun Yat‐sen University, Guangzhou, China, in 2014. He is currently a Professor with the College of Information Science and Engineering, Hunan Normal University, Changsha, China. He has authored over 100 papers in international conferences and journals, such as the IEEE‐TNNLS, the IEEE‐TCYB, the IEEE‐TII, and the IEEE‐TSMCA. He is an Associate Editor of IEEE‐TNNLS. His main research interests include neural networks, robotics, and intelligent information processing.

Lei Jia received the B.S. degree in Information and Computing Science from InnerMongolia Normal University, Hohhot, China, in 2018. She is currently studying toward the Ph.D. degree in Operations Research and Control from College of Mathematics and Statistics, Hunan Normal University, Changsha, China. Her main research interests include neural networks and image processing.