# An Accelerated Double-Integral ZNN with Resisting Linear Noise for Dynamic Sylvester Equation Solving and Its Application to the Control of the SFM Chaotic System

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Based on the novel ZNN design formula, an innovative ADIZNN is constructed for settling the dynamic Sylvester equation under the linear noise.
- The ADIZNN model has a novel double integral structure and activation function, which guarantees accelerated convergence and enhanced anti-noise capacity.
- Theoretical analyses and simulation results are provided to ensure that the ADIZNN model can handle the DSE with excellent convergence and robustness.
- Chaos control schemes of the TFM chaotic system are established to display that the controller based on the ADIZNN has superior performance than that based on the OZNN and IEZNN.

## 2. DSE Description and Models Design

#### 2.1. Description of DSE

#### 2.2. Relevant Models Design

#### 2.3. ADIZNN Model Design

**Remark**

**1.**

**Remark**

**2.**

- Based on the novel ZNN design formula, an innovative ADIZNN is constructed for settling the DSE under the linear noise.
- The novel double integral structure and activation function, which guarantees accelerated convergence and enhanced anti-noise capacity.

## 3. Theoretical Analyses

#### 3.1. Convergence

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

#### 3.2. Robustness

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

## 4. Examples Verification

**Remark**

**3.**

#### 4.1. Experiment 1

**Remark**

**4.**

#### 4.2. Experiment 2

## 5. Application to the Control of the Sine Function Memristor Chaotic System

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DSE | Dynamic Sylvester equation |

ZNN | Zeroing neural network |

OZNN | Original zeroing neural network |

ADIZNN | Accelerated double integral ZNN |

SFM | Sine function memristor |

RNNs | recurrent neural networks |

GNN | Gradient neural network |

IEZNN | integral enhanced ZNN model |

FTAF | fixed-time activation function |

## References

- Wei, Q.; Dobigeon, N.; Tourneret, J.Y.; Bioucas-Dias, J.; Godsill, S. R-FUSE: Robust fast fusion of multiband images based on solving a Sylvester equation. IEEE Signal Process. Lett.
**2016**, 23, 1632–1636. [Google Scholar] [CrossRef] [Green Version] - Huo, L.; Yang, S.; Jiao, L.; Wang, S.; Shi, J. Local graph regularized coding for salient object detection. Infrared Phys. Technol.
**2016**, 77, 124–131. [Google Scholar] [CrossRef] - Shaker, H.R.; Tahavori, M. Control configuration selection for bilinear systems via generalised Hankel interaction index array. Int. J. Control
**2015**, 88, 30–37. [Google Scholar] [CrossRef] - Dolgov, S.; Pearson, J.W.; Savostyanov, D.V.; Stoll, M. Fast tensor product solvers for optimization problems with fractional differential equations as constraints. Appl. Math. Comput.
**2016**, 273, 604–623. [Google Scholar] [CrossRef] - Jin, L.; Yan, J.; Du, X.; Xiao, X.; Fu, D. RNN for solving time-variant generalized Sylvester equation with applications to robots and acoustic source localization. IEEE Trans. Ind. Inform.
**2020**, 16, 6359–6369. [Google Scholar] [CrossRef] - Katsikis, V.N.; Mourtas, S.D.; Stanimirović, P.S.; Zhang, Y. Solving complex-valued time-varying linear matrix equations via QR decomposition with applications to robotic motion tracking and on angle-of-arrival localization. IEEE Trans. Neural Netw. Learn. Syst.
**2021**, 33, 3415–3424. [Google Scholar] [CrossRef] [PubMed] - Li, W.; Han, L.; Xiao, X.; Liao, B.; Peng, C. A gradient-based neural network accelerated for vision-based control of an RCM-constrained surgical endoscope robot. Neural Comput. Appl.
**2022**, 34, 1329–1343. [Google Scholar] [CrossRef] - Li, Z.; Liao, B.; Xu, F.; Guo, D. A new repetitive motion planning scheme with noise suppression capability for redundant robot manipulators. IEEE Trans. Syst. Man Cybern. Syst.
**2018**, 50, 5244–5254. [Google Scholar] [CrossRef] - Liao, B.; Han, L.; Cao, X.; Li, S.; Li, J. Double integral-enhanced Zeroing neural network with linear noise rejection for time-varying matrix inverse. CAAI Trans. Intell. Technol.
**2023**, 1–14. [Google Scholar] [CrossRef] - Yan, X.; Liu, M.; Jin, L.; Li, S.; Hu, B.; Zhang, X.; Huang, Z. New zeroing neural network models for solving nonstationary Sylvester equation with verifications on mobile manipulators. IEEE Trans. Ind. Inform.
**2019**, 15, 5011–5022. [Google Scholar] [CrossRef] - Song, C.; Feng, J.; Wang, X.; Zhao, J. Finite iterative method for solving coupled Sylvester-transpose matrix equations. J. Appl. Math. Comput.
**2014**, 46, 351–372. [Google Scholar] [CrossRef] - Ali Beik, F.P.; Movahed, F.S.; Ahmadi-Asl, S. On the Krylov subspace methods based on tensor format for positive definite Sylvester tensor equations. Numer. Linear Algebra Appl.
**2016**, 23, 444–466. [Google Scholar] [CrossRef] - Wu, H.C.; Chen, T.C.T.; Chiu, M.C. Constructing a precise fuzzy feedforward neural network using an independent fuzzification approach. Axioms
**2021**, 10, 282. [Google Scholar] [CrossRef] - Tuyen, D.N.; Tuan, T.M.; Le, X.H.; Tung, N.T.; Chau, T.K.; Van Hai, P.; Gerogiannis, V.C.; Son, L.H. RainPredRNN: A new approach for precipitation nowcasting with weather radar echo images based on deep learning. Axioms
**2022**, 11, 107. [Google Scholar] [CrossRef] - Su, L.; Zhou, L. Exponential synchronization of memristor-based recurrent neural networks with multi-proportional delays. Neural Comput. Appl.
**2019**, 31, 7907–7920. [Google Scholar] [CrossRef] - Khan, A.H.; Li, S.; Luo, X. Obstacle avoidance and tracking control of redundant robotic manipulator: An RNN-based metaheuristic approach. IEEE Trans. Ind. Inform.
**2019**, 16, 4670–4680. [Google Scholar] [CrossRef] [Green Version] - Jin, L.; Li, S.; Hu, B. RNN models for dynamic matrix inversion: A control-theoretical perspective. IEEE Trans. Ind. Inform.
**2017**, 14, 189–199. [Google Scholar] [CrossRef] - He, X.; Liu, Q.; Yang, Y. MV-GNN: Multi-view graph neural network for compression artifacts reduction. IEEE Trans. Image Process.
**2020**, 29, 6829–6840. [Google Scholar] [CrossRef] - Zhang, Y.; Jiang, D.; Wang, J. A recurrent neural network for solving Sylvester equation with time-varying coefficients. IEEE Trans. Neural Netw.
**2002**, 13, 1053–1063. [Google Scholar] [CrossRef] - Zhang, Z.; Zheng, L.; Weng, J.; Mao, Y.; Lu, W.; Xiao, L. A new varying-parameter recurrent neural-network for online solution of time-varying Sylvester equation. IEEE Trans. Cybern.
**2018**, 48, 3135–3148. [Google Scholar] [CrossRef] [PubMed] - Xiao, L.; Zhang, Z.; Zhang, Z.; Li, W.; Li, S. Design, verification and robotic application of a novel recurrent neural network for computing dynamic Sylvester equation. Neural Netw.
**2018**, 105, 185–196. [Google Scholar] [CrossRef] - Qiu, B.; Zhang, Y.; Yang, Z. New discrete-time ZNN models for least-squares solution of dynamic linear equation system with time-varying rank-deficient coefficient. IEEE Trans. Neural Netw. Learn. Syst.
**2018**, 29, 5767–5776. [Google Scholar] [CrossRef] [PubMed] - He, Y.; Liao, B.; Xiao, L.; Han, L.; Xiao, X. Double accelerated convergence ZNN with noise-suppression for handling dynamic matrix inversion. Mathematics
**2021**, 10, 50. [Google Scholar] [CrossRef] - Xiao, L.; He, Y.; Li, Y.; Dai, J. Design and analysis of two nonlinear ZNN models for matrix LR and QR factorization with application to 3D moving target location. IEEE Trans. Ind. Inform.
**2022**, 1–11. [Google Scholar] [CrossRef] - Katsikis, V.N.; Mourtas, S.D.; Stanimirović, P.S.; Zhang, Y. Continuous-time varying complex QR decomposition via zeroing neural dynamics. Neural Process Lett.
**2021**, 53, 3573–3590. [Google Scholar] [CrossRef] - Xiao, L.; He, Y. A noise-suppression ZNN model with new variable parameter for dynamic Sylvester equation. IEEE Trans. Ind. Inform.
**2021**, 17, 7513–7522. [Google Scholar] [CrossRef] - Tang, G.; Li, X.; Xu, Z.; Li, S.; Zhou, X. An integration-enhanced noise-resistant RNN model with superior performance illustrated via time-varying sylvester equation solving. In Proceedings of the IEEE 2020 Chinese Control And Decision Conference (CCDC), Hefei, China, 22–24 August 2020; pp. 1906–1911. [Google Scholar]
- Gong, J.; Jin, J. A faster and better robustness zeroing neural network for solving dynamic Sylvester equation. Neural Process Lett.
**2021**, 53, 3591–3606. [Google Scholar] [CrossRef] - Han, L.; Liao, B.; He, Y.; Xiao, X. Dual noise-suppressed ZNN with predefined-time convergence and its application in matrix inversion. In Proceedings of the IEEE 2021 11th International Conference on Intelligent Control and Information Processing (ICICIP), Denver, CO, USA, 28–30 September 2021; pp. 410–415. [Google Scholar]
- Xiao, L.; He, Y.; Dai, J.; Liu, X.; Liao, B.; Tan, H. A variable-parameter noise-tolerant zeroing neural network for time-variant matrix inversion with guaranteed robustness. IEEE Trans. Neural Netw. Learn. Syst.
**2020**, 33, 1535–1545. [Google Scholar] [CrossRef] - Guo, D.; Li, S.; Stanimirović, P.S. Analysis and application of modified ZNN design with robustness against harmonic noise. IEEE Trans. Ind. Inform.
**2019**, 16, 4627–4638. [Google Scholar] [CrossRef] - Jin, L.; Zhang, Y.; Li, S. Integration-enhanced Zhang neural network for real-time-varying matrix inversion in the presence of various kinds of noises. IEEE Trans. Neural Netw. Learn. Syst.
**2015**, 27, 2615–2627. [Google Scholar] [CrossRef] - Dzieciol, H.; Sillekens, E.; Lavery, D. Extending phase noise tolerance in UDWDM access networks. In Proceedings of the 2020 IEEE Photonics Society Summer Topicals Meeting Series (SUM), Virtual, 13–15 July 2020; pp. 1–2. [Google Scholar]
- Xiang, Q.; Liao, B.; Xiao, L.; Jin, L. A noise-tolerant Z-type neural network for time-dependent pseudoinverse matrices. Optik
**2018**, 165, 16–28. [Google Scholar] [CrossRef] - Johnson, M.A.; Moradi, M.H. PID Control; Springer: Berlin/Heidelberg, Germany, 2005. [Google Scholar]
- Liao, B.; Han, L.; He, Y.; Cao, X.; Li, J. Prescribed-time convergent adaptive ZNN for time-varying matrix inversion under harmonic noise. Electronics
**2022**, 11, 1636. [Google Scholar] [CrossRef] - Jin, J.; Qiu, L. A robust fast convergence zeroing neural network and its applications to dynamic Sylvester equation solving and robot trajectory tracking. J. Frankl. Inst.
**2022**, 359, 3183–3209. [Google Scholar] [CrossRef] - Zhang, Y.; Yi, C.; Guo, D.; Zheng, J. Comparison on Zhang neural dynamics and gradient-based neural dynamics for online solution of nonlinear time-varying equation. Neural Comput. Appl.
**2011**, 20, 1–7. [Google Scholar] [CrossRef] - Zhang, Y.; Jin, L.; Ke, Z. Superior performance of using hyperbolic sine activation functions in ZNN illustrated via time-varying matrix square roots finding. Comput. Sci. Inf. Syst.
**2012**, 9, 1603–1625. [Google Scholar] [CrossRef] - Yang, Y.; Zhang, Y. Superior robustness of power-sum activation functions in Zhang neural networks for time-varying quadratic programs perturbed with large implementation errors. Neural Comput. Appl.
**2013**, 22, 175–185. [Google Scholar] [CrossRef] - Zhang, Y.; Ding, Y.; Qiu, B.; Zhang, Y.; Li, X. Signum-function array activated ZNN with easier circuit implementation and finite-time convergence for linear systems solving. Inf. Process. Lett.
**2017**, 124, 30–34. [Google Scholar] [CrossRef] - Benner, P. Factorized solution of Sylvester equations with applications in control. Sign (H)
**2004**, 1, 2. [Google Scholar] - Castelan, E.B.; da Silva, V.G. On the solution of a Sylvester equation appearing in descriptor systems control theory. Syst. Control Lett.
**2005**, 54, 109–117. [Google Scholar] [CrossRef] - Wei, Q.; Dobigeon, N.; Tourneret, J.Y. Fast fusion of multi-band images based on solving a Sylvester equation. IEEE Trans. Image Process.
**2015**, 24, 4109–4121. [Google Scholar] [CrossRef] [Green Version] - Diao, H.; Shi, X.; Wei, Y. Effective condition numbers and small sample statistical condition estimation for the generalized Sylvester equation. Sci. China Math.
**2013**, 56, 967–982. [Google Scholar] [CrossRef] - Zhang, R.; Xi, X.; Tian, H.; Wang, Z. Dynamical analysis and finite-time synchronization for a chaotic system with hidden attractor and surface equilibrium. Axioms
**2022**, 11, 579. [Google Scholar] [CrossRef] - Rasouli, M.; Zare, A.; Hallaji, M.; Alizadehsani, R. The synchronization of a class of time-delayed chaotic systems using sliding mode control based on a fractional-order nonlinear PID sliding surface and its application in secure communication. Axioms
**2022**, 11, 738. [Google Scholar] [CrossRef] - He, W.; Luo, T.; Tang, Y.; Du, W.; Tian, Y.C.; Qian, F. Secure communication based on quantized synchronization of chaotic neural networks under an event-triggered strategy. IEEE Trans. Neural Netw. Learn. Syst.
**2019**, 31, 3334–3345. [Google Scholar] [CrossRef] [PubMed] - Xiao, L.; He, Y.; Liao, B. A parameter-changing zeroing neural network for solving linear equations with superior fixed-time convergence. Expert Syst. Appl.
**2022**, 208, 118086. [Google Scholar] [CrossRef] - Su, H.; Luo, R.; Huang, M.; Fu, J. Robust fixed time control of a class of chaotic systems with bounded uncertainties and disturbances. Int. J. Control Autom. Syst.
**2022**, 20, 813–822. [Google Scholar] [CrossRef] - Singer, J.; Wang, Y.; Bau, H.H. Controlling a chaotic system. Phys. Rev. Lett.
**1991**, 66, 1123. [Google Scholar] [CrossRef] - Sun, J.; Zhao, X.; Fang, J.; Wang, Y. Autonomous memristor chaotic systems of infinite chaotic attractors and circuitry realization. Nonlinear Dyn.
**2018**, 94, 2879–2887. [Google Scholar] [CrossRef]

**Figure 1.**State trajectories of OZNN (9), IEZNN (12) and ADIZNN (19) for the DSE with (35) in the absence of the noise. (

**a**) State trajectory of ${p}_{11}\left(t\right)$. (

**b**) State trajectory of ${p}_{12}\left(t\right)$. (

**c**) State trajectory of ${p}_{21}\left(t\right)$. (

**d**) State trajectory of ${p}_{22}\left(t\right)$.

**Figure 2.**State trajectories of OZNN (9), IEZNN (12) and ADIZNN (19) for the DSE with (35) under the linear noise ${z}_{i}\left(t\right)=t/4+4$. (

**a**) State trajectory of ${p}_{11}\left(t\right)$. (

**b**) State trajectory of ${p}_{12}\left(t\right)$. (

**c**) State trajectory of ${p}_{21}\left(t\right)$. (

**d**) State trajectory of ${p}_{22}\left(t\right)$.

**Figure 3.**Error norms ${\u2225W\left(t\right)\u2225}_{\mathrm{F}}$ of OZNN (9), IEZNN (12) and ADIZNN (19) for the DSE with (35) in different noise environments. (

**a**) No noise ${z}_{i}\left(t\right)=0$. (

**b**) Linear noise ${z}_{i}\left(t\right)=t/4+4$. (

**c**) Linear noise ${z}_{i}\left(t\right)=4t+4$. (

**d**) Linear noise ${z}_{i}\left(t\right)=16t+4$.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Han, L.; He, Y.; Liao, B.; Hua, C.
An Accelerated Double-Integral ZNN with Resisting Linear Noise for Dynamic Sylvester Equation Solving and Its Application to the Control of the SFM Chaotic System. *Axioms* **2023**, *12*, 287.
https://doi.org/10.3390/axioms12030287

**AMA Style**

Han L, He Y, Liao B, Hua C.
An Accelerated Double-Integral ZNN with Resisting Linear Noise for Dynamic Sylvester Equation Solving and Its Application to the Control of the SFM Chaotic System. *Axioms*. 2023; 12(3):287.
https://doi.org/10.3390/axioms12030287

**Chicago/Turabian Style**

Han, Luyang, Yongjun He, Bolin Liao, and Cheng Hua.
2023. "An Accelerated Double-Integral ZNN with Resisting Linear Noise for Dynamic Sylvester Equation Solving and Its Application to the Control of the SFM Chaotic System" *Axioms* 12, no. 3: 287.
https://doi.org/10.3390/axioms12030287