A quaternion Sylvester equation solver through noise-resilient zeroing neural networks with application to control the SFM chaotic system

Sondess B. Aoun; Nabil Derbel; Houssem Jerbi; Theodore E. Simos; Spyridon D. Mourtas; Vasilios N. Katsikis; Sondess B. Aoun; Nabil Derbel; Houssem Jerbi; Theodore E. Simos; Spyridon D. Mourtas; Vasilios N. Katsikis

doi:10.3934/math.20231401

AIMS Mathematics

2023, Volume 8, Issue 11: 27376-27395. doi: 10.3934/math.20231401

Previous Article Next Article

Research article Special Issues

A quaternion Sylvester equation solver through noise-resilient zeroing neural networks with application to control the SFM chaotic system

1.
Department of Computer Engineering, College of Computer Science and Engineering, Univ. of Ha'il, Ha'il City 81451, Saudi Arabia
2.
Control & Energy Management Laboratory, National Engineering School of Sfax, Univ. of Sfax, Sfax, Tunisia
3.
Department of Industrial Engineering, College of Engineering, Univ. of Ha'il, Ha'il City 81451, Saudi Arabia
4.
Laboratory of Interdisciplinary Problems in Energy Production, Ulyanovsk State Technical Univ., 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
5.
Department of Medical Research, China Medical Univ. Hospital, China Medical Univ., Taichung City 40402, Taiwan
6.
Center for Applied Math. and Bioinformatics, Gulf Univ. for Science and Technology, West Mishref, 32093 Kuwait
7.
Data Recovery Key Laboratory of Sichun Province, Neijing Normal Univ., Neijiang 641100, China
8.
Section of Mathematics, Dept. of Civil Engineering, Democritus Univ. of Thrace, Xanthi 67100, Greece
9.
Department of Economics, Division of Mathematics-Informatics and Statistics-Econometrics, National and Kapodistrian Univ.of Athens, Sofokleous 1 Street, 10559 Athens, Greece
10.
Laboratory "Hybrid Methods of Modelling and Optimization in Complex Systems", Siberian Federal Univ., Prosp. Svobodny 79, 660041 Krasnoyarsk, Russia

Received: 09 August 2023 Revised: 12 September 2023 Accepted: 23 September 2023 Published: 26 September 2023
MSC : 65F20, 68T05

Dynamic Sylvester equation (DSE) problems have drawn a lot of interest from academics due to its importance in science and engineering. Due to this, the quest for the quaternion DSE (QDSE) solution is the subject of this work. This is accomplished using the zeroing neural network (ZNN) technique, which has achieved considerable success in tackling time-varying issues. Keeping in mind that the original ZNN can handle QDSE successfully in a noise-free environment, but it might not work in a noisy one, and the noise-resilient ZNN (NZNN) technique is also utilized. In light of that, one new ZNN model is introduced to solve the QDSE problem and one new NZNN model is introduced to solve the QDSE problem under different types of noises. Two simulation experiments and one application to control of the sine function memristor (SFM) chaotic system show that the models function superbly.
- Quaternion,
- dynamic Sylvester equation,
- zeroing neural network,
- chaos control
Citation: Sondess B. Aoun, Nabil Derbel, Houssem Jerbi, Theodore E. Simos, Spyridon D. Mourtas, Vasilios N. Katsikis. A quaternion Sylvester equation solver through noise-resilient zeroing neural networks with application to control the SFM chaotic system[J]. AIMS Mathematics, 2023, 8(11): 27376-27395. doi: 10.3934/math.20231401

Related Papers:

Abstract

Dynamic Sylvester equation (DSE) problems have drawn a lot of interest from academics due to its importance in science and engineering. Due to this, the quest for the quaternion DSE (QDSE) solution is the subject of this work. This is accomplished using the zeroing neural network (ZNN) technique, which has achieved considerable success in tackling time-varying issues. Keeping in mind that the original ZNN can handle QDSE successfully in a noise-free environment, but it might not work in a noisy one, and the noise-resilient ZNN (NZNN) technique is also utilized. In light of that, one new ZNN model is introduced to solve the QDSE problem and one new NZNN model is introduced to solve the QDSE problem under different types of noises. Two simulation experiments and one application to control of the sine function memristor (SFM) chaotic system show that the models function superbly.

References

[1]	W. Li, L. Han, X. Xiao, B. Liao and C. Peng, A gradient-based neural network accelerated for vision-based control of an RCM-constrained surgical endoscope robot, Neural Comput. Appl., 34 (2022), 1329–1343. https://doi.org/10.1007/s00521-021-06465-x doi: 10.1007/s00521-021-06465-x
[2]	Z. Li, B. Liao, F. Xu, D. Guo, A new repetitive motion planning scheme with noise suppression capability for redundant robot manipulators, IEEE Trans. Syst. Man Cybern. Syst., 50 (2020), 5244–5254.
[3]	J. Kurzak, A. Buttari, J. J. Dongarra, Solving systems of linear equations on the CELL processor using Cholesky factorization, IEEE Trans. Parallel Distributed Syst., 19 (2008), 1175–1186.
[4]	H. R. Shaker, M. Tahavori, Control configuration selection for bilinear systems via generalised Hankel interaction index array, Int. J. Control, 88 (2015), 30–37. https://doi.org/10.1007/s00521-021-06465-x doi: 10.1007/s00521-021-06465-x
[5]	S. D. Mourtas, V. N. Katsikis, C. Kasimis, Feedback control systems stabilization using a bio-inspired neural network, EAI Endorsed Trans. AI Robotics, 1 (2022), 1–13.
[6]	B. Liao, L. Han, X. Cao, S. Li, J. Li, Double integral‐enhanced zeroing neural network with linear noise rejection for time‐varying matrix inverse, CAAI Trans. Intell. Technol., 1–14.
[7]	Q. Wei, N. Dobigeon, J. Tourneret, J. M. Bioucas-Dias, S. J. Godsill, R-FUSE: Robust fast fusion of multiband images based on solving a Sylvester equation, IEEE Signal Process. Lett., 23 (2016), 1632–1636. https://doi.org/10.1109/LSP.2016.2608858 doi: 10.1109/LSP.2016.2608858
[8]	S. Dolgov, J. W. Pearson, D. V. Savostyanov, M. Stoll, Fast tensor product solvers for optimization problems with fractional differential equations as constraints, Appl. Math. Comput., 273 (2016), 604–623. https://doi.org/10.1016/j.amc.2015.09.042 doi: 10.1016/j.amc.2015.09.042
[9]	L. Huo, S. Yang, L. Jiao, S. Wang, J. Shi, Local graph regularized coding for salient object detection, Infrared Phys. Technol., 77 (2016), 124–131. https://doi.org/10.1016/j.infrared.2016.05.002 doi: 10.1016/j.infrared.2016.05.002
[10]	X. Yan, M. Liu, L. Jin, S. Li, B. Hu, X. Zhang, et al., New zeroing neural network models for solving nonstationary Sylvester equation with verifications on mobile manipulators, IEEE T. Ind. Inform., 15 (2019), 5011–5022.
[11]	T. Sarkar, K. Siarkiewicz, R. Stratton, Survey of numerical methods for solution of large systems of linear equations for electromagnetic field problems, IEEE T. Antenn. Propag., 29 (1981), 847–856.
[12]	F. P. A. Beik, F. Saberi Movahed, S. Ahmadi-Asl, On the Krylov subspace methods based on tensor format for positive definite Sylvester tensor equations., Numer. Linear Algebr., 23 (2016), 444–466.
[13]	C. Song, J. e. Feng, X. Wang, J. Zhao, Finite iterative method for solving coupled Sylvester-transpose matrix equations, J. Appl. Math. Comput., 46 (2014), 351–372.
[14]	L. Xiao, J. Tao, W. Li, An arctan-type varying-parameter ZNN for solving time-varying complex Sylvester equations in finite time, IEEE T. Ind. Inform., 18 (2022), 3651–3660.
[15]	W. R. Hamilton, On a new species of imaginary quantities, connected with the theory of quaternions, P. Royal Irish Acad., 2 (1840), 424–434.
[16]	M. Joldeş, J. M. Muller, Algorithms for manipulating quaternions in floating-point arithmetic, in 2020 IEEE 27th Symposium on Computer Arithmetic (ARITH), IEEE, 2020, 48–55.
[17]	E. Özgür, Y. Mezouar, Kinematic modeling and control of a robot arm using unit dual quaternions, Robot. Auton. Syst., 77 (2016), 66–73. https://doi.org/10.1016/j.robot.2015.12.005 doi: 10.1016/j.robot.2015.12.005
[18]	D. Pavllo, C. Feichtenhofer, M. Auli, D. Grangier, Modeling human motion with quaternion-based neural networks, Int. J. Comput. Vision, 128 (2020), 855–872. https://doi.org/10.1007/s11263-019-01207-y doi: 10.1007/s11263-019-01207-y
[19]	A. M. S. Goodyear, P. Singla, D. B. Spencer, Analytical state transition matrix for dual-quaternions for spacecraft pose estimation, in AAS/AIAA Astrodynamics Specialist Conference, 2019, Univelt Inc., 2020,393–411.
[20]	M. E. Kansu, Quaternionic representation of electromagnetism for material media, Int. J. Geom. Methods M., 16 (2019), 1950105. https://doi.org/10.1142/S0219887819501056 doi: 10.1142/S0219887819501056
[21]	S. Giardino, Quaternionic quantum mechanics in real Hilbert space, J. Geom. Phys., 158 (2020), 103956.
[22]	A. Szynal-Liana, I. Włoch, Generalized commutative quaternions of the Fibonacci type, Boletín de la Sociedad Matemática Mexicana, 28 (2022), 1.
[23]	L. Xiao, S. Liu, X. Wang, Y. He, L. Jia, Y. Xu, Zeroing neural networks for dynamic quaternion-valued matrix inversion, IEEE Trans. Ind. Informatics, 18 (2022), 1562–1571.
[24]	V. N. Kovalnogov, R. V. Fedorov, D. A. Demidov, M. A. Malyoshina, T. E. Simos, S. D. Mourtas, et al., Computing quaternion matrix pseudoinverse with zeroing neural networks, AIMS Math., 8 (2023), 22875–22895. https://doi.org/10.3934/math.20231164 doi: 10.3934/math.20231164
[25]	L. Xiao, P. Cao, W. Song, L. Luo, W. Tang, A fixed-time noise-tolerance ZNN model for time-variant inequality-constrained quaternion matrix least-squares problem, IEEE T. Neur. Net. Lear., 1–10.
[26]	L. Xiao, Y. Zhang, W. Huang, L. Jia, X. Gao, A dynamic parameter noise-tolerant zeroing neural network for time-varying quaternion matrix equation with applications, IEEE T. Neur. Net. Lear., 1–10.
[27]	R. Abbassi, H. Jerbi, M. Kchaou, T. E. Simos, S. D. Mourtas, V. N. Katsikis, Towards higher-order zeroing neural networks for calculating quaternion matrix inverse with application to robotic motion tracking, Mathematics, 11 (2023), 2756.
[28]	N. Tan, P. Yu, F. Ni, New varying-parameter recursive neural networks for model-free kinematic control of redundant manipulators with limited measurements, IEEE T. Instrum. Meas., 71 (2022), 1–14.
[29]	V. N. Kovalnogov, R. V. Fedorov, D. A. Demidov, M. A. Malyoshina, T. E. Simos, V. N. Katsikis, et al., Zeroing neural networks for computing quaternion linear matrix equation with application to color restoration of images, AIMS Math., 8 (2023), 14321–14339. https://doi.org/10.3934/math.2023733 doi: 10.3934/math.2023733
[30]	Y. Zhang, S. S. Ge, Design and analysis of a general recurrent neural network model for time-varying matrix inversion, IEEE T. Neur. Net., 16 (2005), 1477–1490.
[31]	Y. Chai, H. Li, D. Qiao, S. Qin, J. Feng, A neural network for Moore-Penrose inverse of time-varying complex-valued matrices, Int. J. Comput. Intell. Syst., 13 (2020), 663–671.
[32]	W. Wu, B. Zheng, Improved recurrent neural networks for solving Moore-Penrose inverse of real-time full-rank matrix, Neurocomputing, 418 (2020), 221–231. https://doi.org/10.1016/j.neucom.2020.08.026 doi: 10.1016/j.neucom.2020.08.026
[33]	S. Qiao, Y. Wei, X. Zhang, Computing time-varying ML-weighted pseudoinverse by the Zhang neural networks, Numer. Func. Anal. Opt., 41 (2020), 1672–1693.
[34]	X. Wang, P. S. Stanimirovic, Y. Wei, Complex ZFs for computing time-varying complex outer inverses, Neurocomputing, 275 (2018), 983–1001. https://doi.org/10.1016/j.neucom.2017.09.034 doi: 10.1016/j.neucom.2017.09.034
[35]	S. D. Mourtas, V. N. Katsikis, Exploiting the Black-Litterman framework through error-correction neural networks, Neurocomputing, 498 (2022), 43–58. https://doi.org/10.1016/j.neucom.2022.05.036 doi: 10.1016/j.neucom.2022.05.036
[36]	V. N. Kovalnogov, R. V. Fedorov, D. A. Generalov, A. V. Chukalin, V. N. Katsikis, S. D. Mourtas, et al., Portfolio insurance through error-correction neural networks, Mathematics, 10 (2022), 3335.
[37]	S. D. Mourtas, C. Kasimis, Exploiting mean-variance portfolio optimization problems through zeroing neural networks, Mathematics, 10 (2022), 3079. https://doi.org/10.3390/math10173079. doi: 10.3390/math10173079
[38]	W. Jiang, C. L. Lin, V. N. Katsikis, S. D. Mourtas, P. S. Stanimirović, T. E. Simos, Zeroing neural network approaches based on direct and indirect methods for solving the Yang–Baxter-like matrix equation, Mathematics, 10 (2022), 1950.
[39]	H. Jerbi, H. Alharbi, M. Omri, L. Ladhar, T. E. Simos, S. D. Mourtas, V. N. Katsikis, Towards higher-order zeroing neural network dynamics for solving time-varying algebraic Riccati equations, Mathematics, 10 (2022), 4490. https://doi.org/10.3390/math10234490 doi: 10.3390/math10234490
[40]	V. N. Katsikis, P. S. Stanimirović, S. D. Mourtas, L. Xiao, D. Karabasević, D. Stanujkić, Zeroing neural network with fuzzy parameter for computing pseudoinverse of arbitrary matrix, IEEE T. Fuzzy Syst., 30 (2022), 3426–3435.
[41]	H. Alharbi, H. Jerbi, M. Kchaou, R. Abbassi, T. E. Simos, S. D. Mourtas, et al., Time-varying pseudoinversion based on full-rank decomposition and zeroing neural networks, Mathematics, 11 (2023), 600.
[42]	J. Dai, P. Tan, X. Yang, L. Xiao, L. Jia, Y. He, A fuzzy adaptive zeroing neural network with superior finite-time convergence for solving time-variant linear matrix equations, Knowledge-Based Syst., 242 (2022), 108405. https://doi.org/10.1016/j.knosys.2022.108405 doi: 10.1016/j.knosys.2022.108405
[43]	L. Jin, Y. Zhang, S. Li, Integration-enhanced Zhang neural network for real-time-varying matrix inversion in the presence of various kinds of noises, IEEE T. Neur. Net. Lear., 27 (2016), 2615–2627. https://doi.org/10.1109/TNNLS.2015.2497715 doi: 10.1109/TNNLS.2015.2497715
[44]	F. Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl., 251 (1997), 21–57.
[45]	L. Xiao, W. Huang, X. Li, F. Sun, Q. Liao, L. Jia, et al., ZNNs with a varying-parameter design formula for dynamic Sylvester quaternion matrix equation, IEEE T. Neur. Net. Lear., 1–11.
[46]	L. Huang, Q. W. Wang, Y. Zhang, The Moore-Penrose inverses of matrices over quaternion polynomial rings, Linear Algebra Appl., 475 (2015), 45–61. https://doi.org/10.1016/j.laa.2015.02.033 doi: 10.1016/j.laa.2015.02.033
[47]	A. K. Gupta, Numerical methods using MATLAB, MATLAB solutions series, Apress: Berkeley, CA, USA, New York, NY, 2014.
[48]	R. Zhang, X. Xi, H. Tian, Z. Wang, Dynamical analysis and finite-time synchronization for a chaotic system with hidden attractor and surface equilibrium, Axioms, 11 (2022), 579.
[49]	H. Su, R. Luo, M. Huang, J. Fu, Robust fixed time control of a class of chaotic systems with bounded uncertainties and disturbances, Int. J. Control Autom. Syst., 20 (2022), 813–822. https://doi.org/10.1007/s12555-020-0782-1 doi: 10.1007/s12555-020-0782-1
[50]	J. Singer, Y. Wang, H. H. Bau, Controlling a chaotic system, Phys. Rev. Lett., 66 (1991), 1123. https://doi.org/10.1103/PhysRevLett.66.1123 doi: 10.1103/PhysRevLett.66.1123
[51]	W. He, T. Luo, Y. Tang, W. Du, Y. Tian, F. Qian, Secure communication based on quantized synchronization of chaotic neural networks under an event-triggered strategy, IEEE T. Neur. Net. Lear., 31 (2020), 3334–3345.
[52]	J. Sun, X. Zhao, J. Fang, Y. Wang, Autonomous memristor chaotic systems of infinite chaotic attractors and circuitry realization, Nonlinear Dynam., 94 (2018), 2879–2887. https://doi.org/10.1007/s11071-018-4531-4 doi: 10.1007/s11071-018-4531-4

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)