A novel quaternion linear matrix equation solver through zeroing neural networks with applications to acoustic source tracking

Vladislav N. Kovalnogov; Ruslan V. Fedorov; Igor I. Shepelev; Vyacheslav V. Sherkunov; Theodore E. Simos; Spyridon D. Mourtas; Vasilios N. Katsikis; Vladislav N. Kovalnogov; Ruslan V. Fedorov; Igor I. Shepelev; Vyacheslav V. Sherkunov; Theodore E. Simos; Spyridon D. Mourtas; Vasilios N. Katsikis

doi:10.3934/math.20231323

AIMS Mathematics

2023, Volume 8, Issue 11: 25966-25989. doi: 10.3934/math.20231323

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A novel quaternion linear matrix equation solver through zeroing neural networks with applications to acoustic source tracking

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

Received: 13 July 2023 Revised: 28 August 2023 Accepted: 03 September 2023 Published: 08 September 2023
MSC : 65F20, 68T05

Due to its significance in science and engineering, time-varying linear matrix equation (LME) problems have received a lot of attention from scholars. It is for this reason that the issue of finding the minimum-norm least-squares solution of the time-varying quaternion LME (ML-TQ-LME) is addressed in this study. This is accomplished using the zeroing neural network (ZNN) technique, which has achieved considerable success in tackling time-varying issues. In light of that, two new ZNN models are introduced to solve the ML-TQ-LME problem for time-varying quaternion matrices of arbitrary dimension. Two simulation experiments and two practical acoustic source tracking applications show that the models function superbly.
- quaternion,
- linear matrix equation,
- minimum-norm least-squares solution,
- dynamical system,
- zeroing neural network,
- acoustic source tracking
Citation: Vladislav N. Kovalnogov, Ruslan V. Fedorov, Igor I. Shepelev, Vyacheslav V. Sherkunov, Theodore E. Simos, Spyridon D. Mourtas, Vasilios N. Katsikis. A novel quaternion linear matrix equation solver through zeroing neural networks with applications to acoustic source tracking[J]. AIMS Mathematics, 2023, 8(11): 25966-25989. doi: 10.3934/math.20231323

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Abstract

Due to its significance in science and engineering, time-varying linear matrix equation (LME) problems have received a lot of attention from scholars. It is for this reason that the issue of finding the minimum-norm least-squares solution of the time-varying quaternion LME (ML-TQ-LME) is addressed in this study. This is accomplished using the zeroing neural network (ZNN) technique, which has achieved considerable success in tackling time-varying issues. In light of that, two new ZNN models are introduced to solve the ML-TQ-LME problem for time-varying quaternion matrices of arbitrary dimension. Two simulation experiments and two practical acoustic source tracking applications show that the models function superbly.

References

[1]	G. X. Huang, F. Yin, K. Guo, An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB = C, J. Comput. Appl. Math, 212 (2008), 231–244. https://doi:10.1016/j.cam.2006.12.005 doi: 10.1016/j.cam.2006.12.005
[2]	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.
[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]	Z. Zhang, Z. Yan, An adaptive fuzzy recurrent neural network for solving non-repetitive motion problem of redundant robot manipulators, IEEE Trans. Fuzzy Syst., 28 (2020), 684–691. https://doi.org/10.1109/TFUZZ.2019.2914618 doi: 10.1109/TFUZZ.2019.2914618
[5]	T. Sarkar, K. Siarkiewicz, R. Stratton, Survey of numerical methods for solution of large systems of linear equations for electromagnetic field problems, IEEE Trans. Antennas Propag., 29 (1981), 847–856. https://doi.org/10.1109/TAP.1981.1142695 doi: 10.1109/TAP.1981.1142695
[6]	L. Xiao, J. Tao, J. Dai, Y. Wang, L. Jia, Y. He, A parameter-changing and complex-valued zeroing neural-network for finding solution of time-varying complex linear matrix equations in finite time, IEEE T. Ind. Inform., 17 (2021), 6634–6643. https://doi.org/10.1109/TII.2021.3049413 doi: 10.1109/TII.2021.3049413
[7]	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.
[8]	Y. Zhang, S. S. Ge, Design and analysis of a general recurrent neural network model for time-varying matrix inversion, IEEE T. Neur. Network., 16 (2005), 1477–1490. https://doi.org/10.1109/TNN.2005.857946 doi: 10.1109/TNN.2005.857946
[9]	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. https://doi.org/10.2991/ijcis.d.200527.001 doi: 10.2991/ijcis.d.200527.001
[10]	Z. Sun, F. Li, L. Jin, T. Shi, K. Liu, Noise-tolerant neural algorithm for online solving time-varying full-rank matrix Moore-Penrose inverse problems: A control-theoretic approach, Neurocomputing, 413 (2020), 158–172. https://doi.org/10.1016/j.neucom.2020.06.050 doi: 10.1016/j.neucom.2020.06.050
[11]	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
[12]	Y. Zhang, Y. Yang, N. Tan, B. Cai, Zhang neural network solving for time-varying full-rank matrix Moore-Penrose inverse, Computing, 92 (2011), 97–121. https://doi.org/10.1007/s00607-010-0133-9 doi: 10.1007/s00607-010-0133-9
[13]	S. Qiao, X. Z. Wang, Y. Wei, Two finite-time convergent Zhang neural network models for time-varying complex matrix Drazin inverse, Linear Algebra Appl., 542 (2018), 101–117. https://doi.org/10.1016/j.laa.2017.03.014 doi: 10.1016/j.laa.2017.03.014
[14]	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. https://doi.org/10.1080/01630563.2020.1740887 doi: 10.1080/01630563.2020.1740887
[15]	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
[16]	T. E. Simos, V. N. Katsikis, S. D. Mourtas, P. S. Stanimirović, D. Gerontitis, A higher-order zeroing neural network for pseudoinversion of an arbitrary time-varying matrix with applications to mobile object localization, Inf. Sci., 600 (2022), 226–238. https://doi.org/10.1016/j.ins.2022.03.094 doi: 10.1016/j.ins.2022.03.094
[17]	M. Zhou, J. Chen, P. S. Stanimirovic, V. N. Katsikis, H. Ma, Complex varying-parameter Zhang neural networks for computing core and core-EP inverse, Neural Process. Lett., 51 (2020), 1299–1329. https://doi.org/10.1007/s11063-019-10141-6 doi: 10.1007/s11063-019-10141-6
[18]	J. Liu, H. Cai, C. Jiang, X. Han, Z. Zhang, An interval inverse method based on high dimensional model representation and affine arithmetic, Appl. Math. Model., 63 (2018), 732–743. https://doi.org/10.1016/j.apm.2018.07.009 doi: 10.1016/j.apm.2018.07.009
[19]	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
[20]	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.
[21]	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
[22]	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. https://doi.org/10.3390/math10111950 doi: 10.3390/math10111950
[23]	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
[24]	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. https://doi.org/10.1109/TFUZZ.2021.3115969 doi: 10.1109/TFUZZ.2021.3115969
[25]	Y. Zhang, S. Li, J. Weng, B. Liao, GNN model for time-varying matrix inversion with robust finite-time convergence, IEEE T. Neur. Net. Lear., (2022), 1–11. https://doi.org/10.1109/TNNLS.2022.3175899 doi: 10.1109/TNNLS.2022.3175899
[26]	Y. Zhang, Improved GNN method with finite-time convergence for time-varying Lyapunov equation, Inf. Sci., 611 (2022), 494–503. https://doi.org/10.1016/j.ins.2022.08.061 doi: 10.1016/j.ins.2022.08.061
[27]	W. R. Hamilton, On a new species of imaginary quantities, connected with the theory of quaternions, P. Royal Irish Acad., 2 (1840), 424–434. https://www.jstor.org/stable/20520177
[28]	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.
[29]	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
[30]	G. Du, Y. Liang, B. Gao, S. A. Otaibi, D. Li, A cognitive joint angle compensation system based on self-feedback fuzzy neural network with incremental learning, IEEE T. Ind. Inform., 17 (2021), 2928–2937. https://doi.org/10.1109/TII.2020.3003940 doi: 10.1109/TII.2020.3003940
[31]	A. Szynal-Liana, I. Włoch, Generalized commutative quaternions of the Fibonacci type, Boletín de la Sociedad Matemática Mexicana, 28 (2022), 1. https://doi.org/10.1007/s40590-021-00386-4 doi: 10.1007/s40590-021-00386-4
[32]	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
[33]	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.
[34]	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
[35]	S. Giardino, Quaternionic quantum mechanics in real Hilbert space, J. Geom. Phys., 158 (2020), 103956. https://doi.org/10.1142/S0219887819501056 doi: 10.1142/S0219887819501056
[36]	Z. H. Weng, Field equations in the complex quaternion spaces, Adv. Math. Phys., 2014.
[37]	R. Ghiloni, V. Moretti, A. Perotti, Continuous slice functional calculus in quaternionic Hilbert spaces, Rev. Math. Phys., 25 (2013), 1350006. https://doi.org/10.1142/S0219887819501056 doi: 10.1142/S0219887819501056
[38]	L. Xiao, S. Liu, X. Wang, Y. He, L. Jia, Y. Xu, Zeroing neural networks for dynamic quaternion-valued matrix inversion, IEEE Trans. Ind. Inform., 18 (2022), 1562–1571.
[39]	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. Network. Lear., (2022), 1–11.
[40]	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. Network. Lear., (2023), 1–10. https://doi.org/10.1109/TNNLS.2023.3242313 doi: 10.1109/TNNLS.2023.3242313
[41]	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. Network. Lear., (2022), 1–10.
[42]	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. https://doi.org/10.3390/math11122756 doi: 10.3390/math11122756
[43]	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. Instru. Meas., 71 (2022), 1–14. https://doi.org/10.1109/TIM.2022.3161713 doi: 10.1109/TIM.2022.3161713
[44]	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
[45]	P. S. Stanimirović, S. D. Mourtas, V. N. Katsikis, L. A. Kazakovtsev, V. N. Krutikov, Recurrent neural network models based on optimization methods, Mathematics, 10 (2022), 4292. https://doi.org/10.3390/math10224292 doi: 10.3390/math10224292
[46]	F. Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl., 251 (1997), 21–57. https://doi.org/10.1016/0024-3795(95)00543-9 doi: 10.1016/0024-3795(95)00543-9
[47]	J. Groß, G. Trenkler, S. O. Troschke, Quaternions: Further contributions to a matrix oriented approach, Linear Algebra Appl., 326 (2001), 205–213. https://doi.org/10.1016/S0024-3795(00)00283-4 doi: 10.1016/S0024-3795(00)00283-4
[48]	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, Knowl. Based Syst., 242 (2022), 108405. https://doi.org/10.1016/j.knosys.2022.108405 doi: 10.1016/j.knosys.2022.108405
[49]	L. Xiao, H. Tan, J. Dai, L. Jia, W. Tang, High-order error function designs to compute time-varying linear matrix equations, Inform. Sciences, 576 (2021), 173–186. https://doi.org/10.1016/j.ins.2021.06.038 doi: 10.1016/j.ins.2021.06.038
[50]	N. Zhong, Q. Huang, S. Yang, F. Ouyang, Z. Zhang, A varying-parameter recurrent neural network combined with penalty function for solving constrained multi-criteria optimization scheme for redundant robot manipulators, IEEE Access, 9 (2021), 50810–50818. https://doi.org/10.1109/ACCESS.2021.3068731 doi: 10.1109/ACCESS.2021.3068731
[51]	A. K. Gupta, Numerical methods using MATLAB, MATLAB solutions series, Apress: Berkeley, CA, USA, New York, NY, 2014.
[52]	L. Jin, J. Yan, X. Du, X. Xiao, D. Fu, RNN for solving time-variant generalized Sylvester equation with applications to robots and acoustic source localization, IEEE Trans. Ind. Inform., 16 (2020), 6359–6369. https://doi.org/10.1109/TII.2020.2964817 doi: 10.1109/TII.2020.2964817
[53]	K. Kim, S. Wang, H. Ryu, S. Q. Lee, Acoustic-based position estimation of an object and a person using active localization and sound field analysis, Appl. Sci., 10 (2020), 9090. https://doi.org/10.3390/app10249090 doi: 10.3390/app10249090
[54]	P. Du, S. Zhang, C. Chen, A. Alphones, W. D. Zhong, Demonstration of a low-complexity indoor visible light positioning system using an enhanced TDOA scheme, IEEE Photon. J., 10 (2018), 1–10. https://doi.org/10.1109/JPHOT.2018.2840681 doi: 10.1109/JPHOT.2018.2840681
[55]	A. G. Dempster, E. Cetin, Interference localization for satellite navigation systems, Proc. IEEE, 104 (2016), 1318–1326. https://doi.org/10.1109/JPROC.2016.2530814 doi: 10.1109/JPROC.2016.2530814
[56]	J. Tiemann, F. Eckermann, C. Wietfeld, ATLAS - an open-source TDOA-based ultra-wideband localization system, In: Int. Conf. Indoor Positioning Indoor Navigat. (IPIN) (ed. A. de Henares), Spain, 2016.
[57]	Y. Zhang, L. Jin, Robot Manipulator Redundancy Resolution, John Wiley Sons: Hoboken, NJ, USA, 2017.

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