### AIMS Mathematics

2022, Issue 2: 2266-2280. doi: 10.3934/math.2022129
Research article

# Zeroing neural network model for solving a generalized linear time-varying matrix equation

• Received: 11 August 2021 Accepted: 29 October 2021 Published: 10 November 2021
• MSC : 15A09, 15A24

• The time-varying solution of a class generalized linear matrix equation with the transpose of an unknown matrix is discussed. The computation model is constructed and asymptotic convergence proof is given by using the zeroing neural network method. Using an activation function, the predefined-time convergence property and noise suppression strategy are discussed. Numerical examples are offered to illustrate the efficacy of the suggested zeroing neural network models.

Citation: Huamin Zhang, Hongcai Yin. Zeroing neural network model for solving a generalized linear time-varying matrix equation[J]. AIMS Mathematics, 2022, 7(2): 2266-2280. doi: 10.3934/math.2022129

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• The time-varying solution of a class generalized linear matrix equation with the transpose of an unknown matrix is discussed. The computation model is constructed and asymptotic convergence proof is given by using the zeroing neural network method. Using an activation function, the predefined-time convergence property and noise suppression strategy are discussed. Numerical examples are offered to illustrate the efficacy of the suggested zeroing neural network models.

 [1] J. Q. Gong, J. Jin, A better robustness and fast convergence zeroing neural network for solving dynamic nonlinear equations, Neural Comput. Appl., 2021, 1–11. doi: 10.1007/s00521-020-05617-9. [2] L. Xie, J. Ding, F. Ding, Gradient based iterative solutions for general linear matrix equations, Comput. Math. Appl., 58 (2009), 1441–1448. doi: 10.1016/j.camwa.2009.06.047. [3] Z. N. Zhang, F. Ding, X. G. Liu, Hierarchical gradient based iterative parameter estimation algorithm for multivariable output error moving average systems, Comput. Math. Appl., 61 (2011), 672–682. doi: 10.1016/j.camwa.2010.12.014. [4] F. Ding, Combined state and least squares parameter estimation algorithms for dynamic systems, Appl. Math. Modell., 38 (2014), 403–412. doi: 10.1016/j.apm.2013.06.007. [5] H. M. Zhang, Quasi gradient-based inversion-free iterative algorithm for solving a class of the nonlinear matrix equations, Comput. Math. Appl., 77 (2019), 1233–1244. doi: 10.1016/j.camwa.2018.11.006. [6] H. M. Zhang, L. J. Wan, Zeroing neural network methods for solving the Yang-Baxter-like matrix equation, Neurocomputing, 383 (2020), 409–418. doi: 10.1016/j.neucom.2019.11.101. [7] V. N. Katsikis, S. D. Mourtas, P. S. Stanimirovi$\rm\acute{c}$, Y. N. Zhang, Solving complex-valued time-varying linear matrix equations via QR decomposition with applications to robotic motion tracking and on angle-of-arrival localization, IEEE T. Neur. Net. Lear., 2021, 1–10. doi: 10.1109/TNNLS.2021.3052896. [8] L. Xiao, Z. J. Zhang, S. Li, Solving time-varying system of nonlinear equations by finite-time recurrent neural networks with application to motion tracking of robot manipulators, IEEE T. Syst. Man Cy.-S., 49 (2019), 2210–2220. doi: 10.1109/TSMC.2018.2836968. [9] L. Xiao, S. Li, K. L. Li, L. Jin, B. L. Liao, Co-design of finite-time convergence and noise suppression: A unified neural model for time varying linear equations with robotic applications, IEEE T. Syst. Man Cy.-S., 50 (2020), 5233–5243. doi: 10.1109/TSMC.2018.2870489. [10] L. Jin, S. Li, B. L. Liao, Z. J. Zhang, Zeroing neural networks: A survey, Neurocomputing, 267 (2017), 597–604. doi: 10.1016/j.neucom.2017.06.030. [11] Z. B. Sun, T. Shi, L. Jin, B. C. Zhang, Z. X. Pang, J. Z. Yu, Discrete-time zeroing neural network of $O(\tau^4)$ pattern for online time-varying nonlinear optimization problem: Application to manipulator motion generation, J. Frankl. I., 358 (2021), 7203–7220. doi: 10.1016/j.jfranklin.2021.07.006. [12] L. Jin, Y. N. Zhang, Discrete-time Zhang neural network for online time-varying nonlinear optimization with application to manipulattor motion generation, IEEE T. Neur. Net. Lear., 26 (2015), 1525–1531. doi: 10.1109/TNNLS.2014.2342260. [13] L. Jin, S. Li, Distributed task allocation of multiple robots: A control perspective, IEEE T. Syst. Man Cy.-S., 48 (2018), 693–701. doi: 10.1109/TSMC.2016.2627579. [14] Y. M. Qi, L. Jin, Y. N. Wang, L. Xiao, J. L. Zhang, Complex-valued discrete-time neural dynamics for perturbed time-dependent complex quadratic programming with applications, IEEE T. Neur. Net. Lear., 31 (2020), 3555–3569. doi: 10.1109/TNNLS.2019.2944992. [15] L. Jin, Y. N. Zhang, Continuous and discrete Zhang dynamics for real-time varying nonlinear optimization, Numer. Algorithms, 73 (2016), 115–140. doi: 10.1007/s11075-015-0088-1. [16] L. Jin, Y. N. Zhang, S. Li, Y. Y. Zhang, Noise-tolerant ZNN models for solving time-varying zero-finding problems: A control-theoretic approach, IEEE T. Automat. Contr., 62 (2017), 992–997. doi: 10.1109/TAC.2016.2566880. [17] Z. B. Sun, T. Shi, L. Wei, Y. Y. Sun, K. P. Liu, L. Jin, Noise-suppressing zeroing neural network for online solving time-varying nonlinear optimization problem: A control-based approach, Neural Comput. Appl., 32 (2020), 11505–11520. doi: 10.1007/s00521-019-04639-2. [18] Z. B. Sun, F. Li, B. C. Zhang, Y. Y. Sun, L. Jin, Different modified zeroing neural dynamics with inherent tolerance to noises for time-varying reciprocal problems: A control-theoretic approach, Neurocomputing, 337 (2019), 165–179. doi: 10.1016/j.neucom.2019.01.064. [19] S. Z. Qiao, X. Z. Wang, Y. M. Wei, Two finite-time convergent Zhang neural network models for time-varying complex matrix Drazin inverse, Linear Algebra Appl., 542 (2018), 101–117. doi: 10.1016/j.laa.2017.03.014. [20] Z. Li, Y. N. Zhang, Improved Zhang neural network model and its solution of time-varying generalized linear matrix equations, Expert Syst. Appl., 37 (2010), 7213–7218. doi: 10.1016/j.eswa.2010.04.007. [21] S. Li, S. F. Chen, B. Liu, Accelerating a recurrent neural network to finite-time convergence for solving time-varying Sylvester equation by using a sign-bi-power activation function, Neural Process. Lett., 37 (2013), 189–205. doi: 10.1007/s11063-012-9241-1. [22] Y. J. Shen, P. Miao, Y. H. Huang, Y. Shen, Finite-time stability and its application for solving time-varying Sylvester equation by recurrent neural network, Neural Process. Lett., 42 (2015), 763–784. doi: 10.1007/s11063-014-9397-y. [23] L. Xiao, Accelerating a recurrent neural network to finite-time convergence using a new design formula and its application to time-varying matrix square root, J. Frankl. I., 354 (2017), 5667–5677. doi: 10.1016/j.jfranklin.2017.06.012. [24] L. Jia, L. Xiao, J. H. Dai, Z. H. Qi, Z. J. Zhang, Y. S. Zhang, Design and application of an adaptive fuzzy control strategy to zeroing neural network for solving time-variant QP problem, IEEE T. Fuzzy Syst., 29 (2021), 1544–1555. doi: 10.1109/TFUZZ.2020.2981001. [25] V. N. Katsikis, S. D. Mourtas, P. S. Stanimirovi$\rm\acute{c}$, Y. N. Zhang, Continuous-time varying complex QR decomposition via zeroing neural dynamics, Neural Process. Lett., 53 (2021), 3573–3590. doi: 10.1007/s11063-021-10566-y. [26] Z. J. Zhang, L. N. Zheng, J. Weng, Y. J. Mao, W. Lu, L. Xiao, A new varying-parameter recurrent neural-network for online solution of time-varying Sylvester equation, IEEE T. Cybernetics, 48 (2018), 3135–3148. doi: 10.1109/TCYB.2017.2760883. [27] Z. J. Zhang, L. N. Zheng, T. R. Qiu, F. Q. Deng, Varying-parameter convergent-differential neural solution to time-varying overdetermined system of linear equations, IEEE T. Automat. Contr., 65 (2020), 874–881. doi: 10.1109/TAC.2019.2921681. [28] Z. J. Zhang, L. N. Zheng, A complex varying-parameter convergent-differential neural-network for solving online time-varying complex Sylvester equation, IEEE T. Cybernetics, 49 (2019), 3627–3639. doi: 10.1109/TCYB.2018.2841970. [29] Z. J. Zhang, L. D. Kong, L. N. Zheng, P. C. Zhang, X. L. Qu, B. L. Liao, et al., Robustness analysis of a power-type varying-parameter recurrent neural network for solving time-varying QM and QP problems and applications, IEEE T. Syst. Man Cy.-S., 50 (2020), 5106–5118. doi: 10.1109/TSMC.2018.2866843. [30] H. M. Zhang, F. Ding, On the Kronecker products and their applications, J. Appl. Math., 2013 (2013), 1–8. doi: 10.1155/2013/296185. [31] Y. N. Zhang, D. C. Jiang, J. Wang, A recurrent neural network for solving Sylvester equation with time-varying coefficients, IEEE T. Neural Networ., 13 (2002), 1053–1063 doi: 10.1109/TNN.2002.1031938. [32] K. Chen, Recurrent implicit dynamics for online matrix inversion, Appl. Math. Comput., 219 (2013), 10218–10224. doi: 10.1016/j.amc.2013.03.117. [33] Y. N. Zhang, K. Chen, H. Z. Tan, Performance analysis of gradient neural network exploited for online time-varying matrix inversion, IEEE T. Automat. Contr., 54 (2009), 1940–1945. doi: 10.1109/TAC.2009.2023779. [34] F. Ding, G. J. Liu, X. P. Liu, Parameter estimation with scarce measurements, Automatica, 47 (2011), 1646–1655. doi: 10.1016/j.automatica.2011.05.007. [35] F. Ding, Y. J. Liu, B. Bao, Gradient based and least squares based iterative estimation algorithms for multi-input multi-output systems, P. I. Mech. Eng. I-J. Sys., 226 (2012), 43–55. doi: 10.1177/0959651811409491. [36] F. Ding, G. J. Liu, X. P. Liu, Partially coupled stochastic gradient identification methods for non-uniformly sampled systems, IEEE T. Automat. Contr., 55 (2010), 1976–1981. doi: 10.1109/TAC.2010.2050713. [37] L. Xiao, J. H. Dai, L. Jin, W. B. Li, S. Li, J. Hou, A noise-enduring and finite-time zeroing neural network for equality-constrained time-varying nonlinear optimization, IEEE T. Syst. Man Cy.-S., 51 (2021), 4729–4740. doi: 10.1109/TSMC.2019.2944152. [38] L. Xiao, K. L. Li, M. X. Duan, Computing time-varying quadratic optimization with finite-time convergence and noise tolerance: A unified framework for zeroing neural network, IEEE Trans. Neural Netw. Lear. Syst., 30 (2019), 3360–3369. doi: 10.1109/TNNLS.2019.2891252. [39] L. Xiao, Y. S. Zhang, J. H. Dai, K. Chen, S. Yang, W. B. Li, et al., A new noise-tolerant and predefined-time ZNN model for time-dependent matrix inversion, Neural Networks, 117 (2019), 124–134. doi: 10.1016/j.neunet.2019.05.005. [40] F. Yu, L. Liu, L. Xiao, K. L. Li, S. Cai, A robust and fixed-time zeroing neural dynamics for computing time-variant nonlinear equation using a novel nonliear activation function, Neurocomputing, 350 (2019), 108–116. doi: 10.1016/j.neucom.2019.03.053. [41] L. Xiao, Y. K. Cao, J. H. Dai, L. Jia, H. Y. Tan, Finite-time and predefined-time convergence design for zeroing neural network: Theorem, method, and verification, IEEE T. Ind. Inform., 17 (2021), 4724–4732. doi: 10.1109/TII.2020.3021438. [42] L. Xiao, J. H. Dai, R. B. Lu, S. Li, J. C. Li, S. J. Wang, Design and comprehensive analysis of a noise-tolerant ZNN model with limited-time convergence for time-dependent nonlinear minimization, IEEE T. Neur. Net. Lear., 31 (2020), 5339–5348. doi: 10.1109/TNNLS.2020.2966294. [43] L. Xiao, Y. S. Zhang, Q. Y. Zuo, J. H. Dai, J. C. Li, W. S. Tang, A noise-tolerant zeroing neural network for time-dependent complex matrix inversion under various kinds of noises, IEEE T. Ind. Inform., 16 (2020), 3757–3766. doi: 10.1109/TII.2019.2936877. [44] M. Liu, L. M. Chen, X. H. Du, L. Jin, M. S. Shang, Activated gradients for deep neural networks, IEEE T. Neur. Net. Lear., 2021, 1–13. doi: 10.1109/TNNLS.2021.3106044.
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沈阳化工大学材料科学与工程学院 沈阳 110142

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