{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T10:05:00Z","timestamp":1777025100600,"version":"3.51.4"},"reference-count":41,"publisher":"Frontiers Media SA","license":[{"start":{"date-parts":[[2022,11,15]],"date-time":"2022-11-15T00:00:00Z","timestamp":1668470400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004735","name":"Natural Science Foundation of Hunan Province","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004735","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["frontiersin.org"],"crossmark-restriction":true},"short-container-title":["Front. Neurorobot."],"abstract":"<jats:p>Dynamic complex matrix equation (DCME) is frequently encountered in the fields of mathematics and industry, and numerous recurrent neural network (RNN) models have been reported to effectively find the solution of DCME in no noise environment. However, noises are unavoidable in reality, and dynamic systems must be affected by noises. Thus, the invention of anti-noise neural network models becomes increasingly important to address this issue. By introducing a new activation function (NAF), a robust zeroing neural network (RZNN) model for solving DCME in noisy-polluted environment is proposed and investigated in this paper. The robustness and convergence of the proposed RZNN model are proved by strict mathematical proof and verified by comparative numerical simulation results. Furthermore, the proposed RZNN model is applied to manipulator trajectory tracking control, and it completes the trajectory tracking task successfully, which further validates its practical applied prospects.<\/jats:p>","DOI":"10.3389\/fnbot.2022.1065256","type":"journal-article","created":{"date-parts":[[2022,11,15]],"date-time":"2022-11-15T06:48:32Z","timestamp":1668494912000},"update-policy":"https:\/\/doi.org\/10.3389\/crossmark-policy","source":"Crossref","is-referenced-by-count":43,"title":["A robust zeroing neural network and its applications to dynamic complex matrix equation solving and robotic manipulator trajectory tracking"],"prefix":"10.3389","volume":"16","author":[{"given":"Jie","family":"Jin","sequence":"first","affiliation":[]},{"given":"Lv","family":"Zhao","sequence":"additional","affiliation":[]},{"given":"Lei","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Weijie","family":"Chen","sequence":"additional","affiliation":[]}],"member":"1965","published-online":{"date-parts":[[2022,11,15]]},"reference":[{"key":"B1","doi-asserted-by":"crossref","first-page":"3295","DOI":"10.1007\/s00521-019-04586-y","article-title":"A new fixed-time stabilization approach for neural networks with time-varying delays.","volume":"32","author":"Aouiti","year":"2020","journal-title":"Neural Comput. Appl."},{"key":"B2","doi-asserted-by":"crossref","first-page":"5851","DOI":"10.1109\/TSP.2008.2005086","article-title":"Computation of the para-pseudoinverse for oversampled filter banks: forward and backward greville formulas.","volume":"56","author":"Gan","year":"2008","journal-title":"IEEE Trans. Signal Process."},{"key":"B3","doi-asserted-by":"publisher","DOI":"10.1007\/s00521-020-05617-9","article-title":"A better robustness and fast convergence zeroing neural network for solving dynamic nonlinear equations.","author":"Gong","year":"2021","journal-title":"Neural Comput. Appl."},{"key":"B4","doi-asserted-by":"publisher","first-page":"183040","DOI":"10.3389\/fnbot.2020.00054","article-title":"Acceleration-level obstacle avoidance of redundant manipulators.","volume":"7","author":"Guo","year":"2019","journal-title":"IEEE Access"},{"key":"B5","doi-asserted-by":"crossref","first-page":"4627","DOI":"10.1109\/TII.2019.2944517","article-title":"Analysis and application of modified ZNN design with robustness against harmonic noise.","volume":"16","author":"Guo","year":"2020","journal-title":"IEEE Trans. Industr. Inform."},{"key":"B6","doi-asserted-by":"crossref","first-page":"356","DOI":"10.1109\/TII.2020.2970172","article-title":"Repetitive motion planning of robotic manipulators with guaranteed precision.","volume":"17","author":"Guo","year":"2021","journal-title":"IEEE Trans. Industr. Inform."},{"key":"B7","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1007\/s11063-021-10426-9","article-title":"An improved finite time convergence recurrent neural network with application to time-varying linear complex matrix equation solution.","volume":"53","author":"Jin","year":"","journal-title":"Neural Process. Lett."},{"key":"B8","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/s40747-020-00178-9","article-title":"A robust zeroing neural network for solving dynamic nonlinear equations and its application to kinematic control of mobile manipulator.","volume":"7","author":"Jin","year":"","journal-title":"Complex Intelligent Syst."},{"key":"B9","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1016\/j.aej.2020.09.059","article-title":"An interference-tolerant fast convergence zeroing neural network for dynamic matrix inversion and its application to mobile manipulator path tracking.","volume":"60","author":"Jin","year":"2021","journal-title":"Alexandria Eng. J."},{"key":"B10","doi-asserted-by":"crossref","first-page":"3183","DOI":"10.1016\/j.jfranklin.2022.02.022","article-title":"A robust fast convergence zeroing neural network and its applications to dynamic Sylvester equation solving and robot trajectory tracking.","volume":"359","author":"Jin","year":"2022","journal-title":"J. Franklin Instit."},{"key":"B11","doi-asserted-by":"crossref","first-page":"4151","DOI":"10.1007\/s00521-019-04622-x","article-title":"Improved zeroing neural networks for finite time solving nonlinear equations.","volume":"32","author":"Jin","year":"2020","journal-title":"Neural Comput. Appl."},{"key":"B12","doi-asserted-by":"crossref","first-page":"5105","DOI":"10.1109\/TSMC.2021.3114213","article-title":"Neural dynamics for computing perturbed nonlinear equations applied to ACP-based lower limb motion intention recognition.","volume":"52","author":"Jin","year":"","journal-title":"IEEE Trans. Syst. Man Cybernet. Syst."},{"key":"B13","article-title":"A nonlinear zeroing neural network and its applications on time-varying linear matrix equations solving, electronic circuit currents computing and robotic manipulator trajectory tracking.","volume":"41","author":"Jin","year":"","journal-title":"Comput. Appl. Math."},{"key":"B14","article-title":"A fixed-time convergent and noise-tolerant zeroing neural network for online solution of time-varying matrix inversion.","volume":"130","author":"Jin","year":"","journal-title":"Appl. Soft Comput."},{"key":"B15","article-title":"Gradient-based differential neural-solution to time-dependent nonlinear optimization","author":"Jin","year":"","journal-title":"Proceedings of the IEEE Transactions on Automatic Control"},{"key":"B16","doi-asserted-by":"crossref","first-page":"14297","DOI":"10.1007\/s00521-022-06905-2","article-title":"Novel activation functions-based ZNN models for fixed-time solving dynamic Sylvester equation.","volume":"34","author":"Jin","year":"","journal-title":"Neural Comput. Appl."},{"key":"B17","doi-asserted-by":"publisher","DOI":"10.1109\/TCYB.2022.3179312","article-title":"A robust predefined-time convergence zeroing neural network for dynamic matrix inversion","author":"Jin","year":"","journal-title":"Proceedings of the IEEE Transactions on Cybernetics"},{"key":"B18","doi-asserted-by":"publisher","first-page":"2615","DOI":"10.1109\/TNNLS.2015.2497715","article-title":"Integration-enhanced Zhang neural network for real-time-varying matrix inversion in the presence of various kinds of noises.","volume":"27","author":"Jin","year":"2016","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"B19","doi-asserted-by":"crossref","first-page":"992","DOI":"10.1109\/TAC.2016.2566880","article-title":"Noise-tolerant ZNN models for solving time-varying zero-finding problems: a control-theoretic approach.","volume":"62","author":"Jin","year":"2017","journal-title":"IEEE Trans. Automatic Control"},{"key":"B20","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s11063-012-9241-1","article-title":"Accelerating a recurrent neural network to finite-time convergence for solving time-varying Sylvester equation by using a sign-bi-power activation function.","volume":"37","author":"Li","year":"2013","journal-title":"Neural Process. Lett."},{"key":"B21","doi-asserted-by":"crossref","first-page":"4028","DOI":"10.1109\/TSMC.2019.2930763","article-title":"A strictly predefined-time convergent neural solution to equality- and inequality-constrained time-variant quadratic programming.","volume":"51","author":"Li","year":"2021","journal-title":"IEEE Trans. Syst. Man Cybernet. Syst."},{"key":"B22","doi-asserted-by":"crossref","first-page":"8885","DOI":"10.1109\/LRA.2022.3186758","article-title":"A kinematic modeling and control scheme for different robotic endoscopes: a rudimentary research prototype.","volume":"7","author":"Li","year":"2022","journal-title":"IEEE Robot. Autom. Lett."},{"key":"B23","doi-asserted-by":"publisher","first-page":"3195","DOI":"10.1109\/TCYB.2019.2906263","article-title":"A finite-time convergent and noise-rejection recurrent neural network and its discretization for dynamic nonlinear equations solving.","volume":"50","author":"Li","year":"2020","journal-title":"IEEE Trans. Cybernet."},{"key":"B24","doi-asserted-by":"publisher","first-page":"8839","DOI":"10.1109\/TII.2022.3155599","article-title":"Brain-like initial-boosted hyperchaos and application in biomedical image encryption.","volume":"18","author":"Lin","year":"","journal-title":"IEEE Trans. Ind. Inform."},{"key":"B25","article-title":"A memristive synapse control method to generate diversified multi-structure chaotic attractors","author":"Lin","year":"","journal-title":"Proceedings of the IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems"},{"key":"B26","doi-asserted-by":"crossref","first-page":"723","DOI":"10.1109\/ChinaSIP.2014.6889339","article-title":"Deterministic complex-valued measurement matrices based on Berlekamp-Justesen codes","author":"Liu","year":"2014","journal-title":"Proceedings of the 2014 IEEE China Summit &amp; International Conference on Signal and Information Processing (ChinaSIP)"},{"key":"B27","article-title":"Activated gradients for deep neural networks","author":"Liu","year":"2021","journal-title":"Proceedings of the IEEE Transactions on Neural Networks and Learning Systems"},{"key":"B28","doi-asserted-by":"crossref","first-page":"1452","DOI":"10.1109\/JAS.2022.105731","article-title":"Gradient-based differential kWTA network with application to competitive coordination of multiple robots.","volume":"9","author":"Liu","year":"2022","journal-title":"IEEE CAA J. Autom. Sin."},{"key":"B29","first-page":"1","article-title":"Fast Jacobi like algorithms for joint diagonalization of complex symmetric matrices","author":"Maurandi","year":"2013","journal-title":"Proceedings of the 21st European Signal Processing Conference (EUSIPCO 2013)"},{"key":"B30","doi-asserted-by":"crossref","first-page":"1024","DOI":"10.1364\/AO.26.001024","article-title":"Optical implementation of an iterative algorithm for matrix inversion.","volume":"26","author":"Rajbenbach","year":"1987","journal-title":"Appl. Opt."},{"key":"B31","doi-asserted-by":"publisher","first-page":"587","DOI":"10.1109\/TNNLS.2020.3028136","article-title":"Novel discrete-time recurrent neural networks handling discrete-form time-variant multi-augmented Sylvester matrix problems and manipulator application.","volume":"33","author":"Shi","year":"","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"B32","article-title":"Tracking control of cable-driven planar robot based on discrete-time recurrent neural network with immediate discretization method","author":"Shi","year":"","journal-title":"Proceedings of the IEEE Transactions on Industrial Informatics"},{"key":"B33","doi-asserted-by":"publisher","DOI":"10.1109\/TNNLS.2021.3108050","article-title":"Novel discrete-time recurrent neural network for robot manipulator: a direct discretization technical route.","author":"Shi","year":"2021","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"B34","doi-asserted-by":"publisher","first-page":"1932","DOI":"10.1109\/TCYB.2015.2512852","article-title":"Image quality assessment based on gradient complex matrix","author":"Wang","year":"2012","journal-title":"Proceedings of the 2012 International Conference on Systems and Informatics (ICSAI2012)"},{"key":"B35","doi-asserted-by":"publisher","first-page":"1676","DOI":"10.1109\/TNN.2011.2163318","article-title":"Zhang neural network versus gradient neural network for solving time-varying linear inequalities.","volume":"22","author":"Xiao","year":"2011","journal-title":"IEEE Trans. Neural Netw."},{"key":"B36","first-page":"1411","article-title":"Revisit the analog computer and gradient-based neural system for matrix inversion","author":"Zhang","year":"2005","journal-title":"Proceedings of the 2005 IEEE International Symposium on, Mediterrean Conference on Control and Automation Intelligent Control"},{"key":"B37","doi-asserted-by":"publisher","first-page":"1477","DOI":"10.1109\/TNN.2005.857946","article-title":"Design and analysis of a general recurrent neural network model for time-varying matrix inversion.","volume":"16","author":"Zhang","year":"2005","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"B38","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1016\/j.ins.2021.12.084","article-title":"Cluster output synchronizationfor memristive neural networks.","volume":"589","author":"Zhou","year":"","journal-title":"Inform. Sci."},{"key":"B39","article-title":"Observer-based synchronization of memristive neural networks under DoS attacks and actuator saturation and its application to image encryption.","volume":"425","author":"Zhou","year":"","journal-title":"Appl. Math. Comput."},{"key":"B40","article-title":"Design and analysis of anti-noise parameter-variable zeroing neural network for dynamic complex matrix inversion and manipulator trajectory tracking.","volume":"11","author":"Zhou","year":"","journal-title":"Electronics"},{"key":"B41","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/j.matcom.2022.02.019","article-title":"A combined power activation function based convergent factor-variable ZNN model for solving dynamic matrix inversion.","volume":"197","author":"Zhu","year":"2022","journal-title":"Math. Comput. Simul."}],"container-title":["Frontiers in Neurorobotics"],"original-title":[],"link":[{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/fnbot.2022.1065256\/full","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,15]],"date-time":"2022-11-15T06:48:43Z","timestamp":1668494923000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/fnbot.2022.1065256\/full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,15]]},"references-count":41,"alternative-id":["10.3389\/fnbot.2022.1065256"],"URL":"https:\/\/doi.org\/10.3389\/fnbot.2022.1065256","relation":{},"ISSN":["1662-5218"],"issn-type":[{"value":"1662-5218","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,15]]},"article-number":"1065256"}}