{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,27]],"date-time":"2026-07-27T01:11:24Z","timestamp":1785114684072,"version":"3.55.0"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2018,11,10]],"date-time":"2018-11-10T00:00:00Z","timestamp":1541808000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Neural Process Lett"],"published-print":{"date-parts":[[2019,10]]},"DOI":"10.1007\/s11063-018-9953-y","type":"journal-article","created":{"date-parts":[[2018,11,10]],"date-time":"2018-11-10T02:30:34Z","timestamp":1541817034000},"page":"1687-1703","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["A Non-linear and Noise-Tolerant ZNN Model and Its Application to Static and Time-Varying Matrix Square Root Finding"],"prefix":"10.1007","volume":"50","author":[{"given":"Xiaoxiao","family":"Li","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6451-1158","authenticated-orcid":false,"given":"Jiguo","family":"Yu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shuai","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zehui","family":"Shao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lina","family":"Ni","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2018,11,10]]},"reference":[{"issue":"4","key":"9953_CR1","doi-asserted-by":"publisher","first-page":"1603","DOI":"10.2298\/CSIS120121043Z","volume":"9","author":"Y Zhang","year":"2012","unstructured":"Zhang Y, Jin L, Ke Z (2012) Superior performance of using hyperbolic sine activation functions in ZNN illustrated via time-varying matrix square roots finding. Comput Sci Inf Syst 9(4):1603\u20131625","journal-title":"Comput Sci Inf Syst"},{"issue":"2","key":"9953_CR2","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1023\/A:1019150005407","volume":"15","author":"NJ Higham","year":"1997","unstructured":"Higham NJ (1997) Stable iterations for the matrix square root. Numer Algorithms 15(2):227\u2013242","journal-title":"Numer Algorithms"},{"issue":"2","key":"9953_CR3","doi-asserted-by":"publisher","first-page":"362","DOI":"10.1137\/S0895479803426656","volume":"26","author":"B Meini","year":"2004","unstructured":"Meini B (2004) The matrix square root from a new functional perspective: theoretical results and computational issues. SIAM J Matrix Anal Appl 26(2):362\u2013376","journal-title":"SIAM J Matrix Anal Appl"},{"key":"9953_CR4","doi-asserted-by":"publisher","unstructured":"Long J, Hu X, Zhang L (2008) Newton\u2019s method with exact line search for the square root of a matrix. In: International symposium on nonlinear dynamics. \n                    https:\/\/doi.org\/10.1088\/1742-6596\/96\/1\/012034","DOI":"10.1088\/1742-6596\/96\/1\/012034"},{"key":"9953_CR5","doi-asserted-by":"crossref","unstructured":"Zhang Y, Yang Y (2008) Simulation and comparison of Zhang neural network and gradient neural network solving for time-varying matrix square roots. In: Proceedings of the 2nd international symposium on intelligent information technology application, pp 966\u2013970","DOI":"10.1109\/IITA.2008.73"},{"issue":"24","key":"9953_CR6","doi-asserted-by":"publisher","first-page":"1103","DOI":"10.1016\/j.ipl.2010.09.013","volume":"110","author":"Y Zhang","year":"2010","unstructured":"Zhang Y, Ke Z, Xu P, Yi C (2010) Time-varying square roots finding via Zhang dynamics versus gradient dynamics and the former\u2019s link and new explanation to Newton\u2013Raphson iteration. Inf Process Lett 110(24):1103\u20131109","journal-title":"Inf Process Lett"},{"issue":"3","key":"9953_CR7","doi-asserted-by":"publisher","first-page":"453","DOI":"10.1007\/s00521-010-0445-x","volume":"21","author":"Y Zhang","year":"2012","unstructured":"Zhang Y, Yang Y, Cai B, Guo D (2012) Zhang neural network and its application to Newton iteration for matrix square root estimation. Neural Comput Appl 21(3):453\u2013460","journal-title":"Neural Comput Appl"},{"issue":"11","key":"9953_CR8","doi-asserted-by":"publisher","first-page":"4393","DOI":"10.1016\/j.eswa.2013.01.045","volume":"40","author":"Y Zhang","year":"2013","unstructured":"Zhang Y, Li W, Guo D, Ke Z (2013) Different Zhang functions leading to different ZNN models illustrated via time-varying matrix square roots finding. Expert Syst Appl 40(11):4393\u20134403","journal-title":"Expert Syst Appl"},{"issue":"2","key":"9953_CR9","doi-asserted-by":"publisher","first-page":"983","DOI":"10.1007\/s11071-014-1716-3","volume":"79","author":"Y Zhang","year":"2015","unstructured":"Zhang Y, Chen D, Guo D, Liao B, Wang Y (2015) On exponential convergence of nonlinear gradient dynamics system with application to square root finding. Nonlinear Dyn 79(2):983\u20131003","journal-title":"Nonlinear Dyn"},{"key":"9953_CR10","first-page":"1","volume":"10","author":"L Xiao","year":"2017","unstructured":"Xiao L (2017) A finite-time convergent Zhang neural network and its application to real-time matrix square root finding. Neural Comput Appl 10:1\u20138","journal-title":"Neural Comput Appl"},{"key":"9953_CR11","unstructured":"Zhang Y (2005) Revisit the analog computer and gradient-based neural system for matrix inversion. In: IEEE international symposium on intelligent control, pp 1411\u20131416"},{"issue":"8","key":"9953_CR12","doi-asserted-by":"publisher","first-page":"1940","DOI":"10.1109\/TAC.2009.2023779","volume":"54","author":"Y Zhang","year":"2009","unstructured":"Zhang Y, Chen K, Tan H-Z (2009) Performance analysis of gradient neural network exploited for online time-varying matrix inversion. IEEE Trans Autom Control 54(8):1940\u20131945","journal-title":"IEEE Trans Autom Control"},{"issue":"10","key":"9953_CR13","doi-asserted-by":"publisher","first-page":"1710","DOI":"10.1016\/j.neucom.2011.02.007","volume":"74","author":"Y Zhang","year":"2011","unstructured":"Zhang Y, Yang Y, Ruan G (2011) Performance analysis of gradient neural network exploited for online time-varying quadratic minimization and equality-constrained quadratic programming. Neurocomputing 74(10):1710\u20131719","journal-title":"Neurocomputing"},{"issue":"1","key":"9953_CR14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00521-010-0452-y","volume":"20","author":"Y Zhang","year":"2011","unstructured":"Zhang Y, Yi C, Guo D, Zheng J (2011) Comparison on Zhang neural dynamics and gradient-based neural dynamics for online solution of nonlinear time-varying equation. Neural Comput Appl 20(1):1\u20137","journal-title":"Neural Comput Appl"},{"issue":"3","key":"9953_CR15","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1007\/s11063-013-9306-9","volume":"39","author":"L Xiao","year":"2014","unstructured":"Xiao L, Zhang Y (2014) From different Zhang functions to various ZNN models accelerated to finite-time convergence for time-varying linear matrix equation. Neural Process Lett 39(3):309\u2013326","journal-title":"Neural Process Lett"},{"key":"9953_CR16","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1016\/j.neucom.2012.05.012","volume":"97","author":"D Guo","year":"2012","unstructured":"Guo D, Zhang Y (2012) Zhang neural network, Getz\u2013Marsden dynamic system, and discrete-time algorithms for time-varying matrix inversion with application to robots\u2019 kinematic control. Neurocomputing 97:22\u201332","journal-title":"Neurocomputing"},{"issue":"2","key":"9953_CR17","doi-asserted-by":"publisher","first-page":"992","DOI":"10.1109\/TAC.2016.2566880","volume":"62","author":"L Jin","year":"2017","unstructured":"Jin L, Zhang Y, Li S, Zhang Y (2017) Noise-tolerant ZNN models for solving time-varying zero-finding problems: a control-theoretic approach. IEEE Trans Autom Control 62(2):992\u2013997","journal-title":"IEEE Trans Autom Control"},{"issue":"18","key":"9953_CR18","doi-asserted-by":"publisher","first-page":"1639","DOI":"10.1016\/j.physleta.2009.03.011","volume":"373","author":"Y Zhang","year":"2009","unstructured":"Zhang Y, Li Z (2009) Zhang neural network for online solution of time-varying convex quadratic program subject to time-varying linear-equality constraints. Phys Lett A 373(18):1639\u20131643","journal-title":"Phys Lett A"},{"issue":"10","key":"9953_CR19","doi-asserted-by":"publisher","first-page":"2363","DOI":"10.1109\/TCSI.2012.2188944","volume":"59","author":"L Xiao","year":"2012","unstructured":"Xiao L, Zhang Y (2012) Two new types of Zhang neural networks solving systems of time-varying nonlinear inequalities. IEEE Trans Circuits Syst I Regul Pap 59(10):2363\u20132373","journal-title":"IEEE Trans Circuits Syst I Regul Pap"},{"issue":"2","key":"9953_CR20","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1007\/s11063-012-9241-1","volume":"37","author":"S Li","year":"2013","unstructured":"Li S, Chen S, Liu B (2013) 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(2):189\u2013205","journal-title":"Neural Process Lett"},{"issue":"12","key":"9953_CR21","doi-asserted-by":"publisher","first-page":"2615","DOI":"10.1109\/TNNLS.2015.2497715","volume":"27","author":"L Jin","year":"2016","unstructured":"Jin L, Zhang Y, Li S (2016) Integration-enhanced zhang neural network for real-time-varying matrix inversion in the presence of various kinds of noises. IEEE transactions on neural networks and learning systems 27(12):2615\u20132627","journal-title":"IEEE transactions on neural networks and learning systems"},{"key":"9953_CR22","doi-asserted-by":"publisher","first-page":"102","DOI":"10.1016\/j.neunet.2017.11.011","volume":"98","author":"L Xiao","year":"2018","unstructured":"Xiao L, Liao B, Li S, Chen K (2018) Nonlinear recurrent neural networks for finite-time solution of general time-varying linear matrix equations. Neural Netw 98:102\u2013113","journal-title":"Neural Netw"},{"issue":"13","key":"9953_CR23","doi-asserted-by":"publisher","first-page":"5667","DOI":"10.1016\/j.jfranklin.2017.06.012","volume":"354","author":"L Xiao","year":"2017","unstructured":"Xiao L (2017) 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 Inst 354(13):5667\u20135677","journal-title":"J Frankl Inst"},{"issue":"3","key":"9953_CR24","doi-asserted-by":"publisher","first-page":"1581","DOI":"10.1007\/s11071-017-3750-4","volume":"90","author":"L Xiao","year":"2017","unstructured":"Xiao L (2017) A finite-time recurrent neural network for solving online time-varying Sylvester matrix equation based on a new evolution formula. Nonlinear Dyn 90(3):1581\u20131591","journal-title":"Nonlinear Dyn"},{"issue":"1","key":"9953_CR25","doi-asserted-by":"publisher","first-page":"98","DOI":"10.1109\/TII.2017.2717020","volume":"14","author":"L Xiao","year":"2018","unstructured":"Xiao L, Liao B, Li S, Zhang Z, Ding L, Jin L (2018) Design and analysis of FTZNN applied to the real-time solution of a nonstationary lyapunov equation and tracking control of a wheeled mobile manipulator. IEEE Trans Ind Inform 14(1):98\u2013105","journal-title":"IEEE Trans Ind Inform"},{"issue":"6","key":"9953_CR26","doi-asserted-by":"publisher","first-page":"1477","DOI":"10.1109\/TNN.2005.857946","volume":"16","author":"Y Zhang","year":"2005","unstructured":"Zhang Y, Ge SS (2005) Design and analysis of a general recurrent neural network model for time-varying matrix inversion. IEEE Trans Neural Netw 16(6):1477\u20131490","journal-title":"IEEE Trans Neural Netw"},{"issue":"6","key":"9953_CR27","doi-asserted-by":"publisher","first-page":"1606","DOI":"10.1109\/TNN.2006.880580","volume":"17","author":"DV Prokhorov","year":"2006","unstructured":"Prokhorov DV (2006) Training recurrent neurocontrollers for robustness with derivative-free Kalman filter. IEEE Trans Neural Netw 17(6):1606\u20131616","journal-title":"IEEE Trans Neural Netw"},{"issue":"5","key":"9953_CR28","doi-asserted-by":"publisher","first-page":"775","DOI":"10.1109\/TNNLS.2012.2189893","volume":"23","author":"DH Dini","year":"2012","unstructured":"Dini DH, Mandic DP (2012) Class of widely linear complex Kalman filters. IEEE Trans Neural Netw Learn Syst 23(5):775\u2013786","journal-title":"IEEE Trans Neural Netw Learn Syst"},{"issue":"3","key":"9953_CR29","doi-asserted-by":"publisher","first-page":"1036","DOI":"10.1109\/TCST.2016.2577001","volume":"25","author":"W He","year":"2017","unstructured":"He W, Nie S, Meng T, Liu Y-J (2017) Modeling and vibration control for a moving beam with application in a drilling riser. IEEE Transactions on Control Systems Technology 25(3):1036\u20131043","journal-title":"IEEE Transactions on Control Systems Technology"},{"issue":"7","key":"9953_CR30","doi-asserted-by":"publisher","first-page":"1531","DOI":"10.1109\/TNNLS.2016.2531089","volume":"28","author":"Y-J Liu","year":"2017","unstructured":"Liu Y-J, Li S, Tong S, Chen CP (2017) Neural approximation-based adaptive control for a class of nonlinear nonstrict feedback discrete-time systems. IEEE Trans Neural Netw Learn Syst 28(7):1531\u20131541","journal-title":"IEEE Trans Neural Netw Learn Syst"},{"issue":"1","key":"9953_CR31","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1109\/TSMC.2016.2557227","volume":"47","author":"W He","year":"2017","unstructured":"He W, Ge W, Li Y, Liu Y-J, Yang C, Sun C (2017) Model identification and control design for a humanoid robot. IEEE Trans Syst Man Cybern Syst 47(1):45\u201357","journal-title":"IEEE Trans Syst Man Cybern Syst"},{"issue":"7","key":"9953_CR32","doi-asserted-by":"publisher","first-page":"1546","DOI":"10.1109\/TSMC.2016.2633007","volume":"47","author":"Y-J Liu","year":"2017","unstructured":"Liu Y-J, Lu S, Li D, Tong S (2017) Adaptive controller design-based ABLF for a class of nonlinear time-varying state constraint systems. IEEE Trans Syst Man Cybern Syst 47(7):1546\u20131553","journal-title":"IEEE Trans Syst Man Cybern Syst"},{"key":"9953_CR33","doi-asserted-by":"publisher","first-page":"70","DOI":"10.1016\/j.neucom.2018.07.067","volume":"317","author":"X Li","year":"2018","unstructured":"Li X, Yu J, Li S, Ni L (2018) A nonlinear and noise-tolerant ZNN model solving for time-varying linear matrix equation. Neurocomputing 317:70\u201378. \n                    https:\/\/doi.org\/10.1016\/j.neucom.2018.07.067","journal-title":"Neurocomputing"}],"container-title":["Neural Processing Letters"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11063-018-9953-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11063-018-9953-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11063-018-9953-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,11,9]],"date-time":"2019-11-09T19:17:22Z","timestamp":1573327042000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11063-018-9953-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,10]]},"references-count":33,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,10]]}},"alternative-id":["9953"],"URL":"https:\/\/doi.org\/10.1007\/s11063-018-9953-y","relation":{},"ISSN":["1370-4621","1573-773X"],"issn-type":[{"value":"1370-4621","type":"print"},{"value":"1573-773X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,11,10]]},"assertion":[{"value":"10 November 2018","order":1,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}