{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T21:22:09Z","timestamp":1783113729279,"version":"3.54.6"},"reference-count":46,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62333005"],"award-info":[{"award-number":["62333005"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Journal of Computational and Applied Mathematics"],"published-print":{"date-parts":[[2026,12]]},"DOI":"10.1016\/j.cam.2026.117838","type":"journal-article","created":{"date-parts":[[2026,5,28]],"date-time":"2026-05-28T16:06:40Z","timestamp":1779984400000},"page":"117838","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["Parameter identification of Hammerstein-Wiener systems based on hierarchical iterative algorithms"],"prefix":"10.1016","volume":"488","author":[{"given":"Lingling","family":"Lv","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-7170-0475","authenticated-orcid":false,"given":"Yulin","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Huaicheng","family":"Yan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yu","family":"Wu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.cam.2026.117838_bib0001","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2024.115181","article-title":"Online identification methods for a class of Hammerstein nonlinear systems using the adaptive particle filtering","volume":"186","author":"Xu","year":"2024","journal-title":"Chaos Solit. Fractals"},{"key":"10.1016\/j.cam.2026.117838_bib0002","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2024.114502","article-title":"Establishment and identification of MIMO fractional Hammerstein model with colored noise for PEMFC system","volume":"180","author":"Qian","year":"2024","journal-title":"Chaos Solit. Fractals"},{"key":"10.1016\/j.cam.2026.117838_bib0003","doi-asserted-by":"crossref","DOI":"10.1016\/j.compchemeng.2023.108178","article-title":"Pulse response method for the Wiener-type nonlinear process identification","volume":"172","author":"Lim","year":"2023","journal-title":"Comput. Chem. Eng."},{"key":"10.1016\/j.cam.2026.117838_bib0004","doi-asserted-by":"crossref","DOI":"10.1016\/j.ymssp.2024.111329","article-title":"A bias-correction modeling method of Hammerstein\u2013Wiener systems with polynomial nonlinearities using noisy measurements","volume":"213","author":"Hou","year":"2024","journal-title":"Mech. Syst. Signal Process."},{"key":"10.1016\/j.cam.2026.117838_bib0005","doi-asserted-by":"crossref","DOI":"10.1016\/j.isatra.2024.05.007","article-title":"Self-error learning framework-based algorithm for parameter recovery of extended Wiener\u2013Hammerstein systems subject to quantised measurements","author":"Cao","year":"2024","journal-title":"ISA Trans."},{"issue":"2","key":"10.1016\/j.cam.2026.117838_bib0006","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1080\/00207721.2020.1829163","article-title":"Weighted hierarchical stochastic gradient identification algorithms for ARX models","volume":"52","author":"Dong","year":"2021","journal-title":"Int. J. Syst. Sci."},{"issue":"17","key":"10.1016\/j.cam.2026.117838_bib0007","doi-asserted-by":"crossref","DOI":"10.1016\/j.jfranklin.2024.107295","article-title":"Two improved generalized extended stochastic gradient algorithms for CARARMA systems","volume":"361","author":"Lv","year":"2024","journal-title":"J. Frankl. Inst."},{"key":"10.1016\/j.cam.2026.117838_bib0008","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/j.dsp.2018.09.006","article-title":"Robust identification of discrete-time linear systems with unknown time-varying disturbance","volume":"83","author":"Pu","year":"2018","journal-title":"Digit. Signal Process."},{"key":"10.1016\/j.cam.2026.117838_bib0009","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.sigpro.2017.06.025","article-title":"Identification for Hammerstein nonlinear ARMAX systems based on multi-innovation fractional order stochastic gradient","volume":"142","author":"Cheng","year":"2018","journal-title":"Signal Process."},{"key":"10.1016\/j.cam.2026.117838_bib0010","doi-asserted-by":"crossref","DOI":"10.1016\/j.dsp.2025.105024","article-title":"Parameter identification strategy for fractional-order hammerstein MIMO systems with PEMFC experimental validation","volume":"159","author":"Liu","year":"2025","journal-title":"Digit. Signal Process."},{"key":"10.1016\/j.cam.2026.117838_bib0011","first-page":"1","article-title":"Recursive parameter estimation of fractional order Hammerstein output error autoregressive model","author":"Li","year":"2025","journal-title":"Circuits Syst. Signal Process."},{"issue":"9","key":"10.1016\/j.cam.2026.117838_bib0012","doi-asserted-by":"crossref","first-page":"5161","DOI":"10.1007\/s00034-023-02361-y","article-title":"Improved gravitational search and gradient iterative identification for multivariable hammerstein time-delay systems","volume":"42","author":"Li","year":"2023","journal-title":"Circuits Syst. Signal Process."},{"key":"10.1016\/j.cam.2026.117838_bib0013","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1016\/j.sigpro.2015.09.024","article-title":"Nonparametric identification of a Wiener system using a stochastic excitation of arbitrarily unknown spectrum","volume":"120","author":"Lin","year":"2016","journal-title":"Signal Process."},{"key":"10.1016\/j.cam.2026.117838_bib0014","doi-asserted-by":"crossref","DOI":"10.1016\/j.dsp.2020.102716","article-title":"Robust extended recursive least squares identification algorithm for Hammerstein systems with dynamic disturbances","volume":"101","author":"Dong","year":"2020","journal-title":"Digit. Signal Process."},{"issue":"10","key":"10.1016\/j.cam.2026.117838_bib0015","doi-asserted-by":"crossref","first-page":"5366","DOI":"10.1007\/s00034-022-02031-5","article-title":"Parameter and state estimation of nonlinear fractional-order model using luenberger observer","volume":"41","author":"Marzougui","year":"2022","journal-title":"Circuits Syst. Signal Process."},{"issue":"12","key":"10.1016\/j.cam.2026.117838_bib0016","doi-asserted-by":"crossref","first-page":"6007","DOI":"10.1002\/rnc.5576","article-title":"Decomposition-based over-parameterization forgetting factor stochastic gradient algorithm for Hammerstein-Wiener nonlinear systems with non-uniform sampling","volume":"31","author":"Liu","year":"2021","journal-title":"Int. J. Robust Nonlinear Control"},{"issue":"4","key":"10.1016\/j.cam.2026.117838_bib0017","doi-asserted-by":"crossref","first-page":"694","DOI":"10.1007\/s40436-022-00426-w","article-title":"Identification of nonlinear process described by neural fuzzy Hammerstein-Wiener model using multi-signal processing","volume":"11","author":"Li","year":"2023","journal-title":"Adv. Manuf."},{"key":"10.1016\/j.cam.2026.117838_bib0018","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.sigpro.2014.05.030","article-title":"Recursive parameter identification of Hammerstein\u2013Wiener systems with measurement noise","volume":"105","author":"Yu","year":"2014","journal-title":"Signal Process."},{"issue":"3","key":"10.1016\/j.cam.2026.117838_bib0019","doi-asserted-by":"crossref","first-page":"1726","DOI":"10.1007\/s00034-024-02900-1","article-title":"Parameter identification for the Hammerstein-Wiener nonlinear time delay systems with process noises","volume":"44","author":"Li","year":"2025","journal-title":"Circuits Syst. Signal Process."},{"key":"10.1016\/j.cam.2026.117838_bib0020","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/j.dsp.2016.05.005","article-title":"Performance analysis of multi-innovation stochastic Newton recursive algorithms","volume":"56","author":"Cao","year":"2016","journal-title":"Digit. Signal Process."},{"issue":"3","key":"10.1016\/j.cam.2026.117838_bib0021","doi-asserted-by":"crossref","first-page":"505","DOI":"10.3182\/20140824-6-ZA-1003.00496","article-title":"Identification of parallel Wiener-Hammerstein systems with a decoupled static nonlinearity","volume":"47","author":"Schoukens","year":"2014","journal-title":"IFAC Proc. Vol."},{"issue":"7","key":"10.1016\/j.cam.2026.117838_bib0022","first-page":"679","article-title":"Nonlinear system identification with dominating output noise-a case study on the silverbox","volume":"54","author":"Schoukens","year":"2021","journal-title":"IFAC-Pap."},{"issue":"1","key":"10.1016\/j.cam.2026.117838_bib0023","doi-asserted-by":"crossref","first-page":"208","DOI":"10.3182\/20050703-6-CZ-1902.00035","article-title":"The identification of a class of nonlinear systems using a correlation analysis approach","volume":"38","author":"Lang","year":"2005","journal-title":"IFAC Proc. Vol."},{"issue":"10","key":"10.1016\/j.cam.2026.117838_bib0024","doi-asserted-by":"crossref","first-page":"72","DOI":"10.3182\/20090706-3-FR-2004.00011","article-title":"Consistency of weighted least-square estimators for parameter estimation problems based on binary measurements","volume":"42","author":"Juillard","year":"2009","journal-title":"IFAC Proc. Vol."},{"issue":"3","key":"10.1016\/j.cam.2026.117838_bib0025","doi-asserted-by":"crossref","first-page":"4097","DOI":"10.3182\/20140824-6-ZA-1003.01843","article-title":"Identification of gaussian process state-space models with particle stochastic approximation EM","volume":"47","author":"Frigola","year":"2014","journal-title":"IFAC Proc. Vol."},{"key":"10.1016\/j.cam.2026.117838_bib0026","doi-asserted-by":"crossref","DOI":"10.1016\/j.measurement.2022.111951","article-title":"Parameter identification of fractional order Hammerstein model with two-stage piecewise nonlinearity based on iterative algorithms","volume":"203","author":"Rui","year":"2022","journal-title":"Measurement"},{"key":"10.1016\/j.cam.2026.117838_bib0027","doi-asserted-by":"crossref","DOI":"10.1016\/j.cam.2023.115297","article-title":"Iterative parameter identification algorithms for transformed dynamic rational fraction input\u2013output systems","volume":"434","author":"Miao","year":"2023","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.cam.2026.117838_bib0028","first-page":"1","article-title":"Separable synchronous multi-innovation gradient-based iterative signal modeling from on-line measurements","volume":"71","author":"Xu","year":"2022","journal-title":"IEEE Trans. Instrum. Meas."},{"issue":"17","key":"10.1016\/j.cam.2026.117838_bib0029","doi-asserted-by":"crossref","first-page":"7885","DOI":"10.1016\/j.jfranklin.2017.09.014","article-title":"Iterative identification algorithms for bilinear-in-parameter systems with autoregressive moving average noise","volume":"354","author":"Chen","year":"2017","journal-title":"J. Frankl. Inst."},{"key":"10.1016\/j.cam.2026.117838_bib0030","doi-asserted-by":"crossref","DOI":"10.1016\/j.cam.2023.115104","article-title":"Separable synthesis gradient estimation methods and convergence analysis for multivariable systems","volume":"427","author":"Xu","year":"2023","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.cam.2026.117838_bib0031","doi-asserted-by":"crossref","DOI":"10.1016\/j.cam.2023.115107","article-title":"Least squares parameter estimation and multi-innovation least squares methods for linear fitting problems from noisy data","volume":"426","author":"Ding","year":"2023","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.cam.2026.117838_bib0032","doi-asserted-by":"crossref","DOI":"10.1016\/j.dsp.2024.104951","article-title":"Online identification for output-error models with random time delays based on auxiliary model and recursive expectation maximization algorithm","volume":"158","author":"Li","year":"2025","journal-title":"Digit. Signal Process."},{"key":"10.1016\/j.cam.2026.117838_bib0033","doi-asserted-by":"crossref","DOI":"10.1016\/j.arcontrol.2025.100993","article-title":"Hierarchical generalized extended parameter identification for multivariable equation-error ARMA-like systems by using the filtering identification idea","volume":"60","author":"Ding","year":"2025","journal-title":"Annu. Rev. Control"},{"issue":"8","key":"10.1016\/j.cam.2026.117838_bib0034","doi-asserted-by":"crossref","first-page":"3737","DOI":"10.1016\/j.jfranklin.2018.01.052","article-title":"A hierarchical least squares identification algorithm for Hammerstein nonlinear systems using the key term separation","volume":"355","author":"Ding","year":"2018","journal-title":"J. Frankl. Inst."},{"key":"10.1016\/j.cam.2026.117838_bib0035","doi-asserted-by":"crossref","DOI":"10.1016\/j.cam.2022.114794","article-title":"Hierarchical multi-innovation stochastic gradient identification algorithm for estimating a bilinear state-space model with moving average noise","volume":"420","author":"Gu","year":"2023","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.cam.2026.117838_bib0036","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1016\/j.amc.2014.05.059","article-title":"Least squares based iterative identification for multivariable integrating and unstable processes in closed loop","volume":"242","author":"Jin","year":"2014","journal-title":"Appl. Math. Comput."},{"issue":"12","key":"10.1016\/j.cam.2026.117838_bib0037","doi-asserted-by":"crossref","first-page":"825","DOI":"10.1109\/LSP.2012.2221704","article-title":"Hierarchical least squares estimation algorithm for Hammerstein\u2013Wiener systems","volume":"19","author":"Wang","year":"2012","journal-title":"IEEE Signal Process. Lett."},{"key":"10.1016\/j.cam.2026.117838_bib0038","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/j.isatra.2023.10.006","article-title":"Hierarchical iterative identification of output nonlinear Box-Jenkins Wiener model with ARMA noise","volume":"143","author":"Nadi","year":"2023","journal-title":"ISA Trans."},{"issue":"3","key":"10.1016\/j.cam.2026.117838_bib0039","doi-asserted-by":"crossref","first-page":"672","DOI":"10.1016\/j.camwa.2010.12.014","article-title":"Hierarchical gradient based iterative parameter estimation algorithm for multivariable output error moving average systems","volume":"61","author":"Zhang","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"10.1016\/j.cam.2026.117838_bib0040","unstructured":"F. Ding, System identification-auxiliary model identification idea and methods, 2017."},{"issue":"10","key":"10.1016\/j.cam.2026.117838_bib0041","doi-asserted-by":"crossref","first-page":"5510","DOI":"10.1002\/rnc.6657","article-title":"Filtered auxiliary model recursive generalized extended parameter estimation methods for Box\u2013Jenkins systems by means of the filtering identification idea","volume":"33","author":"Ding","year":"2023","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"10.1016\/j.cam.2026.117838_bib0042","doi-asserted-by":"crossref","DOI":"10.1016\/j.arcontrol.2024.100942","article-title":"Recursive identification methods for general stochastic systems with colored noises by using the hierarchical identification principle and the filtering identification idea","volume":"57","author":"Ding","year":"2024","journal-title":"Annu. Rev. Control"},{"issue":"5","key":"10.1016\/j.cam.2026.117838_bib0043","doi-asserted-by":"crossref","first-page":"1200","DOI":"10.1016\/j.camwa.2010.06.001","article-title":"Gradient-based iterative parameter estimation for Box\u2013Jenkins systems","volume":"60","author":"Wang","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"10.1016\/j.cam.2026.117838_bib0044","doi-asserted-by":"crossref","DOI":"10.1002\/oca.3321","article-title":"Auxiliary model maximum likelihood moving-data-window generalized extended gradient-based iterative algorithm for multivariable autoregressive output-Error autoregressive moving-Average systems","author":"Zhang","year":"2025","journal-title":"Optim. Control Appl. Methods"},{"key":"10.1016\/j.cam.2026.117838_bib0045","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.sigpro.2018.01.012","article-title":"The least squares based iterative algorithms for parameter estimation of a bilinear system with autoregressive noise using the data filtering technique","volume":"147","author":"Li","year":"2018","journal-title":"Signal Process."},{"issue":"3","key":"10.1016\/j.cam.2026.117838_bib0046","first-page":"441","article-title":"Auxiliary model maximum likelihood least squares-based iterative algorithm for multivariable autoregressive output-error autoregressive moving average systems","volume":"239","author":"Zhang","year":"2025","journal-title":"Proc. Inst. Mech. Eng. I J. Syst. Control Eng."}],"container-title":["Journal of Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0377042726004802?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0377042726004802?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T20:45:37Z","timestamp":1783111537000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0377042726004802"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,12]]},"references-count":46,"alternative-id":["S0377042726004802"],"URL":"https:\/\/doi.org\/10.1016\/j.cam.2026.117838","relation":{},"ISSN":["0377-0427"],"issn-type":[{"value":"0377-0427","type":"print"}],"subject":[],"published":{"date-parts":[[2026,12]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Parameter identification of Hammerstein-Wiener systems based on hierarchical iterative algorithms","name":"articletitle","label":"Article Title"},{"value":"Journal of Computational and Applied Mathematics","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cam.2026.117838","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"117838"}}