{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T01:14:26Z","timestamp":1778721266865,"version":"3.51.4"},"reference-count":55,"publisher":"Elsevier BV","issue":"6","license":[{"start":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T00:00:00Z","timestamp":1775001600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T00:00:00Z","timestamp":1775001600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62171285"],"award-info":[{"award-number":["62171285"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62327807"],"award-info":[{"award-number":["62327807"]}],"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 the Franklin Institute"],"published-print":{"date-parts":[[2026,4]]},"DOI":"10.1016\/j.jfranklin.2026.108526","type":"journal-article","created":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T00:17:24Z","timestamp":1771373844000},"page":"108526","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"title":["Expectation maximization estimation for multirate state-space models with time-varying delays"],"prefix":"10.1016","volume":"363","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2337-5043","authenticated-orcid":false,"given":"Ya","family":"Gu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-5249-0829","authenticated-orcid":false,"given":"Lujia","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0930-6623","authenticated-orcid":false,"given":"Yonghong","family":"Tan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0167-3341","authenticated-orcid":false,"given":"Shiyi","family":"Yin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"78","reference":[{"issue":"17","key":"10.1016\/j.jfranklin.2026.108526_bib0001","doi-asserted-by":"crossref","DOI":"10.1016\/j.jfranklin.2025.108143","article-title":"Hierarchical maximum likelihood multi-innovation identification methods for a class of multivariable Hammerstein-input-nonlinear systems","volume":"362","author":"Wang","year":"2025","journal-title":"J. Franklin Inst."},{"issue":"9","key":"10.1016\/j.jfranklin.2026.108526_bib0002","doi-asserted-by":"crossref","first-page":"9646","DOI":"10.1109\/TCYB.2021.3063113","article-title":"Robust standard gradient descent algorithm for ARX models using Aitken acceleration technique","volume":"52","author":"Chen","year":"2022","journal-title":"IEEE Trans. Cybern."},{"issue":"9","key":"10.1016\/j.jfranklin.2026.108526_bib0003","doi-asserted-by":"crossref","first-page":"1978","DOI":"10.1002\/acs.4036","article-title":"Hierarchical recursive gradient parameter identification for multi-input ARX systems with partially-coupled information vectors","volume":"39","author":"Ding","year":"2025","journal-title":"Int. J. Adapt. Contr. Signal Process"},{"issue":"5","key":"10.1016\/j.jfranklin.2026.108526_bib0004","doi-asserted-by":"crossref","first-page":"2255","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","volume":"46","author":"Zhang","year":"2025","journal-title":"Optim. Contr. Appl. Methods"},{"key":"10.1016\/j.jfranklin.2026.108526_bib0005","doi-asserted-by":"crossref","first-page":"2089","DOI":"10.1109\/LSP.2025.3567804","article-title":"A novel three-stage filtering identification algorithm for the exponential autoregressive time-series model","volume":"32","author":"Xu","year":"2025","journal-title":"IEEE Signal Process. Lett."},{"issue":"3","key":"10.1016\/j.jfranklin.2026.108526_bib0006","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. Part I-J. Syst. Contr. Eng."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0007","doi-asserted-by":"crossref","DOI":"10.1016\/j.jprocont.2023.103007","article-title":"An identification algorithm of generalized time-varying systems based on the Taylor series expansion and applied to a pH process","volume":"128","author":"Ji","year":"2023","journal-title":"J. Process Contr."},{"issue":"6","key":"10.1016\/j.jfranklin.2026.108526_bib0008","doi-asserted-by":"crossref","first-page":"2198","DOI":"10.1109\/TSMC.2018.2810277","article-title":"Gradient-based particle filter algorithm for an ARX model with nonlinear communication output","volume":"50","author":"Chen","year":"2020","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"issue":"1","key":"10.1016\/j.jfranklin.2026.108526_bib0009","first-page":"206","article-title":"Filtering-based accelerated estimation approach for generalized time-varying systems with disturbances and colored noises","volume":"70","author":"Ji","year":"2023","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"issue":"15","key":"10.1016\/j.jfranklin.2026.108526_bib0010","doi-asserted-by":"crossref","DOI":"10.1016\/j.jfranklin.2025.107992","article-title":"Auxiliary model gradient-based iterative identification with moving data window for Wiener nonlinear output-error systems","volume":"362","author":"Tian","year":"2025","journal-title":"J. Frankl. Inst."},{"issue":"18","key":"10.1016\/j.jfranklin.2026.108526_bib0011","doi-asserted-by":"crossref","DOI":"10.1016\/j.jfranklin.2024.107352","article-title":"Auxiliary model-based maximum likelihood multi-innovation recursive least squares identification for multiple-input multiple-output systems","volume":"361","author":"Wang","year":"2024","journal-title":"J. Frankl. Inst."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0012","doi-asserted-by":"crossref","DOI":"10.1016\/j.sysconle.2025.106166","article-title":"Two-stage parameter estimation methods for linear time-invariant continuous-time systems","volume":"204","author":"Ding","year":"2025","journal-title":"Syst. Contr. Lett."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0013","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1016\/j.automatica.2018.06.027","article-title":"Joint state and parameter estimation of non-linearly parameterized discrete-time nonlinear systems","volume":"97","author":"Ibrir","year":"2018","journal-title":"Automatica"},{"issue":"13","key":"10.1016\/j.jfranklin.2026.108526_bib0014","doi-asserted-by":"crossref","first-page":"5364","DOI":"10.1002\/rnc.7986","article-title":"The Aitken accelerated gradient algorithm for a class of dual-rate volterra nonlinear systems utilizing the self-organizing map technique","volume":"35","author":"Wang","year":"2025","journal-title":"Int. J. Robust Nonlinear Contr."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0015","series-title":"Multirate mixture probability principal component analysis for process monitoring in multimode processes","author":"Lyu","year":"2023"},{"issue":"10","key":"10.1016\/j.jfranklin.2026.108526_bib0016","doi-asserted-by":"crossref","first-page":"1739","DOI":"10.1016\/j.automatica.2004.05.001","article-title":"Combined parameter and output estimation of dual-rate systems using an auxiliary model","volume":"40","author":"Ding","year":"2004","journal-title":"Automatica"},{"issue":"9","key":"10.1016\/j.jfranklin.2026.108526_bib0017","doi-asserted-by":"crossref","first-page":"1436","DOI":"10.1109\/TAC.2005.854654","article-title":"Parameter estimation of dual-rate stochastic systems by using an output error method","volume":"50","author":"Ding","year":"2005","journal-title":"IEEE Trans Autom. Contr."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0018","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.sysconle.2018.07.004","article-title":"Robust sensitive fault detection and estimation for single-rate and multirate nonlinear sampled-data systems","volume":"119","author":"Beikzadeh","year":"2018","journal-title":"Syst. Contr. Lett."},{"issue":"7","key":"10.1016\/j.jfranklin.2026.108526_bib0019","doi-asserted-by":"crossref","first-page":"1155","DOI":"10.1109\/TSMC.2016.2645797","article-title":"An approach to fault detection for multirate sampled-data systems with frequency specifications","volume":"48","author":"Xue","year":"2018","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0020","doi-asserted-by":"crossref","DOI":"10.1016\/j.automatica.2020.109034","article-title":"Modified Kalman filtering based multi-step-length gradient iterative algorithm for ARX models with random missing outputs","volume":"118","author":"Chen","year":"2020","journal-title":"Automatica"},{"key":"10.1016\/j.jfranklin.2026.108526_bib0021","series-title":"Enhanced dynamic dual-latent variable model for multirate process monitoring and its industrial application","author":"He","year":"2023"},{"key":"10.1016\/j.jfranklin.2026.108526_bib0022","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.jprocont.2022.09.008","article-title":"Flexible clockwork recurrent neural network for multirate industrial soft sensor","volume":"119","author":"Chang","year":"2022","journal-title":"J. Process Contr."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0023","doi-asserted-by":"crossref","first-page":"947","DOI":"10.1109\/LSP.2022.3152108","article-title":"Joint parameter and time-delay estimation for a class of nonlinear time-series models","volume":"29","author":"Xu","year":"2022","journal-title":"IEEE Signal Process. Lett."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0024","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1109\/LSP.2024.3519897","article-title":"A delta operator state estimation algorithm for discrete-time systems with state time-delay","volume":"32","author":"Xu","year":"2025","journal-title":"IEEE Signal Process. Lett."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0025","doi-asserted-by":"crossref","DOI":"10.1016\/j.sysconle.2024.105948","article-title":"Stability analysis of systems with time-varying delays for conservatism and complexity reduction","volume":"193","author":"Fan","year":"2024","journal-title":"Syst. Contr. Lett."},{"issue":"7","key":"10.1016\/j.jfranklin.2026.108526_bib0026","doi-asserted-by":"crossref","first-page":"5242","DOI":"10.1109\/TIE.2018.2868008","article-title":"Pade-based stability analysis for a modular multilevel converter considering the time delay in the digital control system","volume":"66","author":"Dong","year":"2019","journal-title":"IEEE Trans. Ind. Electron."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0027","doi-asserted-by":"crossref","DOI":"10.1016\/j.cnsns.2025.109120","article-title":"Stability of nonlinear time-varying systems with delayed stochastic impulses","volume":"151","author":"Song","year":"2025","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0028","article-title":"Input delay compensation of nonlinear systems with uncertain time-varying parameters","volume":"80","author":"Jain","year":"2024","journal-title":"Europe J. Contr."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0029","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1016\/j.automatica.2019.02.037","article-title":"Observation and stabilization of LTV systems with time-varying measurement delay","volume":"103","author":"Sanz","year":"2019","journal-title":"Automatica"},{"key":"10.1016\/j.jfranklin.2026.108526_bib0030","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1016\/j.automatica.2017.07.044","article-title":"Unknown input functional observability of descriptor systems with neutral and distributed delay effects","volume":"85","author":"Javier","year":"2017","journal-title":"Automatica"},{"key":"10.1016\/j.jfranklin.2026.108526_bib0031","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1016\/j.automatica.2018.04.003","article-title":"Variational Bayesian approach for ARX systems with missing observations and varying time-delays","volume":"94","author":"Chen","year":"2018","journal-title":"Automatica"},{"key":"10.1016\/j.jfranklin.2026.108526_bib0032","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":"4","key":"10.1016\/j.jfranklin.2026.108526_bib0033","doi-asserted-by":"crossref","first-page":"1569","DOI":"10.1002\/oca.3279","article-title":"Multi-innovation gradient identification methods for bilinear output-error systems","volume":"46","author":"Li","year":"2025","journal-title":"Optim. Contr. Appl. Methods"},{"issue":"10","key":"10.1016\/j.jfranklin.2026.108526_bib0034","first-page":"1763","article-title":"Highly efficient maximum-likelihood identification methods for bilinear systems with colored noises","volume":"238","author":"Li","year":"2024","journal-title":"Proc. Inst. Mech. Eng. Part I-J Syst. Contr. Eng."},{"issue":"8","key":"10.1016\/j.jfranklin.2026.108526_bib0035","doi-asserted-by":"crossref","first-page":"5185","DOI":"10.1109\/TII.2020.3025581","article-title":"Identification of two-dimensional causal systems with missing output data via expectation-maximization algorithm","volume":"17","author":"Chen","year":"2021","journal-title":"IEEE Trans. Ind. Inf."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0036","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."},{"issue":"16","key":"10.1016\/j.jfranklin.2026.108526_bib0037","doi-asserted-by":"crossref","DOI":"10.1016\/j.jfranklin.2024.107113","article-title":"Multiple-model state-space system identification with time delay using the EM algorithm","volume":"361","author":"Gu","year":"2024","journal-title":"J. Franklin Inst."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0038","doi-asserted-by":"crossref","DOI":"10.1016\/j.automatica.2022.110365","article-title":"Expectation-maximization algorithm for bilinear systems by using the Rauch-Tung-Striebel smoother","volume":"142","author":"Liu","year":"2022","journal-title":"Automatica"},{"key":"10.1016\/j.jfranklin.2026.108526_bib0039","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.automatica.2016.01.036","article-title":"Robust EM kernel-based methods for linear system identification","volume":"67","author":"Bottegal","year":"2016","journal-title":"Automatica"},{"key":"10.1016\/j.jfranklin.2026.108526_bib0040","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."},{"issue":"3","key":"10.1016\/j.jfranklin.2026.108526_bib0041","doi-asserted-by":"crossref","first-page":"1623","DOI":"10.1016\/j.jfranklin.2018.08.030","article-title":"State space model identification of multirate processes with time-delay using the expectation maximization","volume":"356","author":"Gu","year":"2019","journal-title":"J. Frankl. Inst."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0042","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. Contr."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0043","doi-asserted-by":"crossref","DOI":"10.1016\/j.automatica.2021.109663","article-title":"A novel reduced-order algorithm for rational models based on Arnoldi process and Krylov subspace","volume":"129","author":"Chen","year":"2021","journal-title":"Automatica"},{"issue":"3","key":"10.1016\/j.jfranklin.2026.108526_bib0044","doi-asserted-by":"crossref","first-page":"1249","DOI":"10.1002\/oca.3257","article-title":"Auxiliary model-based maximum likelihood multi-innovation forgetting gradient identification for a class of multivariable systems","volume":"46","author":"Wang","year":"2025","journal-title":"Optim. Contr. Appl. Methods"},{"issue":"10","key":"10.1016\/j.jfranklin.2026.108526_bib0045","first-page":"1719","article-title":"Hierarchical extended parameter identification methods and convergence for finite impulse response moving average models based on the hierarchical identification principle","volume":"239","author":"Ding","year":"2025","journal-title":"Proc. Inst. Mech. Eng. Part I-J. Syst. Contr. Eng."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0046","doi-asserted-by":"crossref","DOI":"10.1016\/j.ymssp.2025.112912","article-title":"A robust filter and smoother-based expectation-maximization algorithm for bilinear systems with heavy-tailed noise","volume":"236","author":"Wang","year":"2025","journal-title":"Mech. Syst. Signal Process."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0047","series-title":"Parameter identification and open-circuit voltage estimation of lithium-ion battery circuit model based on multi-innovation theory","author":"Zou","year":"2025"},{"key":"10.1016\/j.jfranklin.2026.108526_bib0048","series-title":"Multi-innovation recursive methods for a class of nonlinear time series models based on the penalty term","first-page":"2494-2510","volume":"39","author":"Niu","year":"2025"},{"key":"10.1016\/j.jfranklin.2026.108526_bib0049","doi-asserted-by":"crossref","DOI":"10.1016\/j.ces.2025.122891","article-title":"Efficient state estimation for post-combustion CO2 capture plants using PODAE-based reduced-order modeling","volume":"321","author":"Liu","year":"2026","journal-title":"Chem. Eng. Sci."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0050","series-title":"Circuits Syst. Signal Process","article-title":"Greedy orthogonal least squares identification for multivariable Hammerstein models","author":"Liu","year":"2025"},{"key":"10.1016\/j.jfranklin.2026.108526_bib0051","doi-asserted-by":"crossref","DOI":"10.1109\/TIM.2025.3617413","article-title":"Dynamic factor and multi-innovation-based output-input feedback Elman network modeling from measurements","volume":"74","author":"Zhou","year":"2025","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0052","series-title":"IEEE Trans. Cybern.","article-title":"Kalman-based joint estimation for generalized time-varying parameter systems with the unknown invariant matrix","author":"Xu","year":"2025"},{"key":"10.1016\/j.jfranklin.2026.108526_bib0053","doi-asserted-by":"crossref","DOI":"10.1016\/j.ymssp.2025.113682","article-title":"Identification of the Silverbox benchmark using extended polynomial kernel-based nonlinear observer canonical models","volume":"243","author":"Li","year":"2026","journal-title":"Mech. Syst. Signal Process."},{"key":"10.1016\/j.jfranklin.2026.108526_bib0054","article-title":"Varying infimum gradient descent algorithm for agent-server systems with uncertain communication network","volume":"70","author":"Chen","year":"2021","journal-title":"IEEE Trans. Instrum. Meas."},{"issue":"10","key":"10.1016\/j.jfranklin.2026.108526_bib0055","doi-asserted-by":"crossref","first-page":"4385","DOI":"10.1109\/TAC.2019.2955030","article-title":"Interval error correction auxiliary model based gradient iterative algorithms for multirate ARX models","volume":"65","author":"Chen","year":"2020","journal-title":"IEEE Trans. Autom. Contr."}],"container-title":["Journal of the Franklin Institute"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0016003226001262?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0016003226001262?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T00:59:53Z","timestamp":1778720393000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0016003226001262"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4]]},"references-count":55,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2026,4]]}},"alternative-id":["S0016003226001262"],"URL":"https:\/\/doi.org\/10.1016\/j.jfranklin.2026.108526","relation":{},"ISSN":["0016-0032"],"issn-type":[{"value":"0016-0032","type":"print"}],"subject":[],"published":{"date-parts":[[2026,4]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Expectation maximization estimation for multirate state-space models with time-varying delays","name":"articletitle","label":"Article Title"},{"value":"Journal of the Franklin Institute","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.jfranklin.2026.108526","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 The Franklin Institute. Published by Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"108526"}}