{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:37:15Z","timestamp":1787330235626,"version":"build-2736575974"},"reference-count":27,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T00:00:00Z","timestamp":1787270400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Distributed maximum correntropy Kalman filters improve robustness to non-Gaussian noise, but existing variants generally introduce consensus through average or weighted fusion without explicitly separating the innovation residual from the state-disagreement residual in a dimensionally consistent objective. This paper proposes a Distributed Maximum Correntropy Kalman Filter with Innovation and Consensus Weighting Terms (DMCKF-IW-CWT). The innovation and consensus residuals are normalized separately and mapped by Gaussian kernels, after which the resulting information matrices are incorporated into a fixed-point local update. Posterior covariance intersection (CI) is then used to fuse neighboring estimates without requiring the unavailable cross-covariances. A sufficient contraction condition is given for the fixed-point iteration. In a five-node benchmark with 500 independent Monte Carlo runs and 1000 sampling steps, the proposed method obtains overall, transient, and steady-state MAEs of 0.172210, 0.188271, and 0.168195, respectively, corresponding to reductions of 0.254%, 0.526%, and 0.178% relative to DMCKF-W; the paired 95% confidence intervals of all three differences remain below zero. The consensus RMS is further reduced by 5.371%. Additional tests involving five noise families, packet loss and communication noise, a four-state nonlinear model, and systems with up to eight states and twenty nodes confirm the numerical convergence and extensibility of the framework.<\/jats:p>","DOI":"10.3390\/e28080941","type":"journal-article","created":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:51:05Z","timestamp":1787327465000},"page":"941","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Distributed Kalman Filter with Maximum Correlation Entropy Criterion and Consensus Weighted Term Fusion"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8798-2312","authenticated-orcid":false,"given":"Xiaoliang","family":"Feng","sequence":"first","affiliation":[{"name":"School of Electrical and Energy Engineering, Shanghai Dianji University, Shanghai 201306, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhouliner","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Electrical and Energy Engineering, Shanghai Dianji University, Shanghai 201306, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Teng","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Electrical and Energy Engineering, Shanghai Dianji University, Shanghai 201306, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,21]]},"reference":[{"key":"ref_1","first-page":"2705","article-title":"Consensus-based labeled multi-bernoulli filter with adaptive event-triggered communication","volume":"70","author":"Shen","year":"2023","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"27818","DOI":"10.1109\/JSEN.2021.3125153","article-title":"Distributed target tracking using maximum likelihood Kalman filter with non-linear measurements","volume":"21","author":"Huang","year":"2021","journal-title":"IEEE Sens. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1109\/TNSE.2022.3216867","article-title":"Distributed state estimation over sensor networks with substate decomposition approach","volume":"10","author":"Xu","year":"2023","journal-title":"IEEE Trans. Netw. Sci. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"111688","DOI":"10.1109\/ACCESS.2022.3215551","article-title":"Koopman Kalman particle filter for dynamic state estimation of distribution system","volume":"10","author":"Wang","year":"2022","journal-title":"IEEE Access"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"104629","DOI":"10.1016\/j.dsp.2024.104629","article-title":"Robust median consensus cubature Kalman filter for distributed sensor networks","volume":"153","author":"Zhou","year":"2024","journal-title":"Digit. Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"110091","DOI":"10.1016\/j.automatica.2021.110091","article-title":"Adaptive distributed Kalman-like filter for power system with cyber attacks","volume":"137","author":"Yang","year":"2022","journal-title":"Automatica"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Olfati-Saber, R. (2005). Distributed Kalman filter with embedded consensus filters. Proceedings of the 44th IEEE Conference on Decision and Control, Seville, Spain, 12\u201315 December 2005, IEEE.","DOI":"10.1109\/CDC.2005.1583486"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4611","DOI":"10.1109\/TSMC.2018.2855741","article-title":"Stability analysis of covariance intersection-based Kalman consensus filtering for time-varying systems","volume":"50","author":"Wei","year":"2020","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3794","DOI":"10.1109\/TAC.2020.3026017","article-title":"Randomized consensus-based distributed Kalman filtering over wireless sensor networks","volume":"66","author":"Qin","year":"2021","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Wang, G., Liu, K., Wang, J., and Xue, R. (2021). Distributed maximum correntropy Kalman filter with consensus strategies. Proceedings of the 2021 7th International Conference on Control, Automation and Robotics (ICCAR), Singapore, 23\u201326 April 2021, IEEE.","DOI":"10.1109\/ICCAR52225.2021.9463508"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1109\/LCSYS.2018.2844734","article-title":"Control-theoretical and topological analysis of covariance intersection based distributed Kalman filter","volume":"2","author":"Chang","year":"2018","journal-title":"IEEE Control Syst. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/j.automatica.2016.01.071","article-title":"Stability of consensus extended Kalman filter for distributed state estimation","volume":"68","author":"Battistelli","year":"2016","journal-title":"Automatica"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2464","DOI":"10.1016\/j.sigpro.2012.03.009","article-title":"Distributed consensus filtering for discrete time nonlinear systems with non-Gaussian noise","volume":"92","author":"Li","year":"2012","journal-title":"Signal Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1109\/TCYB.2015.2409373","article-title":"Weighted average consensus-based unscented Kalman filtering","volume":"46","author":"Li","year":"2016","journal-title":"IEEE Trans. Cybern."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Chen, Q., Wang, W., and Yin, C. (2016). Distributed state estimations based on cubature Kalman filtering. Proceedings of the 2016 Chinese Control and Decision Conference (CCDC), Yinchuan, China, 28\u201330 May 2016, IEEE.","DOI":"10.1109\/CCDC.2016.7531567"},{"key":"ref_16","first-page":"2767","article-title":"Distributed particle filter for nonlinear hybrid systems based on average consensus filter","volume":"44","author":"Samadi","year":"2011","journal-title":"IFAC Proc."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/j.sigpro.2019.02.030","article-title":"A distributed maximum correntropy Kalman filter","volume":"160","author":"Wang","year":"2019","journal-title":"Signal Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"107937","DOI":"10.1016\/j.sigpro.2020.107937","article-title":"Distributed maximum correntropy linear and nonlinear filters for systems with non-Gaussian noises","volume":"182","author":"Wang","year":"2021","journal-title":"Signal Process."},{"key":"ref_19","first-page":"433","article-title":"An efficient distributed Kalman filter over sensor networks with maximum correntropy criterion","volume":"8","author":"Hu","year":"2022","journal-title":"IEEE Trans. Signal Inf. Process. Over Netw."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1016\/j.inffus.2022.11.016","article-title":"Distributed minimum error entropy Kalman filter","volume":"91","author":"Feng","year":"2023","journal-title":"Inf. Fusion"},{"key":"ref_21","unstructured":"Han, C., Zhu, H., and Duan, Z. (2010). Multi-Source Information Fusion, Tsinghua University Press. (In Chinese)."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Principe, J.C. (2010). Information Theoretic Learning: Renyi\u2019s Entropy and Kernel Perspectives, Springer Science.","DOI":"10.1007\/978-1-4419-1570-2"},{"key":"ref_23","first-page":"98","article-title":"UWB indoor positioning algorithm based on maximum correntropy Kalman filter","volume":"51","author":"Zhang","year":"2024","journal-title":"Appl. Sci. Technol."},{"key":"ref_24","first-page":"2703","article-title":"Fixed-point smoothing estimation algorithm based on information entropy criterion in non-Gaussian environments","volume":"39","author":"Ma","year":"2024","journal-title":"Control Decis."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1325","DOI":"10.1109\/9.159570","article-title":"Kalman filtering and Riccati equations for descriptor systems","volume":"37","author":"Nikoukhah","year":"1992","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1254","DOI":"10.1049\/iet-cta.2017.0898","article-title":"Kalman filtering for discrete-time linear fractional-order singular systems","volume":"12","author":"Nosrati","year":"2018","journal-title":"IET Control Theory Appl."},{"key":"ref_27","first-page":"127950","article-title":"Extended fractional singular Kalman filter","volume":"448","author":"Nosrati","year":"2023","journal-title":"Appl. Math. Comput."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/8\/941\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:53:43Z","timestamp":1787327623000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/8\/941"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,21]]},"references-count":27,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2026,8]]}},"alternative-id":["e28080941"],"URL":"https:\/\/doi.org\/10.3390\/e28080941","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,21]]}}}