{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T10:19:46Z","timestamp":1769854786684,"version":"3.49.0"},"reference-count":35,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,1,15]],"date-time":"2022-01-15T00:00:00Z","timestamp":1642204800000},"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","award":["61933013"],"award-info":[{"award-number":["61933013"]}],"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":["61733015"],"award-info":[{"award-number":["61733015"]}],"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":["61733009"],"award-info":[{"award-number":["61733009"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In this paper, a novel design idea of high-order Kalman filter based on Kronecker product transform is proposed for a class of strong nonlinear stochastic dynamic systems. Firstly, those augmenting systems are modeled with help of the Kronecker product without system noise. Secondly, the augmented system errors are illustratively charactered by Gaussian white noise. Thirdly, at the expanded space a creative high-order Kalman filter is delicately designed, which consists of high-order Taylor expansion, introducing magical intermediate variables, representing linear systems converted from strongly nonlinear systems, designing Kalman filter, etc. The performance of the proposed filter will be much better than one of EKF, because it uses more information than EKF. Finally, its promise is verified through commonly used digital simulation examples.<\/jats:p>","DOI":"10.3390\/s22020653","type":"journal-article","created":{"date-parts":[[2022,1,16]],"date-time":"2022-01-16T20:45:21Z","timestamp":1642365921000},"page":"653","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Design Method of High-Order Kalman Filter for Strong Nonlinear System Based on Kronecker Product Transform"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2809-7639","authenticated-orcid":false,"given":"Xiaohan","family":"Liu","sequence":"first","affiliation":[{"name":"School of Automation, Hangzhou Dianzi University, Hangzhou 310018, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chenglin","family":"Wen","sequence":"additional","affiliation":[{"name":"School of Automation, Guangdong University of Petrochemical Technology, Maoming 525000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaohui","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Automation, Hangzhou Dianzi University, Hangzhou 310018, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1016\/j.automatica.2015.02.022","article-title":"Filtering and fault detection for nonlinear systems with polynomial approximation","volume":"54","author":"Liu","year":"2015","journal-title":"Automatica"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Wen, T., Xie, G., Cao, Y., and Cai, B. (2021). A DNN-Based Channel Model for Network Planning in Train Control Systems. IEEE Trans. Intell. Transp. Syst., early access.","DOI":"10.1109\/TITS.2021.3093025"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/j.automatica.2017.03.041","article-title":"Filter design based on characteristic functions for one class of multi-dimensional nonlinear non-Gaussian systems","volume":"82","author":"Wen","year":"2017","journal-title":"Automatica"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1109\/TAC.2006.872771","article-title":"Minimum entropy filtering for multivariate stochastic systems with non-Gaussian noises","volume":"51","author":"Guo","year":"2006","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1049\/cje.2021.02.005","article-title":"A Novel Step-by-Step High-Order Extended Kalman Filter Design for a Class of Complex Systems with Multiple Basic Multipliers","volume":"30","author":"Xiaohui","year":"2021","journal-title":"Chin. J. Electron."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Ye, L., Ma, X., and Wen, C. (2021). Rotating Machinery Fault Diagnosis Method by Combining Time-Frequency Domain Features and CNN Knowledge Transfer. Sensors, 21.","DOI":"10.3390\/s21248168"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1115\/1.3662552","article-title":"A new approach to linear filtering and prediction problems","volume":"82","author":"Kalman","year":"1960","journal-title":"Trans. ASME\u2014J. Basic Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1115\/1.3425006","article-title":"An approximate method of state estimation for nonlinear dynamical systems","volume":"92","author":"Sunahara","year":"1970","journal-title":"J. Basic Eng."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Wang, Q., Sun, X., and Wen, C. (2021). Design Method for a Higher Order Extended Kalman Filter Based on Maximum Correlation Entropy and a Taylor Network System. Sensors, 21.","DOI":"10.3390\/s21175864"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Kim, T., and Park, T.-H. (2020). Extended Kalman filter (EKF) design for vehicle position tracking using reliability function of radar and lidar. Sensors, 20.","DOI":"10.3390\/s20154126"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"e2424","DOI":"10.1002\/stc.2424","article-title":"Improved Kalman filter damage detection approach based on l(p) regularization","volume":"26","author":"Huang","year":"2019","journal-title":"Struct. Control. Health Monit."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1109\/JPROC.2003.823141","article-title":"Unscented filtering and nonlinear estimation","volume":"92","author":"Julier","year":"2004","journal-title":"Proc. IEEE"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1254","DOI":"10.1109\/TAC.2009.2019800","article-title":"Cubature kalman filters","volume":"54","author":"Arasaratnam","year":"2009","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/S0959-1524(98)00052-3","article-title":"Strong tracking filter based adaptive generic model control","volume":"9","author":"Xie","year":"1999","journal-title":"J. Process Control."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2013","DOI":"10.1007\/s12555-016-0589-2","article-title":"Interacting Multiple Model Estimation-Based Adaptive Robust Unscented Kalman Filter","volume":"15","author":"Gao","year":"2017","journal-title":"Int. J. Control. Autom. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2198","DOI":"10.1109\/TAC.2018.2867325","article-title":"A New Continuous Discrete Unscented Kalman Filter","volume":"64","author":"Torben","year":"2019","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"01051","DOI":"10.1051\/itmconf\/20192801051","article-title":"Double hybrid Kalman filtering for state estimation of dynamical systems","volume":"28","author":"Michalski","year":"2019","journal-title":"ITM Web Conf."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Yang, F., Luo, Y., and Zheng, L. (2019). Double-Layer Cubature Kalman Filter for Nonlinear Estimation. Sensors, 19.","DOI":"10.3390\/s19050986"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1152","DOI":"10.1049\/cje.2021.08.010","article-title":"High-Order Extended Strong Tracking Filter","volume":"30","author":"Xiaohui","year":"2021","journal-title":"Chin. J. Electron."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4193","DOI":"10.1049\/iet-rpg.2020.0280","article-title":"Monitoring of subsynchronous oscillation in a series-compensated wind power system using an adaptive extended Kalman filter","volume":"14","author":"Jan","year":"2020","journal-title":"IET Renew. Power Gener."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1557","DOI":"10.1109\/JSYST.2020.2982953","article-title":"Square-Root Sigma-Point Filtering Approach to State Estimation for Wind Turbine Generators in Interconnected Energy Systems","volume":"15","author":"Yu","year":"2021","journal-title":"IEEE Syst. J."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"120805","DOI":"10.1016\/j.energy.2021.120805","article-title":"An Immune Genetic Extended Kalman Particle Filter approach on state of charge estimation for lithium-ion battery","volume":"230","author":"Jiang","year":"2021","journal-title":"Energy"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"59","DOI":"10.3390\/en11010059","article-title":"An Online State of Charge Estimation Algorithm for Lithium-Ion Batteries Using an Improved Adaptive Cubature Kalman Filter","volume":"11","author":"Zhibing","year":"2018","journal-title":"Energies"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Fnadi, M., Plumet, F., and Benamar, F. (2019, January 20\u201324). Nonlinear Tire Cornering Stiffness Observer for a Double Steering Off-Road Mobile Robot. Proceedings of the 2019 International Conference on Robotics and Automation (ICRA), Montreal, QC, Canada.","DOI":"10.1109\/ICRA.2019.8794047"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Fnadi, M., Sandretto, J., Ballet, G., and Pribourg, L. (2021, January 26\u201328). Guaranteed Identification of Viscous Friction for a Nonlinear Inverted Pendulum Through Interval Analysis and Set Inversion. Proceedings of the 2021 American Control Conference (ACC), New Orleans, LA, USA.","DOI":"10.23919\/ACC50511.2021.9483185"},{"key":"ref_26","unstructured":"St-Pierre, M., and Gingras, D. (2004, January 14\u201317). Comparison between the unscented Kalman filter and the extended Kalman filter for the position estimation module of an integrated navigation information system. Proceedings of the IEEE Intelligent Vehicles Symposium, Parma, Italy."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1038","DOI":"10.1016\/j.asr.2018.10.003","article-title":"Nonlinear filtering for sequential spacecraft attitude estimation with real data: Cubature Kalman Filter, Unscented Kalman Filter and Extended Kalman Filter","volume":"63","author":"Garcia","year":"2019","journal-title":"Adv. Space Res."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1007\/s10291-019-0870-y","article-title":"Performance analysis of indoor pseudolite positioning based on the unscented Kalman filter","volume":"23","author":"Liu","year":"2019","journal-title":"GPS Solut."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1106","DOI":"10.1109\/9.618240","article-title":"Polynomial filtering of discrete-time stochastic linear systems with multiplicative state noise","volume":"42","author":"Carravetta","year":"1997","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2059","DOI":"10.1109\/TAC.2005.860256","article-title":"Polynomial extended Kalman filter","volume":"50","author":"Germani","year":"2005","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_31","first-page":"886","article-title":"Polynomial extended Kalman filtering for discrete-time nonlinear stochastic systems","volume":"Volume 1","author":"Germani","year":"2003","journal-title":"Proceedings of the 42nd IEEE International Conference on Decision and Control (IEEE Cat. No. 03CH37475)"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"508","DOI":"10.1049\/cje.2021.04.004","article-title":"High-Order Extended Kalman Filter Design for a Class of Complex Dynamic Systems with Polynomial Nonlinearities","volume":"30","author":"Xiaohui","year":"2021","journal-title":"Chin. J. Electron."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"51035","DOI":"10.1109\/ACCESS.2020.2979735","article-title":"Comparisons on Kalman-Filter-Based dynamic state estimation algorithms of power systems","volume":"8","author":"Liu","year":"2020","journal-title":"IEEE Access"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"4671","DOI":"10.1109\/TIE.2017.2668980","article-title":"Filters design based on multiple characteristic functions for the grinding process cylindrical workpieces","volume":"64","author":"Wen","year":"2017","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_35","first-page":"190","article-title":"Maximum Correntropy High-Order Extended Kalman Filter","volume":"31","author":"Xiaohui","year":"2022","journal-title":"Chin. J. Electron."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/2\/653\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T14:15:04Z","timestamp":1760364904000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/2\/653"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,15]]},"references-count":35,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,1]]}},"alternative-id":["s22020653"],"URL":"https:\/\/doi.org\/10.3390\/s22020653","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1,15]]}}}