{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,8]],"date-time":"2026-10-08T18:12:15Z","timestamp":1791483135721,"version":"4.3.4"},"reference-count":35,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,17]],"date-time":"2023-01-17T00:00:00Z","timestamp":1673913600000},"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":["62263026"],"award-info":[{"award-number":["62263026"]}],"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":["2020LH06006"],"award-info":[{"award-number":["2020LH06006"]}],"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":["2022MS06005"],"award-info":[{"award-number":["2022MS06005"]}],"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":["BYJJ-QNGG2022001"],"award-info":[{"award-number":["BYJJ-QNGG2022001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004763","name":"Natural Science Foundation of Inner Mongolia","doi-asserted-by":"publisher","award":["62263026"],"award-info":[{"award-number":["62263026"]}],"id":[{"id":"10.13039\/501100004763","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004763","name":"Natural Science Foundation of Inner Mongolia","doi-asserted-by":"publisher","award":["2020LH06006"],"award-info":[{"award-number":["2020LH06006"]}],"id":[{"id":"10.13039\/501100004763","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004763","name":"Natural Science Foundation of Inner Mongolia","doi-asserted-by":"publisher","award":["2022MS06005"],"award-info":[{"award-number":["2022MS06005"]}],"id":[{"id":"10.13039\/501100004763","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004763","name":"Natural Science Foundation of Inner Mongolia","doi-asserted-by":"publisher","award":["BYJJ-QNGG2022001"],"award-info":[{"award-number":["BYJJ-QNGG2022001"]}],"id":[{"id":"10.13039\/501100004763","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Baotou medical college young and middle aged backbone","award":["62263026"],"award-info":[{"award-number":["62263026"]}]},{"name":"Baotou medical college young and middle aged backbone","award":["2020LH06006"],"award-info":[{"award-number":["2020LH06006"]}]},{"name":"Baotou medical college young and middle aged backbone","award":["2022MS06005"],"award-info":[{"award-number":["2022MS06005"]}]},{"name":"Baotou medical college young and middle aged backbone","award":["BYJJ-QNGG2022001"],"award-info":[{"award-number":["BYJJ-QNGG2022001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Output probability density function (PDF) tracking control of stochastic systems has always been a challenging problem in both theoretical development and engineering practice. Focused on this challenge, this work proposes a novel stochastic control framework so that the output PDF can track a given time-varying PDF. Firstly, the output PDF is characterised by the weight dynamics following the B-spline model approximation. As a result, the PDF tracking problem is transferred to a state tracking problem for weight dynamics. In addition, the model error of the weight dynamics is described by the multiplicative noises to more effectively establish its stochastic dynamics. Moreover, to better reflect the practical applications in the real world, the given tracking target is set to be time-varying rather than static. Thus, an extended fully probabilistic design (FPD) is developed based on the conventional FPD to handle multiplicative noises and to track the time-varying references in a superior way. Finally, the proposed control framework is verified by a numerical example, and a comparison simulation with the linear\u2013quadratic regulator (LQR) method is also included to illustrate the superiority of our proposed framework.<\/jats:p>","DOI":"10.3390\/e25020186","type":"journal-article","created":{"date-parts":[[2023,1,17]],"date-time":"2023-01-17T04:28:47Z","timestamp":1673929727000},"page":"186","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Tracking Control for Output Probability Density Function of Stochastic Systems Using FPD Method"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9549-3095","authenticated-orcid":false,"given":"Yi","family":"Yang","sequence":"first","affiliation":[{"name":"School of Information Engineering, Inner Mongolia University of Science and Technology, Baotou 014010, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5179-2384","authenticated-orcid":false,"given":"Yong","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Information Engineering, Inner Mongolia University of Science and Technology, Baotou 014010, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuyang","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Computing Engineering and Built Environment, Edinburgh Napier University, Edinburgh EH10 5DT, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1080\/21642583.2019.1588804","article-title":"An introductory survey of probability density function control","volume":"7","author":"Ren","year":"2019","journal-title":"Syst. 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