{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T19:25:17Z","timestamp":1771529117296,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2017,5,11]],"date-time":"2017-05-11T00:00:00Z","timestamp":1494460800000},"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>Stochastic distribution control (SDC) systems are a group of systems where the outputs considered is the measured probability density function (PDF) of the system output whilst subjected to a normal crisp input. The purpose of the active fault tolerant control of such systems is to use the fault estimation information and other measured information to make the output PDF still track the given distribution when the objective PDF is known. However, if the target PDF is unavailable, the PDF tracking operation will be impossible. Minimum entropy control of the system output can be considered as an alternative strategy. The mean represents the center location of the stochastic variable, and it is reasonable that the minimum entropy fault tolerant controller can be designed subjected to mean constraint. In this paper, using the rational square-root B-spline model for the shape control of the system output probability density function (PDF), a nonlinear adaptive observer based fault diagnosis algorithm is proposed to diagnose the fault. Through the controller reconfiguration, the system entropy subjected to mean restriction can still be minimized when fault occurs. An illustrative example is utilized to demonstrate the use of the minimum entropy fault tolerant control algorithms.<\/jats:p>","DOI":"10.3390\/e19050218","type":"journal-article","created":{"date-parts":[[2017,5,11]],"date-time":"2017-05-11T11:06:46Z","timestamp":1494500806000},"page":"218","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Minimum Entropy Active Fault Tolerant Control of the Non-Gaussian Stochastic Distribution System Subjected to Mean Constraint"],"prefix":"10.3390","volume":"19","author":[{"given":"Haokun","family":"Jin","sequence":"first","affiliation":[{"name":"School of Electrical Engineering, Zhengzhou University, Zhengzhou 450001, China"}]},{"given":"Yacun","family":"Guan","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Zhengzhou University, Zhengzhou 450001, China"}]},{"given":"Lina","family":"Yao","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Zhengzhou University, Zhengzhou 450001, China"}]}],"member":"1968","published-online":{"date-parts":[[2017,5,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1016\/S0005-1098(01)00256-4","article-title":"Robust detection of sensor faults by means of a statistical test","volume":"38","author":"Juricic","year":"2002","journal-title":"Automatica"},{"key":"ref_2","first-page":"695","article-title":"Bibliographical review on reconfigurable fault-tolerant control systems","volume":"51","author":"Guo","year":"2006","journal-title":"Annu. Rev. Control"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Wang, H. (2000). Bounded Dynamic Stochastic Systems: Modelling and Control, Springer.","DOI":"10.1007\/978-1-4471-0481-0"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.arcontrol.2004.12.002","article-title":"Model-based fault-detection and diagnosis-status and applications","volume":"29","author":"Isermann","year":"2005","journal-title":"Annu. Rev. Control"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/S0959-1524(97)00016-4","article-title":"Survey of robust residual generation and evaluation methods in observer-based fault detection systems","volume":"7","author":"Frank","year":"1997","journal-title":"J. Process Control"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1101","DOI":"10.1080\/00207170412331293311","article-title":"Fault detection and isolation in nonlinear stochastic systems a combined adaptive Monte Carlo filtering and likelihood ratio approach","volume":"77","author":"Li","year":"2004","journal-title":"Int. J. Control"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1016\/S0005-1098(02)00245-5","article-title":"Optimal stochastic fault detection filter","volume":"39","author":"Chen","year":"2003","journal-title":"Automatica"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1644","DOI":"10.1109\/TCSI.2005.851686","article-title":"Fault detection and diagnosis for general stochastic systems using B-spline expansions and nonlinear filter","volume":"52","author":"Guo","year":"2005","journal-title":"IEEE Trans. Circuits Syst. I"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1504\/IJMIC.2008.019355","article-title":"Fault detection, diagnosis and tolerant control for non-Gaussian stochastic distribution systems using a rational square-root approximation model","volume":"3","author":"Yao","year":"2008","journal-title":"Int. J. Model. Identif. Control"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2305","DOI":"10.1016\/j.automatica.2012.06.036","article-title":"Design of new fault diagnosis and fault tolerant control scheme for non-Gaussian singular stochastic distribution systems","volume":"48","author":"Yao","year":"2012","journal-title":"Automatica"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1109\/TAC.2002.806663","article-title":"Minimum entropy control of closed-loop tracking errors for dynamic stochastic systems","volume":"48","author":"Yue","year":"2003","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_12","first-page":"343","article-title":"Shaping of output PDF based on the rational square-root B-spline model","volume":"31","author":"Zhou","year":"2005","journal-title":"Acta Autom. Sin."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Lan, J.L., and Patton, R.J. (2016). Integrated design of fault-tolerant control for nonlinear systems based on fault estimation and T-S fuzzy modelling. IEEE Trans. Fuzzy Syst., 99.","DOI":"10.1109\/TFUZZ.2016.2598849"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Ciabattoni, L., Fasano, A., Ferracuti, F., Freddi, A., Longhi, S., and Monteriu, A. (2016, January 29\u201331). A thruster failure tolerant control scheme for underwater vehicles. Proceedings of the 12th IEEE\/ASME International Conference on Mechatronic and Embedded Systems and Applications (MESA), Auckland, New Zealand.","DOI":"10.1109\/MESA.2016.7587175"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1016\/j.arcontrol.2012.03.005","article-title":"Fault-tolerant control systems: A comparative study between active and passive approaches","volume":"36","author":"Jiang","year":"2012","journal-title":"Annu. Rev. Control"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1109\/9.983388","article-title":"Minimum entropy control of non-Gaussian dynamic stochastic systems","volume":"47","author":"Wang","year":"2002","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"486","DOI":"10.1109\/9.911429","article-title":"Bounded Stochastic distributions control for pseudo armax stochastic systems","volume":"46","author":"Wang","year":"2001","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_18","unstructured":"Yao, L.N., and Peng, B. (2013, January 17\u201319). Fault diagnosis and fault tolerant control for the non-Gaussian time-delayed stochastic distribution control system. Proceedings of the American Control Conference (ACC), Washington, DC, USA."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"538","DOI":"10.1016\/j.neucom.2015.10.092","article-title":"Fault diagnosis and fault tolerant tracking control for the non-Gaussian singular time-delayed stochastic distribution system with PDF approximation error","volume":"175","author":"Yao","year":"2016","journal-title":"Neurocomputing"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/5\/218\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:35:31Z","timestamp":1760207731000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/5\/218"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,5,11]]},"references-count":19,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2017,5]]}},"alternative-id":["e19050218"],"URL":"https:\/\/doi.org\/10.3390\/e19050218","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,5,11]]}}}