{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:59:18Z","timestamp":1760147958636,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,17]],"date-time":"2023-03-17T00:00:00Z","timestamp":1679011200000},"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":["61903298","62073259","20JS130","21XJZZ0060"],"award-info":[{"award-number":["61903298","62073259","20JS130","21XJZZ0060"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100010228","name":"Natural Science Foundation of Shaanxi Provincial Department of Education","doi-asserted-by":"publisher","award":["61903298","62073259","20JS130","21XJZZ0060"],"award-info":[{"award-number":["61903298","62073259","20JS130","21XJZZ0060"]}],"id":[{"id":"10.13039\/501100010228","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Xi\u2019an Science and Technology Planning Project","award":["61903298","62073259","20JS130","21XJZZ0060"],"award-info":[{"award-number":["61903298","62073259","20JS130","21XJZZ0060"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we developed a probability density function (PDF) shape control method for non-linear stochastic systems using a hybrid logistic function (HLF) as an approximate PDF of the state variable. First, the functional relationship between the hybrid logistic probability density function and the controller was established based on the Fokker\u2013Planck\u2013Kolmogorov (FPK) equation. Then, the optimal PDF shape controller derivation was completed using the optimization method and the inner product definition of Hilbert space. This approach is suitable for any non-linear stochastic system. To evaluate the effectiveness and performance of the proposed method, we conducted a comparison experiment with the multi-Gaussian closure (MGC) method and the exponential polynomial (EP) method. The experimental results show that, for different types of targeted PDFs (symmetric unimodal, asymmetric unimodal, bimodal, and trimodal shapes), the PDF shape controller obtained using the HLF approach can make the PDF shape of the state variable track the targeted PDF effectively. In particular, when the targeted PDF has an asymmetric or complex trimodal shape, the proposed technique has comparatively better control effects. Compared with the EP method, our method requires a much smaller number of parameters, greatly reducing the computational complexity while achieving the same control effects. This study provides another approach for controlling the PDF shape of state variables in non-linear stochastic systems, which has important research significance.<\/jats:p>","DOI":"10.3390\/axioms12030303","type":"journal-article","created":{"date-parts":[[2023,3,17]],"date-time":"2023-03-17T02:59:26Z","timestamp":1679021966000},"page":"303","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Research on PDF Shape Control for Nonlinear Stochastic System Using an Approximate Solution of FPK Equation"],"prefix":"10.3390","volume":"12","author":[{"given":"Lingzhi","family":"Wang","sequence":"first","affiliation":[{"name":"School of Automation, Xi\u2019an University of Posts and Telecommunications, Xi\u2019an 710121, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kun","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Automation, Xi\u2019an University of Posts and Telecommunications, Xi\u2019an 710121, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fucai","family":"Qian","sequence":"additional","affiliation":[{"name":"School of Automation and Information Engineering, Xi\u2019an University of Technology, Xi\u2019an 710048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoli","family":"Zhang","sequence":"additional","affiliation":[{"name":"Shaanxi Key Laboratory of Surface Engineering and Remanufacturing, Xi\u2019an University, Xi\u2019an 710065, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1109\/TAC.1966.1098228","article-title":"Control of linear systems according to the minimal variance criterion: A new approach to the disturbance problem","volume":"11","author":"Sain","year":"1966","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_2","unstructured":"Anderson, B., and Moore, J. (1990). Optimal Control, Linear Quadratic Methods, Prentice-Hall."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0266-8920(95)91891-U","article-title":"A method for the probabilistic analysis of non-linear systems","volume":"10","author":"Paola","year":"1995","journal-title":"Probabilistic Eng. Mech."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1719","DOI":"10.1016\/S0005-1098(96)80009-4","article-title":"Towards fully probabilistic control design","volume":"32","author":"Karny","year":"1996","journal-title":"Automatica"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.apm.2021.09.029","article-title":"A new maximum entropy method for estimation of multimodal probability density function","volume":"102","author":"Li","year":"2022","journal-title":"Appl. Math. Model."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Wang, H. (1998, January 18). Robust control of the output probability density functions for multivariable stochastic systems. Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, FL, USA.","DOI":"10.1109\/CDC.1998.758461"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2103","DOI":"10.1109\/9.802925","article-title":"Robust control of the output probability density functions for multivariable stochastic systems with guaranteed stability","volume":"44","author":"Wang","year":"1999","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_8","first-page":"3369","article-title":"Control for bounded pseudo ARMAX stochastic systems via linear B-spline approximations","volume":"4","author":"Wang","year":"2000","journal-title":"IEEE Conf. Decis. Control"},{"key":"ref_9","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_10","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_11","doi-asserted-by":"crossref","unstructured":"Wang, H., and Sun, X. (2004, January 2\u20134). Neural network based probability density function shape control for unknown stochastic systems. Proceedings of the 2004 IEEE International Symposium on Intelligent Control, Taipei, Taiwan.","DOI":"10.1109\/ISIC.2004.1387669"},{"key":"ref_12","first-page":"274","article-title":"Multi-step Predictive Control of a PDF-shaping Problem","volume":"31","author":"Wang","year":"2005","journal-title":"Acta Autom. Sin."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1109\/TSMCB.2004.839906","article-title":"PID controller design for output PDFs of stochastic systems using linear matrix inequalities","volume":"35","author":"Guo","year":"2005","journal-title":"IEEE Trans. Syst. Man Cybern. Part B Cybern."},{"key":"ref_14","first-page":"493","article-title":"Robust Resilient Optimal Tracking Control for Output Probability Design Function","volume":"15","author":"Luan","year":"2008","journal-title":"Control Eng. China"},{"key":"ref_15","first-page":"1161","article-title":"Finite time stabilization for output probability density function of stochastic system","volume":"24","author":"Luan","year":"2009","journal-title":"Control Decis."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"760","DOI":"10.1109\/TAC.2009.2014934","article-title":"Afshar Puya. ILC-Based Fixed-Structure Controller Design for Output PDF Shaping in Stochastic systems Using LMI Techniques","volume":"54","author":"Wang","year":"2009","journal-title":"IEEE Trans. Autom."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2110","DOI":"10.1109\/TAC.2011.2179844","article-title":"Complete statistical characterization of discrete-time LQG and cumulant control","volume":"57","author":"Qian","year":"2012","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2844","DOI":"10.1002\/rnc.4519","article-title":"Sampled-data output feedback stabilization for a class of switched stochastic nonlinear systems","volume":"29","author":"Mao","year":"2019","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"108588","DOI":"10.1016\/j.automatica.2019.108588","article-title":"Simultaneous input and state estimation for stochastic nonlinear systems with additive unknown inputs","volume":"111","author":"Kim","year":"2020","journal-title":"Automatica"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"8430","DOI":"10.1002\/rnc.5248","article-title":"Event-triggered H\u221e filtering for nonlinear discrete-time stochastic systems with application to vehicle roll stability control","volume":"30","author":"Zhang","year":"2020","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/j.apm.2021.09.026","article-title":"Adaptive neural tracking control for stochastic nonlinear multi-agent periodic time-varying systems","volume":"102","author":"Chen","year":"2022","journal-title":"Appl. Math. Model."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Yuan, T., Yang, H., and Ivanov, I.G. (2021). Reachability and Observability of Positive Linear Electrical Circuits Systems Described by Generalized Fractional Derivatives. Mathematics, 9.","DOI":"10.3390\/math9222856"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Dr\u0103gan, V., Ivanov, I.G., and Popa, I.L. (2023). A Game\u2014Theoretic Model for a Stochastic Linear Quadratic Tracking Problem. Axioms, 12.","DOI":"10.3390\/axioms12010076"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"7805","DOI":"10.1109\/TCYB.2021.3052166","article-title":"Event-Triggered Impulsive Control for Nonlinear Stochastic Systems","volume":"52","author":"Hu","year":"2022","journal-title":"IEEE Trans. Cybern."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"9258","DOI":"10.1002\/rnc.6331","article-title":"Estimation of dual-mode nonlinear stochastic systems with unknown parameters","volume":"32","author":"Ji","year":"2022","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1738","DOI":"10.1002\/rnc.6446","article-title":"Domain stabilization in probability in a fixed time for nonlinear stochastic systems via feedback control","volume":"33","author":"Yin","year":"2023","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1080\/00207179.2021.1982014","article-title":"Optimal regulators for a class of nonlinear stochastic systems","volume":"96","author":"Gashi","year":"2023","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1016\/j.jprocont.2003.07.002","article-title":"Control design for first-order processes: Shaping the probability density of the process state","volume":"14","author":"Forbes","year":"2004","journal-title":"J. Process Control"},{"key":"ref_29","unstructured":"Forbes, M., Forbes, J., and Guay, M. (2003, January 9\u201312). Control design for discrete-time stochastic non-linear processes with a non-quadratic performance objective. Proceedings of the 42nd IEEE International Conference on Decision and Control, Maui, HI, USA."},{"key":"ref_30","unstructured":"Forbes, M. (2003). Performance Characterization and Regulatory Feedback Control Design for Time-Invariant Discrete-Time Stochastic Processes. [Ph.D. Thesis, International University of Alberta]."},{"key":"ref_31","unstructured":"Michael, G., Martin, G., and Forbes, J. (2004, January 2\u20134). Probabilistic control design for continuous time stochastic non-linear systems: A PDF shaping approach. Proceedings of the 2004 IEEE International Symposium on Intelligent Control, Taipei, Taiwan."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1904","DOI":"10.1016\/j.automatica.2007.11.028","article-title":"Optimal probability density function control for NARMAX stochastic Systems","volume":"44","author":"Guo","year":"2008","journal-title":"Automatica"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"197","DOI":"10.3724\/SP.J.1004.2012.00197","article-title":"Target Control Design for Stationary Probability Density Function of Nonlinear Stochastic System","volume":"38","author":"Zhu","year":"2012","journal-title":"Acta Autom. Sinca"},{"key":"ref_34","first-page":"144","article-title":"The study of stochastic non-linear state variable density function control method","volume":"29","author":"Zhao","year":"2013","journal-title":"J. Xi\u2019an Univ. Technol."},{"key":"ref_35","first-page":"669","article-title":"The complete statistical characterization control for a class of stochastic systems","volume":"33","author":"Yang","year":"2016","journal-title":"Control Theory Appl."},{"key":"ref_36","first-page":"2424","article-title":"The shape control of probability density function for a class of stochastic systems","volume":"36","author":"Yang","year":"2016","journal-title":"Syst. Eng. Theory Pract."},{"key":"ref_37","first-page":"1463","article-title":"PDF control of non-linear stochastic systems based on MGC method","volume":"34","author":"Yang","year":"2019","journal-title":"Control Decis."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"144","DOI":"10.1109\/JSEE.2014.00017","article-title":"Shape Control on Probability Density Function in Stochastic Systems","volume":"25","author":"Wang","year":"2014","journal-title":"J. Syst. Eng. Electron."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2231","DOI":"10.1080\/00207721.2013.860201","article-title":"The PDF Shape Control of the State Variable for a Class of Stochastic Systems","volume":"46","author":"Wang","year":"2015","journal-title":"Int. J. Syst. Sci."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1007\/s11427-015-4979-0","article-title":"Shaping PDF of the state variable based on piecewise Linear control for non-linear stochastic systems","volume":"59","author":"Wang","year":"2016","journal-title":"Sci. China Inf. Sci."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/3\/303\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:57:16Z","timestamp":1760122636000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/3\/303"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,17]]},"references-count":40,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["axioms12030303"],"URL":"https:\/\/doi.org\/10.3390\/axioms12030303","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,3,17]]}}}