{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,27]],"date-time":"2025-11-27T10:37:51Z","timestamp":1764239871064,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2014,8,4]],"date-time":"2014-08-04T00:00:00Z","timestamp":1407110400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we propose a steepest descent algorithm based on the natural gradient to design the controller of an open-loop stochastic distribution control system (SDCS) of multi-input and single output with a stochastic noise. Since the control input vector decides the shape of the output probability density function (PDF), the purpose of the controller design is to select a proper control input vector, so that the output PDF of the SDCS can be as close as possible to the target PDF. In virtue of the statistical characterizations of the SDCS, a new framework based on a statistical manifold is proposed to formulate the control design of the input and output SDCSs. Here, the Kullback\u2013Leibler divergence is presented as a cost function to measure the distance between the output PDF and the target PDF. Therefore, an iterative descent algorithm is provided, and the convergence of the algorithm is discussed, followed by an illustrative example of the effectiveness.<\/jats:p>","DOI":"10.3390\/e16084338","type":"journal-article","created":{"date-parts":[[2014,8,4]],"date-time":"2014-08-04T08:06:18Z","timestamp":1407139578000},"page":"4338-4352","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Natural Gradient Algorithm for Stochastic Distribution Systems"],"prefix":"10.3390","volume":"16","author":[{"given":"Zhenning","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Mathematics, Beijing University of Technology, Beijing 100124, China"}]},{"given":"Huafei","family":"Sun","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China"}]},{"given":"Linyu","family":"Peng","sequence":"additional","affiliation":[{"name":"Department of Applied Mechanics and Aerospace Engineering & Research Institute of Nonlinear PDEs, Waseda University, Okubo, Shinjuku, Tokyo 169-8555, Japan"}]},{"given":"Lin","family":"Jiu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, LA 70118,USA"}]}],"member":"1968","published-online":{"date-parts":[[2014,8,4]]},"reference":[{"key":"ref_1","first-page":"81","article-title":"Infromation and accuracy attainable in the estimation of statistical parameters","volume":"37","author":"Rao","year":"1945","journal-title":"Bull. 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