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The conservatism of the system is reduced by using this inequality method and positive definite conditions, and further decreases with the increase of the Legendre parameter\n            <jats:italic>N<\/jats:italic>\n            . The PDC controller considers two control parameters simultaneously, which can represent the actual system more truthfully. Finally, Example 1 proves the effectiveness of the improved method in this paper. Example 2 considers time-varying probability transition perturbations. Example 3 obtains the parameters of PDC controller with different parameters\n            <jats:italic>N<\/jats:italic>\n            . Example 4 illustrates the practical significance of methods in this paper by introducing a time-delay inverted pendulum system with Markovian parameters.\n          <\/jats:p>","DOI":"10.1177\/0142331219889190","type":"journal-article","created":{"date-parts":[[2019,12,18]],"date-time":"2019-12-18T03:36:54Z","timestamp":1576640214000},"page":"1313-1322","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":2,"title":["Parallel distributed compensation controller design for Markovian jump system with time-varying delays using Bessel-Legendre inequality method and improved positive definite rule"],"prefix":"10.1177","volume":"42","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7214-1986","authenticated-orcid":false,"given":"Nuo","family":"Xu","sequence":"first","affiliation":[{"name":"School of Computer Science &amp; Technology, Tianjin Polytechnic University, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6809-5136","authenticated-orcid":false,"given":"Liankun","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Computer Science &amp; Technology, Tianjin Polytechnic University, China"},{"name":"Tianjin Key Laboratory of Autonomous Intelligence Technology and Systems, Tianjin Polytechnic University, China"}]}],"member":"179","published-online":{"date-parts":[[2019,12,18]]},"reference":[{"key":"e_1_3_3_2_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2014.12.023"},{"key":"e_1_3_3_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfranklin.2016.04.014"},{"key":"e_1_3_3_4_1","doi-asserted-by":"publisher","DOI":"10.1080\/03081079.2014.883715"},{"key":"e_1_3_3_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2015.02.011"},{"key":"e_1_3_3_6_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0039-0"},{"issue":"9","key":"e_1_3_3_7_1","first-page":"1021","article-title":"Analytical design of controllers in systems with random attributes","volume":"22","author":"Krasovskii NN","year":"1961","unstructured":"Krasovskii NN, Lidskii EA (1961) Analytical design of controllers in systems with random attributes. 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