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An improved compact form dynamic linearization (iCFDL) is proposed to transfer the original nonlinear system into an affined linear data model including a nonlinear residual term. Both a time-difference estimator and a gradient parameter estimator are designed to estimate the nonlinear residual uncertainties and the unknown parameters in the iCFDL model. Subsequently, a novel improved CFDL based MFA-PID (iCFDL-MFA-PID) control is proposed by incorporating these two estimators. The results are extended by the use of improved partial format dynamic linearization (iPFDL) and full format dynamic linearization (iFFDL). The theoretical results are shown using contraction mapping principle-based mathematical analysis, as well as simulations.<\/jats:p>","DOI":"10.1177\/0142331219896649","type":"journal-article","created":{"date-parts":[[2020,1,28]],"date-time":"2020-01-28T02:32:46Z","timestamp":1580178766000},"page":"1797-1807","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":17,"title":["Model-free adaptive PID control for nonlinear discrete-time systems"],"prefix":"10.1177","volume":"42","author":[{"given":"Shuhua","family":"Zhang","sequence":"first","affiliation":[{"name":"Institute of Artificial Intelligence and Control, School of Automation and Electronics Engineering, Qingdao University of Science & Technology, 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