分数阶控制理论研究摘要进入21世纪以来，随着分数阶微积分理论研究不断取得突破，控制领域中的新的研究热点就是对其进行理论研究，分数阶微积分是整数阶微积分的推广将微积分阶次从我们熟知的整数域推广到实数域，甚至复数域。其理论基础是分数阶微积分算子及方程，这是一个新的研究方向。大量的实践已经证明,在控制理论中应用分数阶微积分，相比整数阶微积分，具有更好的效果。在扩展控制理论的经典研究方法方面，在解释现有结果方面，分数阶微积分都为之提供了非常强劲的支持。论文阐述了分数阶微积分的基本理论，从其定义、导数定义以及性质进行了分析了详细说明。接下来分析了微积分控制理论在实际中的应用，针对分数阶PID进行了研究讨论，在前人研究基础上，对于分数阶PID自整定算法进行了研究分析，最后在matlab里进行仿真讨论。关键词：分数阶，分数系统，分数阶PIDAbstractSincethebeggingofthe21stcentury,thefractionalordercalculustheoryhasachievedlotsofbreakthough.Fractionalcalculusisthecalculuswhoseintegrationordifferentiationorderisnotconventionalintegernumberbutrealorevencomplexone.Itisextensitionofintegercalculus.Farctionalordercontrol,whichisestablishedontheideaoffractionalorderoperatorsandthetheoryoffractionalorderdieffrentialequations,isnowaquitenewresearchdirection.Practicehasprovedthatbetterresultscouldbeobtainedbyintroductionoffractionalcalculusincontroltheory.Fractionalcalculusprovidesapowerfulsupportfortheexpansionoftheclassicresearchmethodsincontroltheoryandabetterexplainationofthecurrentresults.ThisPaperexpoundsthebasictheoryoffractionalordercalculus,fromthedefinitionandnatureofitsdefinition,derivativeisanalyzedindetail.Thenanalyzedthecontroltheoryofcalculusintheactualapplication,inviewofthefractionalorderPIDwiththeresearchanddiscussiononthebasisofpreviousstudies,thefractionalorderPIDself-tuningalgorithmareanalyzed,andfinallyinthematlabsimulationisdiscussed.KeyWords:fractional-order,fractionalsystem,fractionalorderPID目录第一章绪论....................................................................31.1引言.............................................................................................................................................31.2研究背景与现状.........................................................................................................................4第二章分数阶微积分基本理论....................................................62.1分数阶微积分的定义.................................................................................................................62.2.1Gamma函数........................................................................................................................62.2.2Mittag-Leffler函数..............................................................................................................62.2.3Grünwald-Letnikov定义.....................................................................................................72.2.4Riemann-Liouville定义.......................................................................................................72.2.5Caputo定义.........................................................................................................................82.2分数阶导数定义的三种变形.....................................................................................................82.2.1Riemann-Liouville分数阶导数...........................................................................................82.2.2Grunwald-Liouvill...
