4_1_PID_control

1 PIDcontrolleriswidelyusedintheprocessandroboticsindustries 2 4 3 1ProportionalControl P Whenthefeedbackcontrolsignalislinearlyproportionaltothesystemerror wecalltheresultProportionalFeedback 3 Whenismadeverylarge thesteadystateerrorwillbeverysmall Butthedynamicresponsewillbecomeworsebecausethedampingmaybemuchtoolow 4 Example Thecharacteristicequationis 5 6 4 3 2ProportionalplusIntegralControl PI AddinganintegraltermtothecontrollerresultsintheProportionalplusIntegral PI controlequation 7 wherehasnopoleatorigin Theopen looptransferfunctionbecomes 8 Steadystateerrordrops 9 example4 1 ConsiderthesystemwithPIcontrol 10 Solution a 11 b 12 c Thecharacteristicequationis Routh sArray 13 14 d Systemtype1withrespecttoreferencetracking errorconstant 15 4 3 3ProportionalplusDifferentialControl PD AddingandifferentialtermtotheproportionalcontrollerresultsintheProportionalplusDifferential PD controlequation 16 PDcontrollercanmodifythetransientresponseoftheclosed loopsystem Forexample isgivenasastandardsecond ordersystem 17 WhilenoPDcontrollerisadded theclosed loopsystemis 18 Closed looptransferfunctionofstandardsecond ordersystem 19 WhilePDcontrollerisadded theclosed loopsystemis Additionalopen loopzero 20 21 nonstandardsecond ordersystem 22 additionalzero 23 Thenewcharacteristicequationis Thenewundampednaturalfrequencyis Thenewdampingratiois 24 Conclusion 25 example4 2 PDcontrolledsystemisgiven Dynamicperformanceindex FindthePDcontrolparametersand 26 solution WhilenoPDcontrollerisadded theclosed looptransferfunctionis Thecorrespondentparameters Thecorrespondentperformanceindexes 27 where Thecorrespondentparameters 28 Tomeetthedesignrequest Accordingtostandardsecondsystem wecanobtain Solutions 29 Selection Thus Theendofsolution 30 4 3 4Proportional Integral DifferentialControl PIDcontrol 31 Example Thecharacteristicequationis 32 Collectingtermsresultsin Wehavethreefreeparameters Byselectionoftheseparameters therootsofthecharacteristicequationcanbeuniquelyandarbitrarilydetermined 33 4 3 5DifferentialFeedbackControl VelocityFeedback 34 35 Theyhavethesamecharacteristicequation Noadditionalzero Additionalzero 36 37 Consideringstandardsecond ordersystem 38 Theclosed looptransferfunction 39 Theclosed looptransferfunction where Undervelocityfeedback thedampingratioisenhanced Thus thedynamicperformancescanbeimproved 40 example4 3 Giventheblockdiagramofasystem Theovershoot thepeaktime Requirements Findthegainoftheamplifierandthecoefficientofthedifferentialfeedback 41 solution Theopen looptransferfunctionis Theclosed looptransferfunctionis Thusweacquire 42 Accordingtotherequirements yields Becauseoftherelations yields 43 4 3 6Ziegler NicholsTuningofPIDRegulators J G ZieglerandN B Nichols 1942 1943 44 45 Approximatedtransferfunction PIDregulator 46 47 Ziegler NicholsTuningforPIDRegulatorforadecayratioof0 25 48 49 Ziegler NicholsTuningforPIDRegulatorBasedonUltimateSensitivityMethod 50 example4 4 Givenapapermachinemodel