The method is presented for very general setup namely for the case when the closedloop system poles are functions of an unknown parameter In most cases the parameter of interest is the system static gain satisfying However any other unknown and vari ID: 27083 Download Pdf
CEN455: Dr. Nassim Ammour. Root Locus . Techniques. Root . locus is . a graphical presentation of the closed-loop poles as a system . parameter k is varied. . The . graph of all possible roots of this equation (K is the variable parameter) is called the root .
Control . Systems (MCS). Dr. Imtiaz Hussain. Assistant Professor. email: . imtiaz.hussain@faculty.muet.edu.pk. URL :. http://imtiazhussainkalwar.weebly.com/. Lecture-11. Introduction to Compensation.
Control . Systems (MCS). Dr. Imtiaz Hussain. Assistant Professor. email: . imtiaz.hussain@faculty.muet.edu.pk. URL :. http://imtiazhussainkalwar.weebly.com/. Lecture-9-10. Examples of Root Locus. Lecture Outline.
by Frank Owen, PhD, PE. polyXengineering, Inc.. San Luis Obispo, California. general aims for any controller. Reduce percent overshoot. Shorten time to reach peak value after a step change (minimize T.
by Frank Owen, PhD, PE. polyXengineering, Inc.. San Luis Obispo, California. general aims for any controller. Reduce percent overshoot. Shorten time to reach peak value after a step change (minimize T.
by Frank Owen, PhD, PE. polyXengineering, Inc.. San Luis Obispo, California. P-only controller: simple gain adjustment. K. P. ≠ 0, K. I. = 0, K. D. = 0. Controller does not change root locus of existing G.
Lect.8 . Root Locus Techniques. Basil Hamed. Chapter Learning Outcomes. After . completing this chapter the student will be able to. :. Define . a root locus (Sections 8.1-8.2). State . the properties of a root locus (Section 8.3).
By Frank Owen, PhD, PE. polyXengineering, Inc.. San Luis Obispo, California. Purpose of Derivative control. You do not like the dynamics of an existing system. You want to place the closed-loop poles at a point that is not on the current root locus.
If the system has a v ariable loop gain then the location of the closedloop poles depends on the value of the loop gain chosen It is important therefore that the designer know how the closedloop poles move in the splane as the loop gain is varied Fr
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The method is presented for very general setup namely for the case when the closedloop system poles are functions of an unknown parameter In most cases the parameter of interest is the system static gain satisfying However any other unknown and vari
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EssenceoftheRootLocusTechniqueInthischapterwestudyamethodforndinglocationsofsystempoles.Themethodispresentedforaverygeneralset-up,namelyforthecasewhentheclosed-loopsystempolesarefunctionsofanunknownparameter.Inmostcasestheparameterofinterestisthesystemstaticgainsatisfying.However,anyotherunknownandvariablesystemparameteraffectingpolelocationscanbeusedinsteadof.Themethodisknownastherootlocustechniqueforsolvingpolynomialequationswithconstantorvariableparameters.ItwasoriginallypresentedinEwans(1948,1950).Theimportanceoftherootlocusmethodforcontrolsystemtheoryliesinthefactthatthelocationofthesystempolesdeterminesthesystem291 stabilityandthesystemtransientresponse.Insomecases,thedesiredcontrolsystemperformancecanbeobtainedbychangingonlythesystemstaticgain.Itisknownfromthatthechoiceofthesystemstaticgaindeterminestheerrorsofthesystemsteadystateresponseinthesensethatabiggervalueforimpliessmallervaluesforsteadystateerrors(assumingthatthesystemremainsasymptoticallystable).However,changingcausesthesystemtransientresponseparametersalsotochange.Ifoneisnotabletoachieveallthecontrolsystemrequirementsbychangingonlythestaticgain(theessenceoftherootlocusmethod),onehastodesignadynamiccompensator(controller).ThequestionofdesigningdynamiccompensatorsbyusingtherootlocusmethodwillbeaddressedindetailinChapter8. Therootlocustechniqueallowsadjustmentofthesystempolesbychangingthefeedbacksystemstaticgain.Theclosed-loopfeedbacksystem,ingeneral,canberepresentedbyablockdiagramasgiveninFigure7.1.G(s)H(s) U(s)+-Y(s)Controller - PlantFeedback ElementFigure7.1:Blockdiagramofafeedbackcontrolsystem Thecharacteristicequationoftheaboveclosed-loopsystemisgivenbywhichcanbewrittenaswhererepresentsallstaticgainspresentintheloop.Usuallythestaticgainscomesolelyfromthecontroller-plantelement.Thatis,theplantiscontrolledbychangingthestaticgain,whichcanberepresentedbyablockdiagramgiveninFigure7.2. G1(s)H 1(s)U(s)+-Y(s)ControllerPlantFeedback ElementKFigure7.2:Feedbackcontrolsystemwithastaticcontrollerinthedirectpath Theclosed-looptransferfunctionofthissystemisgivenby \n\n\n \r\r\r\r\r\r ThecorrespondingcharacteristicequationisThequestiontobeansweredbytherootlocustechniqueis:whatcanbeachievedbychangingthestaticgain,theoretically,fromto?Canwendthelocationofsystempolesforallvaluesof?Thepositiveanswertothisquestionledtothedevelopmentoftherootlocustechnique.ItwasdiscoveredbyW.Ewansin1948andwasmathematicallyformulatedin1950inhisfamouspaper(Ewans,1950). Themainideabehindtherootlocustechniqueishiddeninequationwhichisanalgebraicequationinvolvingcomplexnumbers.Itactuallyrepresentstwoequations(forrealandimaginaryparts,orformagnitudesandphaseangles).Inthisbook,wewillconsidertherootlocustechniquefor.Thiswillsimplifyderivationsandmaketherootlocusplotsclearer.Thecomplementaryrootlocusfornegativevaluesofcanbesimilarlyderived.Thisequationproducesthefollowingequationsforthemagnitudes andforthephaseanglesIfwefactorasthen,byusingelementaryalgebrawithcomplexnumbers,weget andThelasttwoequationsarecrucialforthedevelopmentoftherootlocustechnique. Example7.1:Giventheopen-looptransferfunctionThelocationsoftheopen-looppolesandzerosaregiveninFigure7.3.Re!{s}s"1=-2+j2-4d#ca$b-3-2-1j%2j%1Im{s}b&3b&2a'1b&1Figure7.3:Apointinthecomplexplainthatdoesnotlieontherootlocus Takeanypoint(inthecomplexplane.Ifthatpointbelongstotherootlocus,itmustsatisfybothequations(forthemagnitudeandforthephase).Forexample,forthepoint(wehave((((Thus,ifthepoint(belongstotherootlocus,thestaticgainatthatpointmustbeequalto.Itfollowsthatforthepoint(thefollowingmustbesatised(((((( whichleadsto))*+,,,,,Wecanconcludethatthepoint)cannotbelongtotherootlocussincethephaseequationisapparentlynotvalid.Thisexampleshowsthatonlyveryselectedpointsfromthecomplexplanecanbelongtotherootlocus. MinumumPhaseSystemsItshouldbeemphasizedthatfromtherootlocusmethoditfollowsthatthesystemshavingunstableopen-loopzerosbecomeunstableforlargevaluesofthestaticgain.Suchkindsofsystemsarecallednonminimumphasesystems,incontrasttominimumphasesystemswhosedenitionisgivenbelow.Denition7.1Systemshavingallopen-loopzerosandpolesintheclosedlefthalfofthecomplexplaneexcludingtheoriginarecalledminimumphasesystems.
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