Lecture 12 Feedback control systems: static analysis

1 S. BoydEE102 Lecture12 Feedbackcontrolsystems:staticanalysis feedbackcontrol:general example open-loop equivalentsystem plantchanges,disturbancerejection, sensor noise12{1 Feedbackcontrolsystemsfeedbackis widelyusedinautomaticcontrolterminology: the systemto be controlledis calledtheplant asensormeasuresthe quantity to be controlled anactuatora ectsthe plant thecontrollerorcontrolprocessorprocesses the sensor signalto drivethe actuator thecontrollaworcontrolalgorithmis the algorithmusedby the controlprocessor to derivethe actuator signalFeedbackcontrolsystems:staticanaly sis12{2 Block diagram(oftenthe sensors and actuators are not shown separately)PSfragreplacementsPlantContro llerdisturbancesensor signalactuator signalcommandsignalactuatorsensorFeedbac kcontrolsystems:staticanalysis12{3 Examplesplants:CD player, disk drivemechanics;aircraftor missile;car suspension,engine;rollingmill;high-riseb uilding,XY stageon stepper machinefor IC lithography;computernetwork; industrialprocess;elevatorsensors:radar altimeter;GPS;shaftencoder; LVDT;straingauge;accelerometer;tachomete r;microphone;pressureand temperaturetransducers;chemicalsensors;m icroswitchactuators:hydraulic,pneumatic, electricmotors; pumps;heaters;aircraftcontrolsurfaces;vo icecoil;solenoid.}}}

2 Piezo-electrictransducerdisturbances:win dgusts;earthquakes; externalshakingand vibration;roadsurfacevariations;variatio nin feedmaterialcontrolprocessors:humanopera tor; mechanical;electro-mechanical;analogelec trical;generalpurpose digitalprocessor; specialpurpose digitalprocessorFeedbackcontrolsystems:s taticanalysis12{4 Examplea plateis to be heatedto a desiredtemperatureTdesby an electricalheater theplantis the plate theactuatoris the electricalheater thecontrollersets the heaterpower, givenTdes the inputuis the heaterpowerP the outputyis the platetemperatureTin steady-state,T=Tamb+ (P+D) Tambis the ambienttemperature,Tamb 70 F is a thermalresistancecoe cient, 1 F=W Dis a thermaldisturbanceto the plate,D 0W(representsotherheat ow, in or out of plate)Feedbackcontrolsystems:staticanaly sis12{5block diagram:PSfragreplacementsPD 0 WTamb 70 F 1 F=WTFeedbackcontrolsystems:staticanalysi s12{6An open-loop controllerobviouscontrollaw.}}}

3 Use power thatyieldsT=TdeswhenTamb= 70 F, = 1 F=W, andD= 0W, ,P= (Tdes 70 F)=(1 F=W)(we assumehereTdes 70 F, soP 0W)calledopen-looporfeedforwardbecausese nsor signalis not usedPSfragreplacementsP70 FD TTdesTambPlantControllerFeedbackcontrols ystems:staticanalysis12{7 How well does it work whenTamb6= 70 F, 6= 1 F=W, andD6= 0W?temperatureerror ise=T Tdes= ( 1 F=W)(Tdes 70 F) + (Tamb 70 F) + Dsomescenarios,withTdes= 150 F:Tamb De70 F1 F=W0W0 F65 F1 F=W0W 5 F70 F 0:9 F=W0W 8 F70 F1 F=W5W5 FFeedbackcontrolsystems:staticanalysis12 {8A closed-loop controlleradd sensor to measureplatetemperatureTmodify heaterpower toP= (Tdes 70 F)=(1 F=W)|{z}open-loop control k(T Tdes)|{z}errorwherek >0W= F calledproportionalcontrolsincewe are feedingbacka signalproportionalto the error kis calledthe proportionalfeedbackgain extraterm`does the rightthing':T < Tdes)e <0)increasePT > Tdes)e >0)decreasePFeedbackcontrolsystems:stati canalysis12{9block diagram:PSfragreplacementsP70 FD TTambTdesekPlantControllerFeedbackcontro lsystems.}}}

4 Staticanalysis12{10 How well does this controllerwork?SolveT=Tamb+ (P+D); P=Tdes 70 F k(T Tdes)for errore=T Tdesto gete=( 1 F=W)(Tdes 70 F) + (Tamb 70 F) + D1 + kThusclosed-loop error=open-loop error1 + ksomescenarios,withTdes= 150 Fandk= 10:Feedbackcontrolsystems:staticanalysis 12{11 Tamb De70 F1 F=W0W0 F65 F1 F=W0W 0:45 F70 F 0:9 F=W0W 1 F70 F1 F=W5W0:45 F closed-loop controllerhandleschangesin ambienttemperature/thermalresistanceand disturbancesmuchbetterthanopen-loop controller improvementis1=(1 + k)| the sensitivity of the feedbackloop | solargek)good performanceFeedbackcontrolsystems: static analysis12{12 FeedbackcontrolCommonsetupfor feedbackcontrol:PSfragreplacementsuryePC Pis the plant;Cis the controller uis the plantinput(actuator signal).}}}

5 Yis the plantoutput(sensorsignal) ris the referenceor commandinput(whatwe'd likeyto be) e=r yis the (tracking)errorcan also be , disturbancesand noisesgoal:makey r, ,esmall(despitevariationsinP, disturbances,.. )Feedbackcontrolsystems:staticanalysis12 {13 Considerstatic,linear case(u,y,P,C, .. are numbers)closed-loop gainfromrtoy,T=P C1 +P C= 1 Sis calledtheclosed-loopinput/output(I/O)gai nclosed-loop gainfromrtoe,S=11 +P Cis the sensitivityfor small P, TT=S PPlarge loop gainL=P C(positiveor negative))Ssmall) T 1 eis small(for a givenr) I/O gainis insensitiveto changesin plantgainFeedbackcontrolsystems:statican alysis12{14example:L 20dB) T 1(withinabout 10%) y rwithinabout 10%(trackingerror is roughly10%)Feedbackcontrolsystems: static analysis12{15 Open-loop equivalentsystemclosed-loop system :PSfragreplacementsuryePCopen-loop equivalent(OLE) system :PSfragreplacements uryPC1 +PC(hassameI/O gainas closed-loop system , ,T)Feedbackcontrolsystems.}}}

6 Staticanalysis12{16 OLEsystemis usedto compare open- & closed-loop arrangementswe'll look at changesinP input& outputdisturbances sensor noiseFeedbackcontrolsystems:staticanalys is12{17 ChangesinPclosed-loop system :PSfragreplacementsuryeP+ PC Tcl=P C+ P C1 +P C+ P C P C1 +P CFeedbackcontrolsystems:staticanalysis12 {18open-loop equivalentsystem:PSfragreplacementsuryP+ PC1 +PC Tole=P C+ P C1 +P C P C1 +P Chence(aftersomealgebra) Tcl=11 +P C+ P C Toleso for small P, Tcl S ToleFeedbackcontrolsystems:staticanalysi s12{19 Input& outputdisturbancessuppose disturbancesdin,doutact on the plantPSfragreplacementsurdoutdinyePCe ectony(withr= 0)ycl=P1 +P Cdin+11 +P CdoutFeedbackcontrolsystems:staticanalys is12{20open-loop equivalentsystem:PSfragreplacementsurydo utdinPC1 +PCe ectony(withr= 0):yole=P din+douthenceycl=Syole, , e ectof disturbancesmultipliedbySLlarge)Ssmall)e ectof disturbancessmallFeedbackcontrolsystems: staticanalysis12{21 sensor noisesuppose sensor has noisedsensPSfragreplacementsurdsensyePCe ectony(withr= 0):ycl= P C1 +P CdsensFeedbackcontrolsystems:staticanaly sis12{22open-loop equivalentsystem:PSfragreplacementsuryds ensPC1 +PCe ectony(withr= 0):yole= 0;muchbetterthanclosed-loop system !}}}}}}}

7 Nally, adisadvantageof Feedback :outputcan be a ectedby sensor noiseFeedbackcontrolsystems:staticanalys is12{23 Summary bene tsof feedbackdeterminedby the sensitivityS= 1=(1 +P C) large loop gainL=P C(positiveornegative)yieldssmallS, hencebene tsof feedbackbene tsof feedbackinclude(whenjSj 1): good tracking(y r) low sensitivity of I/O reductionof e ectof input& outputdisturbanceson outputsomedisadvantagesof feedbackcontrol: cost(or reliability) of sensor sensor noisea ectsoutput(we'll see otherslater)Feedbackcontrolsystems:stati canalysis12{24}}