大学课件：频率响应设计英文版《Design-via-Frequency-Response1》

Chapter6

DesignviaFrequencyResponse2022/12/281Chapter6

DesignviaFrequencyDesignObjectivesProducedesiredtransientresponse.Reducesteady-stateerror.Achieveclosed-loopstability. TotalResponse=NaturalResponse+ ForcedResponse Theclosed-loopcontrolsystem’snaturalresponsemustnotdominate!Theoutputmustfollowtheinput.2022/12/282DesignObjectivesProducedesirCaseStudy:AntennaPositionControlThesearchfor

extraterrestriallifeis

beingcarriedoutwith

radioantennasliketheonepicturedhere.Aradioantennaisan

exampleofasystem

withpositioncontrols.2022/12/283CaseStudy:AntennaPositionCCaseStudy:AntennaAzimuthPositionControlSystemA.SystemConceptB.DetailedLayout2022/12/284CaseStudy:AntennaAzimuthPoCaseStudy:AntennaAzimuthPositionControlSystem(continued)C.SchematicdiagramD.Functionalblockdiagram2022/12/285CaseStudy:AntennaAzimuthPo2022/12/2862022/12/286CaseStudy:AntennaAzimuthPositionControlSystemResponse

Systemnormallyoperatestodrivepointingerrortozero.Motorisdrivenonlywhenthereisapointingerror.Thelargertheerrorthefasterthemotorturns.Toolargeasignalamplifiergaincouldcauseovershoot/instability.Satisfactorydesignrevolvesaroundabalancebetweentransientperformance,steady-stateperformance,andstability.Adjustinggain&addingcompensatorsarethetoolsacontrolengineerhastoachievethisbalance.2022/12/287CaseStudy:AntennaAzimuthPoTheDesignProcessStep1:DetermineaphysicalsystemandspecificationsfromtherequirementsStep2:DrawafunctionalblockdiagramStep3:TransformthephysicalsystemintoaschematicStep4:Usetheschematictoobtainablockdiagram,signal-flowdiagram,orstate-spacerepresentationStep5:Ifmultipleblocks,reducetheblockdiagramtoasingleblockorclosed-loopsystemStep6:Analyze,design,andtesttoseethatrequirementsandspecificationsaremet2022/12/288TheDesignProcessStep1:DeteBasicApproachCompensationinthefrequencydomaincanbeviewedas:

Addinggainatlowfrequenciestoimprovesteady-stateperformance.

Addingphaseangleatthedesiredphasemarginfrequencytoimprovetransientperformance.Phasemarginfrequencyapproximatestheclosed-loopbandwidth.Addingphaseanglecanbeusedtodesignforadesiredbandwidthand/orphasemargin.2022/12/289BasicApproachCompensationinLag&LeadCompensatorFormat2022/12/2810Lag&LeadCompensatorFormat2LeadCompensator

FrequencyResponse2022/12/2811LeadCompensator

FrequencyReMaximumPhaseIncreasefor

LeadCompensationMATLABCode:beta=0:0.01:1;»phi_max=(180/pi)*asin((1-beta)./(1+beta));»plot(beta,phi_max)»grid,title('MaximumPhaseIncreasevsbeta')»ylabel('MaximumPhaseLead-degrees')»xlabel('beta')2022/12/2812MaximumPhaseIncreasefor

LeaKeyElementsof

DesignApproach

Translatespecificationsintoclosed-loopbandwidthand/orphasemarginspecifications.

Controlbandwidthbyselectingfrequencyat0dB.gaincrossover,the“crossoverfrequency”.

Controlphasemarginbyselectingcorrectphaseangleatcrossover.2022/12/2813KeyElementsof

DesignApproaAsymptoticApproximationofClosed-LoopFrequencyResponseClosed-loopbandwidthisapproximatelyatopen-loopgaincrossoverfrequency,|GH(jwgc)|=1(0dB)whichisalsothePhaseMarginfrequency.2022/12/2814AsymptoticApproximationofClDampingRatiovsPhaseMargin

KeyApproximation:PhaseMargin~100*dampingratio2022/12/2815DampingRatiovsPhaseMarginPeakOvershootvs

PhaseMarginNote:PhaseMarginsfrom40to60degcorrespondtoPeakOvershootfrom30to10%.2022/12/2816PeakOvershootvs

PhaseMarginNormalizedBandwidthvs

DampingRatioNote:1)Well-dampedSystem:Bandwidth~NaturalFrequency2)OverdampedSystem:Bandwidth~0.5*NaturalFrequency2022/12/2817NormalizedBandwidthvs

DampinLeadCompensatorDesign1)Adjustcompensatorgain,Kc,toobtaindesiredcrossoverfrequency.2)Selectbetaparametertoobtainenoughphaseleadtomeetphasemarginspecification.3)Selectwmaxapproximatelyequaltocrossoverfrequencysothatadditionalleaddirectlycontributestophasemargin.4)ComplicationsarisesincegainatwmaxisnotunitybutKc/sqrt(beta);therebychangingcrossoverfrequencyandphasemargin.2022/12/2818LeadCompensatorDesign1)AdjuVisualizingLeadCompensationLeadcompensationgetsisnamefromthefactthatitsphaseangleresponseispositive,I.E.,itsoutput“leads”itsinputinphase.1)KcisselectedsothatAisthedesiredclosed-loopbandwidth.2)wmaxisselectednearAsomaximumadditionalphaseisprovidedneargaincrossover.3)SincecompensatorgainatAisnotunity(0dB.)thecompensatorchangescrossoverfrequencytoC.2022/12/2819VisualizingLeadCompensationLDesignProcedure:Step#1

Adjustgaintoobtaincrossoverfrequencythatapproximatesdesiredclosed-loopbandwidth.

Setcrossoverfrequencyequaltobandwidthifoverdampedsystemisdesiredandequalto1/2bandwidthifunderdamped.Ifthereisnobandwidthspecification:A)LowBandwidthAlternative: Adjustgaintomeetphasemarginspecasbestyoucan.B)HighBandwidthAlternative:Adjustgaintomeetsteady-stateerrorspecasbestyoucan.2022/12/2820DesignProcedure:Step#1AdjuDesignProcedure:Step#2

IfphasemarginspecisnotmetgotoSteps#3through7(LeadCompensation).Ifphasemarginspecismet,checksteady-stateerrorspecandgotoSteps#8through10(LagCompensation).Ifbothspecsaremetdesigniscompleteusingproportionalcompensation.2022/12/2821DesignProcedure:Step#2IfpTransient(LeadCompensator)DesignProcedure:Steps#3-7

Step#3:Allowforanextra5to12degofphasemargin.Calculatetheadditionalphaseleadtomeetphasemarginspec:beta_lead=phasemarginspec -Step#1phasemarginresult +(5to12).

Step#4:SetLeadCompensatorgain(Klead)equaltogainobtainedinStep#1.2022/12/2822Transient(LeadCompensator)DDesignProcedure:Step#5

Findthefrequency(wgcc)atwhichtheuncompensated,open-loopfrequencyresponseequals:-20*log10(Klead/sqrt(beta_lead)=-Klead(dB)+10*log10(beta_lead)

wgccisthe“corrected”gaincrossoverfrequency.Itaccountsforthefactthatthegainoftheleadcompensatoratitsmaximumphaseangleisnotunity;therebychangingthegaincrossoverfrequency.2022/12/2823DesignProcedure:Step#5FindDesignProcedure:Steps#6&7Step#6:Setwmax=wgcc=1/(Tlead*sqrt(beta_lead)).SolveforTlead:

Tlead=1/(wgcc*sqrt(beta_lead))Step#7:ObtainBodeplotofcompensated,open-loopsystem.Confirmcrossoverfrequencyandgain&phasemargins.ObtainBodeplotofcompensated,closed-loopsystem.Confirmbandwidth&resonant-peakperformance.Tuneleadcompensatorasnecessary.2022/12/2824DesignProcedure:Steps#6&7Steady-State(LagCompensator)DesignProcedure:Steps#8-10Step#8:

Computetheopen-loopgain,Kol,obtainedaftercompletionofSteps1-7.Step#9:

Computetheopen-loopgain,Kd,neededtomeetthesteady-stateerrorperformancespecifications.Step#10:

Computetheadditionalgainrequired,Kd/Kol.Ifgreaterthanone,setgainoflagcompensator:Klag=beta_lag=Kd/Kol.Then,setTlag~=(2to10)/wgccwiththelargervaluepreferred.Thisincreasesgainwithoutchangingbandwidthorphasemargin.2022/12/2825Steady-State(LagCompensator)ExampleofLead&Lag

CompensationProblem:ConsideraunityfeedbackcontrolsystemwithG(s)=1/s(s+1)whichrepresentsasimplifiedmodelofaDCmotoraswellasasatelliteantenna.Designaseriescompensatortoobtainasteady-stateerroroflessthan0.1forarampinputwithapeakovershootlessthan25%.Specifications:Thevelocityerrorcoefficient,Kv,mustbe>=1/0.1=10&Peakovershoot=<25%requiresaPhaseMargin>=45deg

2022/12/2826ExampleofLead&Lag

CompensaExample:Low-bandwidthDesignStep#1a:Plotuncompensated,open-loopfrequencyresponse.Gp=tf(1,[110]);margin(Gp)PM=51degThus,gaincanbeloweredtoobtainPM=45degbyfindingthefrequencyatwhichtheopen-loopphaseangleis-180+45=-135deg.2022/12/2827Example:Low-bandwidthDesignSExample:Low-bandwidthDesignStep#1a(cont.):[magnitude,angle,frequency]=bode(Gp,1)

magnitude=0.7071angle=-135frequency=1

Kp=1/magnitude

Kp=1.4142

CheckResultingPhaseMargin:[Gm,Pm,Wcg,Wcp]=margin(Kp*Gp)

Gm=InfPm=45.0000Wcg=NaNWcp=1.0000

2022/12/2828Example:Low-bandwidthDesignSExample:Low-bandwidthDesignStep#2:SincethePhaseMarginspeccanbemetwithproportionalcompensation,thereisnoneedforleadcompensation.Checkingthesteady-stateperformance:Kv=Kp*dcgain(conv([10],ng),dg)

Kv=1.4142Thisdoesnotmeetingthespecifiedvalueof10.So,wemustconsideralagcompensatorandmoveontoStep#8.2022/12/2829Example:Low-bandwidthDesignSExample:Low-bandwidthDesignStep#8:Theuncompensatedopen-loopgain,Ko=1.Step#9:Thedesiredopen-loopgain,Kd=10.Step#10:

Kclag=Kd/KoKclag=10

beta_lag=Kclag

beta_lag=10

UsingthecrossoverfrequencyobtainedinStep1,Wcp,Tlag=10/WcpTlag=10.0000LagCompensator:ngc=Kclag*[Tlag1];dgc=[beta_lag*Tlag1];Gc=tf(ngc,dgc)

Transferfunction:100s+10----------100s+12022/12/2830Example:Low-bandwidthDesignSExample:Low-bandwidthDesign

Closed-loopPerformanceG=Gc*Gp;Gclosed_loop=minreal(G/(1+G)2022/12/2831Example:Low-bandwidthDesign

Example:Low-bandwidthDesign

Closed-loopStepPerformancePeak_overshoot=(max(step(Gclosed_loop))-1)*100Peak_overshoot=27.8128

Overshoot>25%duetolagzero@s=-0.1.RedesignwouldincreasePhaseMarginslightly.SettlingTime~15sec2022/12/2832Example:Low-bandwidthDesign

Example:Low-bandwidthDesign

Closed-loopFrequencyResponsebode(Gclosed_loop)Comment:Bandwidthslightlygreaterthan1rad/sec2022/12/2833Example:Low-bandwidthDesign

Example:High-bandwidthDesign2022/12/2834Example:High-bandwidthDesignExample:High-bandwidthDesignPhaseMarginof18degdoesnotmeet45degspec.So,leadcompensationisappropriate.2022/12/2835Example:High-bandwidthDesignExample:High-bandwidthDesignStep#3:

Theadditionalphaseleadneededtomeetspecifications,includinga5to12degreemargin,is:[gm,pm]=margin(G);phimax=45-pm

phimax=27.0358

Withamarginof8degreesphimax=35

phimax=35

So,phimaxr=phimax*pi/180;

beta_lead=(1-sin(phimaxr))/(1+sin(phimaxr))

beta_lead=0.2710

2022/12/2836Example:High-bandwidthDesignExample:High-bandwidthDesignStep#4:Kclead=Kp=10Step#5:

Weneedtofindthefrequencyatwhichthemagnitudeoftheuncompensatedopen-looptransferfunctionis:(Kclead/sqrt(beta_lead))^(-1)ans=0.0521(-26dB)

FromtheBodediagramthiswillbeintheneighborhoodof4rad/sec.Now,magnitude_Gp=bode(Gp,4)magnitude_Gp=0.0606

Searchinginthefrequencydomain,wgcc=4.3;magnitude_Gp=bode(Gp,wgcc)

magnitude_Gp=0.0527

Thisaccuracyisquiteadequatefordesignpurposes2022/12/2837Example:High-bandwidthDesignExample:High-bandwidthDesign2022/12/2838Example:High-bandwidthDesignExample:High-bandwidth

StepResponsePeak_overshoot=(max(step(Gclosed_loop2))-1)*100Peak_overshoot=24.2226

SettlingTime~1.2sec2022/12/2839Example:High-bandwidth

StepRExample:High-bandwidth

FrequencyResponseBandwidth~6rad/sec2022/12/2840Example:High-bandwidth

FrequeExercise1.12022/12/2841Exercise1.12022/12/2841Exercise1.22022/12/2842Exercise1.22022/12/2842Exercise1.32022/12/2843Exercise1.32022/12/2843Exercise1.42022/12/2844Exercise1.42022/12/2844Exercise1.52022/12/2845Exercise1.52022/12/2845Exercise2.12022/12/2846Exercise2.12022/12/2846Exercise2.22022/12/2847Exercise2.22022/12/2847Exercise3.12022/12/2848Exercise3.12022/12/2848Exercise3.22022/12/2849Exercise3.22022/12/2849Exercise4.12022/12/2850Exercise4.12022/12/2850Exercise4.22022/12/2851Exercise4.22022/12/2851Exercise4.32022/12/2852Exercise4.32022/12/2852Exercise5.1五、论述题1．论述：系统校正的主要方法分类及其各自的特点。（5分）2．论述：最小相位系统的基本概念，以及在控制系统性能分析中的意义。（5分）3．论述：系统稳定性的主要判别方法及其应用特点。（5分）

2022/12/2853Exercise5.1五、论述题1．论述：系统校正的主要方Ex.1-2p.72022/12/2854Ex.1-2p.72022/12/2854Ex.1-3p8例1-3为了保持希望的温度，又温控开关接通或断开电加热器电源。在使用热水时，水箱中流出热水并补充冷水。试说明系统工作原理并画出系统原理框图。2022/12/2855Ex.1-3p8例1-3为了保持希望的温度，又温控开关接Ex.2-1p201.列写网络微分方程2.已知：L=1H,C=0.2F,R=2.5.设初始条件为零，Vi(t)=10*1(t)V.试用拉氏变换法求输出信号Vo(t).2022/12/2856Ex.2-1p201.列写网络微分方程2022/12/28Ex.2-3p24例2-3已知参数如图所示，求折算到电动机轴上的等效转动惯量和等效粘性摩擦系数，并导出以M为输入，以2为输出的齿轮系传递函数。2022/12/2857Ex.2-3p24例2-3已知参数如图所示，求折算到Ex.4-3p104例4-3已知系统的传递函数为试绘制系统的概略幅相频特性曲线。2022/12/2858Ex.4-3p104例4-3已知系统的传递函数为试绘制Ex.5-10p146例5-10已知单位反馈系统的开环传递函数试用奈氏判据判断系统的闭环稳定性。2022/12/2859Ex.5-10p146例5-10已知单位反馈系统的开Ex.5-13p1502022/12/2860Ex.5-13p1502022/12/2860Ex.5-14p153例5-14系统开环频率特性如图a、b所示，试判断闭环系统的稳定性。P=0.2022/12/2861Ex.5-14p153例5-14系统开环频率特性如Ex.5-15p154例5-15已知单位反馈系统的开环传递函数试判断系统的闭环稳定性。2022/12/2862Ex.5-15p154例5-15已知单位反馈系统的开环Ex.5-17p1562022/12/2863Ex.5-17p1562022/12/2863Chapter6

DesignviaFrequencyResponse2022/12/2864Chapter6

DesignviaFrequencyDesignObjectivesProducedesiredtransientresponse.Reducesteady-stateerror.Achieveclosed-loopstability. TotalResponse=NaturalResponse+ ForcedResponse Theclosed-loopcontrolsystem’snaturalresponsemustnotdominate!Theoutputmustfollowtheinput.2022/12/2865DesignObjectivesProducedesirCaseStudy:AntennaPositionControlThesearchfor

extraterrestriallifeis

beingcarriedoutwith

radioantennasliketheonepicturedhere.Aradioantennaisan

exampleofasystem

withpositioncontrols.2022/12/2866CaseStudy:AntennaPositionCCaseStudy:AntennaAzimuthPositionControlSystemA.SystemConceptB.DetailedLayout2022/12/2867CaseStudy:AntennaAzimuthPoCaseStudy:AntennaAzimuthPositionControlSystem(continued)C.SchematicdiagramD.Functionalblockdiagram2022/12/2868CaseStudy:AntennaAzimuthPo2022/12/28692022/12/286CaseStudy:AntennaAzimuthPositionControlSystemResponse

Systemnormallyoperatestodrivepointingerrortozero.Motorisdrivenonlywhenthereisapointingerror.Thelargertheerrorthefasterthemotorturns.Toolargeasignalamplifiergaincouldcauseovershoot/instability.Satisfactorydesignrevolvesaroundabalancebetweentransientperformance,steady-stateperformance,andstability.Adjustinggain&addingcompensatorsarethetoolsacontrolengineerhastoachievethisbalance.2022/12/2870CaseStudy:AntennaAzimuthPoTheDesignProcessStep1:DetermineaphysicalsystemandspecificationsfromtherequirementsStep2:DrawafunctionalblockdiagramStep3:TransformthephysicalsystemintoaschematicStep4:Usetheschematictoobtainablockdiagram,signal-flowdiagram,orstate-spacerepresentationStep5:Ifmultipleblocks,reducetheblockdiagramtoasingleblockorclosed-loopsystemStep6:Analyze,design,andtesttoseethatrequirementsandspecificationsaremet2022/12/2871TheDesignProcessStep1:DeteBasicApproachCompensationinthefrequencydomaincanbeviewedas:

Addinggainatlowfrequenciestoimprovesteady-stateperformance.

Addingphaseangleatthedesiredphasemarginfrequencytoimprovetransientperformance.Phasemarginfrequencyapproximatestheclosed-loopbandwidth.Addingphaseanglecanbeusedtodesignforadesiredbandwidthand/orphasemargin.2022/12/2872BasicApproachCompensationinLag&LeadCompensatorFormat2022/12/2873Lag&LeadCompensatorFormat2LeadCompensator

FrequencyResponse2022/12/2874LeadCompensator

FrequencyReMaximumPhaseIncreasefor

LeadCompensationMATLABCode:beta=0:0.01:1;»phi_max=(180/pi)*asin((1-beta)./(1+beta));»plot(beta,phi_max)»grid,title('MaximumPhaseIncreasevsbeta')»ylabel('MaximumPhaseLead-degrees')»xlabel('beta')2022/12/2875MaximumPhaseIncreasefor

LeaKeyElementsof

DesignApproach

Translatespecificationsintoclosed-loopbandwidthand/orphasemarginspecifications.

Controlbandwidthbyselectingfrequencyat0dB.gaincrossover,the“crossoverfrequency”.

Controlphasemarginbyselectingcorrectphaseangleatcrossover.2022/12/2876KeyElementsof

DesignApproaAsymptoticApproximationofClosed-LoopFrequencyResponseClosed-loopbandwidthisapproximatelyatopen-loopgaincrossoverfrequency,|GH(jwgc)|=1(0dB)whichisalsothePhaseMarginfrequency.2022/12/2877AsymptoticApproximationofClDampingRatiovsPhaseMargin

KeyApproximation:PhaseMargin~100*dampingratio2022/12/2878DampingRatiovsPhaseMarginPeakOvershootvs

PhaseMarginNote:PhaseMarginsfrom40to60degcorrespondtoPeakOvershootfrom30to10%.2022/12/2879PeakOvershootvs

PhaseMarginNormalizedBandwidthvs

DampingRatioNote:1)Well-dampedSystem:Bandwidth~NaturalFrequency2)OverdampedSystem:Bandwidth~0.5*NaturalFrequency2022/12/2880NormalizedBandwidthvs

DampinLeadCompensatorDesign1)Adjustcompensatorgain,Kc,toobtaindesiredcrossoverfrequency.2)Selectbetaparametertoobtainenoughphaseleadtomeetphasemarginspecification.3)Selectwmaxapproximatelyequaltocrossoverfrequencysothatadditionalleaddirectlycontributestophasemargin.4)ComplicationsarisesincegainatwmaxisnotunitybutKc/sqrt(beta);therebychangingcrossoverfrequencyandphasemargin.2022/12/2881LeadCompensatorDesign1)AdjuVisualizingLeadCompensationLeadcompensationgetsisnamefromthefactthatitsphaseangleresponseispositive,I.E.,itsoutput“leads”itsinputinphase.1)KcisselectedsothatAisthedesiredclosed-loopbandwidth.2)wmaxisselectednearAsomaximumadditionalphaseisprovidedneargaincrossover.3)SincecompensatorgainatAisnotunity(0dB.)thecompensatorchangescrossoverfrequencytoC.2022/12/2882VisualizingLeadCompensationLDesignProcedure:Step#1

Adjustgaintoobtaincrossoverfrequencythatapproximatesdesiredclosed-loopbandwidth.

Setcrossoverfrequencyequaltobandwidthifoverdampedsystemisdesiredandequalto1/2bandwidthifunderdamped.Ifthereisnobandwidthspecification:A)LowBandwidthAlternative: Adjustgaintomeetphasemarginspecasbestyoucan.B)HighBandwidthAlternative:Adjustgaintomeetsteady-stateerrorspecasbestyoucan.2022/12/2883DesignProcedure:Step#1AdjuDesignProcedure:Step#2

IfphasemarginspecisnotmetgotoSteps#3through7(LeadCompensation).Ifphasemarginspecismet,checksteady-stateerrorspecandgotoSteps#8through10(LagCompensation).Ifbothspecsaremetdesigniscompleteusingproportionalcompensation.2022/12/2884DesignProcedure:Step#2IfpTransient(LeadCompensator)DesignProcedure:Steps#3-7

Step#3:Allowforanextra5to12degofphasemargin.Calculatetheadditionalphaseleadtomeetphasemarginspec:beta_lead=phasemarginspec -Step#1phasemarginresult +(5to12).

Step#4:SetLeadCompensatorgain(Klead)equaltogainobtainedinStep#1.2022/12/2885Transient(LeadCompensator)DDesignProcedure:Step#5

Findthefrequency(wgcc)atwhichtheuncompensated,open-loopfrequencyresponseequals:-20*log10(Klead/sqrt(beta_lead)=-Klead(dB)+10*log10(beta_lead)

wgccisthe“corrected”gaincrossoverfrequency.Itaccountsforthefactthatthegainoftheleadcompensatoratitsmaximumphaseangleisnotunity;therebychangingthegaincrossoverfrequency.2022/12/2886DesignProcedure:Step#5FindDesignProcedure:Steps#6&7Step#6:Setwmax=wgcc=1/(Tlead*sqrt(beta_lead)).SolveforTlead:

Tlead=1/(wgcc*sqrt(beta_lead))Step#7:ObtainBodeplotofcompensated,open-loopsystem.Confirmcrossoverfrequencyandgain&phasemargins.ObtainBodeplotofcompensated,closed-loopsystem.Confirmbandwidth&resonant-peakperformance.Tuneleadcompensatorasnecessary.2022/12/2887DesignProcedure:Steps#6&7Steady-State(LagCompensator)DesignProcedure:Steps#8-10Step#8:

Computetheopen-loopgain,Kol,obtainedaftercompletionofSteps1-7.Step#9:

Computetheopen-loopgain,Kd,neededtomeetthesteady-stateerrorperformancespecifications.Step#10:

Computetheadditionalgainrequired,Kd/Kol.Ifgreaterthanone,setgainoflagcompensator:Klag=beta_lag=Kd/Kol.Then,setTlag~=(2to10)/wgccwiththelargervaluepreferred.Thisincreasesgainwithoutchangingbandwidthorphasemargin.2022/12/2888Steady-State(LagCompensator)ExampleofLead&Lag

CompensationProblem:ConsideraunityfeedbackcontrolsystemwithG(s)=1/s(s+1)whichrepresentsasimplifiedmodelofaDCmotoraswellasasatelliteantenna.Designaseriescompensatortoobtainasteady-stateerroroflessthan0.1forarampinputwithapeakovershootlessthan25%.Specifications:Thevelocityerrorcoefficient,Kv,mustbe>=1/0.1=10&Peakovershoot=<25%requiresaPhaseMargin>=45deg

2022/12/2889ExampleofLead&Lag

CompensaExample:Low-bandwidthDesignStep#1a:Plotuncompensated,open-loopfrequencyresponse.Gp=tf(1,[110]);margin(Gp)PM=51degThus,gaincanbeloweredtoobtainPM=45degbyfindingthefrequencyatwhichtheopen-loopphaseangleis-180+45=-135deg.2022/12/2890Example:Low-bandwidthDesignSExample:Low-bandwidthDesignStep#1a(cont.):[magnitude,angle,frequency]=bode(Gp,1)

magnitude=0.7071angle=-135frequency=1

Kp=1/magnitude

Kp=1.4142

CheckResultingPhaseMargin:[Gm,Pm,Wcg,Wcp]=margin(Kp*Gp)

Gm=InfPm=45.0000Wcg=NaNWcp=1.0000

2022/12/2891Example:Low-bandwidthDesignSExample:Low-bandwidthDesignStep#2:SincethePhaseMarginspeccanbemetwithproportionalcompensation,thereisnoneedforleadcompensation.Checkingthesteady-stateperformance:Kv=Kp*dcgain(conv([10],ng),dg)

Kv=1.4142Thisdoesnotmeetingthespecifiedvalueof10.So,wemustconsideralagcompensatorandmoveontoStep#8.2022/12/2892Example:Low-bandwidthDesignSExample:Low-bandwidthDesignStep#8:Theuncompensatedopen-loopgain,Ko=1.Step#9:Thedesiredopen-loopgain,Kd=10.Step#10:

Kclag=Kd/KoKclag=10

beta_lag=Kclag

beta_lag=10

UsingthecrossoverfrequencyobtainedinStep1,Wcp,Tlag=10/WcpTlag=10.0000LagCompensator:ngc=Kclag*[Tlag1];dgc=[beta_lag*Tlag1];Gc=tf(ngc,dgc)

Transferfunction:100s+10----------100s+12022/12/2893Example:Low-bandwidthDesignSExample:Low-bandwidthDesign

Closed-loopPerformanceG=Gc*Gp;Gclosed_loop=minreal(G/(1+G)2022/12/2894Example:Low-bandwidthDesign

Example:Low-bandwidthDesign

Closed-loopStepPerformancePeak_overshoot=(max(step(Gclosed_loop))-1)*100Peak_overshoot=27.8128

Overshoot>25%duetolagzero@s=-0.1.RedesignwouldincreasePhaseMarginslightly.SettlingTime~15sec2022/12/2895Example:Low-bandwidthDesign

Example:Low-bandwidthDesign

Closed-loopFrequencyResponsebode(Gclosed_loop)Comment:Bandwidthslightlygreaterthan1rad/sec2022/12/2896Example:Low-bandwidthDesign

Example:High-bandwidthDesign2022/12/2897Example:High-bandwidthDesignExample:High-bandwidthDesignPhaseMarginof18degdoesnotmeet45degspec.So,leadcompensationisappropriate.2022/12/2898Example:High-bandwidthDesignExample:High-bandwidthDesignStep#3:

Theadditionalphaseleadneededtomeetspecifications,includinga5to12degreemargin,is:[gm,pm]=margin(G);phimax=45-pm

phimax=27.0358

Withamarginof8degreesphimax=35

phimax=35

So,phimaxr=phimax*pi/180;

beta_lead=(1-sin(phimaxr))/(1+sin(phimaxr))

beta_lead=0