绘未补偿系统根轨迹课件

CH.9DesignviaRootLocusCH.9DesignviaRootLocus1Figure9.1

a.possibledesignpointviagainadjustment(A)

designpointthatcannot

bemetviagainadjustment(B);

需補償器設計9.1IntroductionFigure9.1

a.possibledesign2Figure9.1

a.possibledesignpointviagainadjustment(A)

designpointthatcannot

bemetviagainadjustment(B);

Figure9.1(b)

responsesfrompolesatAandB9.1IntroductionFigure9.1

a.possibledesign3補償器種類

Figure9.2

Compensation

techniques:

a.

cascade;

b.

feedback補償器種類

Figure9.2

Compensatio49.2節:

designofcascadecompensationtoimproveess9.3節:

designofcascadecompensationtoimprove暫態反應9.4節:

designofcascadecompensationtoimproveboth

ess

and暫態反應9.5節:

designoffeedbackcompensation9.2節:59.2cascadecompensationtoimprovesteady-stateerror改善ess

補償器3種

1/S:ideaintegralcompensator理想積分器

→

ess=0亦變transientresponse

Fig.9.3b(i.e.變根軌跡圖)

K1+K2/S:PIcontroller(Proportional-plus-Integral)→

ess=0

可不變

transientresponseFig.9.3c

(S+Zc)/(S+Pc):Pc＜ZcLagCompensator→

ess↓

可不變

transientresponse9.2cascadecompensationto6Figure9.3a.rootlocusb.

notontherootlocuswith1/Scompensatoradded此設計已無從獲致系統AFigure9.3a.rootlocusb.not7補償器1/S:ideaintegralcompensator

→

ess=0

∵systemtype

增1

ess=finite→ess=0(缺點activenetwork)(缺點變transientresponse)補償器1/S:ideaintegralcomp8Figure9.3

c.

onrootlocuswithPIcompensatoradded

當a甚小→0

不變transientresponse

PIcontroller補償Figure9.3c.onrootlocusw9(S

+a)/S:PIcontroller(Proportional-plus-Integral)→

ess=0

∵systemtype

增1可不變transientresponse

當a甚小

Fig.9.3c(缺點activenetwork)

(不變transientresponse)

绘未补偿系统根轨迹课件10绘未补偿系统根轨迹课件11補償器(S+Zc)/(S+Pc):

Pc＜ZcLagCompensator暫態反應okay希望調降穩態誤差暫態反應okay維持P點；Zc→Pc控制器角度貢獻→

0補償器(S+Zc)/(S+Pc):Pc＜Zc12绘未补偿系统根轨迹课件13绘未补偿系统根轨迹课件14绘未补偿系统根轨迹课件15Figure9.4

Closed-loopsystemfor

Example9.1:

設計目標:希望

ess=0；不變transientresponse

(下頁說明控制器選擇)

beforecompensationafterPIcompensationFigure9.4Closed-loopsy16改善ess

補償器3種(控制器選擇)

1/S:ideaintegralcompensator理想積分器

→

ess=0亦變transientresponse

Fig.9.3b(i.e.變根軌跡圖)

K1+K2/S:PIcontroller(Proportional-plus-Integral)→

ess=0

可不變

transientresponseFig.9.3c

(S+Zc)/(S+Pc):Pc＜ZcLagCompensator→

ess↓可不變transientresponse改善ess補償器3種(控制器選擇)17Figure9.5

RootlocusforuncompensatedsystemofFig.9.4(a)3階系統Figure9.6

Rootlocusforcompensated

systemofFigure9.4(b)Example9.1Figure9.5

Rootlocusforunco18Figure9.7timeresponse

PIcompensatedsystemresponseandtheuncompensatedsystemresponseofExample9.1設計目標:

ess=0不變transientresponseFigure9.7timeresponse

19Figure9.8

PIcontrollerFigure9.8

PIcontroller201/111.未補償系統ζ=0.174→根軌跡決定系統位置

2.ess降低10倍→先求未補償系統的ess

1/111.未補償系統ζ=0.174→根軌跡決定21Uncompensatedsystem

先畫根軌跡找出系統現況2/11Uncompensatedsystem

先畫根軌跡找出22Uncompensatedsystem找未補償系統的穩態誤差e(∞)=1/(1+Kp)=0.108

Kp

=lims→0G(s)=164.6/20=8.233/11Uncompensatedsystem找未補償系統的23改善ess

補償器3種(控制器選擇)4/11

1/S:ideaintegralcompensator理想積分器

→

ess=0亦變

transientresponse

Fig.9.3b(i.e.變根軌跡圖)

K1+K2/S:PIcontroller(Proportional-plus-Integral)→

ess=0可不變transient

responseFig.9.3c

(S+Zc)/(S+Pc):Pc＜ZcLagCompensator→

ess↓

可不變

transientresponse改善ess補償器3種(控制器選擇)4/1245/115/1125绘未补偿系统根轨迹课件26e(∞)=1/(1+Kp)

=0.108

Kp

=lims→0G(s)=164.6/20=8.23

ec(∞)=0.0108

由Gc(s)調整7/11Uncompensatedsysteme(∞)7/11Uncompensatedsystem27Figure9.12CompensatedsystemofFigure9.11

8/11ec(∞)=0.0108

=1/(1+Kpc)

Kpc=lims→0Gc(s)G(s)=lims→0Gc(s)Kp

=lims→0

Gc(s)8.23=91.593

lims→0

Gc(s)=Zc/Pc=11.132Gc(s)=(s+Zc)/(s+Pc)→取Pc=0.01→Zc=0.111Figure9.12Compensatedsyst28Table9.1

Predictedcharacteristicsofuncompensatedandlag-compensatedsystemsforExample9.29/11Table9.1

Predictedcharacteri29Figure9.13

Stepresponsesofuncompensatedandlag-compensatedsystemsfor

Example9.210/11

Figure9.13

Stepresponsesof30Figure9.14

StepresponsesofthesystemforExample9.2

lagcompensator更趨近原點→反應較慢最終之ess不變11/11Figure9.14

Stepresponsesof31執行Lagcompensation的相關公式執行Lagcompensation的相關公式329.3ImprovingTransientResponseviaCascadeCompensation改善暫態反應

補償器3種

S:

puredifferentiator純微分器

S

+Zc:

PDcontroller

(Proportional-plus-Derivative)

(S+Zc)/(S+Pc):Pc＞ZcLeadCompensator超前補償器

9.3ImprovingTransientRespon33Figure9.15UsingPDcompensation(S

+Zc)改變暫態反應%OS不變1/3

a.uncompensated;b.compensatorzeroat–2;

c.compensatorzeroat–3;d.compensatorzeroat–4Figure9.15UsingPDcompensa34Figure9.16timeresponse2/3UncompensatedsystemandidealderivativecompensationsolutionsfromTable9.2Figure9.16timeresponse35Table9.2

PredictedcharacteristicsforthesystemsofFigure9.153/3Table9.2

Predictedcharacteri36Example9.3:

S

+Zc:PDcontroller

Figure9.17

Feedbackcontrolsystem1/7設計目標:補償後系統

%OS=16%

i.e.ζ=0.504

tsc=ts/3Example9.3:S+Zc:PDc37Figure9.18Example9.3

Rootlocusforuncompensatedsystem2/7先求未補償系統位置%OS=16%→ζ=0.504

次求未補償系統ts

ts=4/(ζωn)=4/1.205=3.32

Figure9.18Example9.38Figure9.18Example9.3

Rootlocusforuncompensatedsystem2/7Figure9.18Example9.39Figure9.19CompensateddominantpolesuperimposedovertheuncompensatedrootlocusforExample9.33/7希望補償後

dominantpoles=-3.613±j6.192→∠Gc(s)G(s)=1800

目標

設計

Gc(s)=S

+ZcPDcontroller希望補償後tsc=3.32/3=1.107

→(ζωn)c=3.613→%OS不變

→補償後

dominantpoles=-3.613±j6.192Figure9.19Compensatedd40Figure9.204/7

EvaluatingthelocationofthecompensatingzeroforExample9.3希望之角度補償

∠S

+Zc

=1800

－∠G(s)Figure9.2041Figure9.21

RootlocusforthecompensatedsystemofExample9.35/7Figure9.21

Rootlocusforthe42Figure9.22timeresponse

Uncompensatedandcompensatedsystemstepresponsesof

Example9.36/7Figure9.22timeresponse43Table9.3

UncompensatedandcompensatedsystemcharacteristicsforExample9.37/7Table9.3

Uncompensatedandco44Figure9.23

PDcontrollerS

+Zc

Figure9.23

PDcontroller45Example9.4

Leadcompensatordesign

(S+Zc)/(S+Pc):Pc＞Zc

設計目標:

補償後系統

%OS=30%

i.e.ζ=0.358

tsc=ts/2

Figure9.26uncompensatedandcompensateddominantpolesExample9.4Leadcompensato46∠kG(s)=-1800

目標→∠Gc(s)+∠kG(s)

=-1800

∠Gc(s)=∠(S+Zc)/(S+Pc)=θc超前控制器設計→選擇Leadcompensator貢獻的角度∠kG(s)=-1800目標→∠Gc(s)+∠47Figure9.25

ThreeoftheinfinitepossibleleadcompensatorsolutionsLeadcompensator貢獻的角度(S+Zc)/(S+Pc):Pc＞Zc∠(S+Zc)/(S+Pc)=θc

正值

超前控制器目標→∠Gc(s)G(s)=1800

超前控制器設計→選擇Leadcompensator貢獻的角度→∠Gc(s)=∠(S+Zc)/(S+Pc)=θcFigure9.25

Threeoftheinfin48Table9.4

ComparisonofleadcompensationdesignsforExample9.4Whichisthebestdesign?系統階數?是否有2個主要極點?是否有零點?能否抵消?比較規格達成度確認最佳設計Table9.4

Comparisonoflead49Figure9.28

Compensatedsystemrootlocus(whichisthisdesign?)Figure9.28

Compensatedsystem50Figure9.29

Uncompensatedsystemandleadcompensationresponsesfor

Example9.4Figure9.29

Uncompensatedsyst519.4designofcascadecompensationtoimprovebothess

and暫態反應

improveboth暫態反應firstthenimproveessExample9.5自修9.4designofcascadecompens52改善ess

補償器3種1/3

1/S:積分器→

ess=0(S

+a)/S:PI控制器→

ess=0

(S+Zc)/(S+Pc):Pc＜ZcLagCompensator→

ess↓控制器設計原則:選pole,zero

近原點,維持暫態反應不變改善暫態反應

補償器3種

S:微分器

S

+Zc:PD控制器

(S+Zc)/(S+Pc):Pc＞ZcLeadCompensator同時改善暫態反應和

ess

PID

控制器:(S

+Zc)(S

+a)/S

Lag-Lead補償器:[(S+Zc)/(S+Pc)]Lead[(S+Zc)/(S+Pc)]Lag設計原則:先調暫態反應再調穩態誤差改善ess補償器3種531.繪未補償系統根軌跡a.根軌跡條數=3

%OS=20%i.e.ζ=0.456b.實軸上根軌跡c.漸近線(3zerosat∞)Ex.9.6Fig.9.37

%OS=20%i.e.ζ=0.456Tsc=Ts/2Ts=4/(ζωn)essc=ess/10kGH(s)==-16/3

=π/3

；π；

5π/3

2/31.繪未補償系統根軌跡Ex.9.6Fig.9541.繪未補償系統根軌跡d.Breakawaypoint1+kGH(s)=0k(s)=-1/[GH(s)]

→dk/ds=0

→

s=-2.43;-8.24(不適用)e.trial-and-error

∠GH(s)=-1800find未補償系統ats=-1.794±j3.501∣kGH(s)∣=1→k=192.12.

先調暫態反應f.補償後系統