Design via Root Locus -Control engineering

8/9/2019 Design via Root Locus -Control engineering

1/20

' . t l btc$ ]1i \oelearning.utm.my

,ffi outtlnesf his hapter:. How o use he root locus o design ascadecompensatorso improvehe steady-stateerror. How o use he root locus o design ascadecompensatorso improvehe transient

response. How o use he root locuso design ascadecompensatorso improve oth he steady-stateerror and he transient esponse. How o use he root locus o design eedbackcompensatorso improvehe transientresponse

UPNMNIVERSITIERTAHANANASIONALIIALAYSIAEEE321Control ngineeringDesignia RootLocus

Revised y S.PChewlvlarch 7, 2010


2/20


3/20


4/20

elearning.utm.my

ComoensatorsO Integral ompensatorsre used o imprcve teady-statefroro Dervative ompensatorsreused o improveransientesponseOldealcompensatorsustbe mplementedithactivenehvorksuch asactive mplifiers ndpossible dditionalower ources.Oother design echniqueshat precludehe use of activedevices ofcompensationan be adopted.These comPensators,hich can beimplemented ithpassive lements uchas resisto6and capacitors,do not use pure integration nd differentiation nd are not idealcompensators.OAdvantagesf passive etworks re that hey are less expensive nddo not equire dditionalower ourcesor heiropetion.ETheir disadvantages that he steady-siaterror s not driven o zero ncaseswhere deal ompensatorsield eroerrot

lmproving teady-state rror via cascadecomDensationThisdesigns to imorovehesteadv-staterrorwithout lffeqtinqthe ransientesponse.

dr , r , r 4rerr | i3 fPoleatA is noton heroot ocuswith

4


5/20

elearning.utm.my

ldeal ntegralCompensationPl)

0 02 et er lsb=\zt ) t3a '

PoleA s approximatelyn heroot ocuswithcompensatof

. thesteady-staterrorsimprovedithoutaffectinghe ransientresponse. A compensatorithapole at heorigin ndazero closeo hepolescalled n deal ntegralcompensator.

- rysi,r- !9


6/20

elearning.utm.my

uncompensatedystem Compensatedystem

W""::,:"""1"t:1,',n""10"",,"n,"","",pensato,s epresenteds. . x,)c.r,t= , *& = -\-{1

Since he figure has both proportionalnd integral ontrol, he idealintegral ontroller,r compensator,sgivenaltenale 'amePlcontrollet.


7/20

elearning.utm.my

{Mn.3c*re's*r"' Thestatic elocity rrcrconslanr:

fto$( p.,tfcrrsbrl .- Conholler ransfer unciionl'* ' l i ]# 6.(0 = :-:a

+ r^= xn' lL, Y,,,. Type1 uncompensaledystemib. Type1 compensatedyslemi

] LagCompensator>ldeal ntegral ompensation,ith tspoleon the origin, equires nactiventegrator>lf we usepassive etwork,he poleandzeroare movedo the eft,close o theorigin.>The idealcompensatorrives he steady-stateffor o zero,a lagcompensatoritha pole hat s notat the originwill mprovehestaticeror constant y a factotequalo z.tp".>Therealsowill be a minimal ffectupon he kansient esponsefthepole-zeroairofthecompensators placed lose o theorigin.


8/20

elearning.utm.my

Example:compensate he sysiemwhose oot ocu6 s shownbelow toimprovehe steady-staterrorby a factorof 10 f the system s

; =4.174 .+ K =164.6

6 =I=ss23"20

, =--L=--L=n.163"" l+(" l+8.23

CompensatedystemCbmpcattto'

" = -L= o.oror' - l+l{r6 =L9o=l-oo1o8=91.590.0108

p" K,, 4.23Arbitrarilyelecting

P' =0 01

- z.= I t . I3p. E0.11I

hf l)( ,+2)(J_10)


9/20

elearning.utm.my

14.6nFi.d

il-rti:-rnitPredidedcharacteristics fand ag-compensatedystems

n"Step esponsesf uncompensated ..,lar-comoensatedystems

lmprovingransientesponseviacascadeompensation'This s to design responsehat hasadegjlablc-pc@9o19!9l9hoqlandshorterettlinqime han heuncompensatedystem..Acompensatoras he ransferunctions ollows

G,(s)=51'"".This function,he sum of a differentiatornd a puregain, s calledanidealde vative,ot elD391@l!9!..This ypeof compensationequires n activenetworkor ts rcalizationlnstead,t approximatesifferentiationith a passive etwork y addinga zefoanda moredistant ole o the oMard-pathransferunction.


10/20

elearning.utm.my

Usingdealderivativeb. ompensatorero at -2;

c. compnsatoreroat -3;d. compensatorero at - 4

10


11/20

elearninS.utm.my

Predicted haracteristicsor the systemsCompd$adonc Cohpematioid

21.722.34125.13

2.!126121

(J:. i )f* ,ri" + t51.251.5113t0t5

{'+ l-,(, rX" i)

t513

0 1074 !6X

fl

ffi im:mated svstemnddealerivatlve

1.000.750.5n0.25

t, 1.0Time scon.ls)

11


12/20

elearning.utm.my

Example 2: Given the followingsysiem, design an idealcompensatoro yielda 16010vershoot, iih a thrcefold eductionn

' 0.504 59 14"-t.205 : 2.064

Root ocus oruncompensated

Settlingime: r,=--L=:=3.32(a, |.205

Thenewsettlingime s settobe1.107 6 = -= -- =3.613r, 1.107,J, / 3.613tan(59.74")6.193

Evaluatinghe locationof the compensating

d= 3.006, (rL 4+120.26=(21+l)180 180"

3#=t-(rso' es ') =

\2


13/20

elearning.utm.my

compennted system lJncompensatedndcompensatedsystem

:,"

,A^. .,fffi] UncomnensatedndcompensatedystemWT charactenstlcslrnconp.n.n d Slmlrdd Cm!.nr.t d Slmul.don

PtudrndwMlnid0rl\miinn lnls 1.205 i2 061

./1

1522

X(r 1006,.(r +-lX.Q )Ln]] ljd.r9a

t.l.ll 16 .3t6 t .1071.7 i).507

lold-zco

13


14/20

elearning.utm.my

lAo PoController. The deal derivativecompensator sd o improve h translentresponses mplementedwith a proportional-plus-drivativePD)controller.. The eadcompensator5 a passiv etworkused o overcomethe disadvantagesf ideal differentiationandstill retain heability to improve he tranlient response.

C.(r)=K(s+2.)=K,r+KrK

Example: Designhreeead ompensatorsor he ollowingsystemhatwill educehesettlingimebya actor f2 whilemaintaining0% vershoot.omparehesystem haracteristicsbetweenhe hreedesigns.nu. ^_" ti", f-;---l

Lead ompensator esign,showing valuation funcompensatedndcompensatedomlnantpoles

14


15/20

elearning.utm.my

Nole;This lgure s notd|dih to scrle.r"planepictureu5edo calculatethe location f the comPensatorpole

CompeBdted system

Comparisonf leadcompensationdesignsT$r.9,a ciaDdisolb.ddFNrtridatr*ttEt pL9a

Ur..ed.bd CaD.rld.ir

';

:i,;r){r.6i fi;-ii;;il; 3.ir ,

.sinruidbll!es:!a'.M'


16/20

elearning.utm.my

@ :mm*tedsvstemandead

l..lt .2l {)0.8

0.20

Llncornpeosr.d

A.fl&S' lmprovinEteady-StaterrorandTransient esponse98lFfl .Tne esignan se ither ctiverpassrveompensaors.l f we design n activePDcontrollerollowed y an Pl controller,the resulting ompensatofs calleda proportionalplus-integral-plus-derivativePlD)controller.l f we firstdesign passiveeadcompensatornd hendesignpassiveag compensator,he resulting ompensators calledalag-leadompensatot*. l , , ' *

t ' ,*

t t ]' [ , ( , K' )


17/20

elearning.utm.my

f& Thedesign echnique onsistsofthe following\qry steps;1. Evaluate hepClblo?re9ftie-lr@llp9rletedlystem todetermine owmuch mpaovementn ransientesponsesrequrrcc.2. Designhe PDcontrollero meet he ransientesponsespecifications.Thedesigancludeshezero ocation ndthe oopgain.)3. Simulatehesystemo be sureall requirementsavebeenmet.4. Redesignfthe simulationhows hat equirementsavenotDeenmel5. Designhe Pl controlleroyield herequired teady-stateerrot6. Determinehegain6, r '&, and4.7. Simulatehesystemo be sureall requirementsavebeenmet.8. Redsignf the simulationhows hat equirementsavenotbeenmet.

Example4: Given he system shown n the following igure,designa PID controller o that the systemcan operatewith apeak ime hat s two{hirds hatof the uncompensatedystern t200lovershoot ndwithzerosteady-stateffor or a step nputrr,) *o r('r T--_-t-(iiTi---l .t,)-.*qt l (' .,r rr f o) -t-*l -

Uncompensatedeedback ontrol ystem

- - 5.415t 10.57

- Tp= 0.297secRoot ocus or he uncompensatedystem

77


18/20

elearning.utm.my

PD-compensaled

t8.31'

a,= = =t) , t / sec" T, (2',3\(0.291\1 o =---- !+=-913tan 17. l3 'From ncompensatedystem:The um fangles -198.37'= 198.37'-180'=8.37'

X* ClosedloopoleCalculatinghe PD ompensatorero

Thus, he PD controllers

15.87- ,

-=mn la.J/z" -8.13- ze

=55 92G-(s) = s+55.92

PD-compensatedystem Root ocus or P|D-compensated

('rr (s) = - ,'

18


19/20

elearning.utm.my

l. l

, , ; 0.11 t,l

Step esponsesor uncompensated,PD-compensated,ndPID-compensatedystems

T6bl.1U7 A Smmsry ot dlo ChaFqt rltlica ol PfiGtLedand Phs$&g Colnp.nsqtion lietwork.

'1.r-! x

i I irsr \ nr $r( n*il(i.,ri nL ,!i trr ix ll 'l ! \ l! '!re1. r r(.n!N tr$l ili.lLlft)n.dI l'trrNl*itsutrh|'nrli,Ia: l'.rr:r.1:\;i { I ' id'e Lt1(f..-r. lhr,.{t 4hrn.'t!r!'*I R?!N *tJd,!r l ,I '/,r.1 !trtrL lM,cr*i-ndn,.!tr r,{l d!\I {:! trrl.s! !*r. , .rrd! .ltrrnrl|&na I'i /rL r. trokr \r | !r h! trani$ | ' jr!*. ,! d.',iiI \ ihrr nin. l ( .1n!rr l rkrr l r . : r

nifiiln$!r fl 'r{ lx. L, \i!L.l 'n.'!-,lxtr !$! i rs l ' r r ) ,ntr .$L,+\[ .( !anq$a!tr r1i t .n ' "t i ) r ! J, l rsrr i , r r \I llcj rtr\- \* ! t l!3ia{nlrll| !\rt)$\\:,$r' J.!d!*r r.,,I Rdr{.n.rn1 rk.,r,!,I $.\r rntr D rnr*i ,rt,ri-1 ri\,.{! r.l$*c rlr-.i.e!!r. !

t . \1r.n j . ! r . , . rdr. .r ! . r ! r i l i . r t1. \a.r !} l\d Ir;n!$.r r .- :ri'6 *l{er.rltr{; rt , r r '.|+i '!r hlN {I|n'

19


20/20

elearning.utm.my

oFllon.l Anplili4ckut ldcmlardoEc.d-ffi

G.'** i , . , , ',

- nlnd&c']r ,..," ^, ,,,t 4 r"{i,r;::'

Design via Root Locus -Control engineering

Documents

Transcript of Design via Root Locus -Control engineering