VICTORIA UNIVERSITY Controller Controller Design Using Root locus and Robustness obustness test test NEE3201 – Introduction to Control Systems
Submitted Via: VU Collaborate Student ID: 4462542 Submitted to: S RUI !I !ab Su"er#i$or: S RUI !I Date Submitted: 2%&'%&'6
Table o( Content$
1.0 b!ecti"e... b!ecti"e........ ........... ............ ............ ........... ........... ............ ............ ........... ........... ............ ................ ..........3 3 2.0 Introducti Introduction.... on.......... ............ ............ ............ ............ ............ ............ ............ ............ ........... ........... ......... ... 3 3.0 Design S#eci$catio S#eci$cations%.... ns%.......... ........... ........... ............ ............ .............................3 .......................3 &.0 'rocedur 'rocedure...... e............ ............ ............ ............ ............ ............ ............ ............ ............ .....................& ...............& (.0 )nalysis )nalysis and *eri$cation eri$cation%...... %............ ................. .....................................10 ..........................10 (.1 E+#lanatio E+#lanation%.... n%.......... ........... ........... ............ ............ ........... ........... ............ ............ ........... ...........10 ......10 (.2 Controller Controller and Com#ensator Com#ensator%...... %............ ........... ................... ........................10 ..........10 (.3 Need ,or 'ole 'ole -ero -ero Cancellati Cancellation%.... on%......................... ................................10 ...........10 .0 Controller Controller design...... design............ ............ ........... ........... ............ ............ ........... .....................11 ................11 .1 Desired Desired #oint coordinat coordinates es calculation calculation%................... %...........................11 ........11 .2 )ngle o, De$ciency%........................................................12 /.0 ead ead ag Com#ens Com#ensato ator% r%... ...... ...... ........ .......... .......... ........... ........... .......... .......... .......... ......12 .12 /.1 ead Com#ensator Com#ensator%..... %........... ........... ........... ............ ............ ............ ............ .............. .......... ..12 12 /.2 agnit agnitude ude Condi Conditio tion n to $nd $nd te "alue "alue o, o, %......... %.............. ..........13 .....13 /.3 ag Com#ensatio Com#ensation%..... n%.......... ........... ............ ............ ........... ........... .......................1& .................1& 4.0 Robustness obustness test%..... test%........... ............ ............ ........... ........... ............ ............ ............ ............ ........... .....1( 1( 5.0 Conclu Concludin ding g Discus Discussio sion... n...... ...... ........ .......... .......... ........... ........... .......... .......... .......... ........1/ ...1/ 10.0Re,erences%......................................................................14
Table o( )i*ure$
6igure 1 7loc8 Diagram o, te 6ield Controlled Ser"o otor Using Simulin8...........................................................................& 6igure 2 Root locus #lot structure..............................................( 6igure 3 te gra#ical re#resentation o, root9locus design o, closed9loo# system....................................................................( 6igure & Control : Estimation ;ool anager< Ste# Res#onse. .... 6igure ( So=ing Cange in Root ocus.................................... 6igure Control and Estimation Design< =it Integrator res#onse #lot o, Root ocus......................................................./ 6igure / te cange o, ste# res#onse by canging te gain constant in >c?s@ ,rom 1 to &....................................................../ 6igure 4 Closed loo# "ie=er and SIS design ,or estimation on dam#ing ,actor and closed loo# #oles.......................................4 6igure 5 'oles and Aeros o, ead Com#ensator ,or )ngle contribution...............................................................................5 6igure 10 'oles and Aeros o, ag Com#ensator and cange in root locus..................................................................................5 6igure 11 Closed oo# 'ole *ie=er ,or 6inali-ed Design...........10 6igure 12 System ut#ut Res#onse =it Desired S#eci$cations ................................................................................................ 10 6igure 13 Ideal 'oles -eros ,rom gi"en s#eci$ed 'ea8 "ersoot : Settling ;ime........................................................................11 6igure 1& 'oles and Aeros s9#lane angle de$ciency calculations ................................................................................................ 12 6igure 1( )ngle calculations ,or lead com#ensator 'ole #lacement...............................................................................13 6igure 1 'oles o, ead Com#ensator......................................13
6igure 1/ 7loc8 Diagram o, te 6C ser"o motor =it te Controller design.....................................................................1& 6igure 14 ut#ut Res#onse o, te 6inal controller design =it te 'lant..................................................................................1& 6igure 15 Robust Root ocus....................................................1 6igure 20 Robust Root ocus Aoomed......................................1 6igure 21 ;ransient Res#onses.................................................1
'+% Ob,e-ti#e ;e main #ur#ose o, tis laboratory e+#eriment is to use root9locus metod and design a unity ,eedbac8 control system using atab. In order to meet certain closed9loo# system< time9domain #er,ormance s#eci$cations and to analyse te bea"iour o, te obtained system in te #resence o, uncertainty. ;e goals are suc tat% •
•
•
•
Use root9locus metod and design a unity ,eedbac8 control system using atab. Design a controller to closed loo# #er,ormance s#eci$cations including transient #er,ormance and steady error. Use te angle to aid in te #lacement o, a com#ensator #oles and -eros. Robust analysis in te #resence o, (B "ariation o, te system #arameter "alues.
2+% Introdu-tion Root9ocus metod is "ery im#ortant metod ,or designing closed9loo# control systems. ;ime9domain #er,ormance s#eci$cations are gra#ically ma##ed in #ole location regions in te com#le+ #lane< =en all analysis and design considerations are based on te =ell9acce#ted dominating #oles conce#t. >ra#ically it is related to te #ositions o, closed9loo# #oles to te #ositions o, o#en9loo# #oles and o#en9loo# -eros =en a tunable #arameter "aries. 7y ta8ing ad"antage o, te conce#t o, dominant closed loo#9#oles< root locus metod allo=s transient res#onse o, a closed9loo# system to be sa#ed gra#ically and to design an a##ro#riate com#ensator in te s9#lane. ;ere,ore< designing a controller< ,or meeting certain #er,ormance s#eci$cations< by using te tools and te #ro#erties o, te com#le+ #lane< te construction o, te so called gamma9regions =idens te set o, tools in te #lane< by gi"ing ne= so called alternati"es in studying #lant #arametric uncertainties in designing controllers< classi$ed in a category tagged by te notion o, robustness. 1F. In addition );)7 scri#t $le is used ,or te robustness analysis. ;e controller #oles and -eros a"e been manually calculated and are noted in tis re#ort.
.+% De$i*n S"e-i/-ation$: In order to meet te design s#eci$cations ,or te gi"en $eld9controlled ser"omotor and its load
G P ( s ) =
4500
( s + 2 ) ( s + 30 ) te ,ollo=ing #arameters as
to be acie"ed. G P ( s ) =
4500
( s + 2 ) ( s + 30 )
Suc as te ,ollo=ing s#eci$cation to be satis$ed% 1. ;e ste# res#onse o, te re,erence in#ut as -ero steady9state error. 2. Com#letely re!ects ste# disturbances at steady state. 3. ;e "elocity error constant ?or coeGcient@ is
K v ≥
&0.
&. ;e ste# res#onse o, te re,erence in#ut as a &0 ± (B #ea8 o"ersoot. (. ;e ste# res#onse o, te re,erence in#ut as a 2B settling time o, not more tan 3H& o, a second. ;e reduced 7loc8 diagram ,or te #lant is gi"en as%
)i*ure ' 0lo-1 Dia*ram o( te )ield Controlled Ser#o otor U$in* Simulin1
4+% 3ro-edure In command =indo= o, atab< te trans,er ,unction o, te #lant can be de$ned. atab uses sysgt,?numg
sysgt,?numg
4500 2
2
s + 32 s + 60 s
Ne+t ste# is to o#en te C)D tool< ,or suc #ur#ose
)i*ure 2 Root lo-u$ "lot $tru-ture
e,t clic8 on te Control and Estimation ;ools anager in te tas8 bar< in te Control and Estimation tools anager Mindo= tat a##eared< =e selected )rcitecture at te to# o, te =indo= =it a le,t clic8< te le,t clic8 on System Data to re#lace te "alue o, 1 in >< by sysg ,ollo=s by clic8ing . ;e gra#ical re#resentation o, root9locus is gi"en on 6igure 2.
)i*ure . te *ra"i-al re"re$entation o( rootlo-u$ de$i*n o( -lo$ed loo" $$tem
7y selecting o, )nalysis 'lots at te to# o, te Control and Estimation ;ool anager Mindo= and cec8 te bo+ ,or Closed oo# r to y ten sim#ly #ull do=n menu to select S;E' ,or '; 1. ;is is te ste# res#onse o, te closed9 loo# system de$ned by te system data in te s#eci$ed arcitecture. ;e #lot is gi"en on 6igure 3.
)i*ure 4 Control E$timation Tool ana*er7 Ste" Re$"on$e
Rigt clic8 in te #lot area o, te ;I *ie=er =indo= and ten =e selected >rid by a le,t clic8 in te menu tat a##ears to cec8 tat te steady state error is not -ero< because o, te settling time. e,t clic8 on >ra#ic ;unning at te to# o, te Control )nd Estimation ;ools anager =indo=< ten le,t clic8 on te 'lot 1 and ten =e used te rigt most #ull do=n menu to select root– locus to able to see o= a root locus a##ears in te SIS Design Mindo=. ;is is te root9locus o, a unity ,eedbac8 control system< because ?s@ 1< =it #lant trans,er ,unction >#?s@ and a #ro#ortional controller > c?s@ in te ,or=ard #at. ;ere is a small #in8 sOuare located on te root locus. ;at is te #osition o, te closed loo#9#ole =en 1. Me "eri,y tese by selecting )rcitecture ,ollo=s by te System data. Mit te ;I *ie=er and SIS Design< =ic are gi"en on 6igure & belo=< =it le,t clic8 on te #in8 area and drag it =it a small and to a ne= #osition along te root9locus =e =ere able to obser"e te corres#onding cange o, te closed9loo# ste# res#onse in te ;I *ie=er.
)i*ure 5 So8in* Can*e in Root !o-u$
)t te to# o, te Control and Estimation ;ools anager =indo=< ten le,t clic8 on Com#ensator Editor< gi"en on 6igure ( belo=. ;en rigt clic8 on te Dynamics area to add #oles and -eros to te controller. e,t clic8 on te Real #ole o#tion to cange te #osition o, te #ole to be added ,rom 91 to 0< enter 8ey must be #ressed a,ter to ePect tis cange. ;en te controller as to be canged as =ell< ,rom a #ro#ortional to integral controller. )lso te "alue o, te constant term in > c?s@ to 1 as to cange. ;e desired 'ercentage "ersoot can be acie"ed at a "alue o, 80.(33
)i*ure 6 Control and E$timation De$i*n7 8it Inte*rator re$"on$e "lot o( Root !o-u$
Men te SIS Design as been o#ened =e noticed tat te system is no= a tird order system instead o, a second order system. )lso< =en te ;I *ie=er as been o#ened to obser"e te closed9loo# ste# res#onse< it is noted tat te reOuired -ero steady state error as been acie"ed. 7y canging te "alue o, te gain constant in > c?s@ ,rom 1 to 0.(3 =e noticed tat te closed9loo# ste# res#onse in te ;I *ie=er as been s#ed u#< gi"en in 6igure belo=.
)i*ure 9 te -an*e o( $te" re$"on$e b -an*in* te *ain -on$tant in -;$< (rom ' to 4
;e #ea8 o"ersoot in te closed9loo# ste# res#onse as been notes tat is about &0B and
te
± 2 settling time is about &.2 sec< te dam#ing
,actor is 0.24 and te closed loo# #oles are at s90.435 considered idle closed loo# #oles are no longer
± !2.5. te
s =−5.33 ± j 18.3 < because
te system is no= 3 rd order system. )ll results are gi"en in 6igure /.
)i*ure = Clo$ed loo" #ie8er and SISO de$i*n (or e$timation on dam"in* (a-tor and -lo$ed loo" "ole$
)t te to# o, te Control and Estimation ;ools anager =indo= le,t clic8 on Com#ensator Editor te record o, trans,er ,unction and te controller as been made =ic is gi"en on 6igure 4. ;e gain "alue at tis stage% k v =lim s s →0
∴ k v
4500 k
s ( s + 2 ) ( s + 30 )
= 40 ;→
=
4500 k 60
4500 k 4500 k 2400 = =0.533 → k = 40 × 60 2400 4500
)t te to# o, te Control and Estimation ;ools anager =indo= le,t clic8 on te Com#ensator Editor. 7y Rigt clic8ing on te Dynamic area to add #oles and -eros to te controller > C?s@< te le,t clic8 on te Real 'ole o#tion to cange te #osition o, te com#ensation #ole to be added ,rom 91 to 925.&2.
;e #re"ious ste# as been re#eated< but le,t clic8s te real -ero o#tion to cange te #osition o, te com#ensation -ero to be added ,rom 91 to 92 so=n on 6igure 4. ;e SIS Mindo= as been o#en to obser"e te canges in te root9locus.
)i*ure > 3ole$ and ?ero$ o( !ead Com"en$ator (or An*le -ontribution
)gain by adding #oles and -eros ,or te lag com#ensator can meet te desired reOuirements ,or "elocity error constant and settling time< suc tat a #ole at 0.01 is #laced and -ero at 0.015 is #laced as so=n in $gures belo=< te desired '.&0 ± 5 and settling time 0.( is acie"ed by mo"ing te #in8 sOuare in root locus gra#< te gain "alue is 3.< te gi"en "alue ,or gain in te control and estimation tool manager is in time constant ,orm< o=e"er te calculated idle case "alue ,or is 3.45
)i*ure '% 3ole$ and ?ero$ o( !a* Com"en$ator and -an*e in root lo-u$
;e closed loo# #oles a"e been canged to
s =−5.01 ± j 19. because o,
adding te com#ensators< i.e. te root locus as been si,ted. ;e dam#ing ,actor also been reduced to 0.2(( because ,or iger order system te dam#ing ,actordoes not ,ollo= te criteria ,or te 'ea8 o"ersoot.
)i*ure '' Clo$ed !oo" 3ole Vie8er (or )inali@ed De$i*n
S$tem re$"on$e% ;e desired "alue o, 8 is 3. te ste# res#onse is gi"en in $gure 11 belo=.
)i*ure '2 S$tem Out"ut Re$"on$e 8it De$ired S"e-i/-ation$
5+% Anal$i$ and Veri/-ation: 5+'
E"lanation:
Using te root locus te controller design is in continuous domain and is based on te a##ro+imation tat te closed loo# system as a com#le+ con!ugate #ole #air =ic dominates te system bea"iour.
5+2
Controller and Com"en$ator:
6rom #re"ious studies and lab e+#eriments diPerent controllers suc as 'I etc is generally used to im#ro"e te steady state #er,ormance =ere as 'D im#ro"e te relati"e stability or transient res#onse. Similarly a lead com#ensator im#ro"es te dynamic #er,ormance and lag com#ensator im#ro"es te steady state res#onse 2F.
5+.
Need (or 3ole @ero Can-ellation:
Usually in designing controllers undesired #oles or -eros o, a #lant trans,er ,unction is cancelled by -eros and #oles o, controller< but te #ole9-ero
cancellation sceme does not al=ays #ro"ide satis,actory solution. Mereas ne= #oles and -eros also as some ad"antageous locations. 6or e+am#le i, tere is an undesired #ole near
jω a+is< ine+act cancellation<
=ic is almost ine"itable in #ractice< because it may lead to a marginally stable or unstable closed loo# system.
6+% Controller de$i*n ;e e+am#le considered =it te ,ollo=ing #arameters% K a=1 ; K =50 ;
K t =1.5
T e=1 / 30 ; T m=1 / 2 ;
;e goal is to design a controller gi"ing closed9loo# system #er,ormance s#eci$ed by o"ersoot Q &0 ± 5 B and settling9time
3
± 2 B<.
4
6rom te #arameter combination determines te ,ollo=ing indirect #er,ormance s#eci$cations in te com#le+ #lane. i+e+ Dam"in* ratio B %+29> (or 3+O4% 7
;ere,ore< te idle root locus lies at #oint
ω n=18.3
7B
ω n=5.33,
s =−5.33 ± j 18.3
6+' De$ired "oint -oordinate$ -al-ulation: )s te desired s#eci$cation as not met yet< tere,ore< com#ensation is needed ,or te acie"ement o, te gi"en s#eci$cations. Desired "alue< is not #ossible by only altering te E"ans9coeGcient ?te ' controller gain@ in te rigt direction. It is necessary to design a controller tat alters te root locus< in order to ensure tat its brances =ill contain te #oint s#eci,ying te desired #er,ormance 1F. ) #ossible solution may be ,ound by integration. ;e additional integration =ill ensure -ero steady9state error to ste# in#ut. ;e #ole at origin o, te controllers =ill alter te current root locus< im#ro"e stability margins and limit te system s#eed o, res#onse.
)i*ure '. Ideal 3ole$ @ero$ (rom *i#en $"e-i/ed 3ea1 O#er$oot and Settlin* Time
6+2 An*le o( De/-ien-: ;e net angle contribution can be calculated using trigonometry ,rom te ,ollo=ing $gure 13.
)i*ure '4 3ole$ and ?ero$ $"lane an*le de/-ien- -al-ulation$ −1
θ1=¿ tan
18.3 = 36.6 ° , 30−5.330
−1
θ2=180 ° − tan
θ3=180 ° − tan
18.3 = 100.3 ° 5.33− 2
−1 18.3
5.33
=106.3 °
)ngle o, de$ciency
θd =−180 ° −(−36.6 ° −100.3 ° −106.3 ° )
θd =−180 ° + 243.2 ° =63.2 °
9+% !ead !a* Com"en$ator: s + z n
;e com#ensator using root locus design can be gi"en as
9+'
s + p n J
!ead Com"en$ator:
6or te additional angle contribution a lead com#ensator is reOuired< =ic =ill add #ole and -ero to te system< te location o, tese #ole and -ero as an im#ortant ePect on te system< tere,ore te location ,or tese #ole and -ero is "ery im#ortant. )s te angle o, de$ciency is calculated tere,ore te location o, #ole and -ero can be ,ound. ;e -ero caracteri-es te s#eed o, res#onse o, te system. Its "alue is determined by te #er,ormance s#eci,ying #oint< =ic te root locus brances must contain. o=e"er in tis case #lacing te #ole at 92 =ill cancel te #ole and te root locus =ill si,t. ;ere,ore< -ero at 92 is cosen. No= te distance o, te #ole location is reOuired to determine ,rom 'oint '. ;e calculated< angle o, de$ciency is 3.2 ° tere,oreJ by sim#le geometry calculation o, angles te desired location can be ,ound. Consider te gi"en $gure 1& belo= ,or te desired #ole location calculation.
)i*ure '5 An*le -al-ulation$ (or lead -om"en$ator 3ole "la-ement
θ p=100.3 ° − 63.2 ° =37.1 °
ocation o, 'ole ,rom #oint '
l p1=
18.3 tan ( 37.1 ° )
=24.2
;ere,ore te location o, real a+is ,rom #oint '< (.33 2&.2 (.33 25.(
)i*ure '6 3ole$ o( !ead Com"en$ator
;ere,ore
( )=
K s
( ) s ( s + 29.5 ) k s + 2
9+2 a*nitude Condition to /nd te #alue o( : s =−5.33 ± 18.3 at te current #ole and -ero location<
E"aluate
¿ G ( s ) K ( s )∨¿ =− ¿ s
| ( + )( +
|
4500 k ( s + 2 )
s s
2
|(+
5.33 + 18.3
s
29.5
=1
) ( s + 30 ) s=−
5.33+ j 18.3
|
4500 k
s s 29.5 ) ( s + 30 )
||−
=1
=1
s=−−5.33 + j 18.3
4500 k
|=
||−5.33 + j 18.3 + 29.5||−5.33 + j 18.4 + 30|
5.33 + j 18.3
4500 k =1 19.06∗30.21∗31.4 4500 k =16605.72 → k =3.925
K ( s ) =
;ere,ore<
k v =lim s s →0
( 3.696 × 4500 ) ( s +2 ) s ( s + 29.5 )
17750.7 ( s + 2 )
= 17750.7 =20.057 s ( s + 2 ) ( s + 29.5 ) ( s + 30 ) 29.5∗30
1
9+. ;e
!a* Com"en$ation: reOuired
"elocity
com#ensation te
error
k v =20.057
constant
is
k v =¿
&0<
=it
te
lead
< tere,ore a lag com#ensator can be
introduced to #ut te #oles in suc a =ay tat te reOuired transient s#eci$cations and "elocity error constant s#eci$cation can be acie"ed. k req!red= 40 ; k "rrent =20.057
;ere,ore< te #oles ,or te lag com#ensator can be #laced as te ratio o, te
k v
k v ;
reOuired and current z 2 p2
=
k v
req!red
k v
=
i.e. 40 20.057
=1.9943 J
"rrent
z 2=1.9943∗ p2
)s 'ole near origin is necessary at tis stage 3F< tere,ore< etLs coose a #ole at 0.01< tere,ore<
z 2= 1.9943∗0.01=0.019943 ≅0.02
6inalised Controller bloc8 diagram =it te 'lant and ste# res#onse is gi"en in $gure 1/ and $gure 14 belo=.
)i*ure '9 0lo-1 Dia*ram o( te )ield Controlled $er#o motor 8it te Controller de$i*n
)i*ure '= Out"ut Re$"on$e o( te )inal -ontroller de$i*n 8it te 3lant
=+% Robu$tne$$ te$t: ;e notion o, robustness means insensiti"ity o, te system in certain degree to "ariations in te dynamic #arameter "alues. ;e s#eci$cation o, a 9region 3F< =ose boundary is determined by eOuation
s =−# ωn ± j ωn
√ 1−# 2 It sets a desired dynamic bea"iour o, te control system< and guarantees robustness< in te case =en all dominating #oles< re#ositioned in te #lane due to #lant #arameter uncertainties< lie in it% 'lant #arameter uncertainties ,orm an uncertainty region in te com#le+ #lane. ;e analysis o, te relati"e location and o"erla##ing o, tose regions enables te inter#retation o, te robustness #ro#erties o, control systems. In te gi"en sero mecanism motor< te uncertainty in te control system is due to (B "ariations o, te system #arameters< te table o, tese "ariates "alues is gi"en belo= 1F. 'arameters 1.00 K a
k
k t
t e
t m
1
(0
1.(
30
T 3.
(B1.0 ( 9(B0.5( (B(2. ( 9(B&/.( (B1.( /( 9 (B1.&2( (B0.0 353 9 (B0.02/ 3 (B0.( 2( 9 (B0.&/( (B3./
'ea8 "ersoot
Settling ;ime
Rise ;ime
&0
0.//(
0.0(3
&2.1
0./(
0.0/3
3/.5
0./4
0.0/4
&2.1
0./
0.0/25
3/.5
0./4
0.0/4
&2.1
0./(
0.0/25
3/.5
0./4
0.0/4
&3.
0./3
0.0//&
3(.5
0./01
0.0/3&
34.
0.//
0.0//(
&1.2
0./2
0.0/25
&2.1
0./(
0.0/25
4 9(B3.&2
3/.5
0./4
0.0/4
;able 1< 'arameter "ariation o, (B
;e region o, desired #er,ormance gamma is s#eci$ed by u##er and lo=er bounds on te admissible de"iations ,rom te nominal system #er,ormance in $gure 1 belo=< ;e o=er and u##er bounds on te dam#ing ratio e"aluated to be Vn
→ 0.2((J 0.2/5F and te s#eed o, res#onse is → &.&4J (.33F. )s In te #resence o, (B "ariation
o, te system #arameter "alues< te closed9loo# caracteristic eOuation roots dominating te system transient res#onse remain in te #er,ormance s#eci,ying 9region and te o"erall system #ossesses robust #ro#erties in tis sense.
)i*ure '> Robu$t Root !o-u$
)s =e 8no= a system is caracteri-ed by its #oles and -eros in te sense tat tey allo= reconstruction o, te root locus. ;e #oles and -eros are re#resented gra#ically by #lotting teir locations on te com#le+ s9#lane< =ose a+es re#resent te real and imaginary #arts o, te com#le+ "ariable s. ;e location o, te #oles and -eros #ro"ide Oualitati"e insigts into te res#onse caracteristic eOuation. ;e #oles are re#resented by cross K + K and -eros by K o L< #ro"ided in 6igure 1/ =it te robust cange o, s #osition on root locus.
)i*ure 2% Robu$t Root !o-u$ ?oomed
In te time domain it is clearly so=ed tat transient res#onses are s#eci$ed by admissible o"ersoot and settling9time ,or te entire range o, #arameter de"iations in $gure 14.
)i*ure 2' Tran$ient Re$"on$e$
>+% Con-ludin* Di$-u$$ion ;is laboratory e+#eriment =as "ery use,ul in terms o, understanding te root9locus metod< o= to use atlab #rom#t to see root9locus diagram to record te #oles and -eros. )lso< by canging te location o, #oles and -eros =e =ere able to obser"e te ste# res#onse o, te closed loo# system< steady state error< settling time and #ea8 o"ersot more tan &0B and ad!usted #ea8 o"ersoot o, &0 ± 5 B =it
±2
3H& sec settling
time. )lso =e ad a closed9loo# system =it com#le+ #oles to obser"e te ste# res#onse o, te closed –loo# system. ;e com#le+ #lane can be used =en e"aluating te dynamic bea"iour o, control systems in te #resence o, #arametric uncertainties as an alternati"e to oter common a##roaces. It is relati"ely easy to ,orm an uncertainty region< so=ing te combinations o, te #arameter "alues =ic cause canges in system caracteristics. Root loci< and transient res#onse #lotted ,or diPerent dynamic #arameter as E"ans9gain so= "alues o, te res#ecti"e
#arameter< causing "ery lo= loss o, stability. 6or te robustness test analysis #ur#ose< atab scri#t is used ,or entering diPerent #arameters to get te transient and root locus robustness outcome 1F. ter tan tat te system =ill tend to be stable =it additional -eros. o=e"er In many design e+ercises< -eros can be introduced to attract closed9loo# #oles and alter te root locus location. It is also "ery use,ul to a##lied stable #ole9-ero cancellation ,or im#ro"ing system #er,ormance . ;ere,ore in conclusion< root locus metod< in com#arison =it oter design metods< considerably ,acilitates te design o, a controller guaranteeing desired closed9loo# system dynamic bea"iour< since it oPers an adeOuate inter#retation o, te time9domain #er,ormance.
'%+%Re(eren-e$: 1 D. osto"< *. arlo"a and ). ;odoro"< Robust Root ocus )##lication In design F and )nalysis< ;ecnical Uni"ersity< So$a< 2004. 2 7. Robert and D. Ricard C< Robust Control Systems< in Modern Control F System< Ne= Wersey< 'rintice all<< 2011< ##. 532915(. 3 N. Norman S< Design *ia Root ocus< in Control Systems Engineerig< F 'omona< Se"ent Edition< 201(< ##. &&59&4(. & R. . 7. Ricard C. Dor,< odern Control System< Ne= Wersey% 'rintice all< F 2011. ( N. S. Nise< Control System Engineering< 'amona% Won Miley : Son< 2004. F E. Cee"er< inear 'ysical System )nalysis< 201(. nlineF. )"ailable% F tt#%HHl#sa.s=artmore.eduHRootXocusHRocusE+am#les.tml. )ccessed 10 10 201F.
