The transitionbetweenZones1 and2 is the frequencylimit
of a controlsystem.
?aaar
r ^ o tio
Z O NQ
zone I
?
zon?
3
1
(0db)
tinit
freq
ErrorRatio(ER)= ratioof closed-loop
errormagnitude
to
open-loop
errormagnitude
ER=
(1 + G(s).H(s)).(1- c(s).H(s))
Magnitude
of thecomplexnumbergivenby theabove
et438a1O.MCDI
2.) the
Find:1.)the openlooptransferfunction,
Example:
closedlooptransferfunction,3.)the errorratioof a control
systemwherethe forwardgainG(s)is:
G(s) '=
21. 8
1 * 0 .3 7 9 .s+ 0 .0 0 6 3 .s 2
The feedbacksystemgain is:
H(s) =
.356
1 + 0 .4 7 8 .s
valueof the openandclosedlooptransfer
4.) Compute
whenrrl= Q.1, 1, 10and100.
functions
the valueof the errorratiowhenrrr= 0.1
5.)Compute
1, 10and100.
6.) UseMatlabto plotthe openandclosedlooptransfer
functions
on the sameBodeplotaxis.
/ , ) o l a Eru L ooP T€dnrsFEI
I
F u nC/;l/ oN
L
o.3S6
Zr. I
G c" ) /lG )",+ (o.37 9s1 o,o o6l FLI+ o.q?8*t
ft)* l f r p l g d.enao.tN .n-ion
'J
= /t- o,3?9
.s+o.tB//a**o.cnGgsls.orxIo!
s+o.q7e
esst)(+o'4?8s
+o,oo
Q t o.3?9S
?.?6oO
f + ogSZ s To, t8?4lSz+ o'ooto/{s3
et438a1O.MCD9
z4nls
Example(cont.)
Z) C toseD l-ooF TRn rr-FR FaNcl ypxl
?.?aoe
, G(s)HCs)
_
1+o.gs?s*0,tffi
I tG6s)H(s)
l+
7, ?aog
I + 0 rs?s +o,l&)46s2+
Obffi
G(s)H(s\
I * G(s)HG)
? ?<oe
+ 0,8S?sf O, t8'746St+o,oo3of
S:l"Z,ZeoE
c G ) H(s )
I t G(s)HG)
3, ) A R Q o A R n r p
I
/-\ r^ \
rtcT-- ^t <, \\ ' / ' _ t+ G(s)H(s)
I
l t
ER(s)=l,
16
sr,rBsTlrrrTE
/.ru G(s)H(.)
/
I
l
l
ffi
l
l' =
I
F-Qsr.-..r
ilBovr
l/
.
E(6)=l
Qlo63/
-+o, tB:<Ast+o,
Q*o'8s?s
l'
(t+o,es^ts
to,rg?a.tto"oosots3)'_
I
EPG)J,
,=t)t- (z z<ode t
+o,,n ecs'ro.od3o
ft i *",Bs?s
o,oosorss)a
rs?_s
*?,rrz+as'+
+?
__(l
e(*)-i
f
h'\\
l
ll o*
/(;;'qs?s4o'r
et438a1O.MCD 10
lS
Exampfe
(cont.)
4') computethevafr"r-:f
openandctosedrooptransfer
functions
at or=0.1,1,10,11"_
lOOiSJlsec
o P E A / Lds P
7.7608
G ( ar ) . H ( a r=)
1 + 0. 8S 7 .j .rrr
+ O .1 g 7 S(j. . r)2 + O .OO3 (0.
j .r)3
v\vrt'lr-v/-
f* r z , - , ]
l0 lt s- - i
Ld
ol
=eH(w)
l+o,s<?X-
q
Su6" " lrl,^# e ir
I
I
d
q-^dsr"nrpl
14v:o,l
-."I r{v
'J
. , .I.?gu8_. , -.-
6{1o,
l)=.,
l+o.oo/0 7s +o.oSs?d - 3xti6tr'
v
G Hlo, i) =
Z
?
e
o
p
v
_
=
0.999t3 * Xo,oss6g?
q
-**--
- --
? ,? c o ft
l, ouz/t,gf
Arus
r
7, , T eog
-;
Gr {(r)' I +o.8sz40)
j(t)'
- o, tazs(r).- o, oo3
(J0
G t-t(D'
-
J'76a8--- -
-
l-o,1ff?S,+ o'85?t -o'e3:
0d
GH (l ) = -
et438a1O.MCD11
?8 g / + a ,4 2
l,", ?6aP
6,sBl /-494"
?, ?(o I
*(o,8s1
o.Q{aS
Exampfe:
(cont.)
opE^rAoop Gru(,{.
.
7 ?<oB
I + ov,e.'
,g s? .( tn \_ A tfTs(.)._
t
7, 7e08
€;
ry -2
Ga (/a):
7.
--,zeo!
. --
/- /8.7s* . t?^ ,d
GH (ro)=
7, 76oP
ld.co3/taz.s?o
=----
o.ooffi
d,,o)-o,
v
-/7.7s+t,t?
o- 4l7Z ' t 6z . ao
?. r<og
GH(,-) =
rd,.o- d,r07S(oora.y (;t'
1so.Ps
GH( r a);
/+8s.7d-/8Zs-B*d
ftr/.
\
t5 f't ( /cB 1 -
V
V
/> Flz
- 't
^ rr
?'?aog
:*)o,oo?,14
;
. 3.?400._
- tQJq - y 29 t1.S
(
.e**@
3 4 6 { .0 3 4 r r a d - :
CoruurpTTo dl
/ tzz.zld
I Uoi'€ Dho*€
CAanles
JoLT I cH(or)l.=2oI
? r??aaJf _{,erop ho*e
"70.?oa1
zo A"1leH(''o) aoLo1(<ss<)=ta arod{
l"
- 46.4o
pAose,
Zt ,,7 l eH ( r ) ," L,;Q , o / ?=z -) z . s e s J ( - t 6 z , c o1,,,"t<
p
l=
zo L"?lcuQ*){=zo1"7Q,uozzv)
=^s3oJA + tzz.ao
Plr^q
C o rn FuTt
vAtLrES
et438a1O.MCD 12
cr osr|
Jp
Ldbp
'/*uspe
sA,"nE FAsAt
toN.
R fu uq7to4)
valuesof closedlooptransferfunction.
Computing
G(s).H(s) =
Makethis definition
1 + G (s ).H (s )
GH
"(s)
7' ?(o 8
6,ze<rE
+ o.8s?,h.r10,ra>s(7r)'+ o,o)39*)'
G li.(r^,)--
(((
Crt.cr).
- Opo3dt^l3
8. ?aoa* O,SS?6t,'l
- o.tK?Srr.r1
Glt(o.D
'
o.Vs
Q.7 co 8+
v
, o. 886 o qg]
C N. ( o,D
-q -
GH .(/):
GLt.( t) =
= o,FB6- 8.ctxi"'i
, lP
u
o.8?G-o,o89u
d
u
ffiu
V
-vr! r*)-v , - f f
-o,sg3-o,33
_ ?aog
_ .,
: , ; _. i
:-/,ztxnlrerxifr
=
GH.(rq
Co ruveeT To dt
I
t
_ o,s 6" Yhas<_
Ja
-/'osi
une)'
(o
=
o
3
z 4"3/ H.(o't)'/'LoL"1
: - |'9 r r d'6 - s. 67oPhase
zo
Lo](r.9ouq)
Lo1
zu lctlr(r)'
J
et438a1O.MCD 13
I
Example(cont.)
,zo11[ eH.{r.)l= zol.J(o.a?s6)=-:.aetdd
pl"c s e, - / So, 9o
= -s2.esJf
LO L': \ G Hc(r*)\ = ZoL.l (o.ooz.oo)
t2?.62"
doL'<
5.) C..,^p*b f eeoRe4rlo {oe LtJ'o' l, lr /or/oontd/sec
De{,^,1,i'^ d gmon Pqtiro
I = r{1
E'|\
ER=l;:-.-,,t
| ( r- <@[
Af
/{t
(r: o,I dn(o,r)= l.?4 G *gro
* GH<q)(t"'u.) I
AR=l
I
\
l(tr r,l4LFqspY(-?34.A,)
I
ER
'I
s9.o,t /,?, ,c
Ar t* s . l
l=m
G H( r ) : ( ,s g < l /-q a .q
I ,
I
fR'\= !
b
(- assqlg")
{6r+a,ss44s+1
I
ER=l
AT kr-- / o
t
I
l-r
"tj
h,".r@l=1o
G / . 1( r o) =o ,{ t? Z
..(o
K'/* l no,lt\z/Eel
rrr4*g')(
re'cy-o,r
lr,
A R.
o ,8(zg/-L,eP
et438a10.MCD 14
Example(cont.)
the computations
of the errorratio
Complete
& r=o ,I
Z o L " 1 ( o -o r l ) = -3s-3qd(
J
L <s=I
l'^)
: lO
UJ
z, L"3(u'or3): - 3Z-zt Jlo
JL
ZuL"1(l,rsl)= l'28
. / o o ZoL "1
( ,) = @Jt
As thefrequency
increasesthe errorratioincrease.lt peaks
valueof 1.
andthencomesto a constant
Thethreezonesaredefinedas:
Zone1: errorratio< 0 db;goodcontrol;controller
decreases
the error
Zone2. errorratio> 0 db poorcontrol;controller
increase
the error
Zone3: errorration= 0 db no control;controller
doesnotaffectthe error
e t 438 a 1 0 .M C D
l_5
The Matlabcodeto generatethe bodeplotsfollows
t E x am ple bo d e c a l c u l a ti o n s
clear all, '
cl- os e al] - '
I def i- ne t he fo rw a rd g a i n n u me ra tor and denonui nator coeffi ci ents
n um g= I 2L . 8) ;
d em g= [ 0. 0063 0 .3 7 9 ]-l ;
I def ine
t he fe e d b a c k p a th g a i n numerator and denomi nators
n um h= [ 0. 356] ;
d em h= [ 0. 478 ]-l ;
E c ons t r uc t
th e tra n s fe r
fu n c ti o n s
G = t f ( num g, de mg ) ;
H = t f ( num h, de mh ) ;
I f ind G H( s )
GH=G*H
I f ind t he c l -o s e d l o o p tra n s fe r
functi -on
GHc=GHl ( 1+GH)
I c om par e it to th e d e ri v e d
c l o s e d l oop transfer
functi on
I note that
the program does not give the minmum algebraic
modelG Ht = t f ( [ 7. 750 8 ],
0
.1
8
7
5
0.857
8.7608]
[0 .0 0 3 0 1 )
p l o ts
t t his
s t at em e n t
a l l - th re e systems on the same axi s
I Lhe open l o o p s y s te m i s i n g re e n w i t.h ots at every data poi nt
t t he M at lab d e ri v e d
c l o s e d l o o p functoi n
i s i n red
? and t he c lo s e d l o o p fu n c ti o n
d e rj -ved by hand i s i n bl ue w i th
I o' s at ev e ry d a ta p o i n t.
T h e v a l ue i -n curl y brackets
are freq.
l i mi ts
b o de( G H, r go- t rGH c , tr-r rGH tr tb o t, {0. 1, 100} )
Transfer
function:G(S)H(S)
7.761
1 s^3 + 0.1875s^2 + 0.857s + 1
0.00301
+G(s)H(s))
Transferfunction:
G(s)H(s)/(1
machinegenerated
0.02337s^3+ 1.455s^2+ 6.651s + 7.761
9.069e-006
s^6 + 0.001129s^5+ 0.0403s^4 + 0.3507s^3 +
2.564s^2 + 8.365s + 8.761
e t43 8 a 1 0 .M CD
16
The lasttransferfunctioncan be factoredand simplifiedinto
the sameform as the hand-derived
transferfunction
+G(s)H(s))
Transferfunction:G(s)H(s)/(1
handderived
7.761
0.00301s^3 + 0.1875s^2 + 0.857s + 8.761
BodeDagmarns
m
!
(l)
-(f
=
.=
c
o)
(d
o)
o
E
o
(t,
(d
o-
1oo
Frcqrercy(ndsec)
et438a10.MCD 17
rot
Plotthe errorratiofor the examplesystemand determinethe
Maximumfrequencylimit.
System
EnorRatioPlotForExample
10
0
-1 0
-20
3-.0
.9
E -+o
E
b
E -so
-60
-70
-80
lt
I
il
il
il
il
il
I
I
I
I
/
))
-90
1o-t
10'
1oo
(radisec)
Frequency
1ot
limitoccurswhenthe errorratiofirst
frequency
Maximum
crosses0 db.
Maxf is approx.7 radlsecfromthe graph.
et438a10.MCD 18
Anothermethodof findingthe maximum
stablefrequencyis
the deviationratio.
magnitude
closedlooperror
Deviation
Ratio=
magnitude
setpoint
DR(s)=
1+ G (s ).H (s )
PlotDeviation
andErrorratioson samegraphfor the example
system.
'al
E rro r R a ti o A n d D e vi ati on R ati o P l ot For E xam pl e S ystem
bb"'
I
ll
'o
.o
ru
E,
C,
o
.(U
dt
o
o
.U
E,
o
ul
"l
."1
"l
'r o {
I
-roo
10
/
/
./
F
t)
d
-.t/
^/
H
H
./
1oo
1ot
Frequency (radl sec)
Samezonecriteria
as errorratio
et438a10.MCD 19
E tror R ati o
D evi ati on R ati o
10"
Stability
of ControlSystemsFromBodePlots
gainof
A stablecontrolsystemis onethathasan open-loop
phaseanoleis -180
lessthan1 (<0db)whentheopen-loop
degrees.
gainof exactly1 (0 db)
lf controlsystemhasan open-loop
phaseangleis -180degrees,
whentheopen-loop
thenthe
systemis marginally
stable.(willoscillate).
gainof qreaterthan1
lf the controlsystemhasan open-loop
(>0db)whentheopen-loop
phaseanoleis -180degrees,
thenthe systemis unstable.
GainandPhaseMargins
GainMaroin- Gainlevelthatis considered
a safelevelbelow
1 (0 db). Minimum
levelsetat 0.5(-6db)at phaseshiftof
-180degrees.
PhaseMargin- Phaseshiftabove-180degrees
thatis
considered
a safelevel.Minimumlevelset at 40 degrees
Definethetwofrequencies
0 odb
co1go
et438a10.MCD 20
Frequency
wheregainis zerodb
Frequency
wherethe phaseshiftis -180
degrees
GainandPhaseMargins
-m 180
Gainmargin
180 r p 0db
Phase
margin
Open-loop
r
/s^in
p degrees
Fooo
Open
loop
phase
180+Foou
-180deg.
Oodb
0teo
Procedure
1.) Drawverticallinesthroughthe0 db pointof the gainplot
andthe-180degreephaseshiftpointof the phaseplot
2.) Drawhorziontal
linesthroughthe0 db pointof the gain
plotandthe-180degreeso that bothooooandorsolines
are intersected
3.) Readthe gainandphasemarginsfromthe difference
lines
betweenthe 4 horziontal
et438a1O.MCD21
Example:
A firstorderlag processhasa dead-time
delayof 2
secondsandis controlled
by a proportional
controller.The
open-loop
transferfunctionis givenby:
1 --." -2 ' t
G H( s ) = 40 -(1 + 1 0 0 .s )
1.) Findthe magnitude
and phaseangleof the transfer
functionat cD= 0.001,0.01,0.1 and 1 rad/secby hand
calculation.
jrrrfor s and find complexnumber.Forthe
Substitute
exponential
term,rememberEuler'sldentity.
e-A'o- cos(2.rrr)- j.sin(2-rl)
Gainatways1
phaseincreases
with increasing
rrl
For crl:= 0.001
1
. "-4''
'
G H( c o ) = 4 0 .
( 1 r 1 0 0j. crr)
= 0.99_ 0.099j
1 + 1 0 0 j .0 .0 0 1
Convertto polarflrm
| 0 99 - 0.099j| = 0.99S
0r =atanf+:n'
e-l
\
et438a1O.MCD 22
= 1 _2.tO-3j
e-2i.0.001
,
or--5711'des
partto magnitude
(cont.) Convertexponential
Example:
andphaseshift.
1-2.t0-3.jl=1
3
I '1.10\
0o:=atani+-/ 0c=Q.115'des
Multiply
magnitudes
andaddphaseangles
m :=40.0.995.1 m =39.8
Convert
Gainto db
0t :=0t + 0O
db :=20.log(m)
db = 31.998 db
0 t = -$.596'deg Phaseshift
cD:= 0.01
1
'e-2l^
(
:=
40'
GH r,r)
( 1 + 100 j ' rrr)
Now for
- 0 .5 - 0 .5 j
1 * 1 0 0 j .0 .0 1
=1_0. 04
e-2i'0. 01
=
Convert
to polarform I O.S- 0.5j| 0.707
g!^nn
\
(
:=
atan
or
\
or:=atan(:)
et438a10.MCD 23
e-/
I
o l - -5711'des
or =4s.des
gainandphaseshift
Exponential
0 o:=atan(+* ) to=-r.1 4 6 ' d e s
.i
lr -o.o 2 l=r
\/
Multiplymagnitudes
and add phaseangles
m : =4 0 .0 .7 0 7 .1
C onv ertGa i nto d b
m =28. 28 01 : =0l + 0 O
d b : =20. log(m)
d b = 29. 03
db
0 t = 46-146'deg Phaseshift
Usingthe samemethodto findthe valuesat crr=O.1
and 1
rad/secgivesthe followingresults.
At
o :--0.1
d b : = 1 2 .0
At
db
0 t : =-95. 7. de9
crl= 1.0
d b : : - 8 .0
db
0 t : =-204. 0'deg
BodeplotusingMatlabandfindthe
Construct
an open-loop
gainandphasemarginsfor the controlsystem
et438a10.MCD 24
Matlabcodeto find the Bodeplot
I Example bode calculations
cl ear all;
cl o s e all;
gain numerator
t define
the forward
and denominator
coefficients
numgh= [40] ;
d e m gh= [ 100 1] ;
I c ons t r uc t , t h e tra n s fe r
fu n c ti o n
Gh=tf (numgh, demgh) ,'
I Use the bode function
with its
the
argunent,s so that it returns
p and the frequenci es
I m agnit ude, m , th e p h a s e s h i fL ,
so that
t t he ef f ec t
o f th e d e a d ti me d e l a y can be added to the system
[n p w] = bode ( c h , ( 0 .0 0 1 -, 1 ] ) ;
I now c om put e th e v a l u e s o f p h a s e shi ft
for the t,i me del ay usi ng the
E f or m uLa - 2*w * 5 7 .6
p d = - 2*w*57 . 6;
I Add the the phase shift
of the transfer
function
to the deadtime deJ-ay
t tak e t he ph a s e s h i ft
o u t o f th e 3 col umn array
[d p w ]
p h as e= p ( : ) ;
pt=pd+phase;
I compute the gain in db
d b = 20. *log10 (m ) ;
I create
a figure
window
figure,'
I divide
the pJ-ot area in two parts
(
2r
Lr Ll;
su bplot
(
w,
se m ilogx
db , ' g o -' ) ;
xl a bel ( r F r egue n c y (ra d /s e c ) ! ) ;
yl a bel ( t G ain (d b ) ' );
( r E x am ple Bo d e Pl o t' ) ;
ti tl e
g ri d on;
su b plot ( 2, L, 21 ;
se m ilogx ( w, pt , tg o -t ) i
xl abel ( r F r eque n c y (ra d /s e c )'
),'
(D e g re e s ) I );
yl abel ( r P has e S h i ft
g ri d on;
et438a1O.MCD25