Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
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Feder
al
Ur
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogy
LABMANUAL
SI
XTHSEMESTER
LI
NEARCONTROLSYSTEM
ELECTRI
CALMACHI
NES&CONTROLSYSTEM LAB
DEPARTMENTOFELECTRI
CALENGI
NEERI
NG
Pr
epar
edBy
:
Engr
.Saqi
bRi
az
CheckedBy
:
Dr
.Zubai
rKhal
i
d
Appr
ov
edBy
:
Dr
.Nav
eedAl
iKhan
Lect
ur
er(
Lab)
Coor
di
nat
or
DeanEl
ect
r
i
cal
Depar
t
ment
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
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Name:_
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Regi
st
r
at
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onNo:
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Feder
alUr
duUni
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Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
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calEngi
neer
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S.
No.
Ti
t
l
eofPr
act
i
cal
1
I
nt
r
oduct
i
ont
oMat
l
abcommandsusedi
ncont
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ol
sy
st
ems(
Sof
t
war
e)
2
Lapl
aceTr
ansf
or
mat
i
ont
ocheckt
hest
abi
l
i
t
yofaLTISy
st
em (
Sof
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3
I
mpl
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ockDi
agr
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i
onandOper
at
i
ononLTImodel
si
n
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l
ab(
Sof
t
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e)
4
Char
act
er
i
st
i
csofcont
r
ol
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st
em openl
oopandcl
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Sof
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5
I
mpl
ement
at
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onofPI
Dcont
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erv
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acommands
6
I
mpl
ement
at
i
onofPI
Dcont
r
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l
erv
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asi
mul
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nk
7
I
mpl
ement
at
i
onofRootLocusTechni
quei
nMat
l
ab
8
Tounder
st
andt
hef
r
equencyr
esponseofacont
r
ol
sy
st
em usi
ngBodePl
ot&
Ny
qui
stPl
ot
9
Model
i
ngofSpr
i
ngMassDamperSy
st
em
10
Desi
gnandI
mpl
ement
at
i
onofCompensat
or
s
11
Const
r
uct
i
onofapr
act
i
cal
posi
t
i
oncont
r
ol
l
er(
Har
dwar
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12
I
mpl
ement
at
i
onofcl
osedl
oopmot
orspeedcont
r
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(
Temper
at
ur
eBaseusi
ng
Mi
cr
ocont
r
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l
er
)
13
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esponseofposi
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dwar
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14
Toobser
v
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ocusandBodepl
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(
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15
I
mpl
ement
at
i
onofcl
osedl
oopspeedcont
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ol
ofACmot
orusi
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D
Li
nearCont
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ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
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Lab#01
I
nt
r
oduct
i
ont
oMat
l
abi
nCont
r
olSy
st
ems
Obj
ect
i
v
e:
I
nt
r
oduct
i
ont
oMATLAB br
i
ef
l
yi
ncl
udi
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ut
or
i
alofpol
y
nomi
al
s,scr
i
pt
wr
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ngandpr
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r
om cont
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st
emsv
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ewpoi
nt
.To
l
ear
nt
hebasi
coper
at
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on/
codei
nt
heMATLAB.
Whati
sMat
l
ab?
MATLABSt
andsf
orMATr
i
xLABor
at
or
y
.
MATLABi
sacomput
erpr
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am t
hatcombi
nescomput
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stofMATLABcommandsl
i
keot
her
pr
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anguage.
Thewi
ndowsi
nMATLABar
e:
Commandwi
ndow:Commandscanbeent
er
ed,
dat
aandr
esul
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sar
edi
spl
ay
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Wor
kspace:l
i
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ar
i
abl
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ouar
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st
or
ywi
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t
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spl
ay
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ogoft
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aphi
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Fi
gur
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ndow:Di
spl
ay
spl
ot
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aphs,cr
eat
edi
nr
esponset
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gr
aphi
cscommands.
Mf
i
l
eedi
t
or
/
debuggerwi
ndow:Cr
eat
eandedi
tscr
i
pt
sofcommandscal
l
edMf
i
l
es.
MATLAB i
s a hi
ghper
f
or
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anguage f
ort
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calcomput
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t
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nt
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at
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n f
ami
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ar
mat
hemat
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cal
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at
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on.Ty
pi
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usesi
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ude:
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handcomput
at
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on
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gor
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t
hm dev
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si
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i
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Li
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ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
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Var
i
abl
edecl
ar
at
i
on:
Thev
ar
i
abl
esar
edecl
ar
edas:
Mustst
ar
twi
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Maycont
ai
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di
gi
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andt
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_
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Mat
l
abi
scasesensi
t
i
v
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i
.
e.one&OnEar
edi
f
f
er
entv
ar
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abl
es.
Forassi
gni
ngst
at
ement
:
Var
i
abl
e=number
;
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alv
ar
i
abl
es:
ans:
def
aul
tv
ar
i
abl
enamef
ort
her
esul
t
pi
:
π=3.
1415926
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not
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v
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abl
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who:
l
i
st
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i
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i
abl
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whos:
l
i
st
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i
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cl
ear
:
cl
ear
sal
l
v
ar
i
abl
es,
r
esett
hedef
aul
tv
al
uesofspeci
al
v
ar
i
abl
es.
cl
earname:
cl
ear
st
hev
ar
i
abl
ename
cl
c:
cl
ear
st
hecommandwi
ndow
cl
f
:
cl
ear
st
hecur
r
entf
i
gur
eandt
hegr
aphwi
ndow
Mat
l
abi
nCont
r
olEngi
neer
i
ng:
MATLAB i
sv
astt
oolhav
i
ng one ofi
t
s appl
i
cat
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ons i
n CONTROL
ENGI
NEERI
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i
ngi
nv
ol
v
est
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gnofawel
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dev
el
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st
em whi
chcont
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somequant
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t
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asr
oom t
emper
at
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amot
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l
abi
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udesmanyt
ool
sf
orcont
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neer
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ng.
01:Fi
ndi
ngr
oot
sofequat
i
onandmaki
ngofequat
i
onf
r
om r
oot
s?
Ther
oot
sf
unct
i
oncal
cul
at
est
her
oot
sofapol
y
nomi
al
:
1.
01:
>>p=[
1304]
p= 1 3 0 4
>>r
=r
oot
s(
p)
r=
3.
3553
0.
1777+1.
0773i
0.
1777-1.
0773i
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
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_
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1.
02:
>>p=pl
oy
(
r
)
>>p=pol
y
(
r
)
p=
1.
0000 3.
0000 0.
0000 4.
0000
1.
03:
>>r
=[
12]
r=
1 2
>>p=pol
y
(
r
)
p=
1 3 2
>>r
=[
11111]
r=
1.
04:
>>r
=[
11111]
r=
1 1 1 1
>>p=pol
y
(
r
)
p=
1 5 10 10 5 1
02:Mul
t
i
pl
i
cat
i
onoft
wopol
y
nomi
al
sequat
i
on?Conv
ol
ut
i
onandDeconv
ol
ut
i
on
of equat
i
ons.
Pol
y
nomi
almul
t
i
pl
i
cat
i
onanddi
v
i
si
oncor
r
espondt
ot
heoper
at
i
onsconv
ol
ut
i
on
anddeconv
ol
ut
i
on.Thef
unct
i
onsconvanddeconvi
mpl
ementt
heseoper
at
i
ons.
>>a=[
123]
;
>>b=[
456]
;
>>c=conv
(
a,
b)
c=413282718
Usedeconv
ol
ut
i
ont
odi
v
i
debackoutoft
hepr
oduct
:
>>[
q,
r
]=deconv
(
c,
a)
q=456
r=
00000
>>p=[
3201]
;
>>q=[
12]
;
>>n=conv(
p,
q)
Li
nearCont
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ol
Sy
st
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Feder
alUr
duUni
v
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yofAr
t
s,
Sci
ence&Technol
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sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
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n=
3 8 4 1 2
03:Def
i
ni
ngv
al
uei
nanequat
i
on?
Thepol
y
v
al
f
unct
i
onev
al
uat
esapol
y
nomi
al
ataspeci
f
i
edv
al
ue.
>>p=[
3201]
;
>>q=[
14]
;
>>v
al
ue=pol
y
v
al
(
p,
2)
Val
ue=
33
>>v
al
ue=pol
y
v
al
(
n,
2)
v
al
ue=
132
04:Obt
ai
ni
ngdi
gi
t
si
npowerf
or
m?
>>sqr
t
(
7)
ans=
2.
6458
>>sqr
t
(
sy
m(
7)
)
ans=
7^
(
1/
2)
05:Fi
ndi
ngder
i
v
at
i
v
eanduseofsy
mscommandf
ordef
i
ni
ngv
ar
i
abl
e?
5.
01:
>>sy
m(
3/
8)
ans=
3/
8
5.
02:
>>sy
msxy
>>f
=exp(
x*
y
)
f=
exp(
x*
y
)
>>di
f
f
(
f
)
ans=
y
*
exp(
x*
y
)
>>di
f
f
(
f
,
x)
ans=
y
*
exp(
x*
y
)
>>di
f
f
(
f
,
y
)
ans=
x*
exp(
x*
y
)
>>di
f
f
(
f
,
x,
2)
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
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_
_
_
_
_
ans=
y
^
2*
exp(
x)
5.
03:Useofpr
et
t
ycommand?
>>sy
msxyz
>>z=2*
si
n(
2*
x*
y
)
SS
z=
2*
si
n(
2*
x*
y
)
>>di
f
f
(
z,
x,
2)
ans=
8*
si
n(
2*
x*
y
)
*
y
^
2
>>di
f
f
(
z,
y
,
2)
ans=
8*
si
n(
2*
x*
y
)
*
x^
2
>>sy
msxyzt
>>f
=2*
exp(
t
)
2*
exp(
2*
t
)
f=
2*
exp(
t
)
2*
exp(
2*
t
)
>>pr
et
t
y
(
f
)
2exp(
t
)-2exp(
2t
06:Pol
y
nomi
alDer
i
v
at
i
v
es
Thepol
y
derf
unct
i
oncomput
est
heder
i
v
at
i
v
eofanypol
y
nomi
al
.Toobt
ai
nt
he
der
i
v
at
i
v
eoft
hepol
y
nomi
al
>>p=[
1025]>>q=pol
y
der
(
p)
q=302
pol
y
deral
socomput
est
heder
i
v
at
i
v
eoft
hepr
oductorquot
i
entoft
wo
pol
y
nomi
al
s.Forexampl
e,
cr
eat
et
wopol
y
nomi
al
saandb:
>>a=[
135]
;
>>b=[
246]
;
Cal
cul
at
et
heder
i
v
at
i
v
eoft
hepr
oducta*
bbycal
l
i
ngpol
y
derwi
t
hasi
ngl
eout
put
ar
gument
:
>>c=pol
y
der
(
a,
b)
c=
8305638
Cal
cul
at
et
heder
i
v
at
i
v
eoft
hequot
i
enta/
bbycal
l
i
ngpol
y
derwi
t
ht
woout
put
ar
gument
s:
>>[
q,
d]=pol
y
der
(
a,
b)
q=
282
d=
416404836
q/
di
st
her
esul
toft
heoper
at
i
on.
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
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_
07:Mat
r
i
cesOper
at
i
on:
Mat
r
i
cesi
nMATLAB:
I
nMATLAB,
amat
r
i
xi
sar
ect
angul
arar
r
ayofnumber
s.Speci
almeani
ngi
s
somet
i
mesat
t
achedt
o1by
1mat
r
i
ces,whi
char
escal
ar
s,andt
omat
r
i
ceswi
t
h
onl
yoner
ow orcol
umn,whi
char
ev
ect
or
s.MATLABhasot
herway
sofst
or
i
ng
bot
hnumer
i
candnonnumer
i
cdat
a,
buti
nt
hebegi
nni
ng,
i
ti
susual
l
ybestt
ot
hi
nk
ofev
er
y
t
hi
ngasamat
r
i
x.Theoper
at
i
onsi
nMATLABar
edesi
gnedt
obeas
nat
ur
alaspossi
bl
e.Wher
eot
herpr
ogr
ammi
ngl
anguageswor
kwi
t
hnumber
s
oneatat
i
me,
MATLABal
l
owsy
out
owor
kwi
t
hent
i
r
emat
r
i
cesqui
ckl
yandeasi
l
y
.
Def
i
ni
ngaMat
r
i
c:
Amat
r
i
xi
sdef
i
nedasf
ol
l
owsi
nMATLAB:
>>M =[
100;
0j
1;
j
j
+13]
>>k=[
2.
75]
Wher
e‘
M’
i
s3rdor
dermat
r
i
xand‘
k’
i
s1stor
der
.
Remov
i
ngaRoworCol
umn:
Tor
emov
ear
owf
r
om amat
r
i
xweusef
ol
l
owi
ngcommand:
Consi
deramat
r
i
x‘
M’
wher
e
>>M =[
174;
536;
745]
Tor
emov
e2ndr
ow,
>>M(
2,
:
)
=[
]
Nowr
emov
i
ng3r
dcol
umn
>>M(
:
,
3)
=[
]
Mul
t
i
pl
i
cat
i
onofmat
r
i
x(
el
ementbyel
ement
)
Consi
dert
womat
r
i
x‘
x’
&‘
y
’
t
omul
t
i
pl
yt
hem weuse:
>>X=x.
*
y
Wher
e‘
X’
i
st
heanswer
Mul
t
i
pl
i
cat
i
onofmat
r
i
x(
whol
emat
r
i
x)
Consi
der
i
ngt
womat
r
i
x‘
X’
&‘
Y’
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
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_
>>Z=X*
Y
Di
v
i
si
onofmat
r
i
x
Todi
v
i
det
womat
r
i
x‘
A’
&‘
B’
Adi
vB=A/
B(
r
i
ghtdi
v
i
si
on)
Bdi
vA=A\
B(
l
ef
tdi
v
i
si
on)
I
nv
er
seofamat
r
i
x
Let
’
sf
i
ndi
nv
er
seofamat
r
i
x‘
X’
:
i
nv
X=(
A/
B){
def
i
nedabov
e)
LabTask/
LabExer
ci
se
Pr
obl
em #01:
I
fCi
sagi
v
enMat
r
i
x
[ ]
147
963
258
a)Whati
st
hesi
zeofC?
b)Whati
st
hev
al
ueofC(
3,
2)
?
Pr
obl
em #02:
Asy
st
em of3l
i
nearequat
i
onswi
t
h3unknowns(
x1,
x2,
x3)
:
3x1+2x2x3=10
x1+3x2+2x3=5
x1x2x3=1
a)Fi
ndt
heMat
r
i
xA,
b&x?
b)Whati
st
hesi
zeofb?
c)Fi
ndt
hev
al
uesofx1,x2,&x3?
Pr
obl
em #03:
Consi
dert
het
wopol
y
nomi
al
sp(
s)
=s2+2s+1andq(
s)
=s+1.
UseMATLABt
ocomput
e
a)p(
s)
*
q(
s)
b)Root
sofp(
s)andq(
s)
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
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_
_
c)p(
1)andq(
6)
Pr
obl
em #04:
UseMATLABcommandt
of
i
ndt
hepar
t
i
al
f
r
act
i
onoft
hef
ol
l
owi
ng.
B(
s) 2s3+5s2+3s+6
=
A(
s) s3+6s2+11s+6
Pr
obl
em #05:
Fi
ndt
he1st,
2nd&3rdder
i
v
at
i
v
eoft
hef
ol
l
owi
ngequat
i
ons/
f
unct
i
onsw.
r
.
txas
wel
l
asw.
r
.
tyal
so.
a)2
cos3xy
3xy
z
b)e
3
2
x +3xc)2
5x+4
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
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_
Lab#02
Lapl
aceTr
ansf
or
mat
i
ont
ocheckt
he
st
abi
l
i
t
yofaLTISy
st
em
Obj
ect
i
v
e:
Toobt
ai
nt
het
r
ansf
erf
unct
i
onf
ort
hegi
v
enf
unct
i
oni
nt
i
medomai
n,
al
so
t
of
i
ndoutt
hest
abi
l
i
t
yofasy
st
em.
Theor
y
:
ALi
nearTi
meI
nv
ar
i
antSy
st
emsi
sst
abl
ei
ft
hef
ol
l
owi
ngt
wonot
i
onsof
sy
st
em st
abi
l
i
t
yar
esat
i
sf
i
ed
i
) When t
hesy
st
em i
sexci
t
ed byBounded i
nput
,t
heout
puti
sal
so a
Boundedout
put
.
i
i
)I
nt
heabsenceoft
hei
nput
,
t
heout
putt
endst
owar
dszer
o,
i
r
r
espect
i
v
eof
t
hei
ni
t
i
al
condi
t
i
ons.
Thef
ol
l
owi
ngobser
v
at
i
onsar
egener
alconsi
der
at
i
onsr
egar
di
ngsy
st
em
st
abi
l
i
t
yandar
e
i
)I
fal
l
t
her
oot
soft
hechar
act
er
i
st
i
cequat
i
onhav
enegat
i
v
er
eal
par
t
s,
t
hen
t
hei
mpul
ser
esponsei
sboundedandev
ent
ual
l
ydecr
easest
ozer
oand
t
hensy
st
em i
sst
abl
e.
i
i
)I
fanyr
ootoft
hechar
act
er
i
st
i
cequat
i
onhasaposi
t
i
v
er
ealpar
t
,t
hen
sy
st
em i
sunst
abl
e.
i
i
i
)I
ft
hechar
act
er
i
st
i
cequat
i
onhasr
epeat
edr
oot
sont
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hen
sy
st
em i
sunst
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e.
i
v
)I
foneormor
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epeat
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oot
soft
hechar
act
er
i
st
i
cequat
i
onont
hej
ω
axi
s,
Lapl
aceandi
nv
er
seLapl
ace:
L=l
apl
ace(
F)i
st
heLapl
acet
r
ansf
or
m oft
hescal
arsy
mbolFwi
t
hdef
aul
t
i
ndependentv
ar
i
abl
et
.Thedef
aul
tr
et
ur
ni
saf
unct
i
onofs.TheLapl
acet
r
ansf
or
mi
s
appl
i
edt
oaf
unct
i
onoftandr
et
ur
nsaf
unct
i
onofs.
f
(
t
) Lapl
(
s)
ace F
LTImodel
s:
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
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_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
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_
_
_
_
_
_
_
_
_
_
_
_
_
_
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_
_
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_
_
_
_
_
_
_
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_
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_
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The Cont
r
ol Sy
st
em Tool
box sof
t
war
e pr
ov
i
des cust
omi
zed dat
a
st
r
uct
ur
es,knownasLTImodelobj
ect
s,f
oreacht
y
peofmodel
:t
f
,zpk,ss,and
f
r
d.TheseLTImodelobj
ect
sencapsul
at
et
hemodeldat
a.Theyal
l
ow y
out
o
mani
pul
at
eLTIsy
st
emsassi
ngl
eent
i
t
i
esr
at
hert
hancol
l
ect
i
onsofdat
av
ect
or
s
ormat
r
i
ces.
Dependi
ngont
het
y
peofmodel
y
ouuse,
t
hedat
af
ory
ourmodel
cani
ncl
ude:
i
) Oneormor
enumer
at
or
/
denomi
nat
orpai
r
sf
ort
r
ansf
erf
unct
i
ons
i
i
)Zer
osandpol
esf
orzer
opol
egai
nmodel
s
Tr
ansf
erf
unct
i
onModel
s:
Pol
es and Zer
os ofa sy
st
em gi
v
es y
ou t
he i
nf
or
mat
i
on ofsy
st
em
behav
i
ouratanypoi
ntl
i
ke sy
st
em set
t
l
i
ng t
i
me,peak t
i
me and mor
eov
er
pr
ov
i
desav
er
yusef
ul
i
nf
or
mat
i
onaboutt
hest
abi
l
i
t
yofaSy
st
em.
Acont
i
nuoust
i
met
r
ansf
erf
unct
i
oni
npol
y
nomi
al
f
or
mi
sgi
v
enby
:
T.
F=G(
s)
/
H(
s)
Wher
e,G(
s)andH(
s)ar
et
wopol
y
nomi
al
s.Put
tH(
s)=0,t
henr
oot
soft
hi
s
pol
y
nomi
alwi
l
lgi
v
esy
out
hesy
st
em Pol
eandput
tG(
s)=0,t
her
oot
soft
hi
s
pol
y
nomi
al
wi
l
l
gi
v
esy
out
hesy
st
em Zer
os.
01:Tof
i
ndLapl
aceofcont
i
nuoust
i
mef
unct
i
on?
1.
01:
>>sy
mst
>>f
=t
^
4
>>F=l
apl
ace(
f
)
f=
t
^
4
F=
24/
s^
5
1.
02:
>>sy
mst
>>f
=exp(
5*
t
)
>>F=l
apl
ace(
f
)
f=
exp(
5*
t
)
F=
1/
(
s+5)
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
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1.
03:
>>sy
mswt
>>f
=si
n(
w*
t
)
>>F=l
apl
ace(
f
)
f=
si
n(
t
*
w)
F=
w/
(
s^
2+w^
2)
02:Cr
eat
eat
r
ansf
erf
unct
i
onmodeli
npol
y
nomi
al
.
2.
01:
>>G=t
f
(
num,
den)
Forexampl
e,
cr
eat
et
het
r
ansf
erf
unct
i
onG(
s)=s/
(
s2+2s+1)
,
usi
ng:
>>G=t
f
(
[
10]
,
[
121]
)
;
2.
02:
G(
s)=6S2+1/S3+3S2+3S+1
>>num=[
601]
>>denum=[
1331]
>>g=t
f
(
num,
denum)
>>z=r
oot
s(
num)
>>p=r
oot
s(
denum)
>>[
z,
p,
k]
=t
f
2zpk(
num,
denum)
>>h=zpk(
g)
>>[
num,
denum]
=zp2t
f
(
z,
p,
k)
>>pzmap(
num,
denum)
2.
03:
G(
s)=S2+2S+3/
S3+2S23S+1
>>sy
mss
>>num=s^
2+2*
s+3
>>denum=s^
3+2*
s^
23*
s+1
>>g=num/
denum
2.
04:
G(
s)=S2+6S+8/S5+8S4+23S3+35S2+28S+3
>>num=[
168]
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
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_
_
>>denum=[
182335283]
>>f
=t
f
(
num,
denum)
>>pr
i
nt
sy
s(
num,
denum)
>>[
z,
p,
k]
=t
f
2zpk(
num,
denum)
>>pzmap(
num,
denum)
LabTask/
LabExer
ci
se
Pr
obl
em #01:
Fi
ndoutt
heLapl
aceoft
hef
ol
l
owi
ngcont
i
nuoust
i
mesi
gnal
s?
5t
a)f=e
2t
3t
b)f=2e 3e
5coswt
c)f=Pr
obl
em #02:
Fi
ndoutt
hei
nv
er
seLapl
aceoft
hef
ol
l
owi
ngt
r
ansf
erf
unct
i
on?
4w
s+4w2
1
b)F= 2
s
1 3
5
c)F=
- +
s+2 s+5 s3
a)F=
2
Pr
obl
em #03:
Fi
ndt
hepol
eandZer
osofSy
st
em anddr
awi
nSpl
anusi
ngMat
l
ab.I
f
:
a)G(
s)
=6s2+1/
(
s3+3s2+3s+1)
b)H(
s)
=(
s+1)
(
s+2/
(
(
s+2i
)
(
s2i
)
(
s+3)
)
c)E(
s)
=G(
s)
/
H(
s)
d)Fi
ndPol
esandZer
osofG(
s)
,
H(
s)&E(
s)
.
Pr
obl
em #04:
Fi
ndt
hepol
es&zer
osoft
hef
ol
l
owi
ngt
r
ansf
erf
unct
i
oni
nt
heMat
l
ab.
Wher
e
n1=s+1;
Li
nearCont
r
ol
Sy
st
em
H(
s)=(
n1*
n2)
/
(
d1*
d2*
d3)
n2=s+2;
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
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_
_
d1=s+2i
;
d2=s2i
;
d3=s+3;
Andal
sodr
awpol
ezer
omap.
Lab#03
I
mpl
ement
at
i
onofBl
ockDi
agr
am Reduct
i
on
andOper
at
i
ononLTImodel
si
nMat
l
ab
Obj
ect
i
v
e:
Toobt
ai
nt
r
ansf
erf
unct
i
onofgi
v
ensy
st
em usi
ngbl
ockdi
agr
am r
educt
i
on
t
echni
que.
Gener
alDescr
i
pt
i
on:
Bl
ockdi
agr
am consi
stofuni
di
r
ect
i
onaloper
at
i
onalbl
ockt
hatr
epr
esent
t
he t
r
ansf
erf
unct
i
on oft
he v
ar
i
abl
e oft
he i
nt
er
est
.The bl
ockdi
agr
am of
r
epr
esent
at
i
onofagi
v
ensy
st
em of
t
encanber
educedt
oasi
mpl
i
f
i
edbl
ock
di
agr
am wi
t
hf
ewerbl
ockst
hanor
i
gi
nal
bl
ock.
Agr
aphi
calt
oolcanhel
psust
ov
i
sual
i
zet
hemodelofsy
st
em andev
al
uat
et
he
mat
hemat
i
calr
el
at
i
onshi
pbet
weent
hei
rel
ement
s,usi
ngt
her
et
r
ansf
erf
unct
i
on.
I
tr
epr
esent
st
hemat
hemat
i
cal
r
el
at
i
onshi
pbet
weent
heel
ement
soft
hesy
st
em.
Oper
at
i
ononLTImodel
:
Theoper
at
i
onst
hatcanbeper
f
or
medont
heLTImodel
sar
e
i
) Addi
t
i
onofLTIModel
s.
i
i
)Subt
r
act
i
onofLTIModel
s.
i
i
i
)Mul
t
i
pl
i
cat
i
onofLTIModel
s.
i
v
)FeedbackandOt
herI
nt
er
connect
i
onFunct
i
ons.
v
)Cont
i
nuous/
Di
scr
et
eConv
er
si
onsofLTIModel
s.
Par
al
l
elconf
i
gur
at
i
on/Addi
t
i
onofLTIModel
s:
I
ft
het
wobl
ocksar
econnect
edasshownbel
owt
hent
hebl
ocksar
esai
d
t
obei
npar
al
l
el
.I
twoul
dl
i
keaddi
ngt
wot
r
ansf
erf
unct
i
onsAddi
ngLTImodel
si
s
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
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_
_
_
_
_
_
equi
v
al
entt
oconnect
i
ngt
hem i
npar
al
l
el
.Speci
f
i
cal
l
y
,
t
heLTImodel
>>sy
s=sy
s1+sy
s2
Repr
esent
st
hepar
al
l
el
i
nt
er
connect
i
onshownbel
ow
Sy
st
em
1
U(
s)
Y(
s)
Sy
st
em
2
G1(
s)=sy
s1
Y(
s)
T(
s)=
=
U(
s)
G2(
s)=sy
s2
[
sy
s]
=par
al
l
el
(
sy
s1,
sy
s2)
>>sy
s1=1/
(
s+2)
>>sy
s2=2/
(
s+3)
>>num1=[
1]
>>den1=[
12]
>>num2=[
2]
>>den2=[
13]
>>sy
s1=t
f
(
num1,
den1)
>>sy
s2=t
f
(
num2,
den2)
>>sy
s=sy
s1+sy
s2
Subt
r
act
i
onofLTIModel
s:
Thesubt
r
act
i
onoft
woLTImodel
si
sdepi
ct
edbyt
hef
ol
l
owi
ngbl
ock
di
agr
am.
>>sy
s=sy
s1-sy
s2
Sy
st
em
1
y
1
U
Y
Sy
st
em
2
y
2
sy
s
Ser
i
esconf
i
gur
at
i
on/Mul
t
i
pl
i
cat
i
onofLTIModel
s:
I
ft
het
wobl
ocksar
econnect
edasshownbel
owt
hent
hebl
ocksar
esai
d
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
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_
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_
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_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
t
obei
nser
i
es.I
twoul
dl
i
kemul
t
i
pl
y
i
ngt
wot
r
ansf
erf
unct
i
ons.TheMATLAB
commandf
ort
heconf
i
gur
at
i
oni
s“
ser
i
es”
.
Mul
t
i
pl
i
cat
i
onoft
woLTImodel
sconnect
st
hem i
nser
i
es.Speci
f
i
cal
l
y
,
>>sy
s=sy
s1*
sy
s2
r
et
ur
nsanLTImodel
f
ort
heser
i
esi
nt
er
connect
i
onshownbel
ow;
Sy
st
em
1
U(
s)
T(
s)=
Y(
s)
=
U(
s)
Sy
st
em
2
G1(
s)=sy
s1
Y(
s)
G2(
s)=sy
s2
[
sy
s]
=ser
i
es(
sy
s1,
sy
s2)
Feedbacki
nLTIModel
s:
Thecl
osedl
oopmodel
sy
shasuasi
nputv
ect
orandyasout
putv
ect
or
.
TheLTImodel
ssy
s1andsy
s2mustbebot
hcont
i
nuousorbot
hdi
scr
et
ewi
t
h
i
dent
i
cal
.
>>sy
s=f
eedback(
sy
s1,
sy
s2)
u
Toappl
yposi
t
i
v
ef
eedback,
uset
hesy
nt
ax
>>sy
s=f
eedback(
sy
s1,
sy
s2,
+1)
sy
s
1
sy
s
1
Wr
i
t
et
heMat
l
abcodef
orLTImodel
sgi
v
enbel
ow
G
G
>>n1=[
10]
;
>>d1=[
11]
;
>>n2=[
02]
;
Li
nearCont
r
ol
Sy
st
em
y
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
>>d2=[
13]
;
>>sy
s1=t
f
(
n1,
d1)
>>sy
s2=t
f
(
n2,
d2)
>>sy
s=sy
s1*
sy
s2
>>f
eedback(
sy
s1,
sy
s2)
>>f
eedback(
sy
s1,
sy
s2,
+1)
>>f
eedback(
sy
s1,
1)
>>f
eedback(
1,
sy
s1)
>>sy
s=[
sy
s1;
sy
s2]
Modeli
nt
er
connect
i
onFunct
i
on:
>>Sy
s=[
sy
s1,
sy
s2]Hor
i
zont
al
Concat
enat
i
on
>>Sy
s=[
sy
s1;
sy
s2]Ver
t
i
cal
concat
enat
i
on
>>Sy
s=append(
sy
s1,
sy
s2)bl
ockdi
agonal
append
El
ect
r
i
cTr
act
i
onMot
or
:
Wr
i
t
eMatl
abcodef
ort
hef
ol
l
owi
ngEl
ect
r
i
cTr
act
i
onMot
orBl
ockDi
agr
am?
G3
G1
G2
540
10
(
s+1)
1
(
2s+0.
5)
G4
0.
1
>>num1=[
10]
;
>>den1=[
11]
;
>>num2=[
1]
;
>>den2=[
20.
5]
;
>>num3=[
540]
;
>>den3=[
1]
;
>>num4=[
0.
1]
;
>>den4=[
1]
;
>>[
a,
b]
=ser
i
es(
num1,
den1,
num2,
den2)
;
>>[
c,
d]
=f
eedback(
a,
b,
num4,
den4,
1)
;
>>[
e,
f
]
=ser
i
es(
c,
d,
num3,
den3)
;
>>[
g,
h]
=cl
oop(
e,
f
,
1)
;
>>pr
i
nt
sy
s(
g,
h)
num/
den=
Li
nearCont
r
ol
Sy
st
em
5400
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
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_
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_
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_
_
_
_
_
2s^
2+2.
5s+5401.
5
Toseet
hest
epr
esponseandanal
y
zet
hesy
st
em behav
i
our
.
>>t
=[
0:
0.
005:
3]
;
>>[
y
,
x,
t
]
=st
ep(
g,
h,
t
)
;
>>pl
ot
(
t
,
y
)
>>gr
i
don
>>xl
abel
(
'
Ti
mei
nsecond'
)
>>y
l
abel
(
'
wheel
v
el
oci
t
y
'
)
LabTask/
LabExer
ci
se
Pr
obl
em #01:
Wr
i
t
ecodef
oraddi
t
i
on,subt
r
act
i
on,r
i
ghtdi
v
i
si
on,l
ef
tdi
v
i
si
onandmul
t
i
pl
i
cat
i
on
4
0.
5
oft
wosy
st
ems?Sy
s1i
sf(
t
)=tandsy
s2i
st ?
Pr
obl
em #02:
Gi
v
ef
eedbackt
osy
s2ofsy
s1andf
eedbackt
osy
s1ofsy
s2?
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
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_
Pr
obl
em #03:
Fort
hef
ol
l
owi
ngmul
t
i
l
oopf
eedbacksy
st
em,getcl
osedl
oopt
r
ansf
erf
unct
i
on
andt
hecor
r
espondi
ngpol
ezer
omapoft
hesy
st
em.
Wher
e:
G1=1/
s2+s;
s+1/
s+6:
G2=1/
s+1;
H1=s+1/
s+2;
H2=2:
H3=1
G3=s2+1/
s2+4s+4:
G4=
H2
R(
s)
G1
G2
G3
G4
H1
H3
Li
nearCont
r
ol
Sy
st
em
Y(
s)
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
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_
Lab#04
Char
act
er
i
st
i
csofcont
r
olsy
st
em open
l
oopandcl
osedl
oop
ARMATURECONTROLLEDDCMOTORWI
THDI
STRI
BUTI
ONLOADTd,
SPEEDTACHOMETERSYSTEM
+
-
1/
RA(
S)
I
a(
s)
KM
KB
Openl
oopdi
st
ur
bancest
epr
esponse:
>>r
a=1;km=10;
j
=2;
f
=.
5;
kb=.
1;
ka=54;
kt
=1;
n1=[
1]
;
d1=[
j
f
]
;
>>n2=[
(
kb*
km)
/
r
a]
;
>>d2=[
1]
;
>>[
num den]
=f
eedback(
n1,
d1,
n2,
d2)
num =0 1
den=2.
0000 1.
5000
>>num=num
Li
nearCont
r
ol
Sy
st
em
t
m(
s)
+
-
1/
j
s+f
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
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_
_
_
_
_
_
_
_
num =0 1
>>pr
i
nt
sy
s(
num,
den)
num/
den=
1
2s+1.
5
>>t
=0:
1:
10;
>>[
y
,
x,
t
]
=st
ep(
num,
den)
;
>>pl
ot
(
t
,
y
)
>>t
i
t
l
e(
'
openl
oopdi
st
ur
bancest
epr
esponse'
)
>>xl
abel
'
sec'
>>y
l
abel
'
speed'
>>gr
i
don
Cl
osedl
oopdi
st
r
i
but
i
onsy
st
em:
R
a
ka
Va
(
s)
1
r
a
kb
I
NPUT:
>>n1=[
1]
;
>>d1=[
j
f
]
;
>>n2=[
(
ka*
kt
)
]
;
>>d2=[
1]
;
>>n3=[
kb]
;
>>d3=[
1]
;
>>n4=[
km/
r
a]
;
>>d4=[
1]
;
>>[
ab]
=par
al
l
el
(
n2,
d2,
n3,
d3)
a=54.
1000
b=1
>>[
cd]
=ser
i
es(
a,
b,
n4,
d4)
c=541
d=1
>>[
num den]
=f
eedback(
n1,
d1,
c,
d)
num =0 1
den=2.
0000541.
5000
Li
nearCont
r
ol
Sy
st
em
1
j
s+f
km
Td(
s
)
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
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_
_
_
_
_
_
_
_
_
>>num=num
num =0 1
>>pr
i
nt
sy
s(
num,
den)
num/
den=
1
2s+541.
5
>>t
=0:
1:
10;
>>[
y
,
x,
t
]
=st
ep(
num,
den)
;
>>pl
ot
(
t
,
y
)
>>t
i
t
l
e(
'
cl
osel
oopdi
st
ur
bancest
epr
esponse'
)
>>xl
abel
'
t
i
me'
>>y
l
abel
'
speed'
)
;
s>>gr
i
don
LabTask/
LabExer
ci
se
Sy
t
em Sensi
t
i
v
i
t
yofPl
antVar
i
at
i
onEngl
i
shBor
i
ngMachi
nes
I
NPUT:
k=50
num=[
1120]
denum=[
112k]
w=[
0.
1:
0.
05:
20]
s=w*
i
n=s.
^
2+12*
s
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
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_
_
d=s.
^
2+12*
s+k
s=n.
/
d
n2=12*
s
d2=k
s2=n2.
/
d2
subpl
ot
(
1,
2,
1)
pl
ot
(
r
eal
(
s)
,
i
mag(
s)
)
t
i
t
l
e(
'
sy
st
em sensi
t
i
v
i
t
yt
opl
antv
ar
i
at
i
ons'
)
xl
abel
(
'
r
eal
(
s)
'
)
y
l
abel
(
'
i
mag(
s)
'
)
gr
i
don
subpl
ot
(
1,
2,
2)
pl
ot
(
w,
abs(
s)
,
w,
abs(
s2)
)
xl
abel
(
'
w[
r
ad/
sec]
'
)
y
l
abel
(
'
abs(
s)
'
)
t
i
t
l
e(
'
magni
t
udeoft
hesy
st
em'
)
gr
i
don
OUTPUT:
Fi
ndt
het
r
ansf
erf
unct
i
onandshowt
hest
epr
esponseoft
heRLCci
r
cui
t
(
r
esonanceci
r
cui
t
)
.
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
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_
BYUSI
NGSTATESPACEREPRESENTATI
ON:
0
1
1
R X + L U(
X= 1
T)
LC
L
0
⌈ ⌉ ⌊⌋
Y=[
1 0]X +0
R=3,L=1andC=1/
2
I
NPUT
R=3
C=1/
2
L=1
A=[
01;
1/
L*
CR/
L;
]
B=[
0;
1/
L]
C=[
10]
D=[
0]
sy
s=ss(
A,
B,
C,
D)
st
ep(
sy
s)
STEPRESPONSEOFRLCCI
RCUI
T:
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
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_
OBSERVATI
ON:
……………………………………………………………………………
…………………………………………………………………………………………………………………………………………
……………………………………………………………………………………………
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
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_
Lab#05
I
mpl
ement
at
i
onofPI
Dcont
r
ol
l
erv
i
a
commands
Obj
ect
i
v
e:
Toobser
v
et
heef
f
ectofdi
f
f
er
entcont
r
ol
l
eronRi
set
i
me,Ov
er
shoot
,
Set
t
l
i
ngt
i
meandSt
eadySt
at
eer
r
or
.
Pr
ocedur
e:
Thet
r
ansf
erf
unct
i
onofPI
Dcont
r
ol
l
erl
ooksl
i
ket
hef
ol
l
owi
ng
Kp+Ki
/
s+Kd*
s
Wher
eKpi
st
hepr
opor
t
i
onalgai
n,Kdi
st
heder
i
v
at
i
v
egai
nandKii
s
t
hei
nt
egr
algai
n.Theef
f
ecton t
hecl
osedl
oop r
esponseofaddi
ng t
ot
he
cont
r
ol
l
ert
er
m Kp,
Ki
andKdi
sl
i
st
edbel
ow
ClResponse
Kp
Ki
Kd
Ri
seTi
me
Decr
eases
Decr
eases
NoChange
Ov
er
shoot Set
t
l
i
ngTi
me SSEr
r
or
I
ncr
eases
NoChange Decr
eases
I
ncr
eases
I
ncr
eases
El
i
mi
nat
es
Decr
eases
Decr
eases NoChange
OpenLoopSt
epResponse:
Let
’
st
akeasecondor
derpl
antl
i
ke
1/
s^
2+10s+20
Let
sf
i
r
stv
i
ewt
heopenl
oopst
epr
esponse
>>num=1;
>>den=[
11020]
;
>>g=t
f
(
num,
den)
>>st
ep(
g)
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
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_
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_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
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_
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_
_
_
TheDCgai
noft
hepl
anti
s0.
05=1/
20,
so0.
05i
st
hef
i
nal v
al
ueoft
heout
put
f
orauni
tst
epi
nput
.Thi
scor
r
espondst
oast
eadyst
at
eer
r
orof0.
95,
qui
t
el
ar
ge
i
ndeed.Fur
t
her
mor
e,
t
her
i
set
i
mei
saboutonesecondandset
t
l
i
ngt
i
mei
sabout
1.
5second.Mostl
i
kel
y
,
t
her
esponsewi
l
lnotbeadequat
e.Ther
ef
or
e,
weneedt
o
addsomecont
r
ol
.
Pr
opor
t
i
onal
Cont
r
ol
:
AsKp wi
l
lhel
pt
or
educet
hest
eady
st
at
eer
r
orand
decr
easest
her
i
set
i
me5.Let
sf
i
r
staddapr
opor
t
i
onal
cont
r
ol
l
eri
nt
ot
hesy
st
em.
>>num=1;
>>den=[
11020]
;
>>Kp=10;
>>[
numcl
,
denncl
]
=cl
oop(
Kp*
num,
den,
1)
;
>>t
=0:
0.
01:
2;
>>st
ep(
numcl
,
dencl
,
t
)
num=[
1]
;
kp=10;%Changegai
nandseer
esponse
denum=[
11020]
;
[
numcl
,
denumcl
]
=cl
oop(
kp*
num,
denum,
1)
;
t
=0:
0.
01:
2;
st
ep(
numcl
,
denumcl
,
t
)
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
Now t
her
i
set
i
mehasbeenr
educedandt
hest
eadyst
at
eer
r
ori
s
smal
l
er
,i
fweusegr
eat
erKp,t
her
i
set
i
meandst
eadyst
at
eer
r
orwi
l
lbecome
ev
ensmal
l
er
.Changet
heKp=200.Thi
st
i
meweseet
hatt
her
i
set
i
mei
snow
about0.
1secondandt
hest
eadyst
at
eer
r
ori
smuchsmal
l
erbutt
heov
er
shoot
hasbeenv
er
yl
ar
genow.
PDCont
r
ol
:
Ther
i
set
i
mei
snow pr
obabl
ysat
i
sf
act
or
y(
r
i
set
i
mei
sabout0.
1
second)
.
Now l
et
’
s add a der
i
v
at
i
v
e cont
r
ol
l
ert
ot
he sy
st
em t
o see i
ft
he
ov
er
shootcanber
educed.Addanot
herv
ar
i
abl
e,Kd,t
ot
hemf
i
l
e,seti
tequalt
o
10andr
et
ur
nt
hemf
i
l
e:
>>Kp=500;
>>Kd=10;
>>Numc=[
KdKp]
;
>>[
numCL,
denCL]
=cl
oop(
conv
(
num,
den)
,
den)
;
>>st
ep(
numCL,
denCL,
t
)
num=[
1]
;
kp=50;%Changegai
nandseer
esponse
denum=[
11020]
;
[
numcl
,
denumcl
]
=cl
oop(
kp*
num,
denum,
1)
;
t
=0:
0.
01:
2;
st
ep(
numcl
,
denumcl
,
t
)
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
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_
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_
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_
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_
_
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_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
Theov
er
shooti
smuchl
esst
hanbef
or
e.i
ti
snow onl
yt
went
yper
cent
i
nst
eadofal
mostf
or
t
y
f
i
v
eper
cent
.Wecannow t
r
yi
mpr
ov
i
ngt
hatev
enmor
e.
Tr
yi
ncr
easi
ngKdt
o100,
y
ouwi
l
l
seet
heov
er
shootel
i
mi
nat
edcompl
et
el
y
.
We now hav
e a sy
st
em wi
t
haf
astr
i
se t
i
me and no ov
er
shoot
.
Unf
or
t
unat
el
y
,t
her
ei
sst
i
l
labouta5per
centst
eady
st
at
eer
r
or
.I
twoul
dseem
t
hataPDcont
r
ol
l
eri
snotsat
i
sf
act
or
yf
ort
hi
ssy
st
em.Let
’
st
r
yaPIcont
r
ol
l
er
i
nst
ead.
PICont
r
ol
:
Aswehav
eseen,pr
opor
t
i
onalcont
r
olwi
l
lr
educet
hest
eady
st
at
eer
r
or
,
butatt
hecostofl
ar
gerov
er
shoot
.Fur
t
her
mor
e,pr
opor
t
i
onalgai
nwi
l
lnev
er
compl
et
el
yel
i
mi
nat
et
hest
eady
st
at
eer
r
or.
Fort
hatweneedt
ot
r
yi
nt
egr
al
cont
r
ol
.Let
’
si
mpl
ementaPIcont
r
ol
l
erandst
ar
twi
t
hasmal
l
ki
.
>>Kp=500;
>>Ki
=1;
>>Kd=0;
>>Numc[
KdKpKi
]
;
>>Denc=[
10]
;
>>[
numCL,
denCL]
=cl
oop(
conv
(
num,
numc)
,
conv
(
den,
denc)
)
;
>>st
ep(
numCL,
denCL)
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
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_
_
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_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
Tr
yKi
=10,
>>Kp=500;
>>Ki
=10;
>>.
Kd=0;
>>Numc[
KdKpKi
]
;
>>Denc=[
10]
;
>>[
numCL,
denCL]
=cl
oop(
conv
(
num,
numc)
,
conv
(
den,
denc)
)
;
>>st
ep(
numCL,
denCL)
>>axi
s(
[
010001.
5]
)
num=[
1]
;
kp=500;%Changegai
nandseer
esponse
denum=[
11020]
;
[
numcl
,
denumcl
]
=cl
oop(
kp*
num,
denum,
1)
;
t
=0:
0.
01:
2;
st
ep(
numcl
,
denumcl
,
t
)
Fr
om t
het
wocont
r
ol
l
er
sabov
e,weseet
hati
fwewantaf
astr
esponse,
smal
lov
er
shot
,andnost
eady
st
at
eer
r
or
,nei
t
heraPInoraPDcont
r
ol
l
erwi
l
l
suf
f
i
ce.Let
’
si
mpl
ementbot
hcont
r
ol
l
er
sanddesi
gnaPI
Dcont
r
ol
l
ert
oseei
f
combi
ni
ngt
het
wocont
r
ol
l
er
swi
l
ly
i
el
dt
hedesi
r
edr
esponse.Recal
l
i
ngt
hat
,our
PDcont
r
ol
l
ergav
eusapr
et
t
ygoodr
esponseexceptf
oral
i
t
t
l
est
eady
st
at
eer
r
or
.
>>Kp=500;
>>Ki
=1;
>>Kd=100;
>>Numc=[
KdKpKi
]
;
>>Denc[
10]
;
>>[
numCL,
denCL]
=cl
oop(
conv
(
num,
numc)
,
conv
(
den,
denc)
)
:
>>st
ep(
numCL,
denCL)
num=[
1]
;
kp=100;%Changegai
nandseer
esponse
denum=[
11020]
;
[
numcl
,
denumcl
]
=cl
oop(
kp*
num,
denum,
1)
;
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
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_
_
_
_
_
_
_
_
_
_
_
_
t
=0:
0.
01:
2;
st
ep(
numcl
,
denumcl
,
t
)
Todesi
gnaPI
Dcont
r
ol
l
er
,t
hegener
alr
ul
ei
st
oaddpr
opor
t
i
onalcont
r
ol
t
ogett
hedesi
r
edr
i
set
i
me,addder
i
v
at
i
v
econt
r
olt
ogetdesi
r
eov
er
shoot
,and
t
henaddi
nt
egr
al
cont
r
ol
(
i
fneeded)
.
Toel
i
mi
nat
et
hest
eady
st
at
eer
r
or
.
Pr
opor
t
i
onal
Der
i
v
at
i
v
eCont
r
ol
l
er
:
num=[
1]
;
denum=[
11020]
;
kp=100;%Youcansetr
i
set
i
me
kd=10;
num1=[
kdkp]
;
[
numcl
,
denumcl
]
=cl
oop(
conv
(
num1,
num)
,
denum)
;
t
=0:
0.
01:
2;
st
ep(
numcl
,
denumcl
,
t
)
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
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_
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_
_
_
_
_
_
_
_
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_
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_
_
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_
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_
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_
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_
_
_
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_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
num=[
1]
;
denum=[
11020]
;
kp=100;%Youcansetr
i
set
i
me
kd=100;
num1=[
kdkp]
;
[
numcl
,
denumcl
]
=cl
oop(
conv
(
num1,
num)
,
denum)
;
t
=0:
0.
01:
2;
st
ep(
numcl
,
denumcl
,
t
)
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
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_
_
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_
_
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_
_
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_
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_
_
_
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_
_
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_
_
_
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_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
num=[
1]
;
denum=[
11020]
;
kp=500;%Youcansetr
i
set
i
me
kd=100;
num1=[
kdkp]
;
[
numcl
,
denumcl
]
=cl
oop(
conv
(
num1,
num)
,
denum)
;
t
=0:
0.
01:
2;
st
ep(
numcl
,
denumcl
,
t
)
PICont
r
ol
l
er
:
num=[
1]
;
denum=[
11020]
;
kp=500;
ki
=10;
num1=[
kpki
]
;
denum1=[
10]
;
[
numcl
,
denumcl
]
=cl
oop(
conv
(
num1,
num)
,
conv
(
denum1,
denum)
)
;
t
=0:
0.
01:
2;
st
ep(
numcl
,
denumcl
,
t
)
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
PI
DCont
r
ol
l
er
:
num=[
1]
;
denum=[
11020]
;
kp=500;
kd=100;
ki
=10;
num1=[
kdkpki
]
;
denum1=[
10]
;
[
numcl
,
denumcl
]
=cl
oop(
conv
(
num1,
num)
,
conv
(
denum1,
denum)
)
;
t
=0:
0.
01:
2;
st
ep(
numcl
,
denumcl
,
t
)
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
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_
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_
_
_
Exer
ci
se:
Q#1:
G(
s)=F(
s)/X(
s)=1/(
ms^
2+bs+k)
Cal
cul
at
et
hewav
ef
or
msf
ordi
f
f
er
entv
al
uesofm,
bandk,
whenr
ampi
nputi
s
appl
i
ed.
Case1:
El
i
mi
nat
espr
i
ng(
k=0)wher
em =1andb=1
Case2:
El
i
mi
nat
edampi
ng(
b=0)wher
em =1andk=5
Case3:
El
i
mi
nat
emass(
m=0)wher
eb=1andk=5
Obser
v
et
hewav
ef
or
msf
ordi
f
f
er
entv
al
uesofm,
bandk,
whenst
epi
nputi
s
appl
i
ed.
Case1:
El
i
mi
nat
espr
i
ng(
k=0)wher
em =1andb=1
Case2:
El
i
mi
nat
edampi
ng(
b=0)wher
em =1andk=5
Case3:
El
i
mi
nat
emass(
m=0)wher
eb=1andk=5
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
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_
_
Lab#06
I
mpl
ement
at
i
onofPI
Dcont
r
ol
l
erv
i
a
si
mul
i
nk
Obj
ect
i
v
e:
Toknowaboutt
hesi
mul
i
nkbasi
csandknowhowt
ocr
eat
eamodel
.
Pr
ocedur
e:
Si
mul
i
nki
sagr
aphi
calext
ensi
ont
oMATLABf
ormodel
i
ngandsi
mul
at
i
on
ofsy
st
ems.I
nSi
mul
i
nk,sy
st
emsar
edr
awnonscr
eenasbl
ockdi
agr
ams.Many
el
ement
sofbl
ockdi
agr
amsar
eav
ai
l
abl
e,suchast
r
ansf
erf
unct
i
ons,summi
ng
j
unct
i
ons et
c.Si
mul
i
nk i
si
nt
egr
at
ed wi
t
h MATLAB and dat
a can be easi
l
y
t
r
ansf
er
r
edbet
weent
hepr
ogr
ams.
Thesi
mpl
emodelconsi
st
soft
hr
eebl
ocks:St
ep,t
r
ansf
erf
unct
i
on,and
scope.Thest
epi
sasour
cebl
ockf
r
om whi
chast
epi
nputsi
gnalor
i
gi
nat
es.Thi
s
si
gnali
st
r
ansf
er
r
edt
hr
ought
hel
i
nei
nt
hedi
r
ect
i
oni
ndi
cat
edbyt
hear
r
owt
ot
he
t
r
ansf
erf
unct
i
onl
i
nearbl
ock.Thet
r
ansf
erf
unct
i
onmodi
f
i
est
hei
nputsi
gnal
gi
v
esanew si
gnalont
hel
i
net
ot
hescope.Thescopei
sasi
nkbl
ockusedt
o
di
spl
ayasi
gnal
muchl
i
keanosci
l
l
oscope.
Pr
opor
t
i
onal
Cont
r
ol
:
Thepr
opor
t
i
onal
gai
ncanbeusedasi
nt
hemodel
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
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_
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_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
Out
putGr
aph:
Forgai
n10:
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
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_
_
_
_
_
_
_
_
_
_
_
Out
putGr
aph:
ForGai
n50
Out
putGr
aph:
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
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_
_
_
_
_
_
_
_
_
_
_
ForGai
n100:
Out
putGr
aph:
Gai
n150:
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
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_
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_
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_
_
_
_
_
_
_
_
_
_
_
_
Out
putGr
aph:
Nowr
i
set
i
mei
ssmal
l
erbutov
er
shooti
spr
oducedsot
odecr
easet
heov
er
shoot
weuseder
i
v
at
i
v
ecompensat
or
.
PDCont
r
ol
:
beusedas
Thepr
opor
t
i
onalandder
i
v
at
i
v
ecompensat
orcanbeusedascan
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
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_
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_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
Out
putGr
aph:
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
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_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
Out
putGr
aph:
Af
t
ercont
r
ol
l
i
ngr
i
set
i
meov
er
shootandset
t
l
i
ngt
i
menowcomet
owor
ds
t
hei
nt
egr
al
cont
r
ol
t
oel
i
mi
nat
et
hest
eadyst
at
eer
r
or
.
PICont
r
ol
:
Thepr
opor
t
i
onal
andi
nt
egr
al
compensat
orcanbeusedas
Out
putGr
aph:
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
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_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
Out
putGr
aph:
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
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_
_
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_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
Nowcombi
net
het
hr
eecompensat
or
st
oseet
hewhol
er
esul
t
.
PI
DCont
r
ol
:
Out
putGr
aph:
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
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_
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_
_
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_
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_
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_
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_
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_
_
_
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_
_
_
_
_
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_
_
_
_
_
_
_
_
_
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_
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_
_
_
_
_
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_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
Out
putGr
aph:
Out
putGr
aph:
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
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_
_
_
Exer
ci
se:
Q#1:
G(
s)=F(
s)/X(
s)=1/(
ms^
2+bs+k)
Cal
cul
at
et
hewav
ef
or
msf
ordi
f
f
er
entv
al
uesofm,bandk,whenr
ampi
nputi
s
appl
i
ed.
Case1:
El
i
mi
nat
espr
i
ng(
k=0)wher
em =1andb=1
Case2:
El
i
mi
nat
edampi
ng(
b=0)wher
em =1andk=5
Case3:
El
i
mi
nat
emass(
m=0)wher
eb=1andk=5
Obser
v
et
hewav
ef
or
msf
ordi
f
f
er
entv
al
uesofm,bandk,whenst
epi
nputi
s
appl
i
ed.
Case1:
El
i
mi
nat
espr
i
ng(
k=0)wher
em =1andb=1
Case2:
El
i
mi
nat
edampi
ng(
b=0)wher
em =1andk=5
Case3:
El
i
mi
nat
emass(
m=0)wher
eb=1andk=5
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
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_
_
_
_
Lab#07
TOI
MPLEMENTTHEROOTLOCUSTECHNI
QUEI
N
MATLAB
Obj
ect
i
v
e:
Byusi
ngt
heknowl
edgepr
ov
i
dedbyt
heopenl
ooppol
esandopenl
oop
zer
ost
heper
f
or
manceandst
abi
l
i
t
yoft
hesy
st
em canbef
ound
Theor
y
:
I
ncont
r
ol
t
heor
yandst
abi
l
i
t
yt
heor
y
,
r
ootl
ocusanal
y
si
si
sagr
aphi
cal
met
hodf
orexami
ni
nghowt
her
oot
sofasy
st
em changewi
t
hv
ar
i
at
i
onofa
cer
t
ai
nsy
st
em par
amet
er
,
commonl
yagai
nwi
t
hi
naf
eedbacksy
st
em.Thi
s
t
echni
quei
sv
er
yusef
ul
t
of
i
ndt
hest
abi
l
i
t
yi
nf
or
mat
i
onandal
sopr
ov
i
desv
er
y
usef
ul
i
nf
or
mat
i
onaboutsy
st
em par
amet
er
s.Asonesay
si
fy
oucanmeasur
ei
t,
y
oucancont
r
ol
i
tsot
hi
st
echni
quei
sv
er
yusef
ul
i
nt
hi
sr
egar
d.
RootLocusAnal
y
si
s:
1.Ther
oot
soft
heoft
hecl
osedl
oopchar
act
er
i
st
i
cequat
i
ondef
i
net
he
sy
st
em char
act
er
i
st
i
cr
esponses.
2.Thei
rl
ocat
i
oni
nt
hecompl
exspl
anel
eadt
opr
edi
ct
i
onoft
he
char
act
er
i
st
i
csoft
het
i
medomai
nr
esponsesi
nt
er
msof
:
I
.
Dampi
ngr
at
i
on,
I
I
. Nat
ur
al
f
r
equency
,
wn
I
I
I
. Dampi
ngconst
ant
,
f
i
r
st
or
dermodes
I
V. Consi
derhowt
heser
oot
schangeast
hel
oopgai
ni
sv
ar
i
edf
r
om 0t
o
Basi
csofRootLocus:
• Sy
mmet
r
i
cal
aboutr
eal
axi
s
• RLbr
anchst
ar
t
sf
r
om OLpol
esandt
er
mi
nat
esatOLzer
oes
• No.ofRLbr
anches=No.ofpol
esofOLTF
• Cent
r
oi
di
scommoni
nt
er
sect
i
onpoi
ntofal
l
t
heasy
mpt
ot
esont
her
eal
axi
s
• Asy
mpt
ot
esar
est
r
ai
ghtl
i
neswhi
char
epar
al
l
el
t
oRLgoi
ngt
o∞ and
meett
heRLat∞
• No.ofasy
mpt
ot
es=No.ofbr
anchesgoi
ngt
o∞
• AtBr
eakAwaypoi
nt,
t
heRLbr
eaksf
r
om r
eal
axi
st
oent
eri
nt
ot
he
compl
expl
ane
• AtBIpoi
nt
,
t
heRLent
er
st
her
eal
axi
sf
r
om t
hecompl
expl
ane
Equat
i
oncanbewr
i
t
t
enas
1+K*
(
num/
den)
=0
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
Wher
enum i
st
henumer
at
oroft
hepol
y
nomi
al
anddeni
st
hedenomi
nat
or
pol
y
nomi
al
,
andKi
st
hegai
n(
K>0)
.Thev
ect
orKcont
ai
nsal
l
t
hegai
nv
al
ues
f
orwhi
cht
hecl
osedl
ooppol
esar
et
obecomput
ed.
Ther
ootl
oci
i
spl
ot
t
edbyusi
ngt
heMATLAB
commandr
l
ocus(
num,
den)
Thegai
nv
ect
orKi
ssuppl
i
edbyt
heuser
.
Themat
r
i
xrandgai
nv
ect
orKar
eobt
ai
nedbyt
hef
ol
l
owi
ngMATLAB
commands:
[
r
,
k]=r
l
ocus(
num,
den)
[
r
,
k]=r
l
ocus(
num,
den,
k)
[
r
,
k]=r
l
ocus(
A,
B,
C,
D)
[
r
,
k]=r
l
ocus(
A,
B,
C,
D,
K)(
3.
23)
[
r
,
k]=r
l
ocus(
sy
s
Forpl
ot
t
i
ngt
her
ootl
oci
,
t
heMATLABcommandpl
ot(
r
,
‘
‘
)i
sused.Thef
ol
l
owi
ngMATLABcommandar
eused
f
orpl
ot
t
i
ngt
her
ootl
oci
wi
t
hmar
k‘
0’
or‘
x’
:
r=r
l
ocus
(
num,
den)pl
ot(
r
,
‘
0’
)orpl
ot(
r
,
‘
x’
)Exampl
e#01:
G(
s)
H(
s)
=K(
s+1)
/
s(
s+2)
(
s+3)
>>num=[
11]
;
>>denum=[
1560]
;
>>GH=t
f
(
num,
denum)
GH=
s+1
s^
3+5
s^
2+6s
>>pzmap(
num,
denum)
>>r
l
ocus(
num,
denum)
Exampl
e#02:
Consi
dert
hesy
st
em showni
nFi
gur
e.
Pl
otr
ootl
oci
wi
t
hasquar
e
aspectr
at
i
osot
hatal
i
newi
t
hsl
ope1i
sat
r
ue45"l
i
ne.Chooset
her
egi
onofr
oot
l
ocuspl
ott
obe6≤x≤6 &6≤y
≤6wher
exandyar
et
her
eal
axi
scoor
di
nat
eand
i
magi
nar
y
axi
scoor
di
nat
e,
r
espect
i
v
el
y
.
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
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_
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_
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_
_
_
_
_
>>a=[
110]
;
>>b=[
1416]
;
>>c=conv
(
a,
b)
c=
1 5 20 16 0
>>denum=c
denum =
1 5 20 16 0
>>num=[
13]
;
>>denum
denum =
1 5 20 16 0
>>pzmap(
num,
denum)
>>r
l
ocus(
num,
denum)
>>v
=[
6666]
v=
6 6 6 6
>>axi
s(
v
)
;
axi
s(
'
squar
e'
)
;
>>gr
i
d;
>>t
i
t
l
e(
'
RootLocusPl
otofG(
s)
'
)
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
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_
_
_
_
LabTask/
LabExer
ci
se
Pr
obl
em #01:
C(
s)
=k/
s3+8s2+10s+1i
)Pl
ott
her
oot
l
ocusi
i
)Fi
ndt
heVal
uewher
e
dampi
ngr
at
i
oi
s0.
707.
Pr
obl
em #02:
Apl
antt
obecont
r
ol
l
edi
sdescr
i
bedbyat
r
ansf
erf
unct
i
on
2
G(
s)=s+5/
s +7s+25
Obt
ai
nt
her
ootl
ocuspl
otusi
ngMATLAB.
Pr
obl
em #03:
Consi
dert
hesy
st
em whoseopenl
oopt
r
ansf
erf
unct
i
onG(
s)
H(
s)i
s
G(
s)
H(
s)
=K/
s(
s+0.
5)
(
s2+0.
6s+10)
Ther
ear
enoopenl
oopzer
os.Openl
ooppol
esar
el
ocat
edats=0.
3+j
3.
1480,
s=0.
3-j
3.
1480,
s=0.
5,
ands=0.
Obt
ai
nt
her
oot
l
ocuspl
ot
.
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
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_
Lab#08
Tounder
st
andt
hef
r
equencyr
esponseofa
cont
r
olsy
st
em usi
ngBodePl
ot&Ny
qui
stPl
ot
Obj
ect
i
v
e:
Tof
i
ndt
hegai
nmar
gi
nandphasemar
gi
nofasy
st
em.Tounder
st
andt
he
conceptofgai
ncr
ossov
erandphasecr
ossov
erf
orst
abi
l
i
t
y
.
Theor
y
:
ABodedi
agr
am consi
st
soft
wogr
aphs:Onei
sapl
otoft
hel
ogar
i
t
hm of
t
hemagni
t
udeofasi
nusoi
dalt
r
ansf
erf
unct
i
on;t
heot
heri
sapl
otoft
hephase
angl
e;bot
har
epl
ot
t
edagai
nstt
hef
r
equencyonal
ogar
i
t
hmi
cscal
e.Bodepl
ot
s
ar
eav
er
yusef
ulwayt
or
epr
esentt
hegai
nandphaseofasy
st
em asaf
unct
i
on
off
r
equency
.Thi
si
sr
ef
er
r
edt
oast
hef
r
equencydomai
nbehav
i
ourofasy
st
em.
Bodecomput
est
hemagni
t
udeandphaseoft
hef
r
equencyr
esponseof
LTImodel
s.Wheny
oui
nv
oket
hi
sf
unct
i
onwi
t
houtl
ef
t
si
dear
gument
s,bode
pr
oducesaBodepl
otont
hescr
een.Themagni
t
udei
spl
ot
t
edi
ndeci
bel
s(
dB)
,
andt
hephasei
ndegr
ees.Thedeci
belcal
cul
at
i
onf
ormagi
scomput
edas
20l
og10magni
t
udeofH(
j
w)
,wher
eH(
j
w)i
st
hesy
st
em'
sf
r
equencyr
esponse.
Youcanusebodepl
ot
st
oanal
y
zesy
st
em pr
oper
t
i
essuchast
hegai
nmar
gi
n,
phasemar
gi
n,
DCgai
n,
bandwi
dt
h,
di
st
ur
bancer
ej
ect
i
on,
andst
abi
l
i
t
y
.
Thecommandbodecomput
esmagni
t
udesandphaseangl
esoft
hef
r
equency
r
esponseofcont
i
nuoust
i
me,
l
i
near
,
t
i
mei
nv
ar
i
antsy
st
ems.Whent
hecommand
bode (
wi
t
houtl
ef
t
hand ar
gument
s)i
s ent
er
ed i
nt
he comput
er
,MATLAB
pr
oducesaBodepl
otont
hescr
een.Mostcommonl
yusedbodcommandsar
e
bode(
num,
denum)
bode(
num,
denum,
w)
bode(
A,
B,
C,
D)
bode(
A,
B,
C,
D,
w)
bode(
sy
s)
Exampl
e#01:
G(
s)=25/
(
s2+4s+25
>>num=[
25]
;
>>denum=[
1425]
;
>>G=t
f
(
num,
denum)
G=
25
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
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_
_
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_
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_
_
_
_
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_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
s^
2+4s+25
>>bode(
num,
denum)
>>gr
i
don
>>bode(
G)
>>mar
gi
n(
G)
>>[Gm,
Pm,
Wcp,
Wcg]
=mar
gi
n(
G)
Exampl
e#02:
G(
s)=9(
s2+0.
2s+1)/
s(
s2+1.
2s+9)
>>num=[
91.
89]
;
>>denum=[
11.
290]
;
>>G=t
f
(
num,
denum)
G=
9s^
2+1.
8s+9
s^
3+1.
2s^
2+9s
>>bode(
num,
denum)
>>gr
i
don
>>bode(
G)
>>mar
gi
n(
G)
>>[Gm,
Pm,
Wcp,
Wcg]
=mar
gi
n(
G)
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
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_
_
_
_
Exampl
e#03:
G(
s)=1/
(
s2+0.
8s+1)
>>num=[
1]
;
>>denum=[
10.
81]
;
>>G=t
f
(
num,
denum)
G=
1
s^
2+0.
8s+1
>>ny
qui
st
(
num,
denum)
Exampl
e#04:
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
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_
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_
_
_
_
_
_
_
_
_
G(
s)=1/s(
s+1)
>>num=[
1]
;
>>denum=[
110]
;
>>G=t
f
(
num,
denum)
G=
1
s^
2+s
>>ny
qui
st
(
num,
denum)
LabTask/
LabExer
ci
se
Pr
obl
em #01:
Obt
ai
nBodeandNy
qui
stpl
otf
ort
hegi
v
enopenl
oopt
r
ansf
erf
unct
i
oni
s
G(
s)=50/
(
s3+9s2+30s+40)
Pr
obl
em #02:
Obt
ai
nBodeandNy
qui
stpl
otf
ort
hegi
v
enopenl
oopt
r
ansf
erf
unct
i
oni
s
2
2
G(
s)=1/
(
s+4s)
(
s+4s+13)
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
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_
Lab#09
Model
i
ngofSPRI
NGMASSDAMPERSYSTEM
OBJECTI
VE:
Toknowaboutt
hemodel
soft
hemechani
cal
andel
ect
r
i
cal
sy
st
ems:
-
Letf=appl
i
edf
or
ce
f
m =opposi
ngf
or
ceduet
omass
2
dx
ByNewt
on‘
ssecondl
awher
ef
m =f=M 2
dt
Mechani
cal
Rot
at
i
onalSy
st
ems
LetT=appl
i
edt
or
que
Tj
=opposi
ngt
or
queduet
omomentofi
ner
t
i
aoft
hebody
ByNewt
on‘
sl
aw
2
dθ
T=Tj=J 2
dθ
Model
i
ngofel
ect
r
i
calsy
st
em
 El
ect
r
i
calci
r
cui
t
si
nv
ol
v
i
ng r
esi
st
or
s,capaci
t
or
s and i
nduct
or
s ar
e
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
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_
consi
der
ed.Thebehav
i
ourofsuchsy
st
emsi
sgov
er
nedbyOhm‘
sl
awand
Ki
r
chhof
f
‘
sl
aws
 Consi
derar
esi
st
anceofRΩcar
r
y
i
ngcur
r
entiAmpsasshowni
nFi
g,
t
hen
t
hev
ol
t
agedr
opacr
ossi
ti
sV=RI
 Consi
derani
nduct
orLHcar
r
y
i
ngcur
r
enti
Ampsasshowni
nFi
g,
t
hent
he
v
ol
t
agedr
opacr
ossi
tcanbewr
i
t
t
enasv=Ldi
/
dt
 Consi
deracapaci
t
orCFcar
r
y
i
ngcur
r
entiAmpsasshowni
nFi
g,
t
hent
he
∫
i
v
ol
t
agedr
opacr
ossi
tcanbewr
i
t
t
enasv=(
1/
C)
dt
Quest
i
onno01:
Spr
i
ngmasssy
st
em Model
:Wr
i
t
et
heMat
hemat
i
cal
model
si
mul
at
i
onofspr
i
ng
massdampersy
st
em usi
ngMATLAB?
MATLABCODE:
y
o=0.
15
wn=sqr
t
(
2)
zet
a1=3/
(
2*
sqr
t
(
2)
)
zet
a2=1/
(
2*
sqr
t
(
2)
)
t
=[
0:
0.
001:
10]
t
1=acos(
zet
a1)
*
ones(
1,
l
engt
h(
t
)
)
t
2=acos(
zet
a2)
*
ones(
1,
l
engt
h(
t
)
)
c1=y
o.
/
sqr
t
(
1(
zet
a1)
^
2)
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
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_
_
_
_
_
_
_
_
_
_
_
_
c2=y
o.
/
sqr
t
(
1(
zet
a2)
^
2)
y
1=c1.
*
exp(
zet
a1*
wn*
t
)
.
*
si
n(
wn*
sqr
t
(
1(
zet
a1)
^
2)
*
t
+t
1)
y
2=c2.
*
exp(
zet
a2*
wn*
t
)
.
*
si
n(
wn*
sqr
t
(
1(
zet
a2)
^
2)
*
t
+t
2)
pl
ot
(
t
,
y
1,
'
r
'
,
t
,
y
2,
'
b'
)
bu=c2.
*
exp(
zet
a2*
wn*
t
)
pl
ot
(
t
,
y
1,
'
m'
,
t
,
y
2,
'
k'
,
t
,
bu,
'
k'
)
bl
=bu
pl
ot
(
t
,
y
1,
'
r
'
,
t
,
y
2,
'
k'
,
t
,
bu,
'
m'
,
t
,
bl
,
'
y
'
)
t
i
t
l
e(
'
phy
si
cal
sy
st
em'
)
xl
abel
(
'
t
i
me'
)
y
l
abel
(
'
ampl
i
t
ude'
)
Gr
aph:
-
Quest
i
onno02:
Li
nearCont
r
ol
Sy
st
em
LabTask/
LabExer
ci
se
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
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_
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_
_
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_
_
_
_
_
_
_
_
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_
_
_
_
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_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
Consi
deramechani
cal
sy
st
em depi
ct
edshowni
nt
hef
i
gur
et
hei
nputi
sgi
v
enby
f(
t
)andt
heout
puti
sgi
v
enbyy
(
t
)
.Det
er
mi
net
hedi
f
f
er
ent
i
al
equat
i
ongov
er
ni
ng
t
hesy
st
em byusi
ngMATLAB:
Wr
i
t
emf
i
l
eandpl
ott
hesy
st
em r
esponsesucht
hatf
or
ci
ngf
unct
i
on:
FI
GURE:
-
Dat
agi
v
en:
F(
t
)
=1
M=10kg
K=1n/
m
B=0.
5nsec/
m
Sol
ut
i
on:
MATLABCODE:
k=1
m=10
b=0.
5
f
=1
a=[
01;
k/
mb/
m]
b=[
0;
1/
m]
c=[
10]
d=[
0]
sy
s=ss(
a,
b,
c,
d)
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
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_
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_
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_
_
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_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
st
ep(
sy
s,
'
r
'
)
gr
i
don
GRAPH:
-
QUESTI
ONNO03:
Dat
agi
v
en
Byusi
ngdi
f
f
er
ent
i
alequat
i
onsol
v
et
hi
sequat
i
on:
DV/
DT=F/
M –B/
M*
V
M=750
B=30NSEC/
M
R=300N
K=15
ThusEquat
i
on;
My
’
’
+By
’
+Ky
=U
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
SOLUTI
ON:
MATLABCODE:
m=750
b=30
f
=300
k=15
a=[
01;
0b/
m]
b=[
1/
m;
0]
c=[
10]
d=[
0]
sy
s=ss(
a,
b,
c,
d)
st
ep(
sy
s,
'
r
'
)
GRAPH:
-
LabTask/
LabExer
ci
se
Pr
obl
em #01
Consi
dert
hef
ol
l
owi
ngci
r
cui
t
.Fi
ndouti
t
sequat
i
onandt
r
ansf
or
mi
nLapl
ace.
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
Pr
obl
em #02
Consi
dert
hear
mat
ur
econt
r
ol
l
eddcmot
orshowni
nt
hef
ol
l
owi
ngf
i
gur
e.
Fi
ndouti
t
sLapl
ace
Lab#10
Desi
gnandI
mpl
ement
at
i
onofCompensat
or
s
OBJECTI
VE:
Togett
hedesi
r
eper
f
or
manceofacont
r
ol
sy
st
em
Theor
y
:
I
ndesi
gni
ngacont
r
olsy
st
em,i
fot
hert
hanagai
nadj
ust
menti
sr
equi
r
ed,we
mustmodi
f
yt
heor
i
gi
nalr
ootl
ocibyi
nser
t
i
ngasui
t
abl
ecompensat
or
.Oncet
he
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
ef
f
ect
s on t
he r
ootl
ocus oft
he addi
t
i
on ofpol
es and/
orzer
os ar
ef
ul
l
y
under
st
ood,wecanr
eadi
l
ydet
er
mi
net
hel
ocat
i
onsoft
hepol
e(
s)andzer
o(
s)of
t
hecompensat
ort
hatwi
l
lr
eshapet
her
ootl
ocusasdesi
r
ed.I
nessence,i
nt
he
desi
gn byt
her
oot
l
ocusmet
hod,t
her
ootl
ocioft
hesy
st
em ar
er
eshaped
t
hr
ought
heuseofacompensat
orsot
hatapai
rofdomi
nantcl
osedl
ooppol
es
canbepl
acedatt
hedesi
r
edl
ocat
i
on.(
Of
t
en,t
hedampi
ngr
at
i
oandundamped
nat
ur
al
f
r
equencyofapai
rofdomi
nantcl
osedl
ooppol
esar
especi
f
i
ed.
)
Ef
f
ect
soft
heAddi
t
i
onofPol
es.
Theaddi
t
i
onofapol
et
ot
heopenl
oopt
r
ansf
erf
unct
i
onhast
heef
f
ectofpul
l
i
ng
t
her
ootl
ocust
ot
her
i
ght
,t
endi
ngt
ol
owert
hesy
st
em'
sr
el
at
i
v
est
abi
l
i
t
yandt
o
sl
owdownt
heset
t
l
i
ngoft
her
esponse(
Remembert
hatt
headdi
t
i
onofi
nt
egr
al
cont
r
ol
addsapol
eatt
heor
i
gi
n,
t
husmaki
ngt
hesy
st
em l
essst
abl
e.
)
Ef
f
ect
soft
heAddi
t
i
onofZer
os.
Theaddi
t
i
onofazer
ot
ot
heopenl
oopt
r
ansf
erf
unct
i
onhast
heef
f
ectofpul
l
i
ng
t
her
ootl
ocust
ot
hel
ef
t
,t
endi
ngt
omaket
hesy
st
em mor
est
abl
eandt
ospeed
upt
heset
t
l
i
ngoft
her
esponse.(
Phy
si
cal
l
y
,t
headdi
t
i
onofazer
oi
nt
hef
eedf
or
war
dt
r
ansf
erf
unct
i
onmeanst
headdi
t
i
onofder
i
v
at
i
v
econt
r
olt
ot
hesy
st
em.
Theef
f
ectofsuchcont
r
oli
st
oi
nt
r
oduceadegr
eeofant
i
ci
pat
i
oni
nt
ot
hesy
st
em
andspeedupt
het
r
ansi
entr
esponse.
)
LeadCompensat
or
s.
Ther
ear
emanyway
st
or
eal
i
z
econt
i
nuoust
i
me(
oranal
og)l
eadcompensat
or
s,
suchasel
ect
r
oni
cnet
wor
ksusi
ngoper
at
i
onalampl
i
f
i
er
s,
el
ect
r
i
calRCnet
wor
ks,
andmechani
cal
spr
i
ngdashpotsy
st
ems.
R2C2
α=
R1C1
Fr
om Equat
i
onweseet
hatt
hi
snet
wor
ki
sal
eadnet
wor
ki
fR1C1>R2C2,
ora<1.
I
ti
sal
agnet
wor
ki
fR1CI<R2C2.
Leadcompensat
i
onbasi
cal
l
yspeedsupt
her
esponseandi
ncr
easest
hest
abi
l
i
t
y
oft
hesy
st
em.
LagCompensat
or
s:
Theconf
i
gur
at
i
onoft
heel
ect
r
oni
cl
agcompensat
orusi
ngoper
at
i
onalampl
i
f
i
er
s
i
st
hesameast
hatf
ort
hel
eadcompensat
orI
fwechooseR2C2 >RI
C1 i
t
becomesal
agcompensat
or
,t
het
r
ansf
erf
unct
i
onoft
hel
agcompensat
ori
s
gi
v
enbyβ.Lagcompensat
i
oni
mpr
ov
est
hest
eady
st
at
eaccur
acyoft
hesy
st
em,
butr
educest
hespeedoft
her
esponse.
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
Exampl
e:
%Uni
tRampResponse
%Uni
tr
ampr
esponsesofcompensat
edandnoncompensat
edsy
st
em
%Uni
tr
ampr
esponsewi
l
l
beobt
ai
nedasuni
tst
epr
esponseofC(
s)
/
|
sR(
s)
|
%*
*
*
Ent
ert
henumenat
or
sanddenomi
nat
or
sofC1(
s)
/
[
sR(
s)
]and
%C2(
s)
/
[
sR(
s)
]
wher
eC1(
s)
andC2(
s)
ar
eLapl
acet
r
ansf
or
m ofout
putof
%compensat
edandnoncompensat
edsy
st
em r
espect
i
v
el
y
*
*
*
numc=[
00001.
02350.
0512]
;
denc=[
13.
0052.
0151.
033500.
5120]
;
num=[
00001.
06]
;
den=[
1321.
060]
;
%Speci
f
yt
het
i
mer
ange(
suchast
=0:
0.
1:
50;
andent
erst
epandpl
otcommand
t
=0:
0.
1:
50;
[
c1,
x1,
t
]
=st
ep(
numc,
denc,
t
)
;
[
c2,
x2,
t
]
=st
ep(
num,
den,
t
)
;
pl
ot
(
t
,
c1,
'
'
,
t
,
c2,
'
.
'
,
t
,
t
,
'
'
)
gr
i
d
t
ext
(
2.
2,
27,
'
Compensat
edsy
st
em'
)
t
ext
(
26,
21.
3,
'
Uncompensat
edsy
st
em'
)
t
i
t
l
e(
'
Uni
tr
ampr
esponsesofcompensat
edandnoncompensat
edsy
st
em'
)
xl
abel
(
'
Tsec'
)
y
l
abel
(
'
Out
put
sC1andC2'
)
LaLeadCompensat
i
on:
I
fi
mpr
ov
ement
si
n bot
ht
r
ansi
entr
esponse and st
eady
st
at
er
esponse ar
e
desi
r
ed,t
henbot
hal
eadcompensat
orandal
agcompensat
ormaybeused
si
mul
t
aneousl
y
.Rat
hert
hani
nt
r
oduci
ngbot
hal
eadcompensat
orandal
ag
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
compensat
orassepar
at
eel
ement
s,
howev
er
,
i
ti
seconomi
cal
t
ouseasi
ngl
el
agl
eadcompensat
or
.Lagl
eadcompensat
i
oncombi
nest
headv
ant
agesofl
agand
l
eadcompensat
i
ons.Si
ncet
hel
agl
eadcompensat
orpossessest
wopol
esand
t
wozer
os,suchacompensat
i
oni
ncr
easest
heor
deroft
hesy
st
em by2,unl
ess
cancel
l
at
i
onofpol
e(
s)andzer
o(
s)occur
si
nt
hecompensat
edsy
st
em.
Exampl
e:
%Uni
tst
epResponse
%Uni
tst
epr
esponsesofcompensat
edandnoncompensat
edsy
st
em
numc=[
0018.
754.
23]
;
denc=[
17.
429.
554.
23]
;
num=[
004]
;
den=[
124]
;
%Speci
f
yt
het
i
mer
ange(
suchast
=0:
0.
1:
50;
andent
erst
epandpl
otcommand
t
=0:
0.
05:
5;
[
c1,
x1,
t
]
=st
ep(
numc,
denc,
t
)
;
[
c2,
x2,
t
]
=st
ep(
num,
den,
t
)
;
pl
ot
(
t
,
c1,
'
o'
,
t
,
c2,
'
x'
)
gr
i
d
t
ext
(
0.
6,
1.
32,
'
Compensat
edsy
st
em'
)
t
ext
(
1.
3,
0.
68,
'
Uncompensat
edsy
st
em'
)
t
i
t
l
e(
'
Uni
tst
epr
esponsesofcompensat
edandnoncompensat
edsy
st
em'
)
xl
abel
(
'
Tsec'
)
y
l
abel
(
'
Out
put
sC1andC2'
)
Exampl
e7.
1
Exampl
e7.
2
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
Exampl
e7.
3
Exampl
e7.
4
LabTask/
LabExer
ci
se
Pr
obl
em #01:
Desi
gnasui
t
abl
el
agcompensat
orf
ort
hegi
v
enopenl
oopt
r
ansf
erf
unct
i
oni
s
G(
s)=(
K)
/
(s(
s+4)
(
s+80)
)
I
ti
sdesi
r
edt
ohav
et
hePhaseMar
gi
nt
obeatl
east33°
.
>>num=[
9600]
;
>>denum=[
1843200]
;
>>sy
s=t
f
(
num,
den)
>>bode(
sy
s)
>>[
Gm,
Pm,
Wcp,
Wcg]
=mar
gi
n(
sy
s)
>>mar
gi
n(
sy
s)
>>hol
don
Out
put
:
sy
s=
9600
s^
3+84
+320s
Gm =
2.
8000
Pm =
13.
2591
Wcp=
17.
8885
Wcg=
10.
5470
Li
nearCont
r
ol
Sy
st
em
s^
2
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
Nowaf
t
erusi
ngt
heLagCompensat
or
,
>>num=[
00063.
930]
>>denum=[
0.
0423.
5213.
6810]
;
>>sy
s=t
f
(
num,
den)
>>bode(
sy
s)
>>[
Gm,
Pm,
Wcp,
Wcg]
=mar
gi
n(
sy
s)
>>mar
gi
n(
sy
s)
>>hol
don
Out
put
:
sy
s=
63.
9s+30
0.
042s^
4+3.
52s^
3+13.
68s^
2+s
Gm =
15.
761
Pm =
39.
3844
Wcp=
16.
9235
Wcg=
3.
5782
Li
nearCont
r
ol
Sy
st
em
Feder
alUr
duUni
v
er
si
t
yofAr
t
s,
Sci
ence&Technol
ogyI
sl
amabad–Paki
st
an
El
ect
r
i
calEngi
neer
i
ng
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
RESULT:
Thust
hedesi
r
edphasemar
gi
nwasobt
ai
nedf
ort
hegi
v
enopenl
oopt
r
ansf
er
f
unct
i
onusi
ngsui
t
abl
el
agcompensat
or
.
Li
nearCont
r
ol
Sy
st
em