Using the Laplace transform for dynamic policy analysis - Heijdra
3 Control systems - Laplace transform
66
©Dr inŜ. JANUSZ LICHOTA CONTROL SYSTEMS Laplace transform Transfer functions Block algebra Faculty of Mechanical and Power Engineering Wrocław 2007
Transcript of 3 Control systems - Laplace transform
©D
r in
Ŝ.
JA
NU
SZ L
ICH
OT
A
CO
NT
RO
L S
YST
EM
S
Lapla
ce t
ransf
orm
Tra
nsf
er
functi
ons
Blo
ck
alg
ebra
Fa
cu
lty o
f M
ech
an
ica
l a
nd
Po
we
r E
ng
ine
eri
ng
Wro
cła
w 2
007
CO
NT
EN
TS
•L
apla
ce tra
nsf
orm
def
initio
n
–Pro
per
ties
–Pro
ofs
–Exam
ple
s
•T
ransf
er funct
ions
–Exam
ple
s
•B
lock
alg
ebra
–Typic
al co
nnec
tions
–Puzz
les fo
r young a
nd o
ld
Lapla
ce t
ransf
orm
defi
nit
ion
[]
0
()
()
()
stL
ft
Fs
ft
edt
∞−
==∫
C –
com
ple
xnu
mb
ers
set,
s –
com
ple
xnum
ber
,
t –
tim
e.
Th
e L
apla
ce t
ran
sfo
rm e
xis
ts f
or
lin
ear
dif
fere
nti
al e
qu
atio
ns
for
wh
ich
the
tran
sfo
rmat
ion
in
tegra
l co
nv
erg
es.In
tegra
l is
conv
erg
ing
, w
hen
exis
tsfi
nit
ed
nu
mb
ers
K a
ndγ,
su
ch
, ,
Let
be
giv
en f
un
ctio
n, w
her
e f(
t)=
0 f
or
t <
0. T
hen
Lap
lace
tran
sfo
rmis
cal
led
,
:f
→�
�
s∈�
()
tf
tK
eγ≤
t∈�
Re(
)sσ
γ=
>
Inv
erse
Lap
lace
tran
sform
f(t)
(ti
me
fun
ctio
n)
is w
ritt
en a
s
1(
)(
)2
cj
st
cj
ft
Fs
ed
sj
π
+∞
−∞
=∫
j-im
agin
ary
num
ber
,
c-co
nst
ant
Lapla
ce t
ransf
orm
defi
nit
ion
Pie
rre-
Sim
on
mar
qu
isd
e L
apla
ce
23 I
II 1
749
–5 I
II 1
827
Port
rait
pai
nte
daf
ter
a tr
ansf
orm
.
Lapla
ce t
ransf
orm
defi
nit
ion
Pro
pert
ies
Lin
eari
ty
Sca
ling
Tim
esh
ifti
ng
Tt≥
0 (
righ
t)
Tt≤0
(le
ft)
Dam
pin
g
LA
ft
AL
ft
ii
n
ii
n
()
[(
)]=
=∑
∑
=
11
[]
[]
[]
()
()
()
()
Lf
tf
tL
ft
Lf
tα
βα
β+
=+
[]
[]
1(
)(
)L
fat
Lf
ta
=
[]
[]
()
()
tsT
tL
ft
Te
Lf
t−
−=
[]
0
()
()
()
t
t
T
sTs
tL
ft
Te
Fs
fe
dτ
ττ
−−
−
−
=−
∫
[]
()
()
()
sL
ef
tL
fs
aF
sa
τ
=
−=
−
Dif
fere
nti
atio
n (
gen
eral
)
Inte
gra
tion
Co
nv
olu
tion
(Bo
rel
theo
rem
)
Fin
al v
alu
es
()
(0)
Lf
sFs
f•
=
−
12
1
12
()
(0)
(0)
...
(0)
()
nn
nn
n
nn
n
df
td
fd
fL
ss
fs
Fs
dt
dt
dt
−−
−−
−
=−
−−
−+
2
0(
)(0
)(
) tL
fs
Fs
sff
•••
=
=
−−
0
1(
)(
)
t
Lf
tdt
Fs
s
=
∫
12
12
12
00
()*
()
()
()
()
()
tt
ft
ft
ft
fd
ff
td
ττ
ττ
ττ
=−
=−
∫∫
[]
12
12
()*
()
()
()
Lf
tf
tF
sF
s=
0li
m(
)li
m(
)t
sf
tsF
s→
+∞→
=0
lim
()
lim
()
ts
ft
sFs
→+
→∞
=
Pro
pert
ies
Theore
m’s
pro
of
Lin
eari
ty
Co
ncl
usi
on
fro
m l
inea
rity
isth
atL
apla
ce t
ran
sfo
rm c
an’t
be
use
d t
o d
escr
ibe
non-l
inea
r sy
stem
s()
12
0
12
00
12
()
()
()
()
()
()
st
stst
ft
ft
edt
ft
edt
ft
edt
Fs
Fs
αβ
αβ
αβ
∞−
∞∞
−−
+=
+=
+
∫ ∫∫
Theore
m’s
pro
of
Scaling
Sub
stit
uti
ngτ=
at w
e o
bta
in
Theore
m’s
pro
of
Tim
e s
hif
ting
=0
for
Tt>
0, bec
ause
f(t)
=0
fo
r t<
=0
Sub
stit
uti
ngτ=
t-T
tw
e ob
tain
Theore
m’s
pro
of
Dam
pin
g,
frequency s
hif
ting
Sub
stit
uti
ng
p=
s-a
we
obta
in
0
()
()
()
pt
ef
tdt
Fp
Fs
a
∞−
==
−∫
Theore
m’s
pro
of
Dif
fere
nti
ati
on
Inte
gra
tion
by p
arts
del
iver
s
12
1
12
()
(0)
(0)
...
(0)
()
nn
nn
n
nn
n
df
td
fd
fL
ss
fs
Fs
dt
dt
dt
−−
−−
−
=−
−−
−+
Co
nti
nu
ing
inte
gra
tio
ng
ener
alfo
rmu
lais
ob
tain
ed
Theore
m’s
pro
of
Inte
gra
tion
Tw
o c
onti
nuo
usl
y d
iffe
ren
tiab
le f
un
ctio
ns
can
be
inte
gra
te b
y p
arts
So
far
in
tegra
l ex
ists
Theore
m’s
pro
of
Convolu
tion
Sub
stit
uti
ngσ
=t-τ
we
ob
tain
Theore
m’s
pro
of
Init
ial valu
e t
heore
mIn
itia
l v
alu
eth
eore
m
IfF
(s)
= L
[f(t
)]an
dex
ists
lim
it
, th
en0
lim
()
(0)
tf
tf
→+
=+
Fin
alv
alu
eth
eore
m.
If F
(s)
= L
[f(t
)] a
nd
ex
ists
lim
it
Theore
m’s
pro
of
Fin
al valu
e
lim
()
()
tf
tf
→+∞
=+∞
La
pla
ce
est
au
ssi con
nu
pou
r sa
con
ce
ption
d'u
n d
ém
on
(ou
dém
on
de
Lap
lace
) cap
ab
le d
e c
onna
ître
, à
un
insta
nt don
né
, to
us le
s p
ara
mè
tre
s d
e tou
tes le
s p
art
icu
les d
e l'u
niv
ers
. D
an
s c
ett
e p
ers
pe
ctive
, l'a
ute
ur
ad
op
te u
ne
po
sitio
n d
éte
rmin
iste
, so
it u
ne
positio
n p
hilo
soph
ique
et scie
ntifique
cap
ab
le d
'infé
rer
de
ce
qu
i e
st,
ce
qu
i do
it ê
tre
. C
e c
on
ce
pt de
dém
on
se
ra n
ota
mm
en
t re
mis
en
cau
se
pa
r le
prin
cip
e d
'ince
rtitude
d'H
eis
en
be
rg.
La
pla
ce
str
on
gly
belie
ved
in
cau
sa
l de
term
inis
m,
wh
ich
is e
xp
ressed
in
the
fo
llow
ing q
uo
te f
rom
th
e in
tro
du
ction
to
th
e
Essai:
"We
ma
y r
ega
rd the
pre
se
nt
sta
te o
f th
e u
niv
ers
e a
s t
he
eff
ect of
its p
ast and
the
ca
use
of
its f
utu
re.
An
in
telle
ct w
hic
h a
t
a c
ert
ain
mom
en
t w
ou
ld k
no
w a
ll fo
rce
s tha
t se
t na
ture
in
mo
tion
, a
nd
all
po
sitio
ns o
f a
ll item
s o
f w
hic
h n
atu
re is
co
mpo
sed
, if th
is in
telle
ct
we
re a
lso
va
st
en
ou
gh
to
su
bm
it th
ese
data
to
an
aly
sis
, it w
ou
ld e
mb
race
in
a s
ingle
fo
rmula
the
mo
ve
men
ts o
f th
e g
rea
test
bod
ies o
f th
e u
niv
ers
e a
nd
tho
se
of
the
tin
iest a
tom
; fo
r su
ch
an
in
telle
ct
no
thin
g w
ou
ld
be
un
ce
rta
in a
nd
the
futu
re ju
st
like
the
pa
st
wou
ld b
e p
resen
t befo
re its
eye
s."
Laplacescher Dämon
be
ze
ichne
t d
ie e
rkenntn
is-
und
wis
se
nschaft
sth
eo
retisch
e A
uff
assun
g,
de
rgem
äß
es m
öglic
h s
ei,
un
ter
de
r K
enn
tnis
säm
tlic
he
r N
atu
rge
se
tze
un
d a
ller
Initia
lbed
ingun
gen
jede
n v
erg
an
ge
nen
und
je
den
zu
künft
igen
Zu
sta
nd
zu
be
rechne
n. D
er
me
taph
ysis
che
Un
terb
au
die
se
r H
altun
g ist
de
r G
ese
tze
sde
term
inis
mu
s: fü
r Lap
lace
ist
die
Welt d
urc
h A
nfa
ngsbed
ingun
ge
n u
nd
Be
we
gun
gsge
se
tze
vo
llstä
nd
ig d
ete
rmin
iert
, so
da
ss d
ie A
ufg
abe
de
r
Na
turp
hilo
so
ph
ie, d
ie in
de
r H
imm
els
me
ch
an
ikih
r V
orb
ild b
esitzt,
aussch
ließ
lich
in
de
r In
tegra
tion
vo
n
Diffe
ren
tia
lgle
ich
un
gen b
este
ht.
Da
s w
äre
die
Aufg
abe
de
s D
äm
on
s,
de
n L
ap
lace
im
Vo
rwo
rt d
es Essai philosophique
sur les probabilités
vo
n 1
814
en
twirft
; e
r sp
rich
t do
rt jedo
ch
we
nig
er
eff
ekth
eis
che
nd
von
ein
er
Inte
lligen
z (
une
inte
llige
nce
).
Вф
ил
осо
фи
иЛ
апл
асб
ыл
пр
ивер
жен
цем
дет
ерм
ин
изм
а. О
нп
ост
ул
ир
овал
, ч
тоес
ли
бы
как
ое-
ни
буд
ьр
азум
но
есущ
еств
о
смо
гло
узн
ать
по
ло
жен
ия
иск
ор
ост
ивсе
хч
асти
цв
ми
ре
вн
еки
йм
ом
ент,
он
ом
огл
об
ыаб
сол
ютн
ото
чн
оп
ред
сказ
ать
эво
лю
ци
юВ
сел
енн
ой.
Так
ое
гип
оте
тич
еско
есу
щес
тво
вп
осл
едст
ви
ин
азван
од
емо
но
мЛ
апл
аса.
Mom
ent
of
rela
x.
Som
eth
ing
more
com
pre
hensi
ble
.
Tra
nsf
orm
table
Tra
nsfo
rmIn
vers
e
transfo
rm
Tra
nsf
orm
table
Lapla
ce, Louis
iana
Fro
m W
ikip
edia
, th
e fre
e e
ncyclo
pedia
Ju
mp
to
: na
vig
atio
n,
sea
rch
La Place
(som
etim
es s
pe
lled
LaPlace
or Laplace
) is
a c
en
su
s-d
esig
na
ted
pla
ce
loca
ted
in
St.
Joh
n the
Ba
ptist P
arish
,
Lo
uis
iana
, o
n the
Ea
st
Ban
k o
f th
e M
issis
sip
pi R
ive
r. A
s o
f th
e 2
000
cen
su
s,
the
CD
P h
ad
a t
ota
l po
pu
lation
of
27,6
84
.
It is t
he
sou
the
rn te
rmin
us o
f In
ters
tate
55
hig
hw
ay,
wh
ere
it
join
s w
ith
In
ters
tate
10
. La
Pla
ce
is lo
ca
ted
25
mile
s w
est
of
Ne
w O
rle
an
s.
Tra
nsf
orm
table
Fs
kedt
ke
s
k s
st
st
()=
=−
=−
∞−
+∞
∫ 00
Fin
dL
apla
ce t
ran
sfo
rm o
f H
eav
isid
e fu
nct
ion
f(t)
= k
dla
t>
0
So
luti
on
Tra
nsf
orm
table
Exam
ple
1
Dir
acim
pu
lse
δ δ
δ()
,
()
,
()
tt
tt
tdt
=≠
=∞
=
=
−∞
+∞ ∫
00 0
1
Lt
[(
)]δ
=1
Lap
lace
tran
sfo
rm o
f D
irac
im
pu
lse
tt
Geo
met
rica
l m
od
el
tria
ng
les
wit
hu
nit
y s
urf
ace
and t
op p
oin
t →
∞
Of
cou
rse
11
ss
=S
ow
e h
ave
rela
tion
bet
wee
nH
eav
isid
efu
nct
ion
an
dD
irac
im
pu
lse
[]1
11
1()
1(
)d
tL
Ls
ts
dt
δ−
−
=
==
Tra
nsf
orm
table
Exam
ple
2
Pro
of
Tra
nsf
orm
table
Exam
ple
31
te
s
α
α±
→m
0
0
()
0
()
0
()
()
1
1
st
tst
st
st
Fs
ft
edt
ee
dt
edt e
s
s
α
α
α
α
α
∞−
∞−
∞−
−
∞−
−
=
= =
=−
−
=−
∫
∫ ∫
Tra
nsf
orm
Fin
d t
ran
sfo
rm o
f an
equ
atio
n
()
()
yt
kut
=S
olu
tio
n
()
()
Ys
kUs
=
Lap
lace
tra
nsf
orm
conv
erts
tim
efu
nct
ion
f(t)
in
toco
mp
lex
fun
ctio
nF
(s),
th
eref
ore
Dynamic system
u(t)
y(t)
k -
coef
fici
ent
Tra
nsf
orm
table
Exam
ple
4
.
Ty
yku
+=
Tra
nsf
orm
of
dif
fere
nti
atio
n i
s
Fin
d t
ran
sfo
rm o
f an
dif
fere
nti
al e
qu
atio
n
So
luti
on
12
1
12
()
(0)
(0)
()
...
(0)
nn
nn
n
nn
n
df
td
fd
fL
sF
ss
sf
dt
dt
dt
−−
−−
−
=
−−
−−
Th
eref
ore
[]
()
(0)
()
()
TsY
sy
Ys
kUs
−+
=If
in
itia
l val
ue
isg
iven
y(0
) =
0, th
en
()
()
()
()
()
1(
)
TsY
sY
skU
s
Ys
Ts
kUs
+=
+=
Dynamic
system
u(t)
y(t)
Tra
nsf
orm
table
Exam
ple
5
Fin
d t
ran
sfo
rm o
f an
dif
fere
nti
al e
qu
atio
n
So
luti
on
...
12
Ty
Ty
yku
++
=
21
1
21
()
(0)
(0)
()
(0)
....
nn
nn
n
nn
n
df
td
fd
fL
sF
ss
fs
dt
dt
dt
−−
−−
−
=
−−
−−
Tra
nsf
orm
of
dif
fere
nti
atio
n i
s
[]
2
21
(0)
()
(0)
()
(0)
()
()
dy
Ts
Ys
syT
sYs
yY
skU
sd
t
−−
+−
+=
Th
eref
ore
Dynamic
system
u(t)
y(t)
In c
ase
of
init
ial
val
ues
eq
ual
zer
o 2
21
()
1(
)Y
sT
sT
skU
s
+
+=
Tra
nsf
orm
table
Exam
ple
6
..
Ty
yk
uu
+=
+
Fin
d t
ran
sfo
rm o
f an
lin
ear
dif
fere
nti
al e
qu
atio
n
So
luti
on
Dynamic
system
u(t)
y(t)
[]
[]
()
(0)
()
()
(0)
()
TsY
sy
Ys
ksU
su
Us
−+
=−
+
Inca
se o
f in
itia
l v
alu
es e
qu
alze
ro y
(0)=
0, u(0
)=0 w
e o
bta
in
()
()
()
()
TsY
sY
sks
Us
Us
+=
+
[]
[]
()
1(
)1
Ys
Ts
Us
ks+
=+
Tra
nsf
orm
table
Exam
ple
7
Fin
d i
nv
erse
tra
nsf
orm
of
()
()(
)F
ss
ss
()=
+−
++
1
1
20
13
22
3
()
()
()
()
()
()
Fs
s
A
s
B
s
C
s
D
s
E
s(
)=
+−
++
++
++
++
+
1
12
01
13
33
22
32
()
()
()
()
()
()
()
()
sF
ss
s
A
s
B
s
C
s
D
s
E
s+
=+
+−
++
++
++
++
+
11
1
120
11
33
3
22
22
32
() (
)(
)(
)(
)(
)(
)li
m(
)s
sF
sA
sB
sC
s
D
s
E
s→
−+
=−
++
++
++
++
+
1
22
32
11
20
11
33
3
()
()
()(
)(
)li
mli
ms
ss
ss
ss
A→
−→
−+
+−
++
=
−+
=
−1
2
22
31
31
1 1
20
13
120 3
120
12
0 81
20
−=
−A
AB
CD
E=
=−
==
=1 8
3 16
1 4
1 4
3 16
()
()
ft
te
tt
et
t(
).
..
.=
−−
++
−−
375
15
375
52
52
3
So
luti
on.
Ex
pan
din
g i
n a
par
tial
fra
ctio
n e
xp
ansi
on
Co
effi
cien
tA
(re
sid
ue)
is
eval
uat
ed
by m
ult
iply
ing
th
rou
gh
by t
he
den
om
inat
or
fact
or
of
equ
atio
n
corr
esp
on
din
g t
o A
an
d s
etti
ng
s
equ
al t
o t
he
roo
t
Fro
m t
ran
sform
tab
le w
e o
bta
in
Tra
nsf
orm
table
Exam
ple
8
Tra
nsi
ent
resp
onse
of
a fu
nct
ion
f(t)
Sch
eme
in S
imu
lin
k
Dif
fere
nti
atio
ns
giv
esu
s
imp
uls
ere
sponse
of
the
syst
em s
/s
= 1
We
use
step
inp
ut
1/s
, b
ecau
seof
lack
of
Dir
acim
pu
lse
inpro
gra
m
F(s)
u(t)=δ δδδ(t)
f(t)
()
()(
)F
ss
ss
()=
+−
++
1
1
20
13
22
3
()
()
ft
te
tt
et
t(
).
..
.=
−−
++
−−
375
15
375
52
52
3
Tra
nsf
orm
table
Exam
ple
8
Sy
stem
tra
nsf
er f
un
ctio
n
Syst
em
tra
nsf
er
functi
on
Ifw
e w
ill
futh
er t
ran
sform
e.g
. d
iffe
ren
tial
equ
atio
n(e
xam
ple
5)
.
Ty
yku
+=
()
()
1(
)Y
sT
skU
s+
=
we
wil
l fi
nd
inte
rest
ing
rela
tion
bet
wee
no
utp
ut
and
inpu
tsi
gn
al
()
()/
()
/1
Ys
Us
kT
s=
+
On
th
ele
ftsi
de
ther
ear
ed
ivid
edfu
nct
ion
s–
outp
ut
and
inpu
t, o
n t
he
righ
t si
de
ther
e is
rat
io o
f tw
o p
oly
no
mia
ls.
Syst
em
tra
nsf
er
functi
on
Def
initio
n.
Tra
nsf
er f
un
ctio
nG
(s)
isd
efin
ed a
s th
e ra
tio
of
a L
apla
ce
tran
sfo
rm o
f th
e ou
tpu
t v
aria
ble
to t
he
Lap
lace
tra
nsf
orm
of
the
inpu
t v
aria
ble
,
wit
h a
ll i
nit
ial
cond
itio
ns
assu
med
to b
e ze
ro.
()
()
()
Ys
Gs
Us
=Dynamic process
u(t)
y(t)
G(s)
u(t)
y(t)
Tra
nsf
er f
un
ctio
nca
nb
e use
dto
des
crib
eon
lyli
nea
r, s
tati
on
ary (
con
stan
t
par
amet
er)
syst
em.
A t
ran
sfer
fun
ctio
n i
s an
in
put-
outp
ut
des
crip
tio
n o
f th
e
beh
avio
ur
of
a sy
stem
.
Syst
em
tra
nsf
er
functi
on
()
()
()
Ys
Gs
Us
=
Iftr
ansf
er f
un
ctio
no
fa
syst
em i
s
and
inpu
tsi
gn
alis
U(s
), t
hen
outp
ut
sig
nal
can
be
det
erm
ined
()
()
()
Ys
Gs
Us
=
Uti
lizi
ng
Bore
l th
eore
m o
utp
ut
sig
nal
y(t
)in
tim
e-d
om
ain
can
be
com
pu
ted
()
()*
()
yt
gt
ut
=
Ano
ther
way
to g
ety(t
) is
to f
ind
inv
erse
Lap
lace
tra
nsf
om
atio
nfr
om
table
10
1(
)(
),m
ni
ii
ii
ii
Ts
Ys
ks
Us
nm
==
+=
≤
∑
∑ Ms
Ts
iii
im
()=
+
=∑ 1
1L
sk
s i
i
i
n
()=
=∑ 0
()
()
()
()
()
Ys
Ls
Gs
Us
Ms
==
Syst
em
tra
nsf
er
functi
on
Lap
lace
tran
sfo
rmof
lin
ear
dif
fere
nti
aleq
uat
ion
is
Rea
lsy
stem
sfu
lfil
lco
nd
itio
nn
<=
m.
Def
inin
gn
ewv
aria
ble
s
rati
o o
ftw
ora
tion
al p
oly
no
mia
ls i
so
bta
ined
Syst
em
tra
nsf
er
functi
on
Th
e d
eno
min
ato
r po
lyn
om
ial
M(s
), w
hen
set
equ
al t
o z
ero
, is
cal
led
th
e
char
acte
rist
ic e
qu
atio
n, b
ecau
se t
he
roo
ts o
f th
is e
qu
atio
n d
eter
min
e th
e ch
arac
ter
of
the
tim
e re
spon
se
()
()
()
()
()
Ys
Ls
Gs
Us
Ms
==
Th
e ro
ots
of
this
ch
arac
teri
stic
equ
atio
n a
re a
lso c
alle
d t
he
pole
s or
singu
lari
ties
of
the
syst
em.
Th
e ro
ots
of
the
nu
mer
ator
poly
no
mia
l L
(s)
are
call
ed z
eros
of
the
syst
em.
Syst
em
tra
nsf
er
functi
on
Exam
ple
1F
ind
tran
sien
tre
spo
nse
of
ano
utp
ut
sig
nal
y(t
), i
fsy
stem
tra
nsf
er
fun
ctio
nis
Gs
sT
s(
)(
)=
+
1
1
and i
npu
tsi
gn
al i
sli
nea
ru(t
)=t.
Ys
Gs
Us
sT
s(
)(
)(
)(
)=
=+
1
13
Ex
pan
din
gou
tput
in a
par
tial
fra
ctio
n d
eco
mp
osi
tion
Ys
sT
s
A s
B s
C s
D
sT
()
()
=+
=+
++
+
1
11
33
2(
)D
Cs
C TB
sB T
As
A T+
++
++
+−
=
32
10
Co
mp
arin
gco
effi
cien
tat
the
sam
e p
ow
eran
d s
etti
ng
th
em e
qu
al t
o z
ero
AB
TC
TD
T=
=−
==−
12
2
yt
tT
tT
Te
t T(
)=
−+
−−
1 2
22
2
Solu
tion
.
Fro
mtr
ansf
orm
tab
lew
e se
e, t
hat
U(s
)=1
/s2,
ther
efore
Th
eref
ore
outp
ut
sign
alin
tim
e-d
om
ain
is
Gs
sT
s(
)(
)=
+
1
1y
tt
Tt
TT
e
t T(
)=
−+
−−
1 22
22
Fun
ctio
n y
(t),
T=
1 s
econd
chan
ges
ver
sus
tim
e
21
()
12
ty
tt
te−
=−
+−
05
10
15
20
25
30
35
40
45
02
46
81
01
2czas t, sekundy
y(t)Syst
em
tra
nsf
er
functi
on
Exam
ple
1 –
tim
e r
esp
onse
[]
2()
(1)
()
()
()
()
Ys
sT
sU
s
Ts
Ys
sYs
Us
+=
+=
2
2()
()
()
()
dy
td
yt
Tu
tdt
dt
Ty
yu
t
+=
+=
&&&
Rev
erse
op
erat
ion
isposs
ible
too
. S
yst
em t
ran
sfer
fu
nct
ion
allo
ws
to o
bta
in d
iffe
ren
tial
equ
atio
n
Syst
em
tra
nsf
er
functi
on
Exam
ple
1 –
dif
fere
nti
al equati
on
Syst
em c
har
acte
rist
ic e
qu
atio
nis
(1)
0s
Ts+
=
Eq
uat
ion
has
two
po
les
0s=
1s
T=−
and
has
n’t
zero
s. I
fT
=1
sec
ond
, th
enpo
les
locu
sca
nb
e port
rayed
on
Gau
ss
pla
ne
jω=
j Im
(s)
-1/T
0σ
=R
e(s
)
s=σ
+ jω
Po
le l
oci
Syst
em
tra
nsf
er
functi
on
Exam
ple
1 -
pole
s
Syst
em
tra
nsf
er
functi
on
Exam
ple
2 –
tim
e s
olu
tion
Let
go
bac
kto
ex
amp
le5.
Th
isord
inar
ily
dif
fere
nti
aleq
uat
ion
is.
0y
ay
+=
Th
isis
ho
mo
gen
eou
s eq
ua
tion.
Ith
asso
luti
on
()
(0)
at
at
yt
Ce
ye
−−
==
y(0
) is
call
edin
itia
lva
lue.
,o
nho
mo
gen
eous
equ
atio
nis
.
Ty
ay
bu
+=
Ith
asti
me-
dom
ain
solu
tion
uti
lize
d b
y c
onv
olu
tion
()
0
()
()
t
at
at
yt
Ce
be
udt
ττ
−−
−=
+∫
Fir
stco
mp
on
ent
dep
end
s o
n i
nit
ial
con
dit
ion
s, s
eco
nd
on
co
ntr
ol
sig
nal
u(t
). I
fa>
0,
then
syst
em o
utp
ut
sig
nal
y(t
) is
hea
din
gto
war
din
fin
ity
. If
a<
0,
then
syst
emo
utp
ut
sign
alis
hea
din
gto
zer
o.
Syst
em
tra
nsf
er
functi
on
Desc
ripti
on o
f m
any s
yst
em
s at
the s
am
e t
ime
So
luti
on o
fone
tran
sfer
fun
ctio
nis
solu
tio
no
fin
fin
ity
nu
mb
ero
fsy
stem
s
des
cib
edb
y t
he
sam
e st
ruct
ure
of
dif
fere
nti
aleq
uat
ion
.
Even
those
we
don’t k
now
that th
ey e
xists
.
Blo
ck a
lgeb
ra
Blo
ck a
lgebra
Blo
ck d
iagra
m m
odels
U(s)
Y(s)
Th
e im
port
ance
of
the
cause
an
d e
ffec
t re
lati
on
ship
of
the
tran
sfer
fun
ctio
n i
s
evid
ence
db
y t
he
inte
rest
in
rep
rese
nti
ng t
he
rela
tion
ship
of
syst
em v
aria
ble
by
dia
gra
mm
atic
mea
ns.
Th
e b
lock
dia
gra
m r
epes
enta
tion i
s pre
val
ent
in c
on
trol
syst
em e
ng
inee
rin
g.
Blo
ck d
iag
ram
s co
nsi
sts
of
un
idir
ecti
onal
op
erat
ion
al b
lock
s.
Sin
gle
-in
pu
tsy
stem
is
show
n i
n f
igure
. 1
(1)
sT
s+
U(s)
Y(s)
()
Gs
More
gen
eral
ized
str
uct
ure
is
sho
wn
bel
ow
.
Blo
ck
alg
ebra
Blo
ck
model
Sin
gle
-in
pu
t, t
wo-o
utp
uts
syst
em
has
blo
ck d
iag
ram
11
22
()
()
()
()
()
()
Ys
Gs
Us
Ys
Gs
Us
= =
U(s)
Y1(s)
1(
)G
s
Y2(s)
2(
)G
s
Blo
ck
alg
ebra
Blo
ck
model
In o
rder
to
rep
rese
nt
a sy
stem
wit
h s
ever
al v
aria
ble
s und
er c
ontr
ol,
an
inte
rco
nn
ecte
dsy
stem
is
uti
lize
d.
Tra
nsf
er f
un
ctio
ns
G1
2, G
21
are
sho
win
gin
terc
onn
etio
nb
etw
een
inp
uts
and
outp
uts
111
112
2
22
11
22
2
()
()
()
()
()
()
()
()
()
()
Ys
Gs
Us
Gs
Us
Ys
Gs
Us
Gs
Us
=+
=+
U1(s)
Y1(s)
11(
)G
s
Y2(s)
22(
)G
sU2(s)
12(
)G
s
21(
)G
s
+
+++
Blo
ck
alg
ebra
Blo
ck
model U
1(s)
U2(s)
Y1(s)
Y2(s)
Y3(s)
Y4(s)
Blo
ck
alg
ebra
MIM
O s
yst
em
1 2
()
()
()
...
()
p
Us
Us
Us
Us
=
Ys
Ys
Ys
Ys
q
()
()
()
... (
)
=
1 2
Gs
Gs
Gs
Gs
Gs
Gs
Gs
Gs
Gs
Gs
p p
qp
()
()
().
..(
)
()
().
..(
)
...
()
().
..(
)
=
11
12
1
21
22
2
12
1
()
()
()
ss
s=
YG
X
In g
ener
al,
we
wri
te m
ult
i-in
pu
tan
d–
ou
tput
syst
em i
nm
atri
xfo
rm a
s
Or
short
ly
Blo
ck
alg
ebra
Exam
ple
–heat
exchanger
vzi,
Tzi
vp
i,T
pi
vzs
Tzs
vp
s
Tp
s
Gs
T T
T T
v T
v T
T v
T v
v v
v v
T T
T T
v T
v T
T v
T v
v v
v v
ps
pi
zi pi
ps
pi
zi pi
ps
pi
zi pi
ps
pi
zi pi
ps
zs
zi zs
ps
zs
zi zs
ps
zs
zi zs
ps
zs
zi zs
()=
Tp
i
vp
i
Tzs
vzs
Tp
s
vp
s
Tzi
vzi
T –
tem
per
atu
re,
v –
wat
er v
elo
city
Blo
ckd
iagra
m o
f
hea
t ex
chan
ger
Inle
t o
fw
ater
to b
e
hea
ted
Ou
tlet
of
hea
ted
wat
er
Inle
to
f a
war
m
wat
er
Ou
tlet
of
a
coole
d
wat
er
Hea
t ex
chan
ger
tra
nsf
er f
un
ctio
n
Blo
ck
alg
ebra
Exam
ple
–heat
exchanger
Hea
tex
chan
ger
of
typ
eJA
D
Pa
rall
elco
nn
ecti
on
bet
wee
n3
hea
tex
cha
ng
ers
Pa
rall
el c
on
nec
tio
n b
etw
een
2 h
eat
exch
an
ger
s
Blo
ck
alg
ebra
Blo
ck c
onnecti
ons
and r
educti
on
Th
e b
lock
dia
gra
m r
epre
senta
tion
of
a g
iven
syst
em m
ay o
ften
be
red
uce
d b
y
blo
ckd
iagra
m r
edu
ctio
n t
echn
iqu
esto
a s
imp
lifi
ed b
lock
dia
gra
m w
ith
few
er
blo
cks
then
th
e ori
gin
al d
iag
ram
.
Blo
ck
alg
ebra
Blo
ck c
onnecti
ons
1(
)G
s2(
)G
s(
)n
Gs
Ser
ies,
tw
o b
lock
are
con
nec
ted
in
cas
cade
1(
)G
s
2(
)G
s
Par
alle
l
Wit
h f
eed
bac
k 1(
)G
s
2(
)G
s
U(s
)Y
(s)
+ +
U(s
)Y
(s)
Y(s
)-
+U
(s)
1(
)G
s
2(
)G
s
Y(s
)
-
+U
(s)
Dis
turb
an
ce r
elate
d c
on
tro
l lo
op
Set
-po
int
rela
ted c
on
tro
l lo
op
Blo
ck
alg
ebra
Blo
ck
dia
gra
m t
ransf
orm
ati
on
Bas
ic r
ule
of
tran
sfo
rmat
ion
:
Ou
tpu
tsi
gnal
Y(s
) ca
n’t
chan
ge
afte
rm
ov
ing
a b
lock
india
gra
m
(inp
ut
sig
nal
, o
fco
urs
e, c
an’t
chan
ge
too
)
Blo
ck
alg
ebra
Blo
ck
dia
gra
mtr
ansf
orm
ati
on
1(
)G
s2(
)G
s(
)n
Gs
U(s
)Y
(s)
12
12
11
()
()
()
()
()
...
()
().
..(
)(
)(
)(
)(
)n
n
Ys
Ys
Ys
Ys
Gs
Gs
Gs
Gs
Us
Us
Ys
Ys
−
==
=
Eq
uiv
alen
ttr
ansf
er f
un
ctio
nin
seri
esco
nn
ecti
on
ism
ult
ipli
cati
on
of
giv
en
tran
sfer
fu
nct
ion
s
Y1(s
)Y
2(s
)Y
n-1
(s)
Blo
ck
alg
ebra
Blo
ck
dia
gra
mtr
ansf
orm
ati
on
1(
)G
s
2(
)G
s
U(s
)Y
(s)
+ +
Eq
uiv
alen
ttr
ansf
erfu
nct
ion i
n s
erie
s co
nn
ecti
on
is
sum
of
giv
en
tran
sfer
fun
ctio
ns
12
12
()
()
()
()
()
()
()
()
()
Ys
Ys
Ys
Gs
Gs
Gs
Us
Us
Us
==
+=
+
Y1(s
)
Y2(s
)
Blo
ck
alg
ebra
Blo
ck
dia
gra
mtr
ansf
orm
ati
on
1(
)G
s
2(
)G
s
Y(s
)+
+U
(s) Y
1(s
)
Tra
nsf
er f
un
ctio
n o
f po
siti
ve
feed
bac
k i
s
Y2(s
)
21
12 1
2
()
()
()
()
()
()
()
()
()
Ys
Us
Ys
Ys
Gs
Ys
Ys
Gs
Ys
=+
= =
Th
ere
are
rela
tio
ns
in c
on
nec
tio
n
Sub
stit
uti
ng
1-s
tan
d2-n
deq
uat
ion
to t
hir
d
11
2(
)(
)(
)(
)(
)(
)Y
sG
sU
sG
sG
sY
s=
+
1
12
()
()
()
1(
)(
)
Gs
Ys
Us
Gs
Gs
=−
,o
ise-
rela
ted c
on
tro
l sy
stem
Posi
tive
fee
db
ack
Act
ua
tin
g s
ign
al
Dis
turb
ace
sO
utp
ut
sig
na
l
Blo
ck
alg
ebra
Blo
ck
dia
gra
mtr
ansf
orm
ati
on
1(
)G
s
2(
)G
s
Y(s
)-
+U
(s)
More
imp
ort
ant
in p
ract
ice
is n
egat
ive
feed
bac
kco
ntr
ol
syst
em.
Eq
uiv
alen
t tr
ansf
er f
un
ctio
n i
s
1
12
()
()
()
1(
)(
)
Gs
Ys
Us
Gs
Gs
=+
Sig
nin
den
om
inat
or
isch
ang
ed.
,eg
ati
ve f
eed
back
con
tro
lsy
stem
Blo
ck
alg
ebra
Blo
ck
dia
gra
mtr
ansf
orm
ati
on
1(
)G
s
2(
)G
s
Y(s
)
-
+U
(s)
Y1(s
)Y
2(s
)
Eq
uiv
alen
ttr
ansf
erfu
nct
ion o
f n
egat
ive
feed
bac
k s
yst
em i
n s
et-p
oin
t
chan
nel
is
2 12
2
11
()
()
()
()
()
()
()
()
()
Ys
Us
Ys
Ys
Gs
Ys
Ys
Gs
Ys
=−
= =
In t
his
conn
ecti
on
th
ere
are
rela
tion
s
Sub
stit
uti
ng
1-s
ti
2-n
dto
th
ird
equ
atio
n w
e o
bta
in
()
12
12
12
()
()
()
()
()
()
()
()
()
()
()
Ys
Gs
Gs
Us
Ys
Gs
Gs
Us
Gs
Gs
Ys
=−
=−
12
12
()
()
()
()
1(
)(
)
Gs
Gs
Ys
Us
Gs
Gs
=+
Set
-po
int
cha
nge
Blo
ck
alg
ebra
Puzzle
1 f
or
young
and
old
Tra
nsf
orm
atio
n
Ori
gin
ald
iag
ram
Eq
uiv
alen
t d
iag
ram
Co
mb
inin
g b
lock
s in
cas
cad
e
Mo
vin
g a
su
mm
ing
po
int
beh
ind
a
sum
min
g p
oin
t
Mo
vin
g a
pic
koff
po
int
beh
ind
a
pic
ko
ffpo
int
Mo
vin
g a
su
mm
ing
po
int
ahea
da
blo
ck
Mo
vin
g a
su
mm
ing
po
int
ahea
da
blo
ck
Mo
vin
g a
pic
koff
po
int
ahea
da
blo
ck
Blo
ck
alg
ebra
Puzzle
1 f
or
young a
nd o
ldT
ran
sfo
rmat
ion
O
rigin
ald
iag
ram
Eq
uiv
alen
t d
iag
ram
Mo
vin
g a
pic
koff
po
int
beh
ind
a
blo
ck
Mo
vin
g a
pic
koff
po
int
ahea
d
sum
min
g p
oin
t
Mo
vin
g a
pic
koff
po
int
beh
ind
sum
min
g p
oin
t
Fin
dan
equ
ival
ent
tran
sfer
fun
ctio
nof
a sy
stem
G(s
) =
Y(s
)/U
(s)
Blo
ck
alg
ebra
Puzzle
2 f
or
young a
nd o
ld
So
luti
on
G2(s
)
G4(s
)
-
+
G24(s
)
224
24
()
()
1(
)(
)
Gs
Gs
Gs
Gs
=+
we
get
sing
le b
lock
Eli
min
atin
g
a fe
edb
ack l
oop
Uti
liza
tion
of
rule
wh
ich
eli
min
ates
feed
bac
k
giv
es n
ewb
lock
str
uct
ure
G1(s
)
G3(s
)
-
+G
24(s
)
G1(s
)
G3(s
)
-G
24(s
)
Blo
ck
alg
ebra
Puzzle
2 f
or
young a
nd o
ldT
hen
, el
imin
atin
g c
asca
de
conn
ecte
db
lock
s, w
e ob
tain
str
uct
ure
sho
wn
in
fig
.
G124(s
)
G3(s
)
- G124(s
)=G
1(s
) G
24(s
)
Fin
ally
, b
y r
edu
cin
gn
egat
ive
feed
bac
k l
oo
p, w
e ob
tain
124
12
34
124
3
()
()
1(
)(
)
Gs
Gs
Gs
Gs
=+
Th
e cl
ose
d-l
oop
syst
em t
ran
sfer
fu
nct
ion
21
24
1234
21
3
24
()
() 1
()
()
()
()
1(
)(
) 1(
)(
)
Gs
Gs
Gs
Gs
Gs
Gs
Gs
Gs
Gs
Gs
+=
++
Blo
ck
alg
ebra
Puzzle
2 f
or
young a
nd o
ld
12
123
4
24
12
3
()
()
()
1(
)(
)(
)(
)(
)
Gs
Gs
Gs
Gs
Gs
Gs
Gs
Gs
=+
+
24
12
31
()
()
()
()
()
0G
sG
sG
sG
sG
s+
+=
Sim
pli
fyin
g
Th
ed
eno
min
ato
r(c
har
acet
eric
tic
equ
atio
n)
is c
om
pri
sed
of
1 p
lus
the
sum
of
each
loo
ptr
ansf
er f
un
ctio
n
Blo
ck
alg
ebra
Puzzle
3 f
or
young a
nd o
ldF
ind
equ
ival
ent
tran
sfer
fun
ctio
ns
for
a sy
stem
sG
(s)
= Y
(s)/
U(s
)
sho
wn
in
fig
.
Blo
ck
alg
ebra
Hydra
ulic p
rocess
Mix
ing
of
two
flu
ids
wit
hdif
fere
nt
con
cen
trat
ion
s
C –
con
centr
atio
n,
m –
mas
s fl
ow
V –
volu
me
flo
w
t -
tim
e
So
luti
on f
or
a) c
on
cen
trat
ion
b
) m
ass
flow
s
Ca
Ca
we
1/(
Ts+
1)
V
1/m
Ca
-Vaw
y
Va
+V
aw
e1
/s
Th
eref
ore
equiv
alen
td
iag
ram
for
on
e pro
cess
is
For
two
iden
tica
lpro
cess
esco
nn
ecte
din
casc
ade
Ca
Caw
e1
/(T
s+
1)
Ca
Caw
e1
/(T
s+
1)
Blo
ck
alg
ebra
Hydra
ulic p
rocess
for
a) c
on
cen
trat
ion
b
) m
ass
flo
ws
Th
an
k y
ou
for
your
atten
tion
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