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Wro
cła
w 2
007
Fa
cu
lty o
f M
ech
an
ica
l a
nd
Po
we
r E
ng
ine
eri
ng
©D
r in
Ŝ.
JA
NU
SZ L
ICH
OT
A
CO
NT
RO
L S
YST
EM
S
dif
fere
nti
al equati
ons
of
syst
em
s
lineari
zati
on
CO
NT
EN
TS
•T
yp
ica
l in
pu
tsi
gn
als
•P
hy
sica
lsy
stem
s d
iffe
ren
tia
l eq
ua
tio
ns
–m
ech
an
ica
l
–el
ectr
ical
–D
C m
oto
rs
–h
yd
rau
lic
–th
erm
al
–C
on
clu
sio
n
•A
na
lyti
cal
solu
tio
ns
of
dif
fere
nti
al
equ
ati
on
s
•G
ener
ali
zati
on
–tr
an
sfer
fu
nct
ion
•L
inea
ra
pp
rox
ima
tio
n o
fd
iffe
ren
tia
leq
ua
tio
ns
Ob
iek
t
Czu
jnik
Prz
etw
orn
ik
Reg
ula
tor
Sił
ow
nik
Nas
taw
nik
Zad
ajn
ik
PLA
CE O
F A
CT
ION
Pro
cess
-+
PR
OC
ES
S
Contr
olle
r
Sensor
Convert
er
Set-
poin
t
Actu
ato
r
Executive p
art
Ho
wto
investigate
adynam
ic p
ropert
ies o
f a p
rocess
?
TY
PIC
AL IN
PU
T S
IGN
ALS
Sy
gn
ały
sk
ok
ow
eS
yg
nał
y i
mp
uls
ow
eS
yg
nał
y s
ym
etry
czn
e -
fale
Ser
ie i
mp
uls
ów
Sy
gn
ały
zm
od
ulo
wan
eS
zum
bia
ły
Step responses
-squ
are
ste
p
-slo
pin
g s
tep
-sin
usoid
al
Impulse signals
-squ
are
im
pu
lse
-tria
ngle
im
pu
lse
-sin
uso
ida
l im
pu
lse
Symmetrical signals
-squ
are
wa
ve
(re
cta
ngula
r w
ave
)
-tria
ngle
wa
ve
-ra
mp
wa
ve
Con
tro
l in
pu
ts a
re c
alle
d o
ften
sou
rce
te
rms o
r fo
rcin
g inp
uts
Symmetrical signals
-re
cta
ngu
lar-
ram
p w
ave
-sin
e-s
lope
wa
ve
-sin
e w
ave
Impulse series
-re
cta
ngu
lar
imp
uls
e s
erie
s
-tria
ngle
im
pu
lse
se
rie
s
-sin
e im
pu
lse
se
rie
s
Mo
du
late
d s
igna
ls
-sin
uso
ida
l m
odu
latio
n
-squ
are
modu
lation
White
no
ise
(ra
ndo
m s
igna
l w
ith
fla
t
po
we
r d
en
sity)
Diracdelta function (impulse) δ δδδ(t) a
llow
sto
receiv
eimpuls
response
ofa s
yste
m
Sim
ple
rule
: send
imp
uls
to a
n s
yste
min
put and o
bserv
e
outp
ut
sig
nal
Pro
cess
u(t
)
t
y(t
)
t
Input
Outp
ut
TY
PIC
AL IN
PU
T S
IGN
ALS
Com
monly
use
d input
signals
Imp
uls
e h
ead
ing t
ow
ard
s
infi
nit
y a
tt=
0
TY
PIC
AL IN
PU
T S
IGN
ALS
Com
monly
use
d input
signals
u(t
)= δ δδδ(t)
t
u(t
)= δ δδδ(t-t0)
tt 0
Rel
atio
nto
Hea
vis
ide
funct
ion
δ δδδ(t)=d1(t)/dt
(th
est
ep i
np
ut
is t
he
inte
gra
l o
f th
e im
pu
lse
inp
ut)
Bia
sed
imp
uls
eIm
pu
lse
ish
eadin
gto
wa
rds
infi
nit
ya
tt=
0
Th
e H
eav
isid
e fu
nct
ion
isn
’tco
nti
nu
ou
s, t
her
efo
reh
asn
’td
eriv
ativ
e. S
uch
fun
ctio
ns
are
fou
nd
in
dis
trib
uti
on
th
eory
.
Th
est
ep i
sli
mit
of
fun
ctio
nh
(t)
Th
e D
irac
im
pu
lse
is l
imit
of
funct
ion
s
TY
PIC
AL IN
PU
T S
IGN
ALS
Com
monly
use
d input
signals
Are
a=
1
Step (Heavisidefunction) 1(t)
allo
ws
to o
bserv
esyste
m
step response.
Sim
ple
rule
too
:send
ste
p c
hang
eto
an
syste
min
put
and
observ
e o
utp
ut
sig
nal
Pro
cess
u(t
)
t
y(t
)
t
Input
Outp
ut
TY
PIC
AL IN
PU
T S
IGN
ALS
Com
monly
use
d input
signals
1
1(t
-t0)=
1 d
la t
>t 0
1(t
-t0)=
0 d
la t≤t
0
Bia
sed
Heavis
ide f
unction
u(t
) 1t 0
Sinusoidal input signalsin(ω ωωωt)
allo
ws t
o o
bserv
e
frequencyresponse
of
a s
yste
m.
Ste
ady-s
tate
response
ofth
esyste
m t
o a
sin
uso
idalin
pu
t.(C
om
ple
x h
arm
on
ic s
igna
l w
ith
sin
uso
ida
l com
ponen
t w
ith
am
plit
ude
, an
gu
lar
fre
quen
cy
and
pha
se
)
Syste
m
u(t
)
t
y(t
)
t
Input
Outp
ut
TY
PIC
AL IN
PU
T S
IGN
ALS
Com
monly
use
d input
signals
u(t
)=sin
(ω ωωωt)
y(t)=A(ω ωωω)sin(ω ωωωt+
ϕ ϕϕϕ(ω ωωω))
ω-
fre
qu
en
cy, ϕ
-p
ha
se
an
gle
, A
-a
mp
litu
de
TY
PIC
AL IN
PU
T S
IGN
ALS
A s
yste
m c
an b
e inve
stigate
d b
y d
iffe
rent
input
sig
na
ls.
Dif
fere
nt
rea
ctio
ns
on
inp
ut
sig
na
ls i
n c
yber
net
icsy
stem
Do d
iffe
rentsyste
ms h
ave s
om
eth
ing in
com
mon
?
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
Mechanic
al sy
stem
s
Power Dissipated by
Dampers
Damping Components
Elastic Component
Kinematic
relationships
Newton's 2
ndLaw
Conservation Principles for
Linear and Angular Momentum
Rotational Model
Translational
Model
∑=
aF
mJ
=∑
Tα
f(t)
x(t)
m
kc
J
k θ
θ
T
B
2
2
dx
dv
ax
dt
dt
==
=&&
2
2
d dtθ
αθ
==&&
F –
forc
e,
m –
mas
s,
a –
acce
lera
toin
,
x –
dis
pla
cem
ent,
T –
torq
ue,
J –
rota
tion
al i
ner
tia,
α-
ang
ula
rac
cele
rati
on,
θ-an
gula
rdis
pla
cem
ent,
k,B
,c –
coef
fici
ents
spri
ng
Fk
x=
dam
pin
gF
cv
=
dis
sipate
dre
lati
veP
Fv
=
spri
ng
MK
θθ
=
dam
pin
gM
Bω
=
dis
sip
ate
dre
lati
veP
Tω
=
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
Mechanic
al sy
stem
s-bra
ke
Dam
pin
g m
om
ent
J
k θ
θ
T1
B T1M
tłu
mie
nia
Msi
ły
11
sily
MK
θθ
=
11
1T
Jα
Σ=
11
1si
lytlu
mie
nia
MM
TJα
−−
+=
11
11
11
KB
TJ
θθ
ωα
−−
+=
2
11
11
11
2
dd
KB
TJ
dtdt
θ
θθ
θ−
−+
=
2
11
11
11
2
dd
TJ
BK
dtdt
θ
θθ
θ=
++
Fin
dm
oti
on
equ
atio
no
fbra
ke
New
ton
’s2nd
law
Usi
ng
fo
rmu
lafo
r d
amp
ing
mo
men
t
Kin
emat
ic r
elat
ion
s
Eq
uati
on
of
mo
tio
n:
torq
ue-
an
gu
lar
dis
pla
cem
ent
2
12
32
dy
dy
ua
aa
yd
td
t=
++
Dat
a :
J 1,K
θ1
, B
, T
1
Init
ial
cond
itio
ns
are
θ1
(0
), ω
1(0
),tw
ice
inte
gra
l al
low
s to
fin
d a
ngula
r
dis
pla
cem
ent.
Kn
ow
ing
rad
ius
we
can
co
mp
ute
lin
ear
dis
pla
cem
ent.
It i
s eq
uat
ion
of
form
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
Mechanic
al sy
stem
s-gear
Fig
. B
elt c
on
ve
yo
r
(prz
ekła
dnia
cię
gno
wa
)
Pa
ralle
l A
xis
He
lica
l G
ea
rs
Cro
ssed
-He
lica
l G
ea
rsW
orm
gea
rB
eve
l gea
rs
Spur
gears
: g
ears
used to
transm
it r
ota
ry m
otion b
etw
een
para
llel shafts. T
he
y a
re
usually
cylin
dri
cal and h
ave
teeth
that are
str
aig
ht and
para
llel to
the s
haft
axis
. T
heir
majo
r advanta
ge is that th
ey
easy to d
esig
n a
nd
manufa
ctu
re.
Helic
al gears
: H
elic
al gears
are
gears
whic
h m
ay b
e
used for
the tra
nsm
issio
n o
f
motion b
etw
een s
hafts w
ith
either
para
llel or
nonpara
llel
axes. T
hese g
ears
are
often u
sed w
hen loads a
re
very
heavy,
when h
igh
speeds a
re c
alle
d f
or,
or
when it is
im
port
ant to
reduce the n
ois
e level of
the g
earing.
Spu
r gea
rs
Worm
gears
used a
re u
sed w
ith
non-inte
rsecting s
hafts w
hic
h a
re
usually
perp
endic
ula
r to
ea
ch o
ther.
The teeth
on a
worm
gear
are
really
more
sim
ilar
to thre
ads than g
ear
teeth
. W
orm
gears
can p
rovid
e a
larg
e r
eduction in s
haft s
peed
com
pare
d to o
ther
types o
f gears
.
Bevel gears
:
Gears
whic
h a
re u
sed to tra
nsm
it
motion b
etw
een
shaft
s w
hose a
xes insect, u
sually
at
90 d
egre
es
although the
y c
an b
e p
roduced for
alm
ost an
y a
ngle
. S
tra
ight bevel
gears
, lik
e s
pur
gears
, becom
e q
uite
nois
y a
t hig
her
shaft
speeds. S
piral
bevel gears
can
be u
sed to r
educe this
gearing n
ois
e.
Ra
ck a
nd
pin
ion
R=
gea
r ra
diu
s
)=
# o
f g
ear
teet
h
C=
cir
cum
fere
nce
D=
gea
r d
iam
eter
s=
arc
len
gth
θ=
an
gu
lar
dis
pla
cem
ent
ω=
an
gu
lar
vel
oci
ty
α=
an
gu
lar
acce
lera
tio
n
F=
co
nta
ct f
orc
e b
etw
een
gea
rs
T=
to
rqu
e ac
tin
g o
n a
gea
r
P=
po
wer
= T
ω
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
Mechanic
al sy
stem
s-
gear
Gea
r m
oti
on
eq
ua
tio
ns
)2
)1
θ2
s 1 s 2
θ1
R2
R1
12
ss
=G
ear
Pri
nci
ple
1:
Gea
rs i
n c
on
tact
turn
thro
ugh
equ
al a
rc l
eng
ths
11
22
RR
θθ
=2
1
12
R R
θ θ=
12
12
dd
RR
dt
dt
θθ
=1
12
2R
Rω
ω=
22
12
12
22
dd
RR
dt
dt
θθ
=1
12
2R
Rα
α=
22
22
11
11
2 2
RR
C)
RR
C)
π π=
==
12
12
TT
FR
R=
=
R2
R1
T1
FT
2
F
Th
eref
ore
equ
alan
gle
s(r
ad)
Tak
ing
th
e d
eriv
ativ
es–
ang
ula
r v
elo
city
Tak
ing
th
e d
eriv
ativ
es–
ang
ula
r
acce
lera
toin
Sin
ce t
he
nu
mb
er o
f te
eth
is
pro
po
rtio
nal
to t
he
circ
um
fere
nce
of
the
gea
r
Gea
r P
rin
cip
le 2
:G
ears
in
con
tact
ex
ert
equ
al a
nd
opp
osi
te f
orc
es o
n e
ach
oth
er. T
he
equiv
alen
t to
rqu
es o
f th
e co
nta
ct f
orc
e ar
e
ua
y=
This
is
equ
atio
n o
f ty
pe
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
Ele
ctr
onic
al sy
stem
s 1
Systems of real electrical components are modeled as electrical circuit diagrams
v o(t
)
i 1
R
L
C
+ -
i 2
i 3
v 1cv
2cv
1
cv 3
cv 4
11
1
1Ov
vdv
Cv
dt
Rdt
L
−=
+∫
2
11
12
Od
vL
dv
Ldv
LC
vdt
Rdt
Rdt
++
=
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
Ele
ctr
onic
al sy
stem
s 2
Th
en s
ub
stit
ute
in e
ach
of
thes
e in
to t
he
junct
ion
equ
atio
n. C
urr
ent
i 1is
equ
ali 2
+i 3
This
can
be
sim
pli
fied
and
pla
ced
in
to s
tand
ard
fo
rm a
s
eith
er
2
11
23
2
du
dy
dy
ba
aa
yd
td
td
t=
++
This
is
equ
atio
n o
f ty
pe
Det
erm
ine
the
DE
mo
del
of
volt
ag
e v
1v
ersu
s ti
me
t
Kn
ow
ns:
R
, L
, C
, v
o(t
)
Un
kn
ow
ns:
v
1, i 1
, i 2
, i 3
Jun
ctio
n
We
hav
e 4
equ
atio
ns.
Fo
r ju
nct
ion
, re
sist
or,
cap
acit
or
and
co
il. T
o s
olv
e fo
r th
e v
olt
age,
v1
, th
e oth
er
unkn
ow
ns
nee
d t
o b
e el
imin
ated
fro
m t
he
equ
atio
n.
Eli
min
ate
the
curr
ents
, i 1
, i 2
, an
d i
3.S
tart
by s
olv
ing
each
of
the
con
stit
uti
ve
equ
atio
ns
for
the
i te
rms.
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
Ele
ctr
onic
al sy
stem
s 3
Equ
ival
ent
form
s o
f
cond
ense
r d
escr
ipti
on
2
22
12
2(
)d
udu
ut
LC
RC
udt
dt
=+
+
12
()
()
di
ut
LR
it
udt
=+
+
DE
of
circ
uit
Vo
ltag
eu
1d
ecre
ases
on
co
il, re
sist
or
and
cond
ense
r.
2
12
32
dy
dy
ua
aa
yd
tdt
=+
+T
his
is
equ
atio
n o
f ty
pe
•Act
uat
ors
•Po
wer
ran
ge–
wat
to
sev
eral
han
dre
d k
W
•Sup
ply
fro
mel
ectr
icn
et, d
iese
l
gen
erat
or
or
bat
tery
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
DC
Moto
r –
pri
ncip
leof
opera
tion
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
DC
Moto
r –
applicati
on
Rea
din
g h
ead
Har
d d
rive
VC
M
(Vo
ice
Coil
Mo
tor)
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
DC
Moto
r –
applicati
on
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
DC
Moto
r –
applicati
on
Pre
-Am
p -
wzm
acn
iacz
wst
ępn
y
Pea
k d
etec
tor
–u
kła
d w
yk
ryw
ania
mak
sym
aln
ych
zm
ian
nap
ięci
a
Tim
ing
gen
erat
or
–ze
gar
tak
tują
cy
Gre
yco
de
–k
od
Gre
y’a
DS
P –
dig
ital
sig
nal
pro
cess
or
DA
C –
prz
etw
orn
ik c
yfr
ow
o-a
nal
og
ow
y
VC
M D
riv
er –
ster
ow
nik
cew
ki
Spin
con
trol
–re
gula
cja
prę
dk
ośc
i
ob
roto
wej
Spin
dle
dri
ver
–st
erow
nik
prę
dko
ści
ob
roto
wej
Wh
enel
ectr
iccu
rren
tpas
ses
thro
ug
ha
coil
ina
mag
net
icfi
eld
, th
em
agn
etic
forc
ep
rod
uce
sa
torq
ue
wh
ich
turn
the
DC
mo
tor.
W u
zwoje
niu
(p
rzez
któ
re p
łyn
ie p
rąd
) p
rzec
inaj
ącym
po
le
mag
net
ycz
ne
ind
uk
uje
się
siła
wytw
arzaj
ąca
mo
men
t si
ły
ob
raca
jący u
zwoje
nie
Ele
ctri
c cu
rren
t
sup
pli
ed e
xte
rnal
ly
thro
ug
h c
om
uta
tor
Mag
net
ic f
orc
e ac
ts p
erpen
dic
ula
r to
both
wir
e an
d m
agn
etic
fie
ld
F=
BIL
F=
(BxI)
L
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
DC
Moto
r –
pri
ncip
le o
f opera
tion
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
DC
Moto
rE
lect
ric
Mo
tors
:
Dir
ect
Cu
rren
t (D
C)
mo
tors
are
torq
ue
tran
sdu
cers
that
conv
ert
elec
tric
al
ener
gy
to m
ech
an
ica
l en
ergy
.
Th
e to
rqu
e d
evel
op
ed b
y t
he
mo
tor
shaf
t is
dir
ectl
y
pro
port
ion
al t
o t
he
ma
gn
etic
flu
xin
th
e st
ator
fiel
d
and
th
e cu
rren
t in
th
e m
oto
r ar
mat
ure
.
Th
e st
ato
r's
mag
net
ic f
ield
can
be
gen
erat
ed b
y u
se o
f
per
ma
nen
t m
ag
net
so
r b
y p
assi
ng
cu
rren
t th
rough
coil
ed w
ires
that
are
att
ach
ed t
o t
he
mo
tor
hou
sin
g.
Wh
en w
ire
coil
s ar
e u
sed
to
gen
erat
e th
is m
agn
etic
fiel
d, th
e cu
rren
t p
assi
ng
thro
ug
h t
hes
e co
ils
is c
alle
d
the
fiel
d c
urr
ent.
Th
e a
rma
ture
usu
ally
co
nsi
sts
of
an i
ron
-co
red
roto
r
wo
un
d w
ith
man
y w
ire
coil
s. C
urr
ent
is p
asse
d t
o t
he
coil
on
th
e ro
tor
by u
se o
f g
rap
hit
e b
rush
esw
hic
h
conta
ct a
co
mm
uta
tor.
T
he
curr
ent
pas
sing
th
rou
gh
the
coil
s o
n t
he
roto
r p
rodu
ces
a m
agn
etic
fie
ldw
hic
h
push
es a
gai
nst
th
e m
agn
etic
fie
ldg
ener
ated
by t
he
stato
r. T
he
curr
ent
pas
sed
thro
ug
h t
hes
e ro
tor
coil
s is
call
ed t
he
arm
atu
re c
urr
ent.
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
DC
Moto
r –
pri
ncip
le o
f opera
tion
Asi
mp
leD
Cel
ectr
icm
oto
r.
Wh
en t
he
coil
is
po
wer
ed, a
mag
net
ic f
ield
is
gen
erat
ed
arou
nd
th
e ar
mat
ure
.T
he
left
sid
e o
f th
e ar
mat
ure
is
push
ed
away
fro
m t
he
left
mag
net
and
dra
wn
to
war
d t
he
righ
t,
cau
sing
ro
tati
on
Th
e ar
mat
ure
conti
nues
toro
tate
Wh
en t
he
arm
atu
re b
eco
mes
ho
rizo
nta
lly
alig
ned
,th
e co
mm
uta
tor
rev
erse
s th
e dir
ecti
on
of
curr
ent
thro
ugh
th
e co
il,re
ver
sing
th
e
mag
net
ic f
ield
.T
he
pro
cess
th
en r
epea
ts.
If t
he s
haft
of
a D
C m
oto
ris
turn
ed
by
an e
xte
rnal fo
rce,th
em
oto
r w
illact
like
a
genera
tor
and p
roduce a
n E
lectr
om
otive f
orc
e(E
MF
).D
uring n
orm
al opera
tion,th
e
spin
nin
gof
the
moto
rpro
duces
avoltage,
know
nas
the c
ounte
r-E
MF
(C
EM
F)
or
back
EM
F,
because it
opposes t
he a
pplie
d v
oltage
on
the
moto
r.T
his
is t
he
sam
e
EM
Fth
at
is p
roduced w
hen t
he
moto
ris
used
as a
genera
tor
(for
exam
ple
when a
n
ele
ctr
ical lo
ad
(resis
tance)
is p
laced a
cro
ss t
he t
erm
inals
of th
em
oto
rand t
he
moto
rshaft
is d
riven w
ith a
n e
xte
rnal to
rque).
There
fore
,th
e v
oltage
dro
pacro
ss
a
moto
rconsis
ts o
f th
e v
oltage
dro
p,
due
toth
isC
EM
F,
and t
he p
ara
sitic
voltage
dro
pre
sultin
g f
rom
the inte
rnal re
sis
tance o
f th
e a
rmatu
re's
win
din
gs.
Rota
ting m
agnetic fie
ldas a
sum
of
magnetic v
ecto
rs f
rom
3 p
hase
coils
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
DC
Moto
r –
pri
ncip
le o
f opera
tion
Researc
hers
at
Univ
ers
ity o
f C
alif
orn
ia,
Berk
ele
y,
ha
ve d
evelo
ped r
ota
tional
bearings b
ased u
pon m
ultiw
all
carb
on n
anotu
bes.
By
att
achin
ga
gold
pla
te
(with d
imensio
ns o
ford
er
100nm
) to
the o
ute
r shell
of
asuspended m
ultiw
all
carb
on n
anotu
be
(lik
e n
este
d c
arb
on c
ylin
ders
),th
ey a
re a
ble
to
ele
ctr
osta
tically
rota
te t
he o
ute
r shell
rela
tive
toth
e inner
core
.T
hese
bearings a
re v
ery
robust;
devic
es h
ave b
een o
scill
ate
d t
housands o
f tim
es
with
no
indic
ation o
f w
ear.
These n
anoele
ctr
om
echanic
al syste
ms
(NE
MS
)
are
the n
ext
ste
pin
min
iatu
rization t
hat m
ay f
ind t
heir w
ay into
com
merc
ial
aspects
in t
he f
utu
re.
Nanom
oto
r constr
ucte
d a
tU
CB
erk
ele
y.
The
moto
ris
about500nm
acro
ss: 300
tim
es s
malle
r th
an the d
iam
ete
r of
a
hum
an h
air
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
NanoM
oto
r–
pri
ncip
le o
f opera
tion
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
DC
Moto
r –
pri
ncip
le o
f opera
tion
•Ap
ply
av
olt
age
toar
mat
ure
•Arm
atu
re r
ota
tes
in m
agn
etic
fie
ld
•Sp
eed
co
ntr
ol
by
:
–A
rmat
ure
vo
ltag
e
–F
ield
Str
eng
th
•S
pee
d i
s p
rop
ort
ion
alto
volt
age
•P
ręd
ko
ść o
bro
tow
a ≈
nap
ięci
e
•T
orq
ue
is p
rop
ort
ion
alto
cu
rren
tM
om
ent
siły
≈p
rąd
•Po
wer
=sp
eed
x t
orq
ue
m/s
Nm
=m
/s k
g m
/s2=
kg
m2/s
3
W=
J/s=
kg
(m/s
)2/s
=k
g m
2/s
3
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
DC
Moto
r –
pri
ncip
le o
f opera
tion
Sp
eed
contr
ol
thro
ugh
arm
ature
volt
age
and
curr
ent
chan
ge
To
mo
del
lin
g w
e ca
n a
ssu
me
* A
rmat
ure
iner
tia
J =
0.0
1 k
g m
2/s
2
* T
orq
ue
dis
sip
atio
n c
oef
fici
ent
b =
0.1
Nm
s
* c
on
stan
t K
=K
e=K
t =
0.0
1 N
m/A
* r
esis
tance
R =
1 o
hm
* i
nd
uct
ance
L =
0.5
H
* i
np
ut
vo
ltag
eV
:
* o
utp
ut
ang
le t
het
a:
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
DC
Moto
r –
equati
on o
f m
oti
on θ& et
Ke
iK
T
==
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
Ele
ktr
om
echanic
al sy
stem
u
i adc
Ra
La
J
aa
aa
b
di
uR
iL
ed
t=
++
ω ωωω
Mec
han
ical
sub
syst
em
moto
rT
JB
ωω
=+
&
BIn
pu
t: v
olt
age
u
Ou
tput:
Ang
ula
r v
elo
city
ω
Ele
ctri
cal
Sub
syst
em (
loo
p m
etho
d):
e b–
bac
k-e
mf
vo
ltag
e (E
MF
–el
ectr
om
agn
etic
fo
rce)
ui a
dc
Ra
La
ω ωωω
To
rqu
e-C
urr
ent
:
Vo
ltag
e-S
pee
d:
moto
rt
aT
Ki
=
Co
mb
ing p
rev
ious
equat
ions
resu
lts
in t
he
foll
ow
ing
mat
hem
atic
alm
od
el
B
Po
wer
tra
nsf
orm
atio
n
bb
eK
ω=
-0
aa
aa
b
ta
di
LR
iK
ud
t
JB
Ki
ω
ωω
+
+=
+
=
&
wh
ere
Kt:
torq
ue
const
ant,
Kb:
vel
oci
ty c
on
stan
t.F
or
an i
dea
l m
oto
r
tb
KK
=
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
Ele
ctr
om
echanic
alsy
stem
2
12
32
dy
dy
ua
aa
yd
tdt
=+
+
Tim
e d
eriv
atio
n, so
lve
for
i aan
dsu
bst
itute
to
firs
teq
uat
ion
i a, di a/
dt
()
()
...
aa
aa
tb
tL
JJR
BL
BR
KK
Ku
ωω
ω+
++
+=
This
is
equ
atio
n o
f a
typ
e
An
oth
erre
lati
on
bet
wee
nan
gula
r v
elo
city
ωan
d v
olt
age
u
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
DC
Moto
r
Ad
just
th
e v
olt
age
and
cu
rren
t (i
f)
app
lied
to
th
e fi
eld
win
din
g,
or
Ad
just
th
e v
olt
age
and
cu
rren
t ( i A
)
to t
he
arm
atu
re
RA
LA
+θ
M,
ωM
v b
KE,K
T
i A
-
i f
J, c
T
v in
+θ
M,
ωM
v b
KE,K
T
-
i A
J, c
T
Lf
Rf
v in
i f
Armature Controlled DC Motor Field Controlled DC Motor
Ad
va
nta
ges
:
A
dv
an
tag
es:
--go
od
to
rque
at h
igh s
pee
ds
--ener
gy e
ffic
ient
--sm
all
and
inexp
ensi
ve
contr
oll
er
Dis
ad
va
nta
ges
:
Dis
ad
va
nta
ges
:
--la
rger
and
mo
re e
xp
ensi
ve
co
ntr
oll
er
--to
rqu
e d
ecre
ases
at
hig
h s
peed
s
--hig
her
ener
gy l
oss
es
-
-var
iab
le l
oad
aff
ects
sp
eed
--vari
able
lo
ad a
ffec
ts s
pee
d
Mo
del
:
Mo
del
:
--li
near
mo
del
--no
nli
nea
r m
od
el
Ma
them
ati
cal
Mo
del
s:
Ele
ctri
cal
Mo
del
Ele
ctri
cal
Mo
del
Mec
hanic
al M
od
elM
ech
anic
al M
od
el
0A
inA
AA
b
di
vi
RL
vd
t−
−−
=0
f
inf
ff
di
vi
RL
dt
−−
=
bE
vK
ω=
0A
AA
AA
b
di
vi
RL
vd
t−
−−
=
bE
fv
Ki
ω=
d
dT
cT
Jd
tωω
−−
=d
dT
cT
Jd
tωω
−−
=
TA
TK
i=
Tf
AT
Ki
i=
vin
= c
ontr
ol
vo
ltage
v
b=
mo
tor
bac
k e
mf
i A=
arm
atu
re c
urr
ent
i f
= f
ield
curr
ent
T =
mo
tor
torq
ue
Td
= d
istu
rban
ce t
orq
ue
on s
haft
J=
mo
tor
iner
tia
c
= m
oto
r d
am
pin
g c
onst
ant
ω=
sp
eed
of
mo
tor
R
A, R
f=
arm
ature
and
fie
ld r
esis
tan
ce
LA
= A
rmat
ure
ind
uct
ance
L
f=
fie
ld i
nd
uct
ance
KE
= K
T:
Mo
tor
const
ants
(usu
ally
fo
und
in s
pec
ific
atio
ns)
.
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
DC
Moto
r –
equati
on o
f m
oti
on
Ste
pp
erm
oto
r
Clo
sely
rel
ated
in
des
ign
toth
ree-
ph
ase
AC
syn
chro
nou
s m
oto
rs a
re s
tep
per
mo
tors
,w
her
e an
inte
rnal
roto
r
conta
inin
g p
erm
anen
t m
agn
ets
or
ala
rge
iro
n c
ore
wit
h s
alie
nt
po
les
is c
ontr
oll
edb
y a
set
of
exte
rnal
mag
net
s th
at a
re
swit
ched
ele
ctro
nic
ally
. A
step
per
mo
tor
may
als
ob
eth
oug
ht
of
as a
cro
ss b
etw
een
a D
Cel
ectr
icm
oto
ran
da
sole
noid
. A
sea
ch c
oil
is
ener
giz
ed i
n t
urn
,th
ero
tor
alig
ns
itse
lf w
ith
th
e m
agn
etic
fie
ld p
rodu
ced
by
the
ener
giz
ed
fiel
d w
indin
g.U
nli
ke
asy
nch
ronou
sm
oto
r,in
its
app
lica
tion
,th
em
oto
rm
ayn
ot
rota
te c
onti
nu
ou
sly;
inst
ead
,it
"ste
ps"
fro
mo
ne
po
siti
on
toth
e n
ext
asfi
eld
win
din
gs
are
ener
giz
ed a
nd
de-
ener
giz
ed i
n s
equ
ence
.D
epen
din
gon
the
seq
uen
ce,th
ero
tor
may
tu
rn f
orw
ard
s o
r bac
kw
ard
s.
Co
ntr
ol
isd
iscr
ete
inti
me
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
Hydra
ulic
syst
em
Mix
ing
of
two
flu
id w
ith
dif
fere
nt
con
cen
trat
ion
C –
con
centr
atio
n,
m –
mas
s st
ream
V –
volu
me
stre
am
t -
tim
e
m V
dC dt
CC
a
aaw
e.
+=
DE
23
dy
ua
ay
dt
=+
Th
is i
s eq
uat
ion
of
typ
e
DIF
FER
EN
TIA
L E
QU
AT
ION
S O
F S
YST
EM
S
Therm
alsy
stem
V1
V2
V3
V=
abc
Mix
ing t
wo
flu
ids w
ith
diffe
ren
t te
mpe
ratu
res
ρθ
ρθ
ρθ
ρθ
33
3
11
11
22
22
33
33
cV
d dt
cV
cV
cV
pp
pp
=+
−.
..
θθ
θθ
θθ
θθ
θ
110
1
220
21
330
3
=+
=+
=+
∆ ∆ ∆
01
11
12
22
23
33
3=
+−
ρθ
ρθ
ρθ
cV
cV
cV
pp
p
..
.
ρθ
ρθ
ρθ
ρθ
33
3
33
33
11
11
22
22
cV
d
dt
cV
cV
cV
pp
pp
∆∆
∆∆
+=
+.
..
[]
∆∆
∆θ
θθ
31
12
2
1
1(
)(
)(
)s
Ts
ks
ks
=+
+
TV V
kc
V
cV
kc
V
cV
p p
p p
==
=.
. .
. .
30
1
11
10
33
30
1
22
20
33
30
ρ ρ
ρ ρ
En
erg
y c
on
serv
atio
n
Pro
cess
set-
poin
t
Sta
tic
char
acte
rist
ics
in s
tead
y s
tate
Dyn
amic
ch
arac
teri
stic
s
Lap
lace
tra
nsf
orm
2
12
32
dy
dy
ua
aa
yd
tdt
=+
+T
his
is
equ
atio
n o
f ty
pe
Co
ncl
usi
on
CO
NC
LU
SIO
N
Man
yp
roce
sses
can
be
des
crib
edb
y l
inea
r,
tim
e-in
var
ian
teq
uat
ion
s
22
21
00
12
22
...
...
du
du
dy
dy
aa
au
ay
aa
dt
dt
dt
dt
++
+=
++
+
“Mat
hem
atic
s co
mp
ares
dif
fere
nt
ph
eno
men
on
s an
d
dis
cov
ers
secr
etan
alo
gie
sb
etw
een
them
”
Jean Baptiste Joseph Fourier (1768 -1830)
CO
NC
LU
SIO
N
Ho
wto
solv
ediffe
rentia
l eq
uatio
ns?
Analy
tical so
luti
ons
of
dif
fere
nti
al
equati
ons
Classification of Ordinary Differential Equations:
The
rea
son
to
sta
rt w
ith
the
cla
ssific
atio
n o
f diffe
ren
tia
l e
qua
tio
ns is tha
t th
ere
are
diffe
ren
t m
eth
od
s u
sed
to
so
lve
diffe
ren
t ty
pe
s o
f d
iffe
ren
tial e
qua
tion
s.
Som
etim
es y
ou
need
to
know
wh
ich
equa
tio
n t
ype
yo
u h
ave
in
ord
er
to p
ick
an
app
rop
ria
te o
r be
st so
lutio
n m
eth
od
.
To
de
term
ine
the
cla
ssific
ation
of
diffe
ren
tial equ
ation
s,
yo
u s
hou
ld s
tart
by p
utt
ing a
DE
in
to its
sta
nda
rd fo
rm.
Standard Form:
a)
Iden
tify
the
depen
den
t and
ind
epen
den
t va
riab
les.
b)
Pla
ce
all
the
depe
nden
t va
riab
le te
rms o
n the
left
ha
nd
sid
e o
f th
e e
qua
tion
.
c)
Mo
ve
all
oth
er
term
s (
co
nsta
nts
and
in
d.
var.
te
rms)
to t
he
rig
ht
sid
e o
f th
e e
qua
tion
.
d)
Arr
an
ge
all
the
de
riva
tive
s b
y o
rde
r. H
ighe
st
de
riva
tive
goe
sto
the left
.
e)
No
rmaliz
e t
he
coeff
icie
nt
of
the
lo
we
st o
rde
r depe
nden
t va
riab
le t
erm
.
f)
Sim
plif
y t
he
equa
tion
, if p
ossib
le
g)
Asse
mb
le t
he
con
ditio
ns.
The
sta
nda
rd f
orm
of
the
equa
tion
sho
uld
look s
om
eth
ing lik
e th
is.
Som
etim
es e
qua
tion
s,
pa
rtic
ula
rly n
on
linea
r equ
ation
s a
nd
se
ts o
f lin
ked
equa
tion
s, m
ay u
se
a s
ligh
tly d
iffe
ren
t
sta
nda
rd d
iffe
ren
tia
l e
qua
tion
fo
rm.
F
or
the
se
equa
tio
ns,
the
coeff
icie
nt
of
the
hig
he
st
ord
er
de
riva
tive
is n
orm
aliz
ed
to 1
in
ste
ad
of
the
lo
we
st
ord
er
dep
ende
nt
term
.
1
12
1...
()
w
ith
condit
ions
))
))
dx
dx
aa
xf
t)
dt
dt
−
−+
++
=
1
11
...
0
w
ith
c
on
dit
ions
))
))
)
dx
dx
aa
x)
dt
dt
−
−+
++
=
Types of features to classify:
Dependent variable:
The
de
pende
nt
va
riab
le(s
) re
pre
se
nts
the
ou
tpu
t va
riab
le(s
) of
the
mod
el.
The
y a
re the
va
ria
ble
s in
the
nu
me
rato
r of
the
de
riva
tive
s in
the
DE
. If
the
mo
de
l con
sis
ts o
f se
ve
ral lin
ked
equa
tion
s,
then
the
re
ma
y b
e m
ultip
le d
epende
nt
va
ria
ble
s.
Depe
nden
t va
riab
le(s
):
x, y,
an
d z
Indepen
den
t va
riab
le(s
):
t
Independent variable
: T
he
va
riab
le in
the
den
om
ina
tor
of
the
diffe
ren
tia
l e
qua
tion
is t
he
ind
epend
en
t va
riab
le.
Yo
u t
yp
ically
pic
k its
ran
ge
when
solv
ing t
he
equ
ation
. O
rdin
ary
diffe
ren
tia
l e
qua
tion
s h
ave
a s
ingle
inde
pende
nt
va
ria
ble
. P
art
ial d
iffe
ren
tia
l e
qua
tio
ns h
ave
seve
ral in
dep
enden
t va
riab
les.
OD
EP
DE
55
22
dx
xy
td
t+
−=
+
53
0d
yt
yx
zd
t+
−+
=
53
dz
zx
dt−
+=
2
25
52
dx
dx
xx
tdt
dt
+−
=2
22
22
5x
xy
xt
∂Ω
∂Ω
∂Ω
∂Ω+
+=
−∂
∂∂
∂∂
Analy
tical so
luti
ons
of
dif
fere
nti
al
equati
ons
Analy
tical so
luti
ons
of
dif
fere
nti
al
equati
ons
Linear or Nonlinear:
An
equa
tion
is lin
ea
r if a
ll de
pend
en
t va
riab
le te
rms a
re r
ais
ed
to
the p
ow
er
of
1 a
nd
are
no
t in
clu
ded
in
tra
nscend
en
tal fu
nction
s (cos
, exp
, ln
, sqrt
, e
tc.)
Lin
ea
rN
on
linea
r
Homogeneous or Nonhomogeneous
: A
n e
qu
ation
is h
om
ogeneou
s if
the
rig
ht ha
nd
sid
e (
the
drive
te
rm)
of
the
sta
nda
rd f
orm
is e
qu
al to
ze
ro,
f(t)
= 0
.
An
y e
qua
tion
with
a n
on
ze
ro d
rive
te
rm is n
onho
mo
geneou
s.
Ho
mo
ge
neou
sN
on
hom
ogeneo
us
2
25
52
dx
dx
xt
dt
dt
+−
=
2
25
52
dx
dx
xx
td
td
t+
−=
2
25(
1)5
2d
xd
xt
xt
dt
dt
++
−=
2
25
5si
n(2
)2
dx
dx
xt
dt
dt
+−
=
2
2
1(
)2
52
sin
5
dx
xt
dt
t+
=+
3
3
18
02
dx
dx
dt
dt
x+
−=
2
25
50
dx
dx
xd
td
t+
−=
2
25
52
dx
dx
xx
td
td
t+
−=
Constant or Variable Coefficient:
An
equa
tion
wh
ich
ha
s in
depen
den
t va
riab
les w
hic
h c
ann
ot be
sepa
rate
d f
rom
the
de
pende
nt
va
riab
le w
hen
the
equa
tion
is p
laced
in
sta
nd
ard
fo
rm is a
va
riab
le c
oeff
icie
nt
equ
ation
. M
an
y v
aria
ble
coe
ffic
ien
t e
qua
tion
s r
equire
so
lution
s w
hic
h, if s
olv
ab
le,
requ
ire
infin
ite
se
rie
s s
olu
tion
s.
Co
nsta
nt
co
eff
icie
nt :
Va
riab
le c
oeff
icie
nt:
Analytical or Tabular Drive term: T
he
drive
te
rm o
n the
rig
ht
hand s
ide
of
the
equa
tion
ma
y s
om
etim
es b
e d
escrib
ed
usin
g e
xpe
rim
en
tal o
r ta
bula
r d
ata. I
f th
e d
rive
te
rm c
an
be
com
ple
tely
exp
resse
d a
s a
n e
qu
ation
in
te
rms o
f th
e
inde
pend
en
t va
riab
le it is
ca
lled
ana
lytica
l.
An
aly
tica
l D
rive
Te
rmT
ab
ula
r D
rive
Te
rm
wh
ere
t
1
10
1
5
25
f(t)
0
0
1
3
g(t
)
10
5
3
2
2
25
52
dx
dx
xx
td
td
t+
−=
2
25(
1)5
2d
xd
xt
xt
dt
dt
++
−=
2
25
52
5co
s(3
)d
xd
xx
xt
td
td
t+
−=
+2
25
52
()
5(
)d
xd
xx
xf
tg
td
td
t+
−=
+
Analy
tical so
luti
ons
of
dif
fere
nti
al
equati
ons
Analy
tical so
luti
ons
of
dif
fere
nti
al
equati
ons
Condition Type: W
hile
in
itia
l cond
itio
ns a
re t
he
mo
st
com
mo
n,
cond
itio
ns m
ay a
lso
be
of
bo
unda
ry a
nd
m
ixed
typ
e.
Initia
l cond
itio
ns o
ccu
r if a
ll co
nd
itio
ns a
re s
pecifie
d a
t th
e s
am
e v
alu
e o
f th
e indepe
nden
t va
riab
le.
Bou
nda
ry
con
ditio
ns e
xis
t if t
he
con
ditio
ns a
re g
iven
at va
lue
s o
f th
e indep
enden
t va
riab
le w
hic
h b
racke
t th
e r
egio
n o
f in
tere
st.
Initia
l C
ond
itio
ns
Bo
un
da
ry C
ond
itio
ns:
with
x(1
) =
20
, (1
)= -
2,
an
d (
1)=
0
w
ith
x(0
) =
20
and
x(5
)= -
2
x& x&&
Methods of Finding Analytical Solutions to Differential Equations:
1)
Separa
tion o
f V
ariable
s M
eth
od
2)
Undete
rmin
ed C
oeff
icie
nt
Meth
od
3)
Variatio
n o
f P
ara
mete
rs M
eth
od
4)
Opera
tor
Me
thod f
or
Hom
ogene
ous E
qu
ations
Analy
tical so
luti
ons
of
dif
fere
nti
al
equati
ons
Separation of Variables Method:
In t
he
se
pa
ratio
n o
f va
riab
le m
eth
od
you
try
to
iso
late
the
depen
den
t an
d indep
enden
t va
riab
les o
n o
ppo
site
sid
es
of
the
equa
tion
, th
en
inte
gra
te e
ach
sid
e indepe
nden
tly.
Th
is is o
ne m
eth
od
tha
t m
ay o
ften w
ork
fo
r non
linea
r,
ho
mo
ge
neou
s 1
sto
rder
pro
ble
ms.
Exa
mp
le:
Solution
2
02
0
wit
h
(0)
dx
xx
xd
t+
==
Analy
tical so
luti
ons
of
dif
fere
nti
al
equati
ons
22
dx
xd
t=
−
212
dx
dt
x=
−
00
212
xt
xt
dx
dt
x=
−∫
∫ ] 0
0
12
xt t
x
tx
−
=−
()
0
0
11
2t
tx
x
−−
=−
−
()
0
0
11
2t
tx
x=
+−
()
0
0
1
12
x
tt
x
=+
− ()
0
00
12
xx
xt
t=
+−
Analy
tical so
luti
ons
of
dif
fere
nti
al
equati
ons
Undetermined Coefficient Method:
Use
the
und
ete
rmin
ed
coe
ffic
ien
t m
eth
od
to
so
lve
lin
ea
r, o
rdin
ary
, n
on
hom
ogeneo
us e
qu
ation
s w
ith
con
sta
nt
co
eff
icie
nts
.
The
so
lutio
n is fou
nd
by d
ete
rmin
ing t
wo
pa
rts t
o t
he
so
lution
:
The
co
mp
lem
en
tary
so
lution
, x
c,
is s
imp
ly t
he
ho
mo
ge
neou
s s
olu
tion o
bta
ined
wh
en
th
e d
rive
fun
ction
f(t
) is
se
t e
qual to
ze
ro.
The
pa
rtic
ula
r so
lution
, x
p,
accou
nts
fo
r th
e e
ffe
ct
of
the
drive
te
rm.
The
pa
rtic
ula
r so
lution
is
assu
med
to
ha
ve
a fo
rm s
imila
r to
the
drive
fun
ction
bu
t w
ith
a n
um
be
r of
un
kno
wn
coeff
icie
nts
. T
he
assu
med
so
lution
and
its
de
riva
tive
s a
re s
ub
stitu
ted
ba
ck in
to the
nonh
om
ogeneou
s e
qua
tion
and
the
coeff
icie
nts
are
then
de
term
ined
.
1
11
...
()
w
ith
c
on
dit
ion
s)
)
))
)
dx
dx
aa
xf
t)
dt
dt
−
−+
++
=
cp
xx
x=
+
Example:
Solution
:
The
complementary solution
ha
s a
lread
y b
een
found
fro
m the
solu
tion
of
the
hom
ogene
ous e
qua
tion
in
the
pre
vio
us
exa
mp
le.
Th
e p
arti
cula
r so
luti
on
may b
e fo
un
d b
y a
ssu
min
g t
hat
th
e so
luti
on
tak
es o
n t
he
form
then
its
der
ivat
ive
is
Su
bst
itu
tin
g t
hes
e b
ack
in
to t
he
DE
giv
es
Ali
gn
ing t
he
coef
fici
ents
giv
e:
ther
efo
re t
he
par
ticu
lar
solu
tio
n i
s
and
th
e gen
eral
so
luti
on
is
Fo
r th
e in
itia
l co
nd
itio
n o
f ,
C
1m
ay b
e fo
un
d a
s
ther
efo
re
the
gen
eral
so
luti
on
is
22
35
w
ith
(0
)0
dx
xt
xx
dt+
=+
=
20
d
xx
dt+
=2
1
t
cxC
e−
=
2
12
3
px
At
At
A=
++
12
2
px
At
A=
+&
22
35
pp
xx
t+
=+
&(
)(
)2
2
12
12
32
2
35
A
tA
At
At
At
++
++
=+
()
()
22
11
22
32
22
2
30
5
At
AA
tA
At
t+
++
+=
++
12
3
A=
12
22
0
AA
+=
23
25
A
A+
=1
1.5
A=
21
AA
=−
23
5
2AA
−=
= -
1.5
5(
1.5
) =
=
3.2
52
−−
21.5
1.5
3.2
5
px
tt
=−
+
cp
xx
x=
+2
2
11
.51
.53
.25
t
Ce
tt
−=
+−
+
0(0
)x
x=
2(0
)2
01
(0)
1.5
(0)
1.5
(0)
3.2
5
xx
Ce
−=
=+
−+
01
3.2
5
xC
=+
10
3.2
5
Cx
=−
22
0(
3.2
5)
1.5
1.5
3.2
5
tx
xe
tt
−=
−+
−+
Analy
tical so
luti
ons
of
dif
fere
nti
al
equati
ons
Analy
tical so
luti
ons
of
dif
fere
nti
al
equati
ons
Operator Method:
Use
the
ope
rato
r m
eth
od
to
so
lve
lin
ea
r, o
rdin
ary
, h
om
oge
neou
s e
qua
tion
s w
ith
co
nsta
nt
coe
ffic
ien
ts.
Th
is m
eth
od
is im
ple
men
ted
by r
ep
lacin
g e
ach
ith
de
riva
tive
of
the
de
pende
nt
va
riab
le w
ith
D i .
T
he
equa
tio
n is the
n
so
lved
to
fin
d r
oo
ts (
va
lue
s o
f D
) w
hic
h s
atisfy
th
e "
ch
ara
cte
ristic e
qua
tio
n".
The
cha
racte
ristic e
qua
tio
n is a
ge
ne
ral po
lyno
mia
l w
hic
h s
hou
ldha
ve
Nro
ots
, r 1
, r 2
, .
. . r N
. R
oo
ts m
ay b
e r
ea
l,
multip
le, o
r com
ple
x.
The
fo
rm o
f th
e s
olu
tion
de
pend
s u
pon
the
typ
e o
f ro
ots
pre
sen
t.
If allroots arereal
and
unique
If all roots are real
bu
t not unique
, th
en
the
multip
le r
oo
ts r
equ
ire
a m
ultip
le p
ow
er
of t
be
use
d o
n s
ele
cte
d t
erm
s
If some roots are complex
, th
en
ea
ch
com
ple
x p
air s
olu
tion
ma
y b
e w
ritt
en
in
a n
um
be
r of
com
mon
fo
rms.
1
11
...
0
wit
h
co
nd
itio
ns
))
))
)
dx
dx
aa
x)
dt
dt
−
−+
++
=
1
1..
.0
))
)D
aD
a−
++
+=
12
12
...
)rt
rt
rt
)x
Ce
Ce
Ce
=+
++
12
22
12
2..
.)r
tr
tr
tr
t
)x
Ce
Cte
Ct
eC
e=
++
++
()
12
12
12
2
1 1
()
()
12
12
12
22
12
2
...
()
...
=
cos
sin
...
=
(si
n)
...
) )
)
)
rt
rir
tr
irt
)
rt
rt
irt
irt
)
rt
rt
)
rt
rt
)
xC
eC
eC
e
eC
eC
eC
e
eA
rt
Ar
tC
e
eB
rt
BC
e
+−
+−
=+
++
=+
++
++
+
++
+
Example:
Fin
d the
so
lution
of
the D
E u
sin
g t
he
Op
era
tor
me
thod
:
Solution:
Le
t t
his
equa
tion
be
rep
resen
ted
usin
g t
he
Dop
era
tor
as
so
the
cha
racte
ristic e
qu
ation
is
Th
is e
qua
tion
ha
s a
n o
rde
r o
f m
agnitude
of
1 (t
he
refo
re N
= 1
and
the
re is o
ne
ro
ot)
The
roo
t is
Sin
ce
th
ere
is o
ne
rea
l ro
ot,
th
e s
olu
tion
fits the
fo
rm o
f
At
the
in
itia
l cond
itio
n
so
app
lyin
g t
he
ge
ne
ral so
lutio
n w
ith
th
e in
itia
l co
nd
itio
n
giv
es t
he
so
lution
as
02
0
wit
h
(0)
dx
xx
xd
t+
==
20
D
xx
+=
()
20
xD
+=
20
D+
=
12
r
=−
12
11
rt
tx
Ce
Ce
−=
=
0(0
)x
x=
02
()
01
1(0
)t
xx
Ce
C−
==
=
2
0(
)t
xt
xe
−=
Analy
tical so
luti
ons
of
dif
fere
nti
al
equati
ons
Gen
era
liza
tio
n–
tra
nsf
er f
un
cti
on
an
d b
asi
c co
mp
on
en
ts
GEN
ER
ALIZ
AT
ION
Tra
nsf
er f
un
ctio
nit
isra
tio
of
two
com
ple
x p
oly
no
mia
ls
Gs
Ys
Xs
kk
sk
sk
s
Ts
Ts
Ts
nm
n
n
mmm
()
()
()
...
...
,=
=+
++
++
++
+≤
01
2
2
122
21
In m
ath
em
atic
s,th
e fu
nd
amen
tal
theo
rem
of
alg
ebra
stat
es t
hat
ev
ery n
on
-zer
o s
ing
le-v
aria
ble
po
lyn
om
ial,
wit
h c
om
ple
x c
oef
fici
ents
,h
as e
xac
tly
asm
any c
om
ple
x r
oo
tsas
its
deg
ree,
if r
epea
ted
roo
ts a
re c
ou
nte
d u
pto
thei
r m
ult
ipli
city
.E
qu
ival
entl
y,
the
mat
hem
atic
al f
ield
of
com
ple
x n
um
ber
s
is c
lose
d u
nd
er a
lgeb
raic
op
erat
ion
s.In
oth
er w
ord
s, f
or
ever
y c
om
ple
x p
oly
no
mia
lp
of
deg
ree
n >
0
the
equ
atio
np
(z)
= 0
has
ex
actl
yn
com
ple
x s
olu
tio
ns,
cou
nti
ng
mu
ltip
lici
ties
. S
uch
po
lyn
om
ials
can
hav
e fo
rm
()
()
()
()
Ys
Xs
kT
sT
sT
s
sT
sT
sT
s
iiq
iib
idid
idid
j
iri
r
ipip
ipi
p
()
()=
±±
±
±±
±
==
=
==
∏∏
∏
∏∏
11
22
1
1
22
1
12
1
12
1
ξ ξ
Usi
ng
Lap
lace
tra
nsf
orm
DE
is
conv
erte
din
alg
ebra
iceq
uat
ion
an
d i
n
tran
sfer
fu
nct
ion
.
Bec
ause
of
n <
= m
, th
eref
ore
fro
mfo
rmu
lare
sult
sth
atth
ere
are
six
dif
fere
nt
typ
es
of
lin
ear
syst
ems.
Th
ese
wil
l b
e ca
lled
bas
ic t
ran
sfer
fu
nct
ion
s
·ga
ink
(pro
po
rtio
n)
·in
teg
ral
1/s
·fi
rst
ord
er
k/(
Ts+
1)
·d
iffe
ren
tia
ls
·se
con
d o
rde
r k
/(T
12
s2+
2ξT
2s+
1)
(da
mp
ed l
inea
r o
scil
lato
r)
·d
elay
exp(-
sTo)
()
()
()
()
Ys
Xs
kT
sT
sT
s
sT
sT
sT
s
ii
q
iib
idid
idid
j
iri
r
ipip
ipi
p
()
()=
±±
±
±±
±
==
=
==
∏∏
∏
∏∏
11
22
1
1
22
1
12
1
12
1
ξ ξ
GEN
ER
ALIZ
AT
ION
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Ba
sic
tra
nsfe
r fu
nctio
ns p
rop
ert
ies
will
com
e fro
m e
qua
tio
n
Exe
mp
lary
two
step
res
po
nse
sin
cyb
ern
etic
syst
em
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Ga
in-
pro
pert
ies
()
()
yt
kut
=T
ime-
do
mai
neq
uat
ion
Tra
nsf
er f
unct
ion
Ste
p r
esp
on
se
Gs
k(
)=
()
1()
yt
ku
t=
∆
time
k
0
1
Itis
stat
icfu
nct
ion
.
Fig
. S
tep r
esp
onse
Inp
ut
sig
nal
u(t
)
Ou
tput
sign
al y(t
)
Fig
. Sym
bo
l
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Ga
in-
exam
ple
s
U1
U2
12
1
UU
iR−
=
2 2
Ui
R= R
RR
UU
2
12
12
+=
Cu
rren
tth
rou
gh
resi
stor
R1
R1
Cu
rren
t th
rou
gh
res
isto
rR
2R2
Ou
tput
volt
age
U2
Ele
ctri
cal
syst
em-
RR
circ
uit
Mec
han
ical
sy
stem
-lev
erF1
r 1r 2
F2
To
rqu
e F
1r 1
=F
2r 2
Ou
tput
forc
e F
2=
F1r 2
/r1
Mec
han
ical
syst
em-g
ears
12
12
TT
FR
R=
=F
orc
eb
etw
een
two
gea
rsis
equ
al
u(t)
y(t)
Rel
atio
nsh
ip b
etw
een f
orc
e, t
orq
ue
and
mo
men
tum
vec
tors
in a
ro
tati
ng s
yst
em
Inte
gra
l -
pro
pe
rtie
s
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Tim
e-d
om
ain
equ
atio
n
Tra
nsf
er f
unct
ion
Ste
pre
spo
nse
k
0
1
Itis
dy
nam
icp
roce
ss.
Fig
. S
tep
resp
onse
Fig
. Sym
bo
l
0
,(
)
td
yku
yk
ut
dt
dt=
=∫
()
yt
ku
t=
∆
Gs
k s(
)=
time
Inp
ut
sig
nal
u(t
)
Ou
tput
sign
al y(t
)
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Inte
gra
l -
exam
ple
s
Inp
ut
sig
nal
–vo
ltag
e
Ou
tput
sign
al–
shaf
tro
tati
on
angle
Ele
ctri
c sy
stem
AC
act
uat
or
N-z
ero
rob
ocz
e.
eL-
syg
nał
z r
egula
tora
(11
).
aL-
syg
nał
z r
egula
tora
(12
).
N-
zero
rob
ocz
e (N
).
L-
Syg
nał
(13).
Inte
gra
l -
exam
ple
s
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
u(t)
y(t)
PO
MP
A Ł
AD
UJĄ
CA
Z
AS
OB
NIK
PO
MP
A
CY
RK
UL
AC
YJN
A
Inte
gra
l -
exam
ple
s
u(t)
y(t)
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Fig
. R
ud
der
an
del
eva
tor
isco
ntr
oll
edby
Hyd
rau
lic
act
ua
tors
An
oth
erap
pli
cati
on:
bra
ke
intr
uck
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Inte
gra
l -
exam
ple
s
u(t)
y(t)
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Inte
gra
l -
exam
ple
s
Do
ub
leactu
ato
rin
Boe
ing
73
7
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Fir
sto
rder
tran
sfe
r fu
nctio
n
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
k
0
1
Fig
. S
tep r
esp
onse
Fig
. Sym
bo
l
dy
Ty
kud
t+
=
Gs
k
Ts
()=
+1
()
1t T
yt
ku
e−
=∆
−
T
Ta
ng
ent
to a
cu
rve
Tim
e-d
om
ain
equ
atio
n
Tra
nsf
er f
unct
ion
Ste
pre
spo
nse
time
Inp
ut
sig
nal
u(t
)
Ou
tput
sign
al y(t
)
Lap
lace
tran
sfo
rm o
fD
C m
oto
r eq
uat
ion
s
ui a
Kt
Ra
La
ω
Eli
min
atin
g c
urr
ent
I aw
e ob
tain
()
2
()
()
t
aa
aa
tb
Ks
Us
LJs
JRB
Ls
BR
KK
Ω=
++
++
B(
)(
)(
)(
)(
)
()
-(
)0
aa
ab
ta
Ls
RI
sK
sU
s
Js
Bs
KI
s
+
+Ω
=
+Ω
= GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Fir
st o
rder
tran
sfe
r fu
nctio
n–
DC
moto
r
Ass
um
ing
smal
lin
du
ctan
ceL
a≈0
we
ob
tain
()
()
()
()
ta t
ba
KR
s
Us
JsB
KK
R
Ω=
++
Itis
pro
po
rtio
nal
acti
ng
elec
tric
co
mp
on
ent.
ω ωωω
atR
uK
Ba
bt
RK
K
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Fir
st o
rder
tran
sfe
r fu
nctio
n–
DC
moto
r
ui a
Kt
Ra
La
ω
B(
)(
)(
)
()
1
ta t
ba
KR
sk
Us
JsB
KK
RT
s
Ω=
=+
++
Tra
nsf
er f
un
ctio
n,
La=
0:
00.1
0.2
0.3
0.4
0.5
02468
10
12
Tim
e (
secs)
Amplitude
ku
T
u
t
Ass
um
ing
, th
atk
=10
, T
=0
.1w
e
obta
inst
ep r
esp
on
se
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Fir
st o
rder
tran
sfe
r fu
nctio
n–
DC
moto
r
V1
V2
V3
V=
abc
ρθ
ρθ
ρθ
ρθ
33
3
11
11
22
22
33
33
cV
d dt
cV
cV
cV
pp
pp
=+
−.
..
θθ
θθ
θθ
θθ
θ
110
1
220
21
330
3
=+
=+
=+
∆ ∆ ∆
01
11
12
22
23
33
3=
+−
ρθ
ρθ
ρθ
cV
cV
cV
pp
p
..
.
ρθ
ρθ
ρθ
ρθ
33
3
33
33
11
11
22
22
cV
d
dt
cV
cV
cV
pp
pp
∆∆
∆∆
+=
+.
..
[]
∆∆
∆θ
θθ
31
12
2
1
1(
)(
)(
)s
Ts
ks
ks
=+
+
TV V
kc
V
cV
kc
V
cV
p p
p p
==
=.
. .
. .
30
1
11
10
33
30
1
22
20
33
30
ρ ρ
ρ ρ
En
erg
y c
on
serv
atio
n e
qu
atio
n
Op
erat
ing
po
int
Sta
tic
char
acte
rist
ics
Dyn
amic
ch
arac
teri
stic
s
Lap
lace
tra
nsf
orm
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Fir
st o
rder
tran
sfe
r fu
nctio
n–
mix
ing
of
two
flu
ids
wit
hd
iffe
ren
t
tem
per
atu
res
k2
1/(
Ts+
1)
t3
++k1 t2t1
No
men
clat
ure
Fir
st o
rder
tran
sfe
r fu
nctio
n–
mix
ing
of
two
flu
ids
wit
h d
iffe
ren
t
con
cen
trati
on
s.
Ho
wch
ang
eso
utp
ut
con
centr
atio
nat
dif
fere
nt
dis
turb
ance
s?
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Ca
Ca
we
1/(
Ts+
1)
DIS
TU
RB
AN
CE
La
pla
ce
tra
nsfo
rm
L
ap
lace
tra
nsfo
rm
O
UT
PU
T
of
inp
ut
sig
na
l
of o
utp
ut
sig
na
l
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Fir
st o
rder
tran
sfe
r fu
nctio
n–
pn
eum
ati
csy
stem
Fir
st o
rder
tran
sfe
r fu
nctio
n–
pn
eum
ati
csy
stem
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Fir
st o
rder
tran
sfe
r fu
nctio
n–
elec
tric
syst
em
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Fig
. R
C c
ircu
it
Fir
st o
rder
tran
sfe
r fu
nctio
n–
ther
ma
lsy
stem
, te
mp
eratu
re
sen
sor
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
α-w
spó
łczyn
nik
prz
ejm
ow
ania
cie
pła
,
υ-t
emp
erat
ura
,
τ-cz
as,
A-p
ole
po
wie
rzch
ni
czu
jnik
a,
m-m
asa
czu
jnik
a
T-s
tała
cza
sow
a,
c p–
ciep
ło w
łaśc
iwe
mat
eria
łu c
zu
jnik
a
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Hig
he
r ord
er
tra
nsfe
r fu
nctio
ns
1
11
1...
nn
nn
nn
dy
dy
dy
TT
Ty
kudt
dt
dt
−
−−
++
++
=
()
()
1
()
1..
.1
n
kG
sT
sT
s=
++
11
0,
,...,
0n
nT
TT
−>
≥
Iner
cja
3-g
o r
zęd
u
Iner
cja
2-g
o r
zęd
u
Iner
cja
1-g
o r
zęd
u
Ste
p r
esp
on
sein
1 s
,
Tim
e-d
om
ain
equ
atio
n
Tra
nsf
er f
unct
ion
Tra
nsfe
r fu
nctio
n o
f id
ea
l diffe
ren
tia
lsyste
m
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
∞ ∞∞∞
0
1
Fig
. S
tep
resp
onse
Fig
. S
ymb
ol
du
yT
kd
t=
Gs
Tks
()=
yt
Tk
t(
)(
)=
δ
Tim
e-d
om
ain
equ
atio
n
Tra
nsf
er f
unct
ion
Ste
pre
spo
nse
time
Inp
ut
sig
nal
u(t
)
Ou
tput
sign
al y(t
)
Tra
nsfe
rfu
nctio
n o
f re
al d
iffe
ren
tia
lsyste
m
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Fig
. S
tep
resp
onse
Fig
. S
ymb
ol
dy
du
Ty
Tk
dt
dt
+=
Gs
Tks
Ts
()=
+1
()
t Ty
tk
ue
−=
∆
T0
k∆ ∆∆∆u ∆ ∆∆∆u
Tim
e-d
om
ain
equ
atio
n
Tra
nsf
er f
unct
ion
Ste
pre
spo
nse
timeIn
put
sig
nal
u(t
)
Ou
tput
sign
al y(t
)
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Transferfunction of real
differentialsystem,e
xa
mp
le,
ele
ctr
ic s
yste
m
1)
(
)(
)(
12
+=
=T
sTs
sU
sU
sG
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Transferfunction of real
differentialsystem,e
xa
mp
le,
me
ch
an
ica
lsyste
m
dy
du
Ty
Tk
dt
dt
+=
du
dy
cky
dt
dt
=+
Se
co
nd o
rder
tran
sfe
r fu
nctio
n
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
czas
Fig
. S
tep
resp
onse
Fig
. S
ymb
ol
22 2
12
12
02d
ydy
TT
yku
dt
dt
TT+
+=
<<
Gs
k
Ts
Ts
Gs
k
Ts
Ts
()
()
,
=+
+
=+
+
222
1
22
1
21
ξ
TT
T T=
=<
2
1 22
1,ξ
2
2
1(
)1
sin
11
t Tt
yt
ku
eT
ξ
ξϕ
ξ
−
=
∆−
−+
−
ϕ
ξξ
ξ=
−
<
arc
tg1
1
2
,
0
0.51
1.52
2.53
3.5
02
04
0
t
U,Y
Sko
k
Od
p
Tim
e-d
om
ain
equ
atio
n
Tra
nsf
er f
unct
ion
Ste
pre
spo
nse
time
Inp
ut
sig
nal
u(t
)
Ou
tput
sign
al y(t
)
Se
co
nd o
rder
tran
sfe
r fu
nctio
n -
exa
mp
les
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Se
co
nd
ord
er
tran
sfe
rfu
nctio
n-
exa
mp
les 1
di
uiR
Lid
tdt
C=
++
∫V
olt
ageu
dro
ps
con
secu
tiv
ely
on
resi
sto
r, i
nd
uct
or
and
con
den
ser
An
RL
Cci
rcu
it(a
lso
kn
ow
nas
are
son
ant
circ
uit
or
atu
ned
cir
cuit
)is
an
ele
ctri
cal
circ
uit
co
nsi
stin
g o
fa
resi
sto
r(R
),an
in
du
cto
r
(L),
and
aca
pac
ito
r(C
),co
nn
ecte
d i
n s
erie
s o
r in
par
alle
l.
Tu
ned
cir
cuit
sh
ave
man
y a
pp
lica
tio
ns
par
ticu
larl
yfo
ro
scil
lati
ng c
ircu
its
and
in
rad
ioan
d c
om
mu
nic
atio
n e
ngin
eeri
ng.
Th
ey c
an
be
use
dto
sele
cta
cert
ain
nar
row
ran
ge
of
freq
uen
cies
fro
m t
he
tota
l sp
ectr
um
of
amb
ien
tra
dio
wav
es.
Fo
rex
amp
le,
AM
/FM
rad
ios
wit
han
alo
gtu
ner
s ty
pic
ally
use
an
RL
Cci
rcu
itto
tun
ea
rad
iofr
equ
ency.
Mo
stco
mm
on
lya
var
iab
le c
apac
ito
r is
att
ach
ed
toth
e tu
nin
g k
no
b,
wh
ich
all
ow
s yo
u t
och
ang
e th
e v
alu
e o
fC
in t
he
circ
uit
an
d t
un
eto
stat
ion
so
nd
iffe
ren
t fr
equ
enci
es.
An
RL
Cci
rcu
it i
s ca
lled
ase
con
d-o
rder
circ
uit
asan
y v
olt
age
or
curr
ent
in t
he
circ
uit
can
be
des
crib
edb
y a
seco
nd
-ord
er
dif
fere
nti
al e
qu
atio
nfo
rci
rcu
it a
nal
ysi
s.
De
lay
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
T0
0
1
Fig
. S
tep
resp
onse
Fig
. S
ymb
ol
0(
)y
ut
T=
−
Gs
eT
s(
)=
−0
0(
)1(
)y
tu
tT
=∆
−
Tim
e-d
om
ain
equ
atio
n
Tra
nsf
er f
unct
ion
Ste
pre
spo
nse
time
Inp
ut
sig
nal
u(t
)
Ou
tput
sign
al y(t
)
De
lay -
exam
ple
s
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Fig
. G
rate
inw
ast
e u
tili
zati
on
faci
lity
Fu
el b
un
ker
Ste
am
bo
iler
Air
fan
un
der
the
gra
te
Co
mb
ust
ion
ch
am
ber
Co
al
con
veyo
r
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Co
al
conve
yor
inp
ow
erst
ati
on
or
CH
P (
com
bin
ed h
eat
an
d p
ow
er)
or
cog
ener
ati
on
pla
nt
De
lay -
exam
ple
s
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
De
lay –
exa
mp
les, scre
w c
on
ve
yo
r
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
Plumber-artist
at work...
GEN
ER
ALIZ
AT
ION
Basi
c t
ransf
er
functi
ons
pro
pert
ies
De
lay
–e
xa
mp
les, w
ate
r p
ipe
s
ch
imn
eyF
ire
pla
ce
rad
iato
rsu
pp
ly
retu
rnD
om
estic
cen
tra
l he
atin
gp
um
p
Sin
k
By-p
ass
...
...
()
21
01
12
0
...
...
...
t
n
nk
uk
uk
uk
ud
yT
yT
yT
yτ
−+
++
++
=+
++
+∫
GEN
ER
ALIZ
AT
ION
Const
ants
meanin
gD
E o
flin
ea
rsyste
m
Pro
po
rtio
nal
P
Inte
gra
l
I
dif
fere
nti
atio
n
D
Co
ntr
ol
sign
alu
Ou
tpu
t si
gn
aly
Mu
st e
xis
t
Gai
n c
oef
fici
ents
Tim
e co
nst
ants
...
...
()
()
21
01
2..
.1
...
mn
mn
ku
ku
ku
ku
yT
yT
yT
y+
++
=+
++
+
GEN
ER
ALIZ
AT
ION
Ste
ady–s
tate
gain
DE
of lin
ea
rsyste
m
Ifo
utp
utsig
na
lis
ste
ad
y, th
en
0k
uy
=
Co
effic
ient k
0
0
yk
u=
is c
alle
dsteady-state gain
.
GEN
ER
ALIZ
AT
ION
Tra
nsf
er
functi
ons
pro
pert
ies
com
pari
son
LIN
EA
RIZ
AT
ION
OF
NO
NLIN
EA
RSY
ST
EM
S
LIN
EA
RIZ
AT
ION
OF N
ON
LIN
EA
R S
YST
EM
S
Intr
oducti
on
Ob
iekt
Fy
yy
xx
x(
,,
,,
,)
...
.
11
20
=
yd
y dt
yd
y
dt
xd
x dt
...
.
==
=2
2
Fy
xx
(,
,,
,,
)0
00
01
2=
∆∆
∆y
yy
xx
xx
xx
oo
o=
−=
−=
−1
12
2
∆∆
∆y
yx
xx
x.
..
..
.
==
=1
2
Fy
xx
F yy
F y
yF y
yF x
xF x
xF x
xR
(,
,,
,,
).
.
..
..
.
.
00
00
12
1
1
1
1
2
2+
+
+
+
+
+
+
=∂ ∂
∂ ∂
∂ ∂
∂ ∂∂ ∂
∂ ∂∆
∆∆
∆∆
∆
∂ ∂∂ ∂
∂ ∂
∂ ∂∂ ∂
∂ ∂F y
yF y
yF y
yF x
xF x
xF x
x
+
+
+
+
+
=
∆∆
∆∆
∆∆
.
.
..
..
.
.
1
1
1
1
2
20
Ty
Ty
Ty
kx
kx
kx
01
21
12
13
2∆
∆∆
∆∆
∆+
+=
++
...
.
Let
’sta
ke
six
var
iable
eq
uat
ion
of
no
nli
nea
r sy
stem
Inp
ut
sig
nal
u1=
x1
Inp
ut
sig
nal
u2=
x2
Ou
tput
sign
aly
Tim
ed
eriv
ativ
esar
esi
gn
edb
y d
ots
Sta
tic
char
acte
rist
icis
des
crib
edb
y e
qu
atio
nw
ith
tim
ed
eriv
ativ
es
equ
alto
zer
o
Let
’sta
ke
smal
lch
ang
esar
ound
op
erat
ing
poin
t
Rec
all
Tay
lor
seri
es e
xp
ansi
on
Res
t
Tay
lor
seri
es a
llo
ws
to w
rite
equ
atio
n
In t
his
cas
e
Alg
ori
thm
of
lin
eari
zati
on
of
no
nli
nea
r sy
stem
1)
Wri
tedo
wn
all
var
iable
s an
dti
me
der
ivat
ives
2)
Co
mp
ute
equ
atio
n d
eriv
ates
for
all
var
iable
s
3)
Co
mp
ute
der
ivat
ives
val
ues
in
op
erat
ing
po
ints
4)
Wri
te d
ow
n l
inea
rize
d e
qu
atio
n
LIN
EA
RIZ
AT
ION
OF N
ON
LIN
EA
R S
YST
EM
S
Intr
oducti
on
...
.
12
1
,,
,,
,F
FF
FF
F
yx
xy
yx
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂ ∂∂ ∂
∂ ∂
∂ ∂∂ ∂
∂ ∂F y
yF y
yF y
yF x
xF x
xF x
x
+
+
+
+
+
=
∆∆
∆∆
∆∆
.
.
..
..
.
.
1
1
1
1
2
20
...
.
11
2,
,,
,,
yy
yx
xx
Fy
yy
yx
xx
xm
n(
,,
,...
,,
,,
,...
,)
...
()
...
()
11
0=
dF
=0
∂ ∂∂ ∂
∂ ∂∂ ∂
∂ ∂
∂ ∂F y
dy
F y
dy
F ydy
F xdx
F x
dx
F xdx
m
m
n
n
+
+
+
+
+
+
+
=
.
.
()
()
.
.
()
()
...
...
0
∂ ∂∂ ∂
F y
F xk
i(
)(
)
∂ ∂∂ ∂
∂ ∂∂ ∂
F yy
F yy
F xx
F xx
yy
m
yy
m
xx
n
xx
n
mm
nn
++
+
+
+
=
==
==
00
00
0∆
∆∆
∆..
...
.(
)
()
()
()
()
()
()
()
(1)
()
(1)
()
01
01
...
...
mn
mn
Ty
Ty
Ty
kx
kx
kx
∆+
∆+
+∆
=∆
+∆
++
∆
LIN
EA
RIZ
AT
ION
OF N
ON
LIN
EA
R S
YST
EM
S
Genera
lizati
on
Let
’sta
ke
syst
em d
ecri
bed
by D
E
Tota
ld
eriv
ativ
eo
feq
uat
ion
Th
en
Par
tial
der
ivat
ives
in
op
erat
ing
po
int
Sub
stit
uti
ng
par
tial
der
ivat
ives
we
get
hyp
erpla
ne
LIN
EA
RIZ
AT
ION
OF N
ON
LIN
EA
R S
YST
EM
S
Exam
ple
1
Lin
eari
zeeq
uati
on
ino
per
ati
ng
poin
t x
0=
1, y
0=
1 2.
34
yy
x
+
=
Solu
tio
n
1)
Var
iable
s.
,,
yy
x
2)
Der
ivat
ives
2.
34
.
.2
yy
x
y
y
∂
∂
+−
=
2.
34
23
yy
x
yy
∂
∂
+−
=
2.
34
34
yy
x
xx
∂
∂
+−
=
3)
Op
erat
ing
poin
t (d
eriv
ativ
es d
y/d
t=0!)
.
.
0
20
y
y=
=2
13
3y
y=
=3
14
4x
x=
=
4)
Lin
eari
zed
equ
atio
n
34
0y
x∆
−∆
=3(
1)4
(1)
0y
x−
−−
=
LIN
EA
RIZ
AT
ION
OF N
ON
LIN
EA
R S
YST
EM
S
Exam
ple
1
-0.50
0.51
1.52
2.53
00
.51
1.5
22
.5
3(
1)4
(1)
0y
x−
−−
=
34
yx
= Op
erati
ng
poin
t
Sy
stem
sta
tic
cha
ract
eris
tic
Lin
eari
zed
eq
ua
tio
n o
f sy
stem
LIN
EA
RIZ
AT
ION
OF N
ON
LIN
EA
R S
YST
EM
S
Th
an
k y
ou fo
r
your
atten
tion