(A305) Otomatik Kontrol Ders Notu (Slayt)
(A305) Otomatik Kontrol Ders Notu (Slayt)
(A305) Otomatik Kontrol Ders Notu (Slayt)
= =
}
( )
0
0 0
a t
U t
t
s s
=
< <
0
[ ( )] ( ) ( )
st
L u t u t e dt U s
= =
}
0
0
( )
st
st
ae a
U s ae dt
s s
= = =
}
( )
( )
0 0
0
1
[ ]
s a t
at at st s a
e
L e e e dt e dt
s s o o
+
+
= = = =
+ +
} }
Aadaki rampa (ramp) fonksiyonu analitik yntemle znz
( )
0
0 0
bt t
f t
t
s s
=
< <
0 0
( ) ( )
st st
F s f t e dt bte dt
= =
} }
0
0 0
1
(1)
st
st st
e
b te dt bt b e dt
s s
=
} }
2
0
0
0
( ) ( )
st
st
b b e b b
e dt
s s s s s s
= + = = =
}
Properties of Laplace transforms:
1) Linearity : a sabit bir say veya s ve t den bamsz ise
L[af(t)]=aL[f(t)]=aF(s)
2) Sperpozisyon : her iki fonksiyonunda laplace dnm
alnabiliyorsa
1 2 1 2 1 2
[ ( ) ( )] [ ( )] [ ( )] ( ) ( ) L f t f t L f t L f t F s F s = =
3)Translation in time:
[ ( )] ( )
as
L f t a e F s
=
4)Complex Differention:
[ ( )] ( )
d
L tf t F s
ds
=
5)Translation in the s domain:
[ ( ) ( )
at
L e f t F s a
=
6)Real differantiation:
2 2
[ ( )] ( ) (0 )
[ ( )] ( ) (0) (0)
L Df t sF s f
L D f t s F s sf Df
+
=
=
7)Final value Theorem:
0
( ) ( )
lim lim
s s
sF s f t
=
Example:
3
( )
( 2)
Y s
s s
=
+
Solution:
0 0 0
3 3 3
( ) ( ) ( )
( 2) 2 2
lim lim lim lim
s s s s
y t sY s s
s s s
= = = =
+ +
8)Initial value Theorem:
0
( ) ( )
lim lim
s s
sF s f t
=
Laplace Transforms of Most Common Functions of Time
Continuous Function Laplace Transform
Impulse 1
Step
s
1
t
2
1
s
2
t
3
2
s
at
e
a s +
1
at
te
2
) (
1
a s +
Sin(wt)
) (
2 2
w s
w
+
Cos(wt)
) (
2 2
w s
s
+
rnek:
2
3
( )
( 2 5)
f s
s s s
=
+ +
1 2 3
2 2
3
( 2 5) 2 5
K K s K
s s s s s s
+
= +
+ + + +
1 2 3
2 2
3 ( )
( 2 5) 2 5
K K s K
s s s
s s s s s s
+
= +
+ + + +
1
3
5
K =
2
2 3
3 6
3 ( ) ( ) 3
5 5
K s K s = + + + +
1 2 3
2 2
3
( 2 5) 2 5
K K s K
s s s s s s
+
= +
+ + + +
2 2
1 1 1 2 3
3 2 5 K s K s K K s K s = + + + +
1
3
5
K = idi.
2
2 3
3 3
3 ( ) 3 (2 )
5 5
K s x K s = + + + +
2
2 3
3 6
3 ( ) 3 ( )
5 5
K s K s = + + + +
2
3
5
K =
3
6
5
K =
2 2
3
3 3 2
5
( )
( 2 5) 5 2 5
s
f s
s s s s s s
+
= =
+ + + +
2 2
( )
[ cos ]
( )
at
A s a
L Ae wt
s a w
+
=
+ +
2 2
[ sin ]
( )
at
Bw
L Be wt
s a w
=
+ +
2 2
( )
[ cos sin ]
( )
at at
A s a Bw
L Ae wt Be wt
s a w
+ +
+ =
+ +
2 2
2
3 ( 1)
3
5 2
( )
5 ( 1) 2
s
F s
s s
+ +
=
+ +
3 3 1
( ) (cos2 sin 2 )
5 5 2
t
f t e t t
= +
rnek:
2
2
( )
( 1)( 2)
f s
s s
=
+ +
1 2 3
2 2
2
( )
( 1)( 2) ( 1) ( 2) 2
K K K
f s
s s s s s
= = + +
+ + + + +
2
1
2 3
2
( 2) ( 2)
1 1
K
s K s K
s s
= + + + +
+ +
1
2
1
2 2
2
( 1)( 2) ( 1 2)
s
K
s s
= = =
+ + +
1
2 K =
2 s
2
2 K =
1 3
2 2
2 ( 2)
( 1) ( 1)
s s
K K
s s
+
= +
+ +
2 2 2 2
1 2 3
2 2
2
( 2) ( 2) ( 2) ( 2)
( 1)( 2) 1 ( 2) ( 2)
K K K
s x s s s
s s s s s
+ = + + + + +
+ + + + +
1
2 3
2
( 2) ( 2)
( 1) 1
K
s K s K
s s
= + + + +
+ +
leminin trevi alndnda
s = -2ye yaklar.
3
2 K =
1 2 3
2 2
2
( 2) ( 2) ( 2) ( 2)
( 1)( 2) 1 ( 2) ( 2)
K K K
s s s s
s s s s s
+ = + + + + +
+ + + + +
2 s iin ;
3
0 0 K = + +
2
1
2 3
2
( 2) ( 2)
1 ( 1)
K
s K s K
s s
= + + + +
+ +
3 1
2 2
(0)( 1) (1)(2) [0( 1) 1]
0 (2 4)
( 1) ( 1)
s s
K K s
s s
+ +
= + + +
+ +
2 2
2
( 2) 2( 2)( 1) 1( 2)
( 1) ( 1)
s s s s
s s
+ + + +
=
+ +
=
2 2
2
2( 2 2) ( 2)
( 1)
s s s s
s
+ + + +
+
=
2 2
2
2 2 2 4 4
( 1)
s s s s s
s
+ + +
+
2
( 2)
( 1)
s
s
+
+
=
2
2
2
( 1)
s s
s
+
Bir Fonksiyonun Tekil Noktalar ve Kutuplar
S dzleminde tekil noktalar, fonksiyonun yada trevinin bulunmad
noktalardr.Kutup, tekil noktadr.
G(s) s civarnda analitik ve tek deerlidir.
[( ) ( )]
lim
i
r
i
s s
s s G s
2
10( 2)
( )
( 1)( 2)
s
G s
s s s
+
=
+ +
fonksiyonunun sfrlar s=-2 de bir sonlu ve
sonsuzda 3 sfr vardr. s=-3 de katl, s=0 da ve s=-1 de katsz kutbu
vardr.G(s) fonksiyonu bu noktalar dnda analitiktir denir.
3
10
( ) 0
lim lim
s s
G s
s
= =
Adi Dorusal Diferansiyel Denklemler:
Seri RLC devresini ele alalm;
( ) 1
( ) ( ) ( )
di t
Ri t L id t e t
dt C
+ + =
}
.(- )
kinci mertebeden bir diferansiyel denklem:
1
1
1
1
( ) ( ) ( )
... ( ) ( )
n n
n n
n n
d y t a d y t dy t
a a y t f t
dt dt dt
+ + + + = (- - )
Katsaylar y(t)nin bir fonksiyonu olmad srece dorusal adi
diferansiyel denklemdir.
(- )da
1
( ) ( ) x t i t dt =
}
ve
1
2
( )
( ) ( )
dx t
x t i t
dt
= =
2
1 2
( ) 1 1
( ) ( ) ( )
dx t R
x t x t e t
dt LC L L
= +
1. mertebeden durum deikenleri;
1
2
( ) ( )
( )
( )
x t y t
dy t
x t y
dt
=
= =
(- - - ) .
.
.
1
1
1
( )
( )
n
n
n
n
d y t
x t y
dt
= =
1 2
2 3
x x
x x
-
-
=
=
.
.
.
1 n n
x x
-
=
1 1
....
n n n
x a x a x u
-
= +
Dinamik Sistemlerin Matematiksel Modeli
Lineer Sistemler: Bir sisteme sperpozisyon teoremi uygulanyorsa
sistem lineerdir.
1 1
( ) ( ) x t y t
se
1 2 1 2
( ) ( ) ( ) ( ) x t x t y t y t + +
2 2
( ) ( ) x t y t
Lineer zamanla deimeyen ve lineer zamanla deien sistemler:
Bir diferansiyel denklemin katsaylar sabit ise veya fonksiyonlar
bamsz deikenlerden oluuyorsa lineerdir.( Zamanla deien
sistemlere rnek:Uzay arac kontrol sistemidir.Yakt tketiminden
dolay uzay aracnn ktlesi deiir.)
Dorusal olmayan sistemler:Bir sisteme sperpozisyon teoremi
uygulanamyorsa sistem nonlineerdir.
2
2
2
sin
d x dx
x A wt
dt dt
| |
+ + =
|
\ .
2
2
2
( 1) 0
d x dx
x x
dt dt
+ + =
2
3
2
0
d x dx
x x
dt dt
+ + + =
Dinamik Sistemlerin Durum Uzay Gsterimi
1
( ) x t ve
2
( ) x t durum deikenleri olsun;
u(t); Giri,
11 12 21 22 11 21
, , , , , a a a a b b ise sabit katsaylar:
1
11 1 12 2 11
( )
( ) ( ) ( )
dx t
a x t a x t b u t
dt
= + +
2
21 1 22 2 21
( )
( ) ( ) ( )
dx t
a x t a x t b u t
dt
= + +
1
2
( )
( )
( )
x t
x t
x t
(
=
(
Durum denklemleri;
( )
( ) ( ) ( )
dx t
x t Ax t Bu t
dt
-
= = + ile ifade edilir.
1
2
n
x
x
x
x
(
(
(
- (
=
(
-
(
( -
(
,
A =
1 2
0 1 0 0
0 0 1 0
0 0 0 1
n n n n x
a a a a
- - -
(
(
- - -
(
- - - - (
(
- - - -
(
( - - - -
(
- - -
(
(
- - -
B =
0
0
0
1
(
(
(
- (
(
-
(
( -
(
(
(
k ( y= Cx) Y =
| |
1
2
1 0 0
n
x
x
x
x
(
(
(
- (
- - -
(
-
(
( -
(
Filename: kon_sis_tem_2.doc
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Title: LAPLACE TRANSFORMS
Subject:
Author: hp
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Comments:
Creation Date: 09.10.2009 11:01:00
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