[a305] otomatik kontrol ders notu (slayt)
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LAPLACE TRANSFORMS
Definition of the Laplace transform:
0
[ ( )] ( ) ( )stf t f t e dt F s
0
0 0
a tU t
t
0
[ ( )] ( ) ( )stu t u t e dt U s
00
( )st
st ae aU s ae dt
s s
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( )( )
0 0 0
1[ ]
s a tat at st s a e
L e e e dt e dts s
Aadaki rampa (ramp) fonksiyonu analitik yntemle znz
0
0 0
bt tf t t
0 0
( ) ( ) st st F s f t e dt bte dt
0
0 0
1(1)
stst steb te dt bt b e dt
s s
200
0( ) ( )
ststb b e b be dt
s s s s s s
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Properties of Laplace transforms:
1) Linearity : a sabit bir say veya s ve t den bamsz iseL[af(t)]=aL[f(t)]=aF(s)
2) Sperpozisyon : her iki fonksiyonunda laplace dnmalnabiliyorsa
1 2 1 2 1 2[ ( ) ( )] [ ( )] [ ( )] ( ) ( )f t f t L f t L f t F s F s
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3)Translation in time:
[ ( )] ( )asf t a e F s
4)Complex Differention:
[ ( )] ( )d
tf t F sds
5)Translation in the s domain:
[ ( ) ( )ate f t F s a
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6)Real differantiation:
2 2
[ ( )] ( ) (0 )
[ ( )] ( ) (0) (0)
L Df t sF s f
D f t s F s sf Df
7)Final value Theorem:
0
( ) ( )lim lims s
sF s f t
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Example:
3( )
( 2)Y s
s s
Solution:
0 0 0
3 3 3( ) ( ) ( )
( 2) 2 2lim lim lim lims s s sy t sY s s
s s s
8)Initial value Theorem:
0
( ) ( )lim lims s
F s f t
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Laplace Transforms of Most Common Functions of TimeContinuous Function Laplace Transform
Impulse 1
Steps
1
t2
1
s
2t 32
s
ate as1
atte 2)(1
as
Sin(wt))( 22 ws
w
Cos(wt) )( 22 wss
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rnek:2
3( )
( 2 5)f s
s s s
1 2 32 2
3
( 2 5) 2 5
K K s K
s s s s s
1 2 32 2
3 ( )
( 2 5) 2 5
K K s Ks s s
s s s s s s
1
3
5K
22 3
3 63 ( ) ( ) 3
5 5K s K s
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1 2 32 2
3
( 2 5) 2 5
K K s K
s s s s s s
2 21 1 1 2 33 2 5K s K s K K s K s
1
3
5K idi.
22 3
3 33 ( ) 3 (2 )
5 5K s x K s
22 3
3 63 ( ) 3 ( )
5 5K s K s
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2 2
( )[ cos sin ]
( )at at A s a Bw
L Ae wt Be wts a w
2 2
23 ( 1)35 2( )
5 ( 1) 2
sF s
s s
3 3 1( ) (cos2 sin 2 )
5 5 2t
t e t t
rnek:2
2( )
( 1)( 2)f s
s s
1 2 32 2
2( )
( 1)( 2) ( 1) ( 2) 2
K K Kf s
s s s s s
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2 12 3
2( 2) ( 2)
1 1
Ks K s K
s s
1 21
2 22
( 1)( 2) ( 1 2)sK
s s
1 2K
2s 2 2K
1 32 2
2 ( 2)
( 1) ( 1)
s sK K
s s
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2 2 2 21 2 32 2
2( 2) ( 2) ( 2) ( 2)
( 1)( 2) 1 ( 2) ( 2)
K K Ks x s s s
s s s s s
12 32 ( 2) ( 2)
( 1) 1Ks K s K
s s
leminin trevi alndnda
s = -2ye yaklar.
3 2K
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1 2 32 2
2( 2) ( 2) ( 2) ( 2)
( 1)( 2) 1 ( 2) ( 2)
K K Ks s s s
s s s s s
2s iin ; 30 0 K
2 12 3
2( 2) ( 2)
1 ( 1)s K s K
s s
3 12 2
(0)( 1) (1)(2) [0( 1) 1]0 (2 4)
( 1) ( 1)
s sK K s
s s
2 2
2( 2) 2( 2)( 1) 1( 2)( 1) ( 1)
s s s s
s s
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=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
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Bir Fonksiyonun Tekil Noktalar ve Kutuplar
S dzleminde tekil noktalar, fonksiyonun yada trevinin bulunmadnoktalardr.Kutup, tekil noktadr.
G(s) s civarnda analitik ve tek deerlidir.
[( ) ( )]limi
r
is s
s s G s
2
10( 2)( )
( 1)( 2)
sG 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.
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3
10( ) 0lim lim
s s
G ss
Adi Dorusal Diferansiyel Denklemler:
Seri RLC devresini ele alalm;
( ) 1( ) ( ) ( )
di ti t L id t e t
dt C .()
kinci mertebeden bir diferansiyel denklem:
11
11
( ) ( ) ( )... ( ) ( )
n n
n nn n
d y t a d y t dy t a a y t f t
dt dt dt
( )
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Katsaylar y(t)nin bir fonksiyonu olmad srece dorusal adidiferansiyel denklemdir.
()da 1( ) ( )x t i t dt
ve 12( )
( ) ( )dx t
x t i tdt
21 2
( ) 1 1( ) ( ) ( )
dx t Rx t x t e t
dt LC L L
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1. mertebeden durum deikenleri;
1
2
( ) ( )
( )( )
x t y t
dy tx t y
dt
( ) ..
.1
1
1
( )( )
nn
n n
d y tt y
dt
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1 2
2 3
x
x
.
.
.
1n nx
1 1....n n na x a x u
Dinamik Sistemlerin Matematiksel Modeli
Lineer Sistemler: Bir sisteme sperpozisyon teoremi uygulanyorsasistem lineerdir.
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1 1( ) ( )t y t se 1 2 1 2( ) ( ) ( ) ( )x t x t y t y t
2 2( ) ( )t y t
Lineer zamanla deimeyen ve lineer zamanla deien sistemler:
Bir diferansiyel denklemin katsaylar sabit ise veya fonksiyonlarbamsz deikenlerden oluuyorsa lineerdir.( Zamanla deiensistemlere rnek:Uzay arac kontrol sistemidir.Yakt tketimindendolay uzay aracnn ktlesi deiir.)
Dorusal olmayan sistemler:Bir sisteme sperpozisyon teoremi
uygulanamyorsasistem nonlineerdir.
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22
2sin
d x dxx A wt
dt dt
22
2 ( 1) 0
d x dx
x xdt dt
23
20
d x dxx x
dt dt
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Dinamik Sistemlerin Durum Uzay Gsterimi
1( )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:
111 1 12 2 11
( )( ) ( ) ( )
dx ta x t a x t b u t
dt
221 1 22 2 21
( )( ) ( ) ( )
dx ta x t a x t b u t
dt
1
2
( )( )
( )
x tx t
x t
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Durum denklemleri;
( )( ) ( ) ( )
dx tx t Ax t Bu t
dt
ile ifade edilir.
1
2
n
x
x
x
,
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A =
1 2
0 1 0 0
0 0 1 0
0 0 0 1
n n n n xa a a a
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B =
0
0
0
1
k ( y= Cx) Y =
1
2
1 0 0
n
x
x
x
x
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Filename: kon_sis_tem_2.docDirectory: C:\Documents and
Settings\Administrator\Desktop\FUNDAMENTALS OF CONTROLSYSTEMS\kontrol_temelleri
Template: C:\Documents and Settings\Administrator\Application
Data\Microsoft\Templates\Normal.dotmTitle: LAPLACE TRANSFORMSSubject:Author: hpKeywords:Comments:Creation Date: 09.10.2009 11:01:00Change Number: 39Last Saved On: 08.07.2010 15:34:00Last Saved By: PERFECTTotal Editing Time: 541 Minutes
Last Printed On: 08.07.2010 15:40:00As of Last Complete Printing
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