The Pulse Transfer Function -...
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The Pulse Transfer Function • Convolution Summation – For the continuous time-system – For the discrete-time system • For a physical system a response cannot precede the input k k k z kT y z Y t y 0 ) ( ) ( ) ( Z t t d g t x d x t g t y 0 0 ) ( ) ( ) ( ) ( ) ( 0 0 * ) ( ) ( ) ( ) ( ) ( k k kT t kT x kT t t x t x kT t ht x hT t g t y h 0 ) ( ) ( ) ( 0
Transcript of The Pulse Transfer Function -...
The Pulse Transfer Function
• Convolution Summation
– For the continuous time-system
– For the discrete-time system
• For a physical system a response cannot precede the input
k
k
kzkTyzYty0
)()()(Z
tt
dgtxdxtgty00
)()()()()(
00
* )()()()()(kk
kTtkTxkTttxtx
kTthtxhTtgtyh
0 )()()(0
The Pulse Transfer Function
• Convolution Summation (cont.) – The value of the output y(t) at the sampling instants t=kT are
given by
– Since we assume that x(t)=0 for t <0
– It is noted that if G(s) is a ratio of polynimials in s and if the degree of the denominator polynomial exceeds that of the numerator polynomial only by 1 the output y(t) is discontinuous.
k
h
k
h
hTghTkTxhTxhTkTgkTy00
)()()()()( Convolution summation
)(*)()( kTgkTxkTy
00
)()()()()(hh
hTghTkTxhTxhTkTgkTy
The Pulse Transfer Function
• Convolution Summation (cont.) – In analyzing discrete-time control systems it is important to
remember that the system response to the impulse-sampled signal may not portray the correct time-response behavior of the actual system unless the transfer function G(s) of the continuous-time part of the system has at least two more poles than zeros, so that
0)(lim
ssGs
The Pulse Transfer Function
• Pulse Transfer Function
– The z transform of y(kT)
Pulse transfer function
,2,1,0 )()()(0
khTxhTkTgkTyh
)()(
)()()()(
)()()()(
0 00 0
)(
0 00
zXzG
zhTxzmTgzhTxmTg
zhTxhTkTgzkTyzY
m h
hm
m h
hm
k h
k
k
k
)(
)()(
zX
zYzG
)()( zGzY
to the Kronecker delta input
The Pulse Transfer Function
• Starred Laplace Transform of the Signal involving both Ordinary and Starred Laplace Transform
)()()( * sXsGsY 2,1,0 ),()( ** kkjsXsX s
)()()()()()()( ******* sXsGsXsGsXsGsY
)()()()()(
)()()()()()()()(
00 0
0 00
**1
kTxkTtgdkTxtg
dkTxtgdxtgsXsGty
kk
t
t
k
t
L
)()(
)()()()()()(0 0
)(
0 0
zXzG
zkTxmTgkTxkTnTgtyzYm k
mk
n k
n-zZ
)()()( *** sXsGsY
The Pulse Transfer Function
• General Procedures for Obtaining Pulse Transfer Functions
)()()(
)(sGzG
zX
zYZ
)()(
)(sG
sX
sY
)()()( * sXsGsY )()()( *** sXsGsY
*** )()()()( sGXsXsGsY
)()()()()()()()( zXzGzGZsGXsXsGsYzY ZZZ
The Pulse Transfer Function
• Pulse Transfer Function of Cascaded Elements
)()()( ),()()( ** sUsHsYsXsGsU
)()()( ),()()( ****** sUsHsYsXsGsU
)()()()()()( ****** sXsGsHsUsHsY
)()()()( zXzHzGzY )()()(
)(zHzG
zX
zY
)()()()()()( ** sXsGHsXsHsGsY
)()()( *** sXsGHsY
)()()( zXzGHzY
)()()(
)(sGHzGH
zX
zYZ Note that )()()()( sGHzGHzHzG Z
The Pulse Transfer Function
Pulse Transfer Function of Closed-Loop Systems
)()()(
)()()()(
* sEsGsC
sCsHsRsE
)()()()()( * sEsGsHsRsE
)()()()( **** sEsGHsRsE )(1
)()(
*
**
sGH
sRsE
)()()( *** sEsGsC
)(1
)()()(
*
***
sGH
sRsGsC
)(1
)()()(
zGH
zRzGzC
)(1
)(
)(
)(
zGH
zG
zR
zC
Refer to Table 3-1
The Pulse Transfer Function
• Table 3-1: Five typical configurations for closed-loop discrete-time control systems
The Pulse Transfer Function
• Pulse Transfer Function of a Digital Controller – The input to the digital controller is e(k) and the output is m(k)
– The z transform of the equation
)()1()(
)()2()1()(
10
21
nkebkebkeb
mkmakmakmakm
n
n
)()()(
)()()()(
1
10
2
2
1
1
zEzbzEzbzEb
zMzazMzazMzazM
n
n
n
n
)()()()1( 1
10
2
2
1
1 zEzbzbbzMzazaza n
n
n
n
n
n
n
nD
zazaza
zbzbb
zE
zMzG
2
2
1
1
1
10
1)(
)()(
The Pulse Transfer Function
• Closed-loop Pulse Transfer Function of a Digital Control System
)()(1
sGsGs
ep
Ts
)()()( ** sEGsGsC D )()()()( **** sEsGsGsC D
)()()()( zEzGzGzC D
)()()()()(
)()()(
zCzRzGzGzC
zCzRzE
D
)()(1
)()(
)(
)(
zGzG
zGzG
zR
zC
D
D
The Pulse Transfer Function
• Pulse Transfer Function of a Digital PID Controller – The PID control action in analog controllers
– Discretization of the equation to obtain the pulse transfer function
t
d
i dt
tdeTte
TteKtm
0
)()(
1)()(
T
TkekTeT
kTeTkeTeTeTee
T
TkTeKkTm d
i
))1(()(
2
)())1((
2
)2()(
2
)()0()()(
))1(()(2
)())1(()()(
1
TkekTeT
ThTeThe
T
TkTeKkTm d
k
hi
Define 0)0( ),(2
)())1((
fhTf
hTeThe
k
h
k
h
hTfhTeThe
11
)(2
)())1((
The Pulse Transfer Function
• Pulse Transfer Function of a Digital PID Controller(cont.)
))1(()(2
)())1(()()(
1
TkekTeT
ThTeThe
T
TkTeKkTm d
k
hi
)()1(1
)()1(1
1
21
)()1(1
1
21)(
)(
1
1
1
1
1
1
1
zEzKz
KK
zEzT
T
zT
T
T
TK
zEzT
T
z
z
T
TKzM
zG
DI
P
d
ii
d
i
D
)(1
1)0()(
1
1)(
2
)())1((11
11
zFz
fzFz
hTfhTeThe k
h
k
h
ZZ
)(2
1)()(
1
zEz
hTfzF
Z )()1(2
1
2
)())1((1
1
1
zEz
zhTeThek
h
Z
The Pulse Transfer Function
• Obtaining response between consecutive sampling instants – Laplace transform method
– Modified z transform method
– State-space method
• Laplace Transform Method
)(1
)()()()()(
*
**
sGH
sRsGsEsGsC
)(1
)()()()(
*
*11
sGH
sRsGsCtc LL
