Fundamentals of Transfer Function in control system.pptx

Transfer Function
• Transfer Function is the ratio of Laplace transform of the
output to the Laplace transform of the input. Consider
all initial conditions to zero.
• Where is the Laplace operator.
Plant
y(t)
u(t)
)
(
)
(
)
(
)
(
S
Y
t
y
and
S
U
t
u
If





1
PPD By: Mr. Abhishek K Ranjan, Lecturer,
G.P Sheikhpura

Transfer Function
• The transfer function G(S) of the plant is given as
G(S) Y(S)
U(S)
)
(
)
(
)
(
S
U
S
Y
S
G 
2
G.P Sheikhpura

Why Laplace Transform?
• Using Laplace transform, we can convert many common
functions into algebraic function of complex variable s.
• For example
• Where s is a complex variable (complex frequency) and
is given as
2
2





s
t
sin

a
s
e at


 1


 j
s 

3
G.P Sheikhpura

Laplace Transform of Derivatives
• Not only common function can be converted into
simple algebraic expressions but calculus operations
can also be converted into algebraic expressions.
• For example
)
0
(
)
(
)
(
x
s
sX
dt
t
dx



dt
dx
x
s
s
X
s
dt
t
x
d )
0
(
)
0
(
)
(
)
( 2
2
2





4
G.P Sheikhpura

Laplace Transform of Derivatives
• In general
• Where is the initial condition of the system.






n
k
k
k
n
n
n
n
x
s
s
X
s
dt
t
x
d
1
)
1
(
)
0
(
)
(
)
(

)
(0
x
5
G.P Sheikhpura

Laplace Transform of Integrals
)
(
1
)
( s
X
s
dt
t
x 


• The time domain integral becomes division by
s in frequency domain.
6
G.P Sheikhpura

Calculation of the Transfer Function
dt
t
dx
B
dt
t
dy
C
dt
t
x
d
A
)
(
)
(
)
(


2
2
• Consider the following ODE where y(t) is input of the system and
x(t) is the output.
• or
• Taking the Laplace transform on either sides
)
(
'
)
(
'
)
(
'
' t
Bx
t
Cy
t
Ax 

)]
(
)
(
[
)]
(
)
(
[
)]
(
'
)
(
)
(
[ 0
0
0
0
2
x
s
sX
B
y
s
sY
C
x
sx
s
X
s
A 





7
G.P Sheikhpura

Calculation of the Transfer Function
• Considering Initial conditions to zero in order to find the transfer
function of the system
• Rearranging the above equation
)]
(
)
(
[
)]
(
)
(
[
)]
(
'
)
(
)
(
[ 0
0
0
0
2
x
s
sX
B
y
s
sY
C
x
sx
s
X
s
A 





)
(
)
(
)
( s
BsX
s
CsY
s
X
As 

2
)
(
]
)[
(
)
(
)
(
)
(
s
CsY
Bs
As
s
X
s
CsY
s
BsX
s
X
As




2
2
B
As
C
Bs
As
Cs
s
Y
s
X



 2
)
(
)
(
8
G.P Sheikhpura

Transfer Function
• In general
• Where x is the input of the system and y is the output of
the system.
9
G.P Sheikhpura

Transfer Function
• When order of the denominator polynomial is greater
than the numerator polynomial the transfer function is
said to be ‘proper’.
• Otherwise ‘improper’
10
G.P Sheikhpura

Transfer Function
• Transfer function can be used to check
– The stability of the system
– Time domain and frequency domain characteristics of the
system
– Response of the system for any given input
11
G.P Sheikhpura

Stability of Control System
• There are several meanings of stability, in general
there are two kinds of stability definitions in control
system study.
– Absolute Stability
– Relative Stability
12
G.P Sheikhpura

• Roots of denominator polynomial of a transfer
function are called ‘poles’.
• The roots of numerator polynomials of a
transfer function are called ‘zeros’.
13
G.P Sheikhpura

• Poles of the system are represented by ‘x’ and
zeros of the system are represented by ‘o’.
• System order is always equal to number of
poles of the transfer function.
• Following transfer function represents nth
order plant (i.e., any physical object).
14
G.P Sheikhpura

• Poles is also defined as “it is the frequency at which
system becomes infinite”. Hence the name pole
where field is infinite.
• Zero is the frequency at which system becomes 0.
15
G.P Sheikhpura

• Poles is also defined as “it is the frequency at which
system becomes infinite”.
• Like a magnetic pole or black hole.
16

Example
• Consider the Transfer function calculated in previous
slides.
• The only pole of the system is
17
B
As
C
s
Y
s
X
s
G



)
(
)
(
)
(
0

 B
As
is
polynomial
r
denominato
the
A
B
s 

G.P Sheikhpura

Examples
• Consider the following transfer functions.
– Determine
• Whether the transfer function is proper or improper
• Poles of the system
• zeros of the system
• Order of the system
18
)
(
)
(
2
3



s
s
s
s
G
)
)(
)(
(
)
(
3
2
1 



s
s
s
s
s
G
)
(
)
(
)
(
10
3
2
2



s
s
s
s
G
)
(
)
(
)
(
10
1
2



s
s
s
s
s
G
i) ii)
iii) iv)
G.P Sheikhpura

Stability of Control Systems
• The poles and zeros of the system are plotted in s-plane
to check the stability of the system.
19
s-plane
LHP RHP


j

 j
s 

Recall
G.P Sheikhpura

• If all the poles of the system lie in left half plane the
system is said to be Stable.
• If any of the poles lie in right half plane the system is said
to be unstable.
• If pole(s) lie on imaginary axis the system is said to be
marginally stable.
20
s-plane
LHP RHP


j
G.P Sheikhpura

• For example
• Then the only pole of the system lie at
21
10
3
1 



 C
and
B
A
B
As
C
s
G if ,
,
)
(
3


pole
s-plane
LHP RHP


j
X
-3
G.P Sheikhpura

Examples
• Consider the following transfer functions.
 Determine whether the transfer function is proper or improper
 Calculate the Poles and zeros of the system
 Determine the order of the system
 Draw the pole-zero map
 Determine the Stability of the system
22
)
(
)
(
2
3



s
s
s
s
G
)
)(
)(
(
)
(
3
2
1 



s
s
s
s
s
G
)
(
)
(
)
(
10
3
2
2



s
s
s
s
G
)
(
)
(
)
(
10
1
2



s
s
s
s
s
G
i) ii)
iii) iv)
G.P Sheikhpura

The Other Definition of Stability
• The system is said to be stable if for any bounded
input the output of the system is also bounded
(BIBO).
• Thus for any bounded input the output either
remain constant or decrease with time.
23
u(t)
t
1
Unit Step Input
Plant
y(t)
t
Output
1
overshoot
G.P Sheikhpura

The Other Definition of Stability
• If for any bounded input the output is not
bounded the system is said to be unstable.
24
u(t)
t
1
Unit Step Input
Plant
y(t)
t
Output
at
e
G.P Sheikhpura

BIBO vs Transfer Function
• For example
3
1
)
(
)
(
)
(
1



s
s
U
s
Y
s
G
3
1
)
(
)
(
)
(
2



s
s
U
s
Y
s
G
-4 -2 0 2 4
-4
-3
-2
-1
0
1
2
3
4
Pole-Zero Map
Real Axis
Imaginary
Axis
-4 -2 0 2 4
-4
-3
-2
-1
0
1
2
3
4
Pole-Zero Map
Real Axis
Imaginary
Axis
stable
unstable
G.P Sheikhpura
25

• For example
3
1
)
(
)
(
)
(
1



s
s
U
s
Y
s
G
3
1
)
(
)
(
)
(
2



s
s
U
s
Y
s
G
)
(
)
(
3
1
)
(
)
(
)
(
3
1
1
1
1
t
u
e
t
y
s
s
U
s
Y
s
G
t








 


)
(
)
(
3
1
)
(
)
(
)
(
3
1
1
2
1
t
u
e
t
y
s
s
U
s
Y
s
G
t




 





G.P Sheikhpura
26

• For example
)
(
)
( 3
t
u
e
t
y t

 )
(
)
( 3
t
u
e
t
y t

0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
exp(-3t)*u(t)
0 2 4 6 8 10
0
2
4
6
8
10
12
x 10
12
exp(3t)*u(t)
G.P Sheikhpura
27

• Whenever one or more than one poles are in
RHP the solution of dynamic equations
contains increasing exponential terms.
• That makes the response of the system
unbounded and hence the overall response of
the system is unstable.
G.P Sheikhpura
28

Summary
• Transfer Function
• The Order of Control Systems
• Poles, Zeros
• Stability
• BIBO
29
G.P Sheikhpura

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