Transfer Function, Concepts of stability(critical, Absolute & Relative) Poles, Zeros Stability calculation, BIBO stability, Transient Response Characteristics

TOPICS
 Transfer Function
 The Order of Control Systems
 Concepts of stability(critical, Absolute & Relative)
 Poles, Zeros
 Stability calculation
 BIBO stability
 Transient Response Characteristics
1

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

Transfer Function
 When order of the denominator polynomial is greater
than the numerator polynomial the transfer function
is said to be ‘proper’.
 Otherwise ‘improper’
3

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
4

“Concept of Stability”
The concept of stability can be illustrated by a cone placed on a
plane horizontal surface.
. . 14

A linear time invarient system is stable if following
conditions are satisfied:
A bounded input is given to the system, the response
of the system is bounded and controllable.
In the absence of the inputs, the output should tend to
zero as time increases.
Stable System
. . 6

Stable System
Bounded i/p bounded o/p
for stable system
Location of roots
for stable system
. . 7

 A linear time invarient system comes under the class of
unstable system if the system is excited by a bounded
input, response is unbounded.
 This means once any input is given system output goes
on increasing & designer does not have any control on
it
Unstable System
. . 8

Unstable System
Bounded i/p Unbounded o/p
for unstable system
Location of roots
for unstable system
. . 9

 When the input is given to a linear time invarient system,
for critically stable systems the output does not go on
increasing infinitely nor does it go to zero as time increases.
 The output usually oscillates in a finite range or remains
steady at some value.
 Such systems are not stable as their response does not
decay to zero. Neither they are defined as unstable because
their output does not go on increasing infinitely.
Critically Stable System
. . 10

Critically stable System
Bounded i/p & o/p response
for critically stable system
Location of roots
for critically stable system
. . 11

 A system may be absolutely stable i.e. it may have
passed the Routh stability test.
 Asa result their response decays to
zero under zero input conditions.
 The ratio at which these decay to zero is important to
check the concept of “Relative stability”
Relative Stability
. . 12

 When the poles are located far away from jw axis in
LHP of s-plane, the response decays to zero much
faster, as compared to the poles close to jw-axis.
 The more the poles are located far away from jw-axis
the more is the system relatively stable.
Relative Stability
. . 13

Relative Stability
Response comparison
. . 14

Relative Stability
Location of poles comparison

. . 15

j
j




System B
Complex conjugate
poles
System A
Complex conjugate
poles

Relative Stability

. . 16

j
j
Relative Stability improves as one moves
away from jw axis

Zeros
 The zeros of a Laplace function are the values of s
that make the Laplace function evaluate to zero.
They are therefore the zeros of the numerator
polynomial
 10 (s + 2)/[(s + 1)(s + 3)] has a zero at s = -2
 Complex zeros always appear in complex-conjugate
pairs

Poles
 The poles of a Laplace function are the values of s
that make the Laplace function evaluate to infinity.
They are therefore the roots of the denominator
polynomial
 10 (s + 2)/[(s + 1)(s + 3)] has a pole at s = -1 and a
pole at s = -3
 Complex poles always appear in complex-conjugate
pairs
 The transient response of system is determined by
the location of poles

Stability of Control System
 Roots of denominator polynomial of a transfer
function are called ‘poles’.
 The roots of numerator polynomials of a transfer
function are called ‘zeros’.
19

 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).
20

 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.
21

 Poles is also defined as “it is the frequency at which
system becomes infinite”.
 Like a magnetic pole or black hole.
22

 Poles and Zeros of a transfer function are the frequencies
for which the value of the denominator and numerator of
transfer function becomes zero respectively. The values of
the poles and the zeros of a system determine whether the
system is stable, and how well the system performs.
Control systems, in the most simple sense, can be designed
simply by assigning specific values to the poles and zeros of
the system.
 Physically realizable control systems must have a number
of poles greater than or equal to the number of zeros
23

 Poles are frequencies near which the magnitude of
transfer function actually shoots up to hypothetically
to infinity. Zeros are frequencies at which the response
magnitude becomes zero. Poles determine the
transient response of the system, while the zero
determines the speed of response to be more general.
Zeros become important when there are delays, or
non-minimum phase
24

Stability
 A system is stable if bounded inputs produce bounded
outputs
 Stable: If all poles lies in Left Half Plane
 Unstable: If any pole lies in Right Half Plane
 Marginally Stable: one Pole lies at origin and all other in LHP
s-plane
Stable(LHP) Unstable(RHP)
x
x
x
x x
x j


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
26
)
(
)
(
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)

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.
27
u(t)
t
1
Unit Step Input
Plant
y(t)
t
Output
1
overshoot

The Other Definition of Stability
 If for any bounded input the output is not bounded the
system is said to be unstable.
28
u(t)
t
1
Unit Step Input
Plant
y(t)
t
Output
at
e

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

 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




 






 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)

 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.

Stability
BIBO Stability:
Output must be bounded for bounded input.
Asymptotic Stability: If system input is remove from the system, then
output of system is reduced to zero.
Critical Stability: The output usually oscillates in a finite range or
remains steady at some value
absolute Stability: A system is stable for all values of system
parameters for bounded output. (poles)
[Routh Hurwitz, Rout locus & NyquistPlot]
Relative Stability: This is a quantitative measure of how fast system
oscillation die out with time and how fast steady state reached.
(Damping Ratio, Gain Margin and Phase Margin)
System having poles away from the imaginary axis of S-Plane, in
negative direction, has higher stability.
[Bode Plot & Nyquist Plot]

Transient Response
Characteristics
 .
34
state
steady
of
%
specified
within
stays
time
Settling
:
reached
is
peak value
at which
Time
:
value
state
steady
reach
first
until
delay
time
Rise
:
value
state
steady
of
50%
reach
until
Delay time
:


s
p
r
d
t
t
t
t

Transient Response Characteristics
state
steady
of
%
specified
within
stays
time
Settling
:
reached
is
peak value
at which
Time
:
value
state
steady
reach
first
until
delay
time
Rise
:
value
state
steady
of
50%
reach
until
Delay time
:


s
p
r
d
t
t
t
t
0.5 1 1.5 2 2.5 3
0.25
0.5
0.75
1
1.25
1.5
1.75
2

j

Transfer Function, Concepts of stability(critical, Absolute & Relative) Poles, Zeros Stability calculation, BIBO stability, Transient Response Characteristics

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