nyquist plot steps
TRANSCRIPT
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CONTROL SYSTEM(part 2)
EEE 350
EN. MUHAMMAD NASIRUDDIN
MAHYUDDIN
FREQUENCY DOMAIN ANALYSIS
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Nasiruddin's Horizons 2
n Frequency Response Technique
Continues.
n NYQUIST PLOT
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Nasiruddin's Horizons 3
NYQUIST PLOT
The basis of Nyquist Plot is the polar plot (Plot Kutub).
Polar plot of a transfer function )()( sHsG is a magnitude plot for )()( jHjG
against its phase plot with frequency, , acts as a parameter that changes from
0 to infinity aftersis replaced withj inG(s)H(s).
Mathematically, plotting a polar plot for )()( jHjG is a process of mappingthe positive side of the S-planes imaginary into a )()( jHjG -plane.
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Nasiruddin's Horizons 4
NYQUIST PLOT
Generally a polar plot or nyquist plot of a system is done by the aid of computer.
However, a sketch can be done if the following information:
The behaviour of the magnitude and phase for )()( jHjG at 0 frequency (w=0)
and infinite frequency (w=).
The intersection point between the polar plot and the real, imaginary axis in the
G(jw)H(jw)-plane, and the values of w at the intersection point.
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Nasiruddin's Horizons 5
NYQUIST PLOT
n Worked Example:
Sketch a polar plot for the following transfer function.
)5)(1(
10
)(
++= ssssG
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Nasiruddin's Horizons 6
NYQUIST PLOT
Solution:
First, substitute s withjwin the transfer function,
)5(6
)5(6*
)5(6-
10
)55(
10
)5)((
10)5)(1(
10)(
32
32
32
223
2
www
www
www
wwww
www
www
w
---
---
-+=
+---
=
++-=
++
=
j
j
j
jj
jj
jjj
jG
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Nasiruddin's Horizons 7
NYQUIST PLOT
234
22
)5(36
)5(1060)(
www
www
w--
---=
jjG
At frequency 0=w , we only observe the most significant terms that take the effect. For
this case,
000
2510)(
=== ==w
ww
wjj
jG .
Magnitude for G(jw) at frequency 0=w ,
===== ww
ww
wwww
2lim
2lim)(lim)(
0000 j
jGjG
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Nasiruddin's Horizons 8
NYQUIST PLOT
Phase for G(jw) at frequency w=0,
o902
lim)(0
0-==
= ww
ww j
jG
At w , we shall look at the most significant term that takes effect when the frequency
approaches infinity. The term of G(jw) is3)(
10)(
w
ww
jjG = .
For magnitude,
010
lim)(
10lim)(
33===
www
www
jjG
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Nasiruddin's Horizons 9
NYQUIST PLOTFor phase,
( )o270
10lim|)(
3-=
=
w
w
w
wj
jG
The point of intersection of the plot with the real axis,
5
5
0)-10(5
0)5(36
)-(510-
0)(Im
2
2
234
2
=
=
=
=-+
=
w
w
w
www
ww
wjG
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Nasiruddin's Horizons 10
NYQUIST PLOT
The intersection point between the polar plot and the real axis is
when 5=w at,
3
1|)(
5 -==wwjG
The intersection between the polar plot with the imaginary axis
can be obtained by setting the real part of 0)( =wjG .
Re 0)( =wjG
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Nasiruddin's Horizons 11
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Therefore,
0)( =jG
=
=
-+
-
w
www
w0
)5(36
60
234
2
Nyquist Diagram0 2
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Nasiruddin's Horizons 12
Real Axis
-0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
B
-20 dB
-10 dB-6 dB-4 dB-2 dB
System: Open Loop L
Real: -0.327
Imag: -0.000358
Frequency (rad/sec): -2.27
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Nasiruddin's Horizons 13
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Nyquist stability criterion
yquist stability criterion is a graphical method to determine the stability of
a closed-loop system by examining the behaviour of the frequency domain in
response to the open-loop system.
yquist stability criterion determines the stability of the closed-loop system
based on the open-loop transfer function of that system.
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Nasiruddin's Horizons 14
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The stability of a closed-loop system can be determined by means of characteristicequation, that is )()(1)( sHsGsF += in the S-plane when s equals to the points on the
Nyquist path. Then, we need to study the behaviour of the plot, comparing with
the origin in the S-plane. This plot is called theNyquist Plotfor 1+G(s)H(s).
However, to simplify things, it is easy to construct a Nyquist plot for G(s)H(s) in
the G(s)H(s)-plane rather than in 1+G(s)H(s)-plane like what we did for Polar plot
(remember?)
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Nasiruddin's Horizons 15
NYQUIST PLOTThere are two types of stability to be examined in any control system:
Open-loop stability
Closed-loop stability
By using the Nyquist criterion,
1. The stability of open loop system can be found by studying the behaviour of the
Nyquist plot forG(s)H(s) in relative to the origin of G(s)H(s)-plane although the
poles ofG(s)H(s)are not known.
2. The stability of closed loop system can be found by studying the behaviour of
Nyquist plot forG(s)H(s)in relative to the (-1,j0) point.
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Nasiruddin's Horizons 16
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Nyquist Path what is it?
-a path that goes in counterclockwise direction (arah lawan jam) that encloses
the right-half S-plane and does not pass through the poles of F(s)=1+G(s)H(s)=0,
located on the imaginary axis(instead, the Nyquist path encircles half way and
roceed downwards)
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Nasiruddin's Horizons 17
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The Nyquist stability criterion methods can be summarized as follows:
1. The Nyquist path is determined in S-plane.
2. Nyquist plot for G(s)H(s) is sketched in the G(s)H(s)-plane with s value equals to
the points value along the Nyquist-path.
3. The open-loop and closed-loop stability for a system can be determined by
observing the behaviour of the Nyquist plot for G(s)H(s) relative to the origin
(0,j0) and point (-1,j0).
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Nasiruddin's Horizons 18
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The followings are the symbols used to determine the system stability by using
Nyquist Criterion:
0N : The number of encirclement around the origin (0,j0) by theNyquist plotfor
G(s)H(s)(positive if the encirclement(kepungan) is counterclockwise direction.
:0Z The number of zeros forG(s)H(s)that have been enclosed (dikepung)by the
Nyquist pathor on the right half of s-plane.
:0P The number of poles forG(s)H(s)that have been enclosed by theNyquist
path or on the right half of s-plane.
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:1-N The number of encirclement around the point (-1,j0) by the Nyquist plot forG(s)H(s)(positive if the encirclement is in counterclockwise direction)
:1-Z The number of zeros forF(s)=1 + G(s)H(s)that have been enclosed by theNyquist path or on the right half of S-plane.
:1- The number of poles forF(s)=1+G(s)H(s)that have been enclosed by theNyquist path or on the right half of s-plane.
Since poles forG(s)H(s)is the same as poles forF(s)=1+G(s)H(s), then
10 -=
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By Nyquist Criterion, for open-loop system stability, the following should be adhered,
000 PZN -=
with
00=P
for closed-loop stability, then,
111 --- -= PZN
with
01=-Z
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Nasiruddin's Horizons 21
NYQUIST PLOT
Nyquist Stability Criterion can be stated as follow:
i. For open-loop system to be stable, the Nyquist plot for G(s)H(s) must encloses or
encircles(mengepung) origin (0,j0) as many as the number of zeros of G(s)H(s) that
situates on the right half of S-plane. The encirclement must be in counterclockwise
direction ,hence00
ZN = .
ii. For closed-loop system to be stable, the Nyquist plot for G(s)H(s)must encircles the
point (-1,j0) in clockwise direction with number of encirclements as many as the
number of poles of G(s)H(s) that located on the right-half of S-plane, hence
011 PPN -=-= -- .
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Nasiruddin's Horizons 22
NYQUIST PLOT
Steps in determining the stability using Nyquist Stability Criterion:
i. From the characteristic equation, F(s)=1+G(s)H(s)=0, the Nyquist path on the S-
plane is constructed from the behaviour of zero-pole ofG(s)H(s)at first.
ii. Sketch the Nyquist plot forG(s)(s) on the G(s)H(s) plane.
iii. Determine the value of 10 -NandN from the behaviour of Nyquist plot for G(s)H(s)
with respect to origin point (0,j0) and point (-1,j0).
iv. Obtain the value of 0P (if not known) from
000 PZN -= ( 0Z is known)
If 00=P , then the open loop system is stable.
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v. Then, after 0P is known, obtain the value of 1-P by 0P= 1-P .
vi. Obtain 1-Z from 111 --- -= PZN .
If 1-Z =0, then, the closed-loop system is stable.
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Examples 1
)5()()(
+=
ss
KsHsG
Determine the system stability when K changes from 0 to infiniti.
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n Gain margin and phase margin from Nyquist plot.
Gain cross-over frequency is the frequency at which the
point on the Nyquist Plot for G(s)H(s) has magnitude equals
to 1.
1)()(1
==ww
sHsG
Phase cross-over frequency is the frequency at which thepoint on the Nyquist plot for G(s)H(s) has phase difference
of 180
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n The gain margin can be obtained from the Nyquist plot
by the followings,
n In designing a control system, phase margin is chosen
such that it is in range between 30 to 60.
XGain
jHjGX
1Margin
)()(
=
= ww
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Nasiruddin's Horizons 28
NYQUIST PLOT