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8/10/2019 NYQUIST PLOT Dr Nasiruddin
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CONTROL SYSTEM(part 2)
EEE 350
DR. MUHAMMAD NASIRUDDIN
MAHYUDDIN
FREQUENCY DOMAIN ANALYSIS
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Frequency Response Technique
Continues.
NYQUIST PLOT
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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 afters is replaced with j in G(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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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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NYQUIST PLOT
Worked Example:
Sketch a polar plot for the following transfer function.
)5)(1(
10
)(
ssssG
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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
j
j
j
jj
jj
jjj
jG
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NYQUIST PLOT
234
22
)5(36
)5(1060)(
jjG
At frequency 0 , we only observe the most significant terms that take the effect. For
this case,
000
2510)(
jjjG .
Magnitude for G(jw) at frequency 0 ,
2lim
2lim)(lim)(
0000 j
jGjG
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NYQUIST PLOT
Phase for G(jw) at frequency w=0,
902
lim)(0
0
j
jG
At , we shall look at the most significant term that takes effect when the frequency
approaches infinity. The term of G(jw) is3)(
10)(
jjG
.
For magnitude,
010
lim)(
10lim)(
33
jjG
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NYQUIST PLOTFor phase,
270
10lim|)(
3
jjG
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
jG
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NYQUIST PLOT
The intersection point between the polar plot and the real axis is
when 5 at,
3
1|)(
5 jG
The intersection between the polar plot with the imaginary axis
can be obtained by setting the real part of 0)( jG .
Re 0)( jG
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Therefore,
0)( jG
0)5(36
60
234
2
DR NASIRUDDIN
Nyquist Diagram
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Nyquist Diagram
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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NYQUIST PLOT
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
yquist path. Then, we need to study the behaviour of the plot, comparing with
the origin in the S-plane. This plot is called the Nyquist Plot for 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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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 for G(s)H(s)in relative to the origin of G(s)H(s)-plane although the
poles of G(s)H(s)are not known.
2.
The stability of closed loop system can be found by studying the behaviour of
Nyquist plot for G(s)H(s)in relative to the (-1,j0) point.
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NYQUIST PLOT
yquist Pathwhat is it?
-a path that goes in counterclockwise direction (arah lawan jam) that encloses
the ri ght-hal f S-plane and does not pass thr ough the poles of F (s)=1+G(s)H (s)=0,
located on the imaginary axis(instead, the Nyqui st path encir cles hal f way and
proceed downwards)
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NYQUIST PLOT
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 tothe 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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NYQUIST PLOT
The followings are the symbols used to determine the system stability by using
yquist Criterion:
0N : The number of encirclement around the origin (0,j0) by the Nyquist plotfor
G(s)H(s)(positive if the encirclement(kepungan)is counterclockwise direction.
:0Z The number of zeros for G(s)H(s)that have been enclosed (dikepung)by the
Nyquist path or on the right half of s-plane.
:0P The number of poles for G(s)H(s)that have been enclosed by the Nyquist
path or on the right half of s-plane.
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NYQUIST PLOT
:1N 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)
:1Z 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.
:1P 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 for G(s)H(s)is the same as poles forF(s)=1+G(s)H(s), then
10 PP
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NYQUIST PLOT
By Nyquist Criterion, for open-loop system stability, the following should be adhered,
000 PZN
with
00P
for closed-loop stability, then,
111 PZN
with
01Z
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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 ,hence 00 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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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 of G(s)H(s) at first.
ii. Sketch the Nyquist plot for G(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 00P , then the open loop system is stable.
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NYQUIST PLOT
v. Then, after 0Pis known, obtain the value of 1P by 0P= 1P .
vi. Obtain 1Z from 111 PZN .
If 1Z =0, then, the closed-loop system is stable.
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NYQUIST PLOT
Examples 1
)5()()(
ss
KsHsG
Determine the system stability when K changes from 0 to infiniti.
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NYQUIST PLOT
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
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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NYQUIST PLOT
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NYQUIST PLOT
The gain margin can be obtained from the Nyquist plot
by the followings,
In designing a control system, phase margin is chosen
such that it is in range between 30to 60.
XGain
jHjGX
1Margin
)()(