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16.06 Principles of Automatic Control Lecture 28
The Nichols Chart
The Nichols chart may be thought of as a Nyquist plot on a log scale. A Nyquist plot is a plot in the complex plane of
Gpjωq “ RepGpjωqq ` jImpGpjωqqlooooomooooon looooomooooon x-coordinate y-coordinate
Instead, on a Nichols chart, we plot
log Gpjωq “ log |Gpjωq| `j =pGpjωqqlooooomooooon loooomoooon y-coordinate x-coordinate
Notice that we reverse the coordinates - the real part is plotted on the vertical, and the imaginary part is plotted on the horizontal.
In addition, the chart has contours of constant closed-loop magnitude and phase,
ˇ ˇ
ˇ G ˇ
ˇ ˇM “ ̌ˇ1 ` G
G N “=
1 ` G
The Nichols chart template is shown below. Usually, we are interested in the range of frequencies where the phase is greater than ́ 180˝ . The Nichols chart is often expanded (see plot below).
1
−360 −315 −270 −225 −180 −135 −90 −45 0−20
−15
−10
−5
0
5
10
15
20
25
30
−20 dB
−15 dB
−12 dB
−9 dB
−6 dB
−5 dB
−4 dB
−3 dB
−2 dB
−1 dB
0 dB1 dB
2 dB
3 dB
4 dB
5 dB
6 dB
9 dB
12 dB
−350
−340
−330
−300
−270
−240
−210 −180 −150
−120
−90
−60
−30
−20
−10
−5
−2
Phase, deg
Magnitude, dB
2
−225 −180 −135 −90 −45 0−20
−15
−10
−5
0
5
10
15
20
25
30
−20 dB
−15 dB
−12 dB
−9 dB
−6 dB
−5 dB
−4 dB
−3 dB
−2 dB
−1 dB
0 dB1 dB
2 dB
3 dB
4 dB
5 dB
6 dB
9 dB
12 dB
−350
−340
−330
−300
−270
−240
−210 −180 −150
−120
−90
−60
−30
−20
−10
−5
−2
Phase, deg
Magnitude, dB
3
The Nichols chart was once very useful, since computers were not available to do the kids of calculations that are now done by e.g., Matlab.
However, Nichols chart may be used to give insight into the closed-loop behavior of systems. Consider first the system
-
+G
1
r
where
which has ωc “ 10 r/s, P M “ 45˝ .
G1 “
? 2 10
sp1 ` s{10q
Bode of G1:
10−1
100
101
102
103
10−5
100
105
Magnitude of G
1
10−1
100
101
102
103
−200
−150
−100
−50
0
Phase, degrees
ω
ω
-1
-2
The Nichols plot can be made by lifting points of the Bode plot, at individual frequencies, and plotting on the Nichols chart. See plot below for the plot of G1 :
4
−225 −180 −135 −90 −45 0−20
−15
−10
−5
0
5
10
15
20
25
30
−20 dB
−15 dB
−12 dB
−9 dB
−6 dB
−5 dB
−4 dB
−3 dB
−2 dB
−1 dB
0 dB1 dB
2 dB
3 dB
4 dB
5 dB
6 dB
9 dB
12 dB
−350
−340
−330
−300
−270
−240
−210 −180 −150
−120
−90
−60
−30
−20
−10
−5
−2
Phase, deg
Magnitude, dB
ωr=9.57, M
r=1.31 (2.4 dB)
ωc=10
Note that ωr « ωc, so the peak in the frequency response (CL) is very close to crossover.
Note also that
Mp “ 0.23
5
For PM “ 45˝ , we expect
ζ “0.45
ñ Mp “0.21
Mr “1.24
In pretty good agreement with the actual results.
Now consider the plant
100 1 ` s{10 G2 “ ?
s22
in a similar unity feedback control. For this system, we have
PM “ 45˝, ωc “ 10 r/s, also.
Bode plot:
10−1
100
101
102
103
10−5
100
105
Ma
gn
itu
de
of
G1
10−1
100
101
102
103
−200
−150
−100
−50
0
Ph
ase
, de
gre
es
ω
6
Since the crossover and phase margin are the same, we expect to get similar performance. Do we?
One clue can be seen in the Nichols chart, below.
−225 −180 −135 −90 −45 0−20
−15
−10
−5
0
5
10
15
20
25
30
−20 dB
−15 dB
−12 dB
−9 dB
−6 dB
−5 dB
−4 dB
−3 dB
−2 dB
−1 dB
0 dB1 dB
2 dB
3 dB
4 dB
5 dB
6 dB
9 dB
12 dB
−350
−340
−330
−300
−270
−240
−210 −180 −150
−120
−90
−60
−30
−20
−10
−5
−2
Phase, deg
Magnitude, dB
ωr=7.44, M
r=1.61 (4.1 dB)
ωc=10
Note that, in this case, ωr is significantly smaller than ωc, and Mr is larger than might be
7
expected from the PM. So we would expect that the closed-loop system
G2T2 “
1 ` G2
would be a bit slower, and have more overshoot, than the system
G1T1 “
1 ` G1
even though they have the same PM and ωc.
In fact, this is the case, as seen from the step responses below.
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Time, t (sec)
Ste
p R
esp
on
se
y2(t) M
p=0.34
y1(t) M
p=0.23
Counting Encirclements on a Nichols Chart
Counting encirclements on a Nichols chart can be tricky, because
1. The ́ 1{k point can be on either the ́ 180˝ line or the 0˝line.
2. CW and CCW are reversed, because the orientation of the axes is reversed.
Will demonstrate with examples.
8
Example 1. s ´ 0.1
Gpsq “ 3000 ps ´ 1qps ´ 2qps ` 10q2
Nyquist and Nichols plots are shown below.
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0−5
−4
−3
−2
−1
0
1
2
3
4
5
Nyquist Diagram
Real Axis
Ima
gin
ary
Axi
s
N=-2
N=0
N=+1
9
−270 −225 −180 −135 −90−100
−80
−60
−40
−20
0
20
Nichols Chart
Open−Loop Phase (deg)
Op
en
−L
oo
p G
ain
(d
B)
N=+1
N=0
N=-219 dB
6.27 dB
3.52 dB
Example 2. 1 ps ` 0.1q2
Gpsq “ s3 ps ` 10q2
-10 -.1
Nichols chart is shown below. Note that care must be used to properly close contour near ω “ 0.
10
Nichols Chart
Open−Loop Phase (deg)
Op
en
−L
oo
p G
ain
(d
B)
−360 −315 −270 −225 −180 −135 −90 −45 0−200
−150
−100
−50
0
50
100
N=0
N=2
N=2
N=1
11
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16.06 Principles of Automatic ControlFall 2012
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