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

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−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

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−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.

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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.

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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

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16.06 Principles of Automatic ControlFall 2012

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