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Three-Dimensional Visualization of
Nichols, Hall, and Robust-Performance Diagrams
Kent H. Lundberg and Zachary J. Malchano
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology, Cambridge, MA 02139
Abstract Three-dimensional models and animations of
Nichols charts, Hall charts, and robust-performance diagrams
are presented. Using these models, students can visualize the
implications and importance of these charts and diagrams.
By viewing these animations, students develop better intuition
concerning the connection between open-loop gain/phase plots,
open-loop polar plots, and closed-loop frequency response.
I. I NTRODUCTION
Upon first presentation, the Nichols chart and the Hall chart
are often confusing to first-term control students. This paper
describes several educational animations that help students
visualize the implications and importance of Nichols charts
and Hall charts by showing the magnitude of the closed-loop
frequency response as the third dimension.
Robust performance, in the face of multiplicative plant uncertainty, can be illustrated with two disjoint sets of frequencydependent circles on a Nyquist diagram (one set for plant
uncertainty and the other set for specifications on sensitivity).
However, distinct circles must be plotted at all frequencies.
By using the third dimension for frequency, a Nyquist plot
can be rendered that includes the effect of plant uncertainty
at all frequencies. Examining this model for intersections of
the corresponding solids indicates the success or failure of a
robust-performance design.
II. N ICHOLS C HART
The Nichols chart [1] has been used for many years to
bridge the gap between open-loop frequency response and
closed-loop frequency response. Plotting the open-loop frequency response on gain/phase coordinates allows the closedloop frequency response to be read from the chart. Unfortunately, students are often bewildered by the complicated chart
when it is first presented, likening it to a plate of spaghetti or
a sadistic dart board, as shown in Figure 1.
A change in point of view can help students appreciate the
importance of the chart. The Nichols chart can be thought of as
a contour map of Mount Nichols (Figure 2) where the height
of the mountain corresponds to the magnitude of the closedloop frequency response for all possible values of open-loop
frequency response. When the frequency response of a specific
loop transfer function for a unity-feedback system, such as
L1 (s) =
1
,
s(s + 1)
(1)
Fig. 1. The original Nichols chart, reprinted from the 1947 textbook by
James, Nichols, and Phillips [1]. The curves show contours of constant closedloop magnitude and constant closed-loop phase. To use the chart, the openloop frequency response is plotted on the rectilinear gain/phase coordinates,
and the closed-loop frequency response is read from the contours of closedloop magnitude and phase.
is plotted on the gain/phase plane as shown in Figure 3,
the height of the surface of Mount Nichols corresponds to
the magnitude of the closed-loop frequency response for that
L(j) as shown in Figure 4.
Showing Mount Nichols and its infinite peak also reinforces
the danger of getting too close to the s = 1 point, as
illustrated by
10
(2)
L2 (s) =
s(s + 1)
shown by the upper curves in Figures 5 and 6. As the openloop frequency response gets closer the the peak of Mount
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Nichols Chart
40
0 dB
30
20
0.25 dB
0.5 dB
15
1 dB
10
5
OpenLoop Gain (dB)
magnitude of L/(1+L) (dB)
20
0
5
10
15
20
300
1 dB
3 dB
10
3 dB
6 dB
0
6 dB
10
12 dB
20
20 dB
40
30
200
30
20
10
100
0
10
0
phase of L (degrees)
20
40
180
magnitude of L (dB)
Fig. 2.
Mount Nichols. The height of the mountain corresponds to
the magnitude of the closed-loop frequency response. When the open-loop
frequency response is plotted on the rectilinear gain/phase coordinates on
the horizontal plane, the magnitude of the closed-loop frequency response is
shown by the height of the surface.
135
90
OpenLoop Phase (deg)
45
40 dB
0
Fig. 5. Nichols chart for L1 (s) and L2 (s). The change in gain between
L1 (s) and L2 (s) corresponds to a shift up the vertical axis. Since L2 (j)
approaches closer to the s = 1 point, the closed-loop frequency response
will appear much higher up the foothills of Mount Nichols.
Closedloop Bode Diagram
20
Nichols Chart
40
0 dB
30
10
0.25 dB
0.5 dB
1 dB
1 dB
0
Magnitude (dB)
OpenLoop Gain (dB)
20
3 dB
10
3 dB
6 dB
0
6 dB
10
12 dB
20
20 dB
10
20
30
30
40
180
135
90
OpenLoop Phase (deg)
45
40
1
10
40 dB
0
Fig. 3. Nichols chart for L1 (s). The closed-loop frequency response is read
from the contours of closed-loop magnitude.
10
Frequency (rad/sec)
10
Fig. 6. Closed-loop Bode plot for L1 (s) and L2 (s). The extreme increase
in the magnitude peaking for L2 (s) is due to its height on Mount Nichols.
This correlation between height on Mount Nichols and closed-loop magnitude
peaking is demonstrated by the animations on the next page.
Closedloop Bode Diagram
20
10
Magnitude (dB)
10
20
30
40
1
10
10
Frequency (rad/sec)
10
Fig. 4. Closed-loop Bode plot for L1 (s). The slight peaking in the response
agrees with the 1-dB contour in Figure 3.
Nichols, the peaking in the closed-loop frequency response
gets larger.
This transformation from the open-loop frequency response
to the closed-loop frequency response is illustrated by a
computer animation that starts with the gain/phase plot on
the surface of Mount Nichols as viewed from above. The
camera view is rotated to show the elevation of the curve on
Mount Nichols to demonstrate that the height of the open-loop
frequency response corresponds to the closed-loop frequency
response.
The animation can be repeated for multiple loop transfer
functions, such as (1) and (2). Still frames from these animations are shown in Figures 7 and 8, respectively.
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Fig. 7.
Still images from Nichols animation for L1 (s) =
1
.
s(s + 1)
Fig. 8.
Still images from Nichols animation for L2 (s) =
10
.
s(s + 1)
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Hall Chart
4
0 dB
10
3
2 dB
2 dB
2
6
4 dB
1
Imaginary Axis
magnitude of L/(1+L)
4 dB
6 dB
10 dB
6 dB
10 dB
1
0
2
2
1
0
1
1
imaginary axis
2
2
4
6
real axis
Fig. 9. Mount Hall. The height of the mountain corresponds to the magnitude of the closed-loop frequency response. When the open-loop frequency
response is plotted on the Nyquist coordinates on the horizontal plane, the
magnitude of the closed-loop frequency response is shown by the height of
the surface.
2
Real Axis
Fig. 10. Hall chart for L1 (s). The closed-loop frequency response is read
from the contours of closed-loop magnitude.
Hall Chart
4
0 dB
III. H ALL C HART
3
2 dB
2 dB
2
4 dB
1
Imaginary Axis
The Hall chart [2] predates the Nichols chart by several
years, but is not as popular with control educators. Pedagogically, the Hall chart is a powerful tool, since it reinforces the
relationship between the Nyquist plot and the Bode plot. In
addition, the M -circles of the Hall chart can be used in robust
Nyquist diagrams, and early introduction of these concepts can
improve student understanding and intuition.
A similar change in perspective can help students appreciate
the importance of the Hall chart. The Hall chart is a contour
map of Mount Hall (Figure 9) where the height of the
mountain corresponds to the magnitude of the closed-loop
frequency response for all possible values of the open-loop
frequency response. Mount Hall has an infinite peak at the
s = 1 point, and it has a dimple to zero at the origin, as we
expect from the magnitude of the closed-loop transfer function
4 dB
6 dB
10 dB
6 dB
10 dB
4
6
2
Real Axis
Fig. 11. Hall chart for L1 (s) and L2 (s). The frequency response L2 (j)
is closer to the s = 1 point and the peak of Mount Hall.
Closedloop Bode Diagram
20
.
M =
1 + L
0
Magnitude (dB)
Figures 10 and 11 show the loop transfer functions (1) and
(2) plotted with the Hall chart. Reading the M -circles on the
Hall chart provides the magnitude of the closed-loop frequency
response, as shown in Figure 12.
Animations similar to those shown in Figures 7 and 8 are
made to help students visualize the connection between the
polar plot of the open-loop frequency response and the Bode
plot of the closed-loop frequency response. The animation
starts with the polar plot on the surface of Mount Hall as
viewed from above. The camera view is rotated to show the
elevation of the curve on Mount Hall to demonstrate that the
height of the open-loop frequency response corresponds to the
closed-loop frequency response.
10
10
20
30
40
1
10
10
Frequency (rad/sec)
10
Fig. 12. Closed-loop Bode plot for L1 (s) and L2 (s). The 10-dB peaking in
the response agrees with the 10-dB contour in Figure 11 and the associated
height on the foothills of Mount Hall.
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Bode Diagram
Nyquist Diagram
10
Imaginary Axis
Magnitude (dB)
2
5
10
2
15
20
25
3
10
10
10
10
Frequency (rad/sec)
10
6
2
10
Fig. 13. Magnitude plot of nominal-plant sensitivity transfer function and
sensitivity bound W1 (j). For the system to meet the specified requirements
on dynamic-tracking error, the sensitivity transfer function must always be
below the bound W1 (j).
4
Real Axis
10
Fig. 14. Performance chart showing nominal-plant loop transfer function
L(j) and frequency-dependent sensitivity bound W1 (j). At each frequency, the loop transfer function L (shown by each coloredx) must be
outside the circle of radius |W1 | (of the same color) centered at s = 1.
Nyquist Diagram
6
IV. ROBUST-P ERFORMANCE D IAGRAMS
10
L(s) =
(s + 1)(0.1s + 1)
The requirements for a small dynamic-tracking error can be
specified in terms of a sensitivity bound
1
1
1 + L(j) |W1 (j)|
For a particular set of dynamic-tracking requirements, the
sensitivity bound could be expressed as
W11 (s) =
0.1(20s + 1)
s+1
as shown in the Bode plot in Figure 13.
The sensitivity bound can also be written in terms of a
vector length from the s = 1 point [3] as
|1 + L(j)| > |W1 (j)|
Therefore, the sensitivity bound can also be displayed on a
Nyquist plot, as shown in Figure 14. At each frequency k ,
the loop transfer function L(jk ) must be outside the circle
of radius |W1 (jk )| centered at s = 1.
Now consider a perturbed plant transfer function with an
unmodeled high-frequency pole
1
10
L (s) =
(s + 1)(0.1s + 1) s + 1
where it is known that < m = 0.04 second.
2
Imaginary Axis
Three-dimensional visualizations can also help students
understand the frequency dependence of robust-performance
diagrams. A robust-performance diagram shows the competing
bounds on sensitivity and uncertainty, which must not intersect
for a successfully robust design. However, the usual way of
plotting these diagrams [3] can be confusing.
As an example, consider a nominal-plant transfer function
6
2
4
Real Axis
10
Fig. 15. Robust-stability chart showing loop transfer function and multiplicative uncertainty bound. As long as the uncertainty smear along the plot of
L(j) does not include the s = 1 point, the system stability is robust to
uncertainty.
A bound on the multiplicative uncertainty can be developed
from the perturbed and nominal plants
L (j)
L(j) 1 |W2 (j)|
For robust stability, the s = 1 point must lie outside the
circles of radius |W2 (jk )L(jk )| centered at the corresponding points L(jk ) for all frequencies k . In above example,
the uncertainty of the unmodeled high-frequency pole can be
bounded by
s m s
1
1
s + 1 m s + 1
s + 1
As long as the uncertainty circles along of the plot of L(j) do
not include the s = 1 point (as demonstrated in Figure 15),
the system stability is robust to uncertainty.
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Nyquist Diagram
6
Imaginary Axis
6
2
4
Real Axis
10
Fig. 16. Robust-performance chart. At each frequency, the robust-stability
circle of radius |W2 L| and centered at L must be outside the performance
circle of radius |W1 | and centered at s = 1.
frequency, thus the bounds are displayed on the plot for all
frequencies. Robust-performance is guaranteed if the uncertainty snake does not intersect the sensitivity cone. Using a
three-dimensional model, the absence of this intersection can
be readily examined.
Nyquist Diagram
1
0.8
0.6
0.4
Imaginary Axis
Fig. 18.
Three-dimensional model of the robust-performance chart in
Figure 16. Examining this model for intersections of the corresponding solids
indicates the success or failure of the robust-performance design.
0.2
0
ACKNOWLEDGMENTS
0.2
0.4
0.6
0.8
1
1
0.8
0.6
0.4
0.2
0
Real Axis
0.2
0.4
0.6
0.8
Fig. 17. Magnification of robust-performance chart in Figure 16. The black
circles in this plot show the closest approach, but no intersection. These circles
must be drawn and checked for all frequencies.
For robust performance the circles created by the sensitivity
bound are included in the plot, as shown in Figure 16. The
concept of a smeared frequency response (as shown in
Figure 15) cannot be used.
Therefore, it is required that at each frequency k , the
disc of uncertainty (with radius |L(jk )W2 (jk )|) around the
loop transfer function L(jk ) must not intersect the circle of
radius |W1 (jk )| centered at s = 1. The robust-performance
diagram in Figure 16 shows that there is no intersection, even
at the closest approach. However, the circles at all frequencies
must be examined.
Unless these circles are drawn and checked for all frequencies, robust performance cannot be guaranteed. A threedimensional model of the robust-performance chart (as shown
in Figure 18) clears up the potential ambiguity. The bounds
on the sensitivity function and multiplicative plant uncertainty
become solids that must not intersect. The sensitivity bound
becomes a cone centered on s = 1, and the multiplicative
plant uncertainty is represented by the thickness of the loop
transfer function snake. The vertical axis corresponds to
The authors understanding of robust control has been
greatly enhanced by many helpful conversations with Professor Leonard Gould.
Special thanks to Hubert James, Nathaniel Nichols, Ralph
Phillips, and McGraw-Hill for putting their classic text [1] in
the public domain ten years after the original publication, thus
allowing us to freely reprint Figure 1.
The visualizations presented here were rendered using the
Visualization Toolkit [4], a freely available open-source graphics package (http://www.vtk.org/).
R EFERENCES
[1] Hubert M. James, Nathaniel B. Nichols, and Ralph S.
Phillips. Theory of Servomechanisms, volume 25 of MIT
Radiation Laboratory Series. McGraw-Hill, New York,
1947.
[2] Albert C. Hall. The Analysis and Synthesis of Linear
Servomechanisms. Technology Press, M.I.T., Cambridge,
MA, 1943.
[3] John C. Doyle, Bruce A. Francis, and Allen R. Tannenbaum. Feedback Control Theory. Macmillan, New York,
1992.
[4] W. J. Schroeder, K. M. Martin, and W. E. Lorensen. The
design and implementation of an object-oriented toolkit for
3d graphics and visualization. In IEEE Visualization, pages
93100, 1996.
