Nichols plot
plot
The Nichols plot is a plot used in signal processing and control design, named after American engineer Nathaniel B. Nichols. It plots the phase response versus the response magnitude of a transfer function for any given frequency, and as such is useful in characterizing a system's frequency response.
Use in control design
Given a transfer function,
G
(
s
)
=
Y
(
s
)
X
(
s
)
{\displaystyle G(s)={\frac {Y(s)}{X(s)}}}
with the closed-loop transfer function defined as,
M
(
s
)
=
G
(
s
)
1
+
G
(
s
)
{\displaystyle M(s)={\frac {G(s)}{1+G(s)}}}
the Nichols plots displays
20
log
10
(
|
G
(
s
)
|
)
{\displaystyle 20\log _{10}(|G(s)|)}
versus
arg
(
G
(
s
)
)
{\displaystyle \arg(G(s))}
. Loci of constant
20
log
10
(
|
M
(
s
)
|
)
{\displaystyle 20\log _{10}(|M(s)|)}
and
arg
(
M
(
s
)
)
{\displaystyle \arg(M(s))}
(so-called Hall circles) are overlaid to allow the designer to obtain the closed loop transfer function directly from the open loop transfer function. Thus, the frequency
ω
{\displaystyle \omega }
is the parameter along the curve. This plot may be compared to the Bode plot in which the two inter-related graphs -
20
log
10
(
|
G
(
s
)
|
)
{\displaystyle 20\log _{10}(|G(s)|)}
versus
log
10
(
ω
)
{\displaystyle \log _{10}(\omega )}
and
arg
(
G
(
s
)
)
{\displaystyle \arg(G(s))}
versus
log
10
(
ω
)
{\displaystyle \log _{10}(\omega )}
) - are plotted.
In feedback control design, the plot is useful for assessing the stability and robustness of a linear system. This application of the Nichols plot is central to the quantitative feedback theory (QFT) of Horowitz and Sidi, which is a well known method for robust control system design.
In most cases,
arg
(
G
(
s
)
)
{\displaystyle \arg(G(s))}
refers to the phase of the system's response. Although similar to a Nyquist plot, a Nichols plot is plotted in a Polar coordinate system while a Nyquist plot is plotted in a Cartesian coordinate system.
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