Nichols plot
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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.