# Nichols plot

The Nichols plot is a plot used in signal processing and control design, named after American engineer Nathaniel B. Nichols.

## Use in control design

Given a transfer function,

$G(s)={\frac {Y(s)}{X(s)}}$ with the closed-loop transfer function defined as,

$M(s)={\frac {G(s)}{1+G(s)}}$ the Nichols plots displays $20\log _{10}(|G(s)|)$ versus $\arg(G(s))$ . Loci of constant $20\log _{10}(|M(s)|)$ and $\arg(M(s))$ are overlaid to allow the designer to obtain the closed loop transfer function directly from the open loop transfer function. Thus, the frequency $\omega$ is the parameter along the curve. This plot may be compared to the Bode plot in which the two inter-related graphs - $20\log _{10}(|G(s)|)$ versus $\log _{10}(\omega )$ and $\arg(G(s))$ versus $\log _{10}(\omega )$ ) - 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, $\arg(G(s))$ refers to the phase of the system's response. Although similar to a Nyquist plot, a Nichols plot is plotted in a Cartesian coordinate system while a Nyquist plot is plotted in a Polar coordinate system.