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.