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Nichols plot

From Quire, the open reference·Chart of a transfer function's phase response vs. magnitude·1 min read
A Nichols plot.
A Nichols plot. Image: source and license

The Nichols plot is a plot used in signal processing and control design, named after American engineer Nathaniel B. Nichols.[1][2][3] 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#edit

Given a transfer function,

{\displaystyle G(s)={\frac {Y(s)}{X(s)

with the closed-loop transfer function defined as,

{\displaystyle M(s)={\frac {G(s)}{1+G(s)

the Nichols plots displays {\displaystyle 20\log _{10}(|G(s)|)} versus {\displaystyle \arg(G(s))}. Loci of constant {\displaystyle 20\log _{10}(|M(s)|)} and {\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 {\displaystyle \omega } is the parameter along the curve. This plot may be compared to the Bode plot in which the two inter-related graphs - {\displaystyle 20\log _{10}(|G(s)|)} versus {\displaystyle \log _{10}(\omega )} and {\displaystyle \arg(G(s))} versus {\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, {\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.

See also#edit

References#edit

  1. ^ Isaac M. Howowitz, Synthesis of Feedback Systems, Academic Press, 1963, Lib Congress 63-12033 p. 194-198
  2. ^ Boris J. Lurie and Paul J. Enright, Classical Feedback Control, Marcel Dekker, 2000, ISBN 0-8247-0370-7 p. 10
  3. ^ Allen Stubberud, Ivan Williams, and Joseph DeStefano, Shaums Outline Feedback and Control Systems, McGraw-Hill, 1995, ISBN 0-07-017052-5 ch. 17
Categories Plots (graphics)Signal processingClassical control theory

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