control//frequency response//Bode plot

A Bode plot draws a system's frequency response as two curves against frequency on a logarithmic axis, the gain in decibels above and the phase in degrees below, and its main practical use is to read how close a feedback loop is to oscillating. Drawn for the open loop (controller and plant in series, the path once round the loop), it shows the two stability margins that tuning specifications and tuning tools are written in.

A Bode plot draws a system's frequency response as two curves against frequency on a logarithmic axis, the gain in decibels above and the phase in degrees below, and its main practical use is to read how close a feedback loop is to oscillating. Drawn for the open loop (controller and plant in series, the path once round the loop), it shows the two stability margins that tuning specifications and tuning tools are written in.

The phase margin is how much extra lag the loop could take before it oscillates: find the frequency where the open-loop gain crosses 0 dB (the crossover frequency) and measure how far the phase there is above −180 degrees. The gain margin is how much extra gain it could take: find the frequency where the phase crosses −180 degrees and measure how far the gain there is below 0 dB. Reasonable designs keep a phase margin of 30 to 60 degrees and a gain margin of 2 to 5, about 6 to 14 dB; MATLAB's pidtune aims at 60 degrees of phase margin by default.

The logarithmic axes make the plot quick to sketch. Each lag contributes a straight-line asymptote, flat and then falling at 20 dB per decade, with its phase sliding from 0 to −90 degrees, and the curves of elements in series simply add. Engineers could shape loops with a ruler before computers, and the plot survived the computers that now draw it.

It is not outdated. Current tools compute and plot exactly this (MATLAB's bode and margin, python-control's bode_plot and margin, scipy.signal's bode), and single-loop tuning, robustness checks and filter design are still argued on it; state-space methods took over the coupled, multivariable and optimal designs (LQG).

Features of the plant have signatures on it. A dead time shows as a phase that falls faster and faster with no change in gain, the mark of the plant that limits every loop; a mechanical resonance shows as a peak in gain with a sharp drop in phase, the mark of a flexible mechanism.

The margins assume a linear loop and one thing changing at a time. A loop can have good margins and still be fragile to a combined change of gain and phase, and the margins say nothing about saturation.

?Is a loop with a gain margin of 6 dB stable at twice its gain?

It sits exactly at the edge. 6 dB is a factor of two, so doubling the loop gain brings it to a sustained oscillation. Designs keep more than that because the plant's gain is rarely known to a factor of two, and because it may change with load.