control//frequency response//Nyquist stability criterion

The Nyquist stability criterion is a test that decides whether a feedback loop is stable by looking only at its open-loop frequency response, the loop gain \(L(j\omega)\) traced as a curve in the complex plane, and counting how many times that curve circles the point \(-1\). It is the theorem underneath the stability margins, and the tool when the loop contains what a polynomial cannot hold: a pure dead time, a measured frequency response, a sensor whose dynamics are only known as a curve.

The Nyquist stability criterion is a test that decides whether a feedback loop is stable by looking only at its open-loop frequency response, the loop gain L(jω)L(j\omega)L(jω) traced as a curve in the complex plane, and counting how many times that curve circles the point −1-1−1. It is the theorem underneath the stability margins, and the tool when the loop contains what a polynomial cannot hold: a pure dead time, a measured frequency response, a sensor whose dynamics are only known as a curve.

The point −1-1−1 is where a signal returns from one trip round the loop with the same size and inverted sign; with the minus of negative feedback, the correction then arrives exactly in step with the error and sustains it. The criterion turns that into a count. Draw L(jω)L(j\omega)L(jω) for all frequencies (the Nyquist plot), count its net clockwise encirclements NNN of −1-1−1, and know PPP, the number of unstable open-loop poles:

Z=N+P,Z=N+P,Z=N+P,

with ZZZ the number of unstable closed-loop poles. For the common case of a plant and controller that are stable on their own (P=0P=0P=0), the loop is stable exactly when the curve does not encircle −1-1−1.

The margins are distances to −1-1−1. The gain margin says how far the curve could be stretched before it reaches the point, the phase margin how far it could be turned, and the peak of the sensitivity function is one over the closest approach. A loop whose curve passes near −1-1−1 is stable and fragile at once.

Delay is where it earns its keep. A dead time e−jωτe^{-j\omega\tau}e−jωτ leaves the gain unchanged and rotates the curve by ωτ\omega\tauωτ radians, more at high frequency, so a loop that was comfortably stable spirals toward −1-1−1 as the delay grows; the delay margin is the delay at which it touches it. Pole-based tools such as the Routh-Hurwitz criterion or the root locus need a polynomial approximation of the delay to say the same.

It handles unstable plants honestly. A balancing robot or a pendulum held upright has an unstable open-loop pole (P>0P>0P>0), so its stable closed loop must encircle −1-1−1 the right number of times counterclockwise; the Bode plot margins, read naively, can mislead there, and the Nyquist plot does not.

It can be fed with measurements. A sine sweep on the real loop gives L(jω)L(j\omega)L(jω) point by point, and the criterion applies to that curve with no model at all, which is how a loop on a test bench is certified against its specification.