The Nyquist Stability Criterion
Closed-loop stability follows from counting how many times the open-loop Nyquist curve encircles the critical point minus one.
Counting Encirclements
The Nyquist criterion is a precise rule derived from Cauchy's argument principle in complex analysis. It relates the number of unstable closed-loop poles to the number of encirclements the open-loop locus makes around the point minus one plus zero j in the complex plane.
The formula
Let P be the number of open-loop poles in the right half-plane, N the number of clockwise encirclements of minus one by the Nyquist curve, and Z the number of closed-loop poles in the right half-plane. Then Z = N + P. For closed-loop stability we require Z = 0, so the curve must encircle minus one counterclockwise exactly P times.
- Stable open-loop plant (P = 0): stability requires zero encirclements of minus one.
- Unstable open-loop plant (P > 0): the curve must encircle minus one P times counterclockwise to stabilize.
- Any net clockwise encirclement when P = 0 means the closed loop is unstable.
Why it works
The closed-loop poles are the zeros of 1 + L(s). Mapping the imaginary axis through 1 + L(s) and counting encirclements of the origin is equivalent to mapping through L(s) and counting encirclements of minus one. The argument principle then converts encirclements into a count of right-half-plane zeros minus poles.
Practical strength
Unlike the Routh-Hurwitz test, the Nyquist criterion works with measured frequency-response data and with pure time delays, since it needs only the shape of the locus, not a rational polynomial. This makes it the standard rigorous stability test for unstable plants stabilized by feedback, a category that includes magnetically confined plasma position loops.
The criterion also generalizes: how closely the curve approaches minus one quantifies robustness, leading directly to gain margin, phase margin, and the vector-margin measures used in robust control.