 Nyquist stability criterion

The Nyquist stability criterion applies the argument principle to a specified negative-feedback return ratio $L(s)$. Assume a proper rational loop with a well-posed interconnection, no poles or closed-loop characteristic zeros on the imaginary axis, and no singularity at the closing part of the contour. Let $P$ count poles of $L$ in the open right half-plane and let $Z$ count zeros of $1+L$ there, both with multiplicity. For a positively oriented contour enclosing that half-plane, let $N$ be the signed counterclockwise winding of its image under $L$ around $-1$. ThenA clockwise contour convention reverses the sign of $N$, so the orientation must accompany the formula. Closed-loop input-output stability requires $Z=0$ and exclusion of imaginary-axis zeros; simply passing through or encircling $-1$ is not a conclusion without the pole count and convention.

Poles on the imaginary axis require an indented contour and a corresponding boundary analysis. Also, cancellations in a reduced transfer function can conceal unstable internal states, so a Nyquist conclusion about $1+L$ alone does not prove internal stability of an arbitrary nonminimal realization. The bounded-input bounded-output stability example exhibits that distinction.

The Nyquist stability criterion is a fundamental principle in control theory used to determine the stability of a linear time-invariant (LTI) system based on its frequency response. Specifically, it relates the open-loop frequency response of a system to the stability of the closed-loop system.