Nyquist stability criterion

The Nyquist criterion is a graphical test that decides closed-loop stability just by looking at the open-loop frequency response — without ever computing the closed-loop poles. You plot G(jω)H(jω) as a curve in the complex plane while sweeping frequency ω from −∞ to +∞, then count how many times that curve wraps around the single point −1 + j0. The genius is that this one count, made entirely from the open-loop data you can actually measure, reveals whether the closed-loop system you haven't built yet will be stable.

The bookkeeping rule is Z = N + P: the number of unstable closed-loop poles (Z) equals the clockwise encirclements of −1 (N) plus the number of unstable open-loop poles (P). For the common case of a stable open-loop plant (P = 0), the verdict is beautifully simple — if the curve doesn't encircle −1 at all, the closed loop is stable; encircle −1 and it's unstable. Unlike the Bode margin test, Nyquist handles tricky cases that trip up simpler methods: loops with multiple gain crossovers, conditionally-stable systems that become unstable if you lower the gain, and open-loop-unstable plants. It's the rigorous foundation that gain and phase margins are really just shortcuts for.