Nyquist plot

The Nyquist plot is a parametric curve in the complex plane that represents the frequency response of a linear time-invariant system as frequency varies from zero to infinity. Each point on the curve corresponds to the complex value \(G(j\omega)\) at a particular frequency, with the real part plotted against the imaginary part. The Nyquist plot is the geometric foundation of the Nyquist stability criterion, one of the most powerful tools in control theory because it determines closed-loop stability from open-loop data without explicitly computing closed-loop poles.

The criterion states that the number of clockwise encirclements of the critical point \(-1 + j0\) by the Nyquist plot equals the number of unstable closed-loop poles minus the number of unstable open-loop poles. A system is stable if and only if this count matches the number of open-loop unstable poles — typically zero for stable plants. The distance from the Nyquist curve to the \(-1\) point is a direct measure of stability margin: the closer the curve comes to \(-1\), the less robust the system is to gain and phase variations.

The Nyquist plot compresses all frequency information into a single curve, making it less intuitive for reading bandwidth or roll-off than the Bode plot, but more powerful for stability analysis. The plot's behavior at infinity — how it approaches the origin as frequency increases — reveals the system's relative degree and high-frequency roll-off. Its behavior near the origin — low-frequency asymptotes — reveals the system's type (number of integrators) and steady-state error characteristics.

The Nyquist plot is the control theorist's Rorschach test. Stare at it long enough, and you see not just stability but robustness, performance, and the ghost of every unmodeled dynamic that might destabilize your loop. The Bode plot tells you what the system does at each frequency. The Nyquist plot tells you whether your design will survive contact with reality.