NyquistPlot

The Nyquist plot is a graphical method in control theory that represents the complex open-loop transfer function L(jω) = G(jω)H(jω) as frequency ω varies. It plots the locus of L(jω) in the complex plane, providing a compact view of both magnitude and phase response. For systems with real coefficients, L(−jω) is the complex conjugate of L(jω), so the plot for ω ≥ 0 captures the essential information, with symmetry for negative frequencies. When poles lie on the imaginary axis or delays are present, the frequency path must be deformed in the complex plane to avoid singularities.

The Nyquist plot is used in conjunction with the Nyquist stability criterion to assess the stability of unity feedback systems. By observing how the plot encircles the critical point −1 in the complex plane and accounting for any open-loop poles in the right half-plane, one can determine the number of unstable closed-loop poles. If the open-loop transfer function L(s) has no right-half-plane poles and the plot does not encircle −1, the closed-loop system is stable. In general, the number of encirclements of −1 relates to the number of unstable closed-loop poles.

Nyquist plots are widely employed to evaluate gain and phase margins and robustness. They are implemented in many software tools and can be constructed from a transfer function or measured frequency response data. Extensions exist for multi-input, multi-output systems and for more general stability analyses.