Nyquistplott

Nyquistplott, commonly known as the Nyquist plot, is a graphical method used in control theory to assess the stability of feedback systems. It plots the complex values of the open-loop transfer function L(jω) = G(jω)H(jω) as frequency ω runs from -∞ to ∞ (or equivalently from 0 to ∞ with the corresponding complex conjugate for negative frequencies). The resulting locus in the complex plane conveys how the system’s gain and phase vary with frequency.

Construction and interpretation: To build the plot, evaluate L(jω) along the imaginary axis. If the open-loop transfer function L(s) has poles in the right-half plane, the Nyquist contour must encircle those poles, and their contribution is added to the plot. The critical reference point is −1 in the complex plane, which is where the closed-loop characteristic equation 1 + L(s) = 0 is satisfied for a unit feedback system.

Stability criterion: The Nyquist stability criterion relates the number of encirclements N of the point −1 by the Nyquist plot to P, the number of L(s) poles in the right-half plane, and Z, the number of closed-loop poles (zeros of 1 + L(s)) in the right-half plane. The relation is Z = N + P. In particular, if P = 0 and the plot does not encircle −1, the closed loop is stable (Z = 0). The criterion extends to systems with RHP poles by accounting for their contributions.

Margins and applications: From the plot one can extract gain margin and phase margin, indicating how much gain or phase can change before the system becomes unstable. Nyquist plots are widely used for stability analysis and robustness checks, especially when frequency-domain data is available or when time-domain methods are impractical.