Nyquistkriterium

NyquistKriterium, known in English as the Nyquist criterion, is a stability test used in control theory to assess the closed-loop stability of systems with unity feedback. It relates the stability of the closed-loop system to the frequency response of the open-loop transfer function L(s) = G(s)H(s).

The core idea is that the closed-loop poles are the solutions of 1 + L(s) = 0. To apply the criterion, one examines the Nyquist plot, which traces L(jω) as ω runs from −∞ to ∞, typically with a contour that loops around the right-half of the complex plane to avoid any poles on the imaginary axis. The key geometric object is the point −1 in the complex plane, the critical point.

Let P denote the number of open-loop poles of L(s) in the right half-plane, and let N denote the number of times the Nyquist plot encircles the point −1. If Z is the number of closed-loop poles in the right half-plane, the Nyquist criterion states Z = N − P. Consequently, for a stable closed loop (Z = 0), the Nyquist plot must encircle −1 exactly P times. In the common case where the open-loop transfer function has no right-half-plane poles (P = 0), the criterion reduces to requiring that the Nyquist plot of L(jω) does not encircle −1.

Extensions of the criterion apply to multi-input multi-output systems through the loop transfer matrix and eigenvalue encirclement considerations. The Nyquist criterion is widely used to assess gain and phase margins and to analyze robustness against model uncertainties, while noting limitations such as poles on the imaginary axis or nonminimum-phase behavior.