A frequency‑domain test for linear time‑invariant (LTI) control systems that determines the number of closed‑loop poles in the right‑half complex plane by mapping the open‑loop transfer function evaluated on a Nyquist contour into the complex plane and counting encirclements of the critical point −1.

Principle

The count N of clockwise encirclements of −1 by the open‑loop Nyquist plot, together with the number P of open‑loop poles in the right‑half plane (RHP), gives the number Z of closed‑loop RHP poles via Z = P + N; closed‑loop stability requires Z = 0.

Demonstration

Illustrative scenario: An LTI single‑input single‑output (SISO) plant has no open‑loop RHP poles (P = 0). If the Nyquist plot of L(jω) for ω from 0 to ∞ (with the standard contour accounting for infinity and negative frequencies) does not encircle −1 (N = 0), then Z = 0 and the closed‑loop system is stable. If the plot encircles −1 clockwise once (N = 1), then Z = 1 and the closed‑loop is unstable.

Misapplication

Counting encirclements from a plotted frequency response without accounting for open‑loop RHP poles, poles on the imaginary axis, or the correct orientation of the Nyquist contour; or applying the criterion to nonlinear or time‑varying systems without appropriate linearization.

Consequence

Provides a practical method to assess closed‑loop stability from open‑loop frequency response data without explicitly computing closed‑loop characteristic roots; it also underlies design measures for gain and phase margins.

Reversal

The criterion requires an LTI, rational transfer function and a contour that excludes poles on the Nyquist path; when the open‑loop has poles on the imaginary axis or for nonlinear/time‑varying systems, modifications or alternative methods (e.g., root‑locus, Lyapunov) are required.

Boundary

Clearly within: SISO LTI systems with rational transfer functions and no imaginary‑axis poles. Boundary case: open‑loop has simple poles on jω axis — requires indentation of contour and special counting. Clearly outside: inherently nonlinear systems or stochastic/time‑varying systems without linearization. Discrete‑time systems require the z‑plane variant using the unit circle.

Semantic Tension

Trade‑off between frequency‑domain (Nyquist/Bode) and time‑domain (root‑locus, state‑space) stability methods: frequency methods use measured or modelled response directly but require contour assumptions; time‑domain methods work from state matrices but need accurate models.

Synthesis

Nyquist stability reduces closed‑loop pole counting to a topological mapping of the open‑loop frequency response: stability is determined by how the open‑loop plot winds around −1 after accounting for open‑loop RHP poles, making frequency‑response measurements directly relevant to stability conclusions.