Nyquist stability criterion

The Nyquist stability criterion determines closed-loop stability by relating a system’s open-loop frequency response to the locations of its unstable poles and the encirclements of a critical point.

Nyquist plot: A plot of a transfer function’s complex values as frequency varies along a specified contour. The criterion reads stability from this mapped contour’s encirclements of the critical point.
Feedback: A process that returns part of a system’s output to influence its input. The criterion analyzes stability specifically in a feedback loop.
Gain margin: The multiplicative gain change required to bring a feedback system to the stability boundary. It can be read from how the Nyquist plot approaches the critical point.
Routh–Hurwitz stability criterion: An algebraic test that determines whether a polynomial has roots in the right half-plane. It tests characteristic-polynomial stability without requiring a frequency-response contour.
Argument principle: A complex-analysis theorem relating contour integrals to zeros and poles enclosed by a contour. It mathematically justifies counting closed-loop poles through encirclements.
Transfer function: A ratio of output to input transforms that describes a linear time-invariant system. The open-loop response is commonly represented as a transfer function.
Phase margin: The additional phase lag required at unity gain to reach the stability boundary. It quantifies angular separation from the critical point at unity loop gain.
Root locus: A plot of closed-loop pole locations as a feedback parameter varies. It tracks pole movement directly, while Nyquist counts unstable poles from open-loop data.
Open-loop transfer function: A transfer function describing a system before its feedback path is closed. Its frequency response is the input from which the Nyquist contour is constructed.
Pole–zero analysis: A method for studying system behavior through the poles and zeros of its transfer function. The Nyquist count is ultimately a statement about closed-loop poles.

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