Nyquist Stability Criterion¶
Infer the number of unstable closed-loop poles from the winding of an open-loop frequency-response contour around the critical point, corrected by the known count of unstable open-loop poles.
Core Idea¶
For a single-loop linear time-invariant system with negative feedback and loop transfer function \(L(s)\), closed-loop poles are zeros of \(1+L(s)\). The Nyquist stability criterion maps a contour enclosing the unstable half-plane through \(L\) and uses the argument principle to infer how many zeros of \(1+L\) lie there from the winding of the image around the critical point \(-1\).
The winding count is combined with the known number of right-half-plane poles of \(L\). With orientation and sign conventions declared, closed-loop stability is exactly the condition that the resulting count of right-half-plane closed-loop poles is zero.
Scope of Application¶
Nyquist analysis applies to continuous-time LTI feedback systems, including rational plants, time-delay models, and open-loop unstable plants. Modified contours cover poles on the imaginary axis. Discrete-time versions use the unit circle after the appropriate mapping. Multivariable generalizations use determinants, characteristic loci, or related operator forms and require additional care.
The criterion answers stability; robustness design additionally interprets distance from the critical point, uncertainty sets, gain margin, and phase margin.
Clarity¶
State the feedback sign, contour orientation, definition of positive winding, inclusion of the infinite semicircle, treatment of boundary poles, and counts \(P\), \(N\), and \(Z\). Encirclement sign conventions differ across texts, so the argument-principle equation—not a memorized clockwise rule—must control the conclusion.
Manages Complexity¶
The method converts root-finding for a potentially high-order or delayed closed-loop characteristic equation into a topological count along a frequency contour. The winding number is stable under continuous deformations that do not cross the critical point, making qualitative robustness visible.
Abstract Reasoning¶
- Form the loop transfer function and verify feedback well-posedness.
- Count its poles in the unstable region.
- Construct the correct Nyquist contour, indenting boundary singularities.
- Map the complete oriented contour through \(L(s)\).
- Count oriented windings about the convention-specific critical point.
- Apply the argument principle to obtain the number of unstable closed-loop poles.
- Declare stability only when that number is zero and boundary poles are excluded.
- Read margins only after the stability count is correct.
Knowledge Transfer¶
The portable pattern is infer interior failures from boundary winding around a forbidden value, corrected for failures already present in the open system. The proposed immediate parent is Stability.
Relationships to Other Abstractions¶
Current abstraction Nyquist Stability Criterion Domain-specific
Parents (1) — more general patterns this builds on
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Nyquist Stability Criterion is a kind of Stability Prime
Stability is the proposed immediate parent.
Hierarchy path (1) — routes to 1 parentless root
- Nyquist Stability Criterion → Stability
Neighborhood in Abstraction Space¶
Nyquist Stability Criterion sits in a sparse region of the domain-specific corpus (96th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (1565 abstractions)
Nearest neighbors
- Closed-loop transfer function — 0.78
- H-infinity loop-shaping — 0.77
- Hurwitz-Stable Matrix — 0.76
- Redheffer Star Product — 0.76
- Full state feedback — 0.75
Computed from structural-signature embeddings · 2026-09-08