A frequency-domain stability test for closed-loop feedback systems which maps the open-loop transfer function around a contour encircling the right half-plane and counts net encirclements of the critical point −1 to determine the number of closed-loop poles in the right half-plane, taking into account poles of the open-loop in the right half-plane via the argument principle.

Principle

Argument-principle mapping: the net number of clockwise encirclements of −1 by the Nyquist plot of the open-loop transfer function equals the number of open-loop RHP poles minus the number of closed-loop RHP poles, so counting encirclements yields closed-loop stability information.

Demonstration

For a SISO linear feedback loop with open-loop L(s) = G(s)H(s), plot L(iω) for ω from −∞ to +∞ (with appropriate detours for poles on the axis) and count encirclements of −1; if encirclements match the RHP pole count so that closed-loop RHP poles are zero, the closed-loop is stable. Illustrative scenario: assessing stability and gain/phase margins when tuning a controller.

Misapplication

Neglecting to include open-loop right-half-plane poles in the encirclement accounting, using an incorrect contour or orientation, or failing to deform the contour correctly around branch points or time-delay singularities leads to wrong stability conclusions.

Consequence

Enables determination of closed-loop stability and computation of robustness margins (gain and phase margins) from frequency response data, and guides controller design with direct insight into how loop shape affects pole placement.

Reversal

One can invert the reasoning to design open-loop shapes that produce desired encirclement counts, but misinterpreting encirclements as direct pole locations (instead of counts relative to RHP poles) reverses the logic and yields errors.

Boundary

Applies primarily to linear time-invariant SISO systems and to rational transfer functions possibly extended to systems with delays after careful treatment; it requires a proper understanding of branch cuts, pole-zero cancellations, and multiplicities and is less direct for MIMO systems where generalized Nyquist approaches are needed.

Semantic Tension

Tightly connected to Bode, root-locus, and time-domain pole tests: the Nyquist method emphasizes global frequency-shape encirclement counts whereas Bode gives local margin measures and root-locus shows parametric pole trajectories; tension is practical: which method offers the clearest design insight for a given problem.

Synthesis

The Nyquist Stability Criterion translates closed-loop pole-counting into a frequency-domain encirclement problem using the argument principle: by plotting the open-loop response around the Nyquist contour and correctly accounting for open-loop RHP poles and contour deformations, one deduces closed-loop stability and robustness margins, with careful attention to contour choices and system class limitations.