##  [Nyquist Stability Criterion](/nyquist-stability-criterion-0) 

 Definition

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

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

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

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

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

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

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

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

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.