AI Glossary / Reinforcement learning and robotics / Advanced
NYQUIST PLOT
Definition
A Nyquist plot traces a system's complex frequency response as frequency varies. In feedback control, it commonly displays the loop transfer function and supports assessment of closed-loop stability through the Nyquist criterion. Its horizontal axis is the real component and its vertical axis is the imaginary component; frequency is followed along the curve rather than shown as a separate axis.
What it means
A controller can correct an error yet still produce oscillation or instability if its response arrives with an unsuitable combination of strength and phase delay. The Nyquist plot helps engineers examine that combination across frequencies.
For the usual negative-feedback convention, the critical location is the point −1 on the real axis. The plot's relationship to this point matters, but an image alone is insufficient: engineers also need the loop's unstable-pole count and a clear convention for following the contour. Stability is one design requirement, not proof that a robot performs its task well.
How it works
For a continuous-time linear model, let L(s) represent the transfer function around the feedback loop. Evaluate L(jω), where j is the imaginary unit, and plot the resulting complex numbers over the relevant frequency range. Real-coefficient models produce conjugate symmetry between positive and negative frequencies.
The closed-loop characteristic equation under negative feedback is 1 + L(s) = 0. Nyquist analysis relates how the complete mapped contour surrounds −1 to the number of unstable closed-loop poles, accounting for unstable poles already present in L.
With the conventional clockwise traversal around the right-half-plane region and clockwise encirclements counted positive, Z = P + N. Here P counts enclosed open-loop poles, N is the signed encirclement count, and Z counts enclosed zeros of 1 + L. Different sign conventions must be translated consistently.
Poles on the imaginary axis require an appropriately indented contour. A few sampled frequency points cannot establish the full encirclement count, and unmodeled dynamics or delay can change the conclusion.
Three practical examples
- Joint control: examine whether a proposed arm controller remains stable around a modeled operating point.
- Motor speed regulation: compare controller gains while considering the frequency response of the motor and sensing path.
- Remote operation: investigate how communication delay changes the phase of a feedback loop and reduces the room available for stable control.
Think of it this way
Think of walking around a landmark while drawing your route on a map. To say how many times you circled the landmark, you need the whole route and its direction, not just a few positions. In a Nyquist plot, the landmark is the critical complex point and the route represents changing frequency, not physical movement.
Worked example
Consider the deliberately simple loop L(s) = 4/(s + 2), with unity negative feedback and no delay. Its only open-loop pole is s = −2, so P = 0. Substituting s = jω gives L(jω) = 8/(4 + ω²) − j4ω/(4 + ω²).
At ω = 0, the plotted point is 2 + j0. At ω = 2 radians per second, it is 1 − j1. At ω = 4, it is 0.4 − j0.8. As positive frequency increases without bound, the curve approaches the origin from below. Negative frequencies give the mirror image above the real axis.
Writing the coordinates as x and y, they satisfy (x − 1)² + y² = 1. Thus the full frequency-response locus is a circle centered at (1, 0) with radius 1. It does not surround −1, giving N = 0 and Z = 0 for the complete Nyquist contour in this strictly proper example.
A separate algebraic check gives the closed-loop transfer function L/(1 + L) = 4/(s + 6), whose pole is −6. This confirms stability for the stated model. Its steady-state gain is 4/6, or two-thirds: a stable system can still have tracking error. The numerical points illustrate the curve, while the circle equation establishes its shape.
Illustrative case study
Fictional case: BirchMotion changes the sampling and communication setup for a positioning stage. A controller that appeared satisfactory in an earlier model now produces visible oscillation. Engineers update the loop model to include the changed delay, examine the frequency response, and compare the result with measured behavior.
They also test tracking error, actuator limits, and operation across relevant loads. This fictional case shows why a stability plot should be connected to the actual implementation and operating conditions. It does not claim that any particular controller adjustment guarantees a successful deployment.
Common misconceptions and mistakes
A Nyquist plot is not a plot of position against time, and its imaginary axis does not represent an imaginary physical motion. Avoid concluding that every loop with no encirclements is stable: the open-loop unstable-pole count matters. Passing through −1 is a critical boundary case, not comfortable stability. The Nyquist plot is also distinct from the Nyquist sampling frequency, despite the shared name.
Business use
For robotics and automation teams, this analysis can inform controller review, commissioning, and investigation of oscillations. Ask which loop is plotted, which operating point is modeled, and whether delays and important dynamics are included. Review robustness and performance alongside nominal stability. A claim that a system is stable should identify the model and assumptions supporting that claim, rather than relying on an unlabeled curve.
People also ask
Does it require AI?
No. It is a classical control tool that may be used within an AI-enabled robotic system.
Is it the same as a Bode plot?
No. Bode plots display magnitude and phase separately against frequency; Nyquist plots combine them in the complex plane.
Can it analyze every nonlinear behavior directly?
No. Linear analysis may apply locally and needs appropriate interpretation.
Check your understanding
Where is the critical point for the stated negative-feedback convention?
Answer: At −1 + j0.
What is P in the worked example?
Answer: Zero, because the open-loop pole is in the left half-plane.
Does the example track a constant reference perfectly?
Answer: No. Its closed-loop steady-state gain is two-thirds despite stability.
Related terms
- Transfer function
- Stability
- Feedback control
- frequency response
- complex number
- Bode plot
- gain margin
- phase margin
- time delay
- linearization
Learn next
Next, study how gain and phase changes move a loop's frequency response. Then compare nominal stability with robustness to modeling errors and delayed measurements.
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Sources
IEEE ROBOTS, Robotics Glossary, Nyquist Plot: https://robotsguide.com/learn/robotics-glossary MIT, Nyquist Criterion applet documentation, loop definition and frequency response: https://web.mit.edu/jorloff/www/jmoapplets/nyquist/nyquistCrit.html Joel L. Dawson, MIT 6.302 Recitation 10, Nyquist Stability, contour conventions and pole accounting: https://www.mit.edu/~6.302/notes/6.302rec10.pdf
Independent review checked clockwise encirclement and pole-count conventions.
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