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Circle criterion

A frequency-domain sufficient condition for absolute stability of a feedback interconnection between a linear time-invariant plant and a memoryless nonlinearity constrained to a sector, expressed as a Nyquist-plot exclusion of a sector-dependent circle or disk.

Version
v1 · 2026-09-28 · History
Domain-specific #
8459
Domain group
Applied Sciences & Engineering
Origin domain
Robotics & Automation
Subdomains
Nonlinear Control Theory, Absolute Stability → Robotics & Automation

Core Idea

The circle criterion is a frequency-domain sufficient condition for absolute stability of a feedback interconnection between an LTI plant and a memoryless nonlinearity constrained to a sector. The sector determines a forbidden circle or disk that the plant's Nyquist locus must avoid under the theorem's pole and encirclement conditions. Sector endpoints determine a forbidden circle or disk in the complex plane.

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Stay-Out-of-the-Circle Test

Engineers build machines that keep checking themselves and fixing their own mistakes. Sometimes one part is a bit unpredictable, but they know it never pushes too hard or too soft. The circle criterion is a safety check: they draw a special picture of the machine, and if the picture stays out of a 'no-go' circle, the machine is sure to settle down calmly. If the picture touches the circle, the machine might still be fine; the check just can't promise it.

The Forbidden Circle Test

Many machines use feedback: they measure what they're doing and correct themselves, like a thermostat. Sometimes the loop has a well-understood steady part plus a tricky part that doesn't act in a simple straight-line way. Engineers may not know the tricky part exactly, only that its output always stays between two limits. The circle criterion turns those two limits into a forbidden circle on a special graph. If the graph of the steady part keeps clear of that circle, in the right way, the whole system is guaranteed to calm down. If it doesn't keep clear, that doesn't prove the system will go wild; the test just gives no guarantee.

Sector-Bounded Stability Test

The circle criterion is a stability test for a specific kind of feedback loop, called a Lur'e system: a linear time-invariant part connected in a loop with a nonlinear part that has no memory and whose output always stays within a known 'sector,' meaning between two straight lines through the origin. The two sector edges define a forbidden circle (or disk) in the complex plane. You then plot the linear part's Nyquist curve, its response across all frequencies; if the curve stays out of that disk and meets the required encirclement conditions, the loop is guaranteed to be stable. It extends ordinary frequency-domain stability reasoning beyond a single linear gain. The test is sufficient but not necessary: failing it doesn't prove the system is unstable. Special cases, like a sector with an infinite edge or an unstable open-loop plant, need the proper general version of the theorem rather than the simple picture.

 

The circle criterion certifies stability of a Lur'e feedback system, in which a linear time-invariant plant is interconnected with a memoryless nonlinearity whose input-output graph lies in a known sector [k1, k2]. It extends Nyquist-style frequency-domain reasoning from a single linear loop gain to this whole class of nonlinearities. The sector endpoints define a critical circle or disk in the complex plane; under the chosen sign convention and theorem form, the plant's Nyquist locus must avoid that region and satisfy the required encirclement and open-loop pole conditions. Degenerate sectors (for example, an endpoint of zero) and open-loop unstable plants require the appropriate generalized statement rather than mechanically copying the elementary picture. The result is sufficient only and can be conservative: a failed test does not demonstrate instability. A sound application reports the equilibrium shift, plant transfer function, pole count, sector inequality and its domain, whether the nonlinearity is memoryless or time-varying, any loop transformation, the frequency grid or analytic proof used, strict margins, and whether global or only local asymptotic stability is established.

Scope of Application

The criterion is used in nonlinear control, saturation analysis, actuator limits, absolute stability, robust feedback design, adaptive/nonlinear systems teaching, and passivity-related analysis. Use it with equilibrium and loop sign, LTI transfer function/realization/poles, well-posedness, nonlinearity inputs/outputs and memorylessness, sector inequality/endpoints/domain and ordinary versus incremental form, loop transformation, exact theorem variant and exclusion formula, Nyquist contour/encirclement/resolution, strict margin, stability type/region, and model uncertainty. A failed test is inconclusive, not proof of instability; distinguish it from linear Nyquist, Popov, describing-function, and small-gain analyses.

  • Saturation. Bounds static actuator maps.
  • Robust design. Certifies a sector family.
  • Analysis. Uses transfer-function geometry.
  • Teaching. Connects Nyquist and nonlinear stability.
  • Margins. Quantifies distance to forbidden region.

Clarity

Report state/equilibrium and loop sign, LTI realization/transfer function and poles, well-posedness, nonlinearity input/output and memoryless/time-varying status, sector inequality/endpoints/domain and incremental versus ordinary form, loop transformation, exact criterion variant, forbidden circle/disk formula, Nyquist contour/encirclement and numerical resolution, strict margin, stability type/region, uncertainty/model limits, and comparison with Popov, passivity, small-gain, or simulation. The closest near miss sets the boundary: The Popov criterion is nearest: it can certify some sector systems with a different frequency inequality and multiplier.

Manages Complexity

The criterion compresses an infinite family of nonlinear maps into a sector and a geometric frequency exclusion, gaining tractability at the cost of conservatism and strict modeling assumptions. The central robust guarantee–conservatism tradeoff is this: One sector covers many nonlinearities while ignoring helpful shape information. A second frequency geometry–model fidelity tension matters because A plot simplifies proof while unmodeled dynamics and memory may dominate.

Abstract Reasoning

Use three linked moves: put the system in an explicit Lur'e feedback form; prove the nonlinearity's sector and domain; choose the theorem consistent with poles and sign convention. As a collapse test, the certificate cannot be claimed if nonlinearity memory, sector domain, sign convention, open-loop poles, or strictness conditions depart from the selected theorem. A fourth check is to construct the sector-dependent exclusion region and verify Nyquist conditions with margin.

Knowledge Transfer

The sector-exclusion pattern transfers to other feedback uncertainties only with newly justified multipliers, supply rates, or dynamic bounds; the basic circle cannot be copied to hysteretic or dynamic nonlinearities. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. Target property, not the certificate itself.

Neighborhood in Abstraction Space

Circle criterion sits in a sparse region of the domain-specific corpus (69th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Dynamical Systems & Differential Structures (37 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08