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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 certifies stability for a Lur'e feedback system: a linear time-invariant block interconnected with a memoryless nonlinearity whose input–output relation lies in a known sector. It extends frequency-domain reasoning beyond a single linear loop gain.

Sector endpoints determine a forbidden 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/pole conditions. Degenerate sectors and open-loop unstable plants need the appropriate generalized statement, not the elementary picture copied mechanically.

The result is sufficient and potentially conservative. A failed circle test does not prove instability. A valid report specifies equilibrium shift, plant transfer function, pole count, sector inequality and domain, memorylessness/time variation, loop transformation, frequency grid or analytic proof, strict margins, and whether the theorem establishes global or only local asymptotic stability.

How would you explain it like I'm…

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.

Structural Signature

Sig role-phrases:

  • LTI forward dynamics. Provides the transfer function and pole/analytic conditions used in the frequency test. Constitutive linear block. If altered: Unmodeled dynamics can invalidate the certificate.
  • memoryless nonlinearity. Maps feedback signal pointwise, possibly time-varying, under a declared sign convention. Constitutive nonlinear block. If altered: Dynamic hysteresis is outside the basic form.
  • sector bound. Constrains the nonlinear input–output slope/value between two lines over the claimed domain. Identity-bearing uncertainty set. If altered: Local and global sectors yield different conclusions.
  • Nyquist exclusion region. Maps sector endpoints into the circle/disk and imposes nonintersection plus encirclement conditions. Constitutive test. If altered: Formula depends on convention and special cases.
  • stability conclusion and scope. States global/local asymptotic stability, strictness, equilibrium, and robustness supported by the theorem. Necessary verdict. If altered: Sufficiency is not necessity.

What It Is Not

  • Not linear Nyquist alone. Sector uncertainty changes the exclusion geometry.
  • Not necessary for stability. Failure can reflect conservatism.
  • Not for arbitrary nonlinear memory. Basic theorem assumes pointwise sector behavior.
  • Not a describing function. It is a rigorous sufficient certificate under assumptions.

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.

  • 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.

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.

Abstract Reasoning

  1. Put the system in an explicit Lur'e feedback form.
  2. Prove the nonlinearity's sector and domain.
  3. Choose the theorem consistent with poles and sign convention.
  4. Construct the sector-dependent exclusion region and verify Nyquist conditions with margin.
  5. State the exact stability conclusion and residual model risk.

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.

Examples

Canonical

A stable plant with a static saturation is shifted to its equilibrium, the saturation is proved globally sector-bounded, and an analytic Nyquist argument shows strict separation from the corresponding disk, certifying global asymptotic stability.

Mapped back: LTI forward dynamics → declared stable transfer function; memoryless nonlinearity → static saturation; sector bound → proved global endpoints; Nyquist exclusion region → computed disk and strict separation; stability conclusion and scope → global asymptotic certificate.

Applied / In Practice

A servo design models actuator nonlinearity in a measured sector, evaluates the circle margin across parameter corners, and reports that an intersecting Nyquist locus means the test is inconclusive rather than that the prototype is unstable.

Mapped back: LTI forward dynamics → corner plant models; memoryless nonlinearity → bounded actuator map; sector bound → measured operating envelope; Nyquist exclusion region → corner-specific circle checks; stability conclusion and scope → certificate/inconclusive distinction.

Structural Tensions

T1: robust guarantee vs. conservatism. One sector covers many nonlinearities while ignoring helpful shape information. Diagnostic: Would a tighter sector or multiplier change the decision?

T2: frequency geometry vs. model fidelity. A plot simplifies proof while unmodeled dynamics and memory may dominate. Diagnostic: Does the physical nonlinearity satisfy the selected form?

T3: numerical plot vs. strict theorem. Dense sampling looks persuasive while missed crossings defeat proof. Diagnostic: What analytic or interval bound certifies exclusion?

Structural–Framed Character

The circle criterion is structural. Feedback decomposition, sector, Nyquist image, and exclusion define a mathematical certificate; engineering choices frame the model. Evaluative weight is low; institutional dependence is low; origin is control theory; vocabulary travels with equations; transfer recognizes the same theorem conditions. Its portable skeleton is Uncertainty-to-Exclusion Certificate, a prospective future-prime candidate. Its character: transform a bounded nonlinear family into a forbidden region for a linear frequency response.

Structural Core vs. Domain Accent

Skeletal core. Map an uncertainty set into an exclusion geometry and certify separation under topology/count conditions.

Domain-bound accent. LTI plant, Lur'e loop, sector nonlinearity, Nyquist locus, poles, and asymptotic stability define the criterion.

Why not prime. Exclusion certificates travel; circle criterion is a named control theorem.

  • Stability. Target property, not the certificate itself.
  • Feedback. Structural setting for the interconnection.

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

Not to Be Confused With

  • Nyquist criterion. Tell: Linear loop gain or sector-bounded nonlinear family?
  • Popov criterion. Tell: Circle exclusion or multiplier inequality?
  • Describing function. Tell: Rigorous guarantee or approximation?
  • Small-gain theorem. Tell: Sector geometry or norm bound?

References

  • Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Circle_criterion (revision 1361044933).
  • Preserved source candidate: https://web.archive.org/web/20110721081050/http://www.nt.ntnu.no/users/skoge/prost/proceedings/cdc03/pdffiles/papers/FrA02.1.pdf
  • Preserved source candidate: http://www-control.eng.cam.ac.uk/jmm/4f3/handout4.pdf
  • Preserved source candidate: http://reference.wolfram.com/mathematica/ref/FeedbackSector.html

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.