small-gain theorem
E189566
The small-gain theorem is a fundamental result in control theory that provides a sufficient condition for the stability of feedback interconnections by requiring the product of system gains to be less than one.
All labels observed (4)
| Label | Occurrences |
|---|---|
| integral small-gain theorem | 1 |
| nonlinear small-gain theorem | 1 |
| small-gain theorem canonical | 1 |
| stochastic small-gain theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1675166 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: small-gain theorem Context triple: [Nyquist stability criterion, relatedConcept, small-gain theorem]
-
A.
Lyapunov stability theory
Lyapunov stability theory is a fundamental framework in dynamical systems and control theory that uses energy-like functions to assess the stability of equilibrium points without explicitly solving differential equations.
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B.
Ulam stability
Ulam stability is a concept in the theory of functional equations that studies when approximate solutions imply the existence of exact solutions nearby, forming the basis of what is now called Hyers–Ulam stability.
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C.
control theory
Control theory is a branch of engineering and mathematics that studies how to model, analyze, and design systems that regulate their own behavior using feedback.
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D.
Nyquist stability criterion
The Nyquist stability criterion is a graphical frequency-domain method in control theory used to determine the stability of feedback systems by analyzing how their open-loop transfer function encircles a critical point in the complex plane.
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E.
Poincaré–Bendixson theorem
The Poincaré–Bendixson theorem is a fundamental result in the qualitative theory of dynamical systems that characterizes the possible long-term behaviors of trajectories in two-dimensional continuous flows, ruling out chaotic dynamics in the plane.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: small-gain theorem Target entity description: The small-gain theorem is a fundamental result in control theory that provides a sufficient condition for the stability of feedback interconnections by requiring the product of system gains to be less than one.
-
A.
Lyapunov stability theory
Lyapunov stability theory is a fundamental framework in dynamical systems and control theory that uses energy-like functions to assess the stability of equilibrium points without explicitly solving differential equations.
-
B.
Ulam stability
Ulam stability is a concept in the theory of functional equations that studies when approximate solutions imply the existence of exact solutions nearby, forming the basis of what is now called Hyers–Ulam stability.
-
C.
control theory
Control theory is a branch of engineering and mathematics that studies how to model, analyze, and design systems that regulate their own behavior using feedback.
-
D.
Nyquist stability criterion
The Nyquist stability criterion is a graphical frequency-domain method in control theory used to determine the stability of feedback systems by analyzing how their open-loop transfer function encircles a critical point in the complex plane.
-
E.
Poincaré–Bendixson theorem
The Poincaré–Bendixson theorem is a fundamental result in the qualitative theory of dynamical systems that characterizes the possible long-term behaviors of trajectories in two-dimensional continuous flows, ruling out chaotic dynamics in the plane.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
result in control theory
ⓘ
theorem ⓘ |
| appliesTo |
LTI systems
ⓘ
feedback interconnections ⓘ input-output stable systems ⓘ nonlinear systems ⓘ time-varying systems ⓘ |
| appliesToConfiguration |
feedback loop
ⓘ
interconnection of two stable systems ⓘ |
| assumes |
bounded-input bounded-output stability of components
ⓘ
causal systems ⓘ well-defined input-output operators ⓘ |
| basedOn |
gain of systems
ⓘ
input-output gain ⓘ operator norm ⓘ |
| concernedWith |
feedback stability
ⓘ
robust stability ⓘ stability ⓘ |
| conditionType | sufficient condition ⓘ |
| coreRequirement |
loop gain less than one
ⓘ
product of gains less than one ⓘ |
| ensures |
BIBO stability of closed-loop system
ⓘ
robustness to model uncertainty ⓘ |
| field |
control theory
ⓘ
systems theory ⓘ |
| framework |
input-output framework
ⓘ
operator-theoretic approach ⓘ |
| generalizes | classical gain margin ideas ⓘ |
| gives | sufficient condition for stability ⓘ |
| hasVariant |
small-gain theorem
self-linksurface differs
ⓘ
surface form:
integral small-gain theorem
small-gain theorem self-linksurface differs ⓘ
surface form:
nonlinear small-gain theorem
small-gain theorem self-linksurface differs ⓘ
surface form:
stochastic small-gain theorem
|
| relatedTo |
Nyquist stability criterion
ⓘ
circle criterion ⓘ input-output stability theory ⓘ passivity theorem ⓘ |
| typicalFormulation |
norm of closed-loop operator less than one
ⓘ
||G1||·||G2|| < 1 implies stability ⓘ |
| typicalNorm |
H-infinity norm
ⓘ
L2-induced norm ⓘ L∞-induced norm ⓘ |
| usedFor |
compositional stability analysis
ⓘ
interconnected ISS systems analysis ⓘ |
| usedIn |
H-infinity control
ⓘ
large-scale interconnected systems ⓘ networked control systems ⓘ passivity-based control analysis ⓘ robust control design ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: small-gain theorem Description of subject: The small-gain theorem is a fundamental result in control theory that provides a sufficient condition for the stability of feedback interconnections by requiring the product of system gains to be less than one.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
nonlinear small-gain theorem
this entity surface form:
integral small-gain theorem
this entity surface form:
stochastic small-gain theorem