Routh–Hurwitz stability criterion
E189565
The Routh–Hurwitz stability criterion is a mathematical test in control theory that determines whether all roots of a system’s characteristic polynomial lie in the left half of the complex plane, ensuring system stability without explicitly computing the roots.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Hurwitz polynomials | 1 |
| Jury stability criterion | 1 |
| Routh array | 1 |
| Routh–Hurwitz stability criterion canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1675143 — 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: Routh–Hurwitz stability criterion Context triple: [Nyquist stability criterion, comparedWith, Routh–Hurwitz stability criterion]
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A.
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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B.
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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C.
Lyapunov equation
The Lyapunov equation is a fundamental matrix equation in control theory and dynamical systems used to analyze the stability of equilibrium points and design stable controllers.
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D.
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.
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E.
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.
- 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: Routh–Hurwitz stability criterion Target entity description: The Routh–Hurwitz stability criterion is a mathematical test in control theory that determines whether all roots of a system’s characteristic polynomial lie in the left half of the complex plane, ensuring system stability without explicitly computing the roots.
-
A.
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.
-
B.
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.
-
C.
Lyapunov equation
The Lyapunov equation is a fundamental matrix equation in control theory and dynamical systems used to analyze the stability of equilibrium points and design stable controllers.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
control theory concept
ⓘ
mathematical test ⓘ stability criterion ⓘ |
| appliesTo |
characteristic polynomials
ⓘ
linear time-invariant systems ⓘ |
| appliesToPolynomialsOf |
finite degree
ⓘ
real coefficients ⓘ |
| assumes |
linear system dynamics
ⓘ
time-invariant system parameters ⓘ |
| basedOn | location of polynomial roots in the complex plane ⓘ |
| canDetect | marginal stability through zero entries in the Routh array ⓘ |
| canIndicate | oscillatory behavior when roots lie on the imaginary axis ⓘ |
| characterizes |
Routh–Hurwitz stability criterion
self-linksurface differs
ⓘ
surface form:
Hurwitz polynomials
|
| conditionForStability |
all coefficients of the characteristic polynomial must be positive
ⓘ
all leading principal minors of the Hurwitz matrix must be positive ⓘ no sign changes in the first column of the Routh array ⓘ |
| developedBy |
Adolf Hurwitz
ⓘ
Edward Routh ⓘ
surface form:
Edward John Routh
|
| doesNotRequire | explicit computation of polynomial roots ⓘ |
| ensures | asymptotic stability of a linear time-invariant system ⓘ |
| field |
applied mathematics
ⓘ
control theory ⓘ systems theory ⓘ |
| goal |
determine stability of a linear system
ⓘ
determine whether all roots of a polynomial lie in the left half-plane ⓘ |
| historicalPeriod | late 19th century ⓘ |
| input | characteristic polynomial of the closed-loop system ⓘ |
| mathematicalDomain |
complex analysis
ⓘ
polynomial theory ⓘ |
| namedAfter |
Adolf Hurwitz
ⓘ
Edward Routh ⓘ
surface form:
Edward John Routh
|
| output |
number of roots in the right half of the complex plane
ⓘ
stability or instability decision ⓘ |
| relatedConcept |
Hurwitz matrix
ⓘ
Jury test ⓘ
surface form:
Jury stability criterion
Lyapunov stability ⓘ Nyquist stability criterion ⓘ root locus method ⓘ |
| stabilityDefinition | all roots have strictly negative real parts ⓘ |
| typicalUse |
analysis of feedback control systems
ⓘ
design of stable controllers ⓘ stability assessment of differential equation models ⓘ |
| usedIn |
aerospace engineering
ⓘ
chemical process control ⓘ classical control theory ⓘ electrical engineering ⓘ mechanical engineering ⓘ |
| uses |
Hurwitz determinants
ⓘ
Routh–Hurwitz stability criterion self-linksurface differs ⓘ
surface form:
Routh array
|
How these facts were elicited
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Input
Subject: Routh–Hurwitz stability criterion Description of subject: The Routh–Hurwitz stability criterion is a mathematical test in control theory that determines whether all roots of a system’s characteristic polynomial lie in the left half of the complex plane, ensuring system stability without explicitly computing the roots.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Jury stability criterion
Routh–Hurwitz stability criterion
→
uses
→
Routh–Hurwitz stability criterion
self-linksurface differs
ⓘ
this entity surface form:
Routh array
Routh–Hurwitz stability criterion
→
characterizes
→
Routh–Hurwitz stability criterion
self-linksurface differs
ⓘ
this entity surface form:
Hurwitz polynomials