Hurwitz determinants
E714448
Hurwitz determinants are specific determinants constructed from a polynomial’s coefficients that are used to test whether all roots of the polynomial lie in the left half of the complex plane, thereby assessing system stability.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
determinant
ⓘ
mathematical concept ⓘ stability criterion component ⓘ tool in control theory ⓘ |
| advantage | avoid explicit computation of polynomial roots ⓘ |
| appliesTo |
characteristic polynomials of linear time-invariant systems
ⓘ
continuous-time linear systems ⓘ univariate polynomials with real coefficients ⓘ |
| assumes |
polynomial degree is finite
ⓘ
polynomial has real coefficients ⓘ |
| basedOn | coefficients of a polynomial ⓘ |
| condition | all leading principal minors of the Hurwitz matrix must be positive for stability ⓘ |
| constructedFrom | Hurwitz matrix of the polynomial NERFINISHED ⓘ |
| criterionFor |
all roots having negative real parts
ⓘ
location of polynomial zeros in the complex plane ⓘ |
| field |
algebra
ⓘ
applied mathematics ⓘ complex analysis ⓘ systems and control ⓘ |
| generalizationOf | stability tests based on principal minors ⓘ |
| influenced | development of algebraic stability criteria in control theory ⓘ |
| introducedIn | 19th century ⓘ |
| mathematicalNature | leading principal minors of a structured matrix ⓘ |
| namedAfter | Adolf Hurwitz NERFINISHED ⓘ |
| notTypicallyUsedFor | discrete-time stability in the z-plane ⓘ |
| propertyTested |
Hurwitz stability
ⓘ
location of roots relative to the imaginary axis ⓘ |
| purpose |
assess asymptotic stability of linear time-invariant systems
ⓘ
check Hurwitz stability of a polynomial ⓘ test whether all roots of a polynomial lie in the open left half-plane ⓘ |
| relatedTo |
Hurwitz matrix
ⓘ
Hurwitz polynomial NERFINISHED ⓘ Lyapunov stability NERFINISHED ⓘ Routh array NERFINISHED ⓘ characteristic equation ⓘ eigenvalues of system matrix ⓘ |
| requires | arranging polynomial coefficients in a specific structured matrix ⓘ |
| usedFor |
design and analysis of feedback control systems
ⓘ
stability margins assessment ⓘ verification of closed-loop stability without computing roots explicitly ⓘ |
| usedIn |
Hurwitz stability criterion
ⓘ
Routh–Hurwitz stability criterion NERFINISHED ⓘ automatic control ⓘ control theory ⓘ linear systems theory ⓘ signal processing ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.