Hurwitz matrix
In mathematics, a Hurwitz matrix, or Routh–Hurwitz matrix, in engineering stability matrix, is a structured real square matrix constructed with coefficients of a real polynomial.
Hurwitz matrix and the Hurwitz stability criterion
Namely, given a real polynomial
the square matrix
is called Hurwitz matrix corresponding to the polynomial . It was established by Adolf Hurwitz in 1895 that a real polynomial with is stable (that is, all its roots have strictly negative real part) if and only if all the leading principal minors of the matrix are positive:
and so on. The minors are called the Hurwitz determinants. Similarly, if then the polynomial is stable if and only if the principal minors have alternating signs starting with a negative one.
Hurwitz stable matrices
In engineering and stability theory, a square matrix is called a Hurwitz matrix if every
for each eigenvalue . is also called a stable matrix, because then the differential equation
is asymptotically stable, that is, as
If is a (matrix-valued) transfer function, then is called Hurwitz if the
has a Hurwitz transfer function.
Any hyperbolic
The Hurwitz stability matrix is a crucial part of control theory. A system is stable if its control matrix is a Hurwitz matrix. The negative real components of the eigenvalues of the matrix represent negative feedback. Similarly, a system is inherently unstable if any of the eigenvalues have positive real components, representing positive feedback.
See also
- Liénard–Chipart criterion
- M-matrix
- P-matrix
- Perron–Frobenius theorem
- Z-matrix
- Jury stability criterion, for the analogue criterion for discrete-time systems.
References
- Asner, Bernard A. Jr. (1970). "On the Total Nonnegativity of the Hurwitz Matrix". JSTOR 2099475.
- Dimitrov, Dimitar K.; Peña, Juan Manuel (2005). "Almost strict total positivity and a class of Hurwitz polynomials". hdl:11449/21728.
- Gantmacher, F. R. (1959). Applications of the Theory of Matrices. New York: Interscience.
- Hurwitz, A. (1895). "Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Teilen besitzt". S2CID 121036103.
- Khalil, Hassan K. (2002). Nonlinear Systems. Prentice Hall.
- Lehnigk, Siegfried H. (1970). "On the Hurwitz matrix". S2CID 123380473.
This article incorporates material from Hurwitz matrix on