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 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.
Hurwitz stable matrices
In engineering and stability theory, a square matrix is called stable matrix (or sometimes Hurwitz matrix) if every eigenvalue of has strictly negative real part, that is,
for each eigenvalue . is also called a stability 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 poles of all elements of have negative real part. Note that it is not necessary that for a specific argument be a Hurwitz matrix — it need not even be square. The connection is that if is a Hurwitz matrix, then the dynamical system
has a Hurwitz transfer function.
Any hyperbolic fixed point (or equilibrium point) of a continuous dynamical system is locally asymptotically stable if and only if the Jacobian of the dynamical system is Hurwitz stable at the fixed point.
The Hurwitz stability matrix is in crucial part on 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
References
- Hurwitz, A. (1895). "Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Teilen besitzt". Mathematische Annalen, Leipzig (Nr. 46): 273–284.
- Gantmacher, F.R. (1959). "Applications of the Theory of Matrices". Interscience, New York. 641 (9): 1–8.
- Hassan K. Khalil (2002). Nonlinear Systems. Prentice Hall.
- Siegfried H. Lehnigk, On the Hurwitz matrix, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), May 1970
- Bernard A. Asner, Jr., On the Total Nonnegativity of the Hurwitz Matrix, SIAM Journal on Applied Mathematics, Vol. 18, No. 2 (Mar., 1970)
- Dimitar K. Dimitrov and Juan Manuel Peña, Almost strict total positivity and a class of Hurwitz polynomials, Journal of Approximation Theory, Volume 132, Issue 2 (February 2005)
This article incorporates material from Hurwitz matrix on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.