Hurwitz matrix

Summary

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 stable matrix (or sometimes a 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 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

References

  • Asner, Bernard A., Jr. (1970). "On the Total Nonnegativity of the Hurwitz Matrix". SIAM Journal on Applied Mathematics. 18 (2): 407–414. doi:10.1137/0118035. JSTOR 2099475.
  • Dimitrov, Dimitar K.; Peña, Juan Manuel (2005). "Almost strict total positivity and a class of Hurwitz polynomials". Journal of Approximation Theory. 132 (2): 212–223. doi:10.1016/j.jat.2004.10.010.
  • 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". Mathematische Annalen. 46 (2): 273–284. doi:10.1007/BF01446812.
  • Khalil, Hassan K. (2002). Nonlinear Systems. Prentice Hall.
  • Lehnigk, Siegfried H. (1970). "On the Hurwitz matrix". Zeitschrift für Angewandte Mathematik und Physik. 21 (3): 498–500. Bibcode:1970ZaMP...21..498L. doi:10.1007/BF01627957.

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