Hurwitz matrix

In mathematics, a square matrix $A$ is called a Hurwitz matrix if every eigenvalue of $A$ has strictly negative real part, that is,: for each eigenvalue $lambda\_i$. $A$ is also called a stability matrix, because then the differential equation:$dot\; x\; =\; A\; x$is stable , that is, $x(t)\; o\; 0$ as $t\; oinfty.$

If $G(s)$ is a (matrix-valued) transfer function, then $G$ is called Hurwitz if the poles of all elements of $G$ have negative real part. Note that it is not necessary that $G(s),$ for a specific argument $s,$ be a Hurwitz matrix — it need not even be square. The connection is that if $A$ is a Hurwitz matrix, then the dynamical system:$dot\; x(t)=A\; x(t)\; +\; B\; u(t)$ : $y(t)=C\; x(t)\; +\; D\; u(t),$ 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.

References

* Hassan K. Khalil (2002). "Nonlinear Systems". Prentice Hall.
* Siegfried H. Lehnigk, [http://www.springerlink.com/content/h192106tq8nl2274/ "On the Hurwitz matrix"] , "Zeitschrift für Angewandte Mathematik und Physik (ZAMP)", May 1970
* [http://www.springerlink.com/content/n58t70x478p843p1/ "Hurwitz-Radon matrices revisited: From effective solution of the Hurwitz matrix equations to Bott periodicity"] , in "Mathematical Survey Lectures 1943–2004", Springer Berlin Heidelberg, 2006
* 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, [http://portal.acm.org/citation.cfm?id=1063186.1063190 "Almost strict total positivity and a class of Hurwitz polynomials"] , Journal of Approximation Theory, Volume 132, Issue 2 (February 2005)

External links

*planetmath reference|id=5395|title=Hurwitz matrix