# Hurwitz matrix

A square matrix $A$ is called a Hurwitz matrix if all eigenvalues of $A$ have strictly negative real part, $Re[\lambda_{i}]<0$; $A$ is also called a stability matrix, because the feedback system

 $\dot{x}=Ax$

is stable.

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

 $\displaystyle\dot{x}(t)$ $\displaystyle=$ $\displaystyle Ax(t)+Bu(t)$ $\displaystyle y(t)$ $\displaystyle=$ $\displaystyle Cx(t)+Du(t)$

has a Hurwitz transfer function.

Reference: Hassan K. Khalil, Nonlinear Systems, Prentice Hall, 2002

Title Hurwitz matrix HurwitzMatrix 2013-03-22 14:02:45 2013-03-22 14:02:45 lha (3057) lha (3057) 4 lha (3057) Definition msc 93D99 Hurwitz transfer function stability matrix