Hurwitz Matrix - Hurwitz Stable Matrices

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 Hurwitz matrix is named after Adolf Hurwitz.

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.