Hurwitz matrix

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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[edit]

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[edit]

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 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[edit]

References[edit]

This article incorporates material from Hurwitz matrix on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

External links[edit]