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
for each eigenvalue . is also called a stable 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.
- Liénard–Chipart criterion
- Perron–Frobenius theorem
- Jury stability criterion, for the analogue criterion for discrete-time systems.
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