Hurwitz-stable matrix explained

In mathematics, a Hurwitz-stable matrix,^[1] or more commonly simply Hurwitz matrix,^[2] is a square matrix whose eigenvalues all have strictly negative real part. Some authors also use the term stability matrix.^[2] Such matrices play an important role in control theory.

Definition

A is called a Hurwitz matrix if every eigenvalue of A has strictly negative real part, that is,\operatorname{Re}[ λ _{i]} < 0 for each eigenvalue λ _{i} . A is also called a stable matrix, because then the differential equation• x = A x is asymptotically stable, that is, x(t)\to 0 as t\to inf ty.

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• 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.

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

• M-matrix
• Perron–Frobenius theorem, which shows that any Hurwitz matrix must have at least one negative entry
• Z-matrix

Notes and References

1. Duan . Guang-Ren . Patton . Ron J. . 1998 . A Note on Hurwitz Stability of Matrices . Automatica . 34 . 4 . 509–511 . 10.1016/S0005-1098(97)00217-3 .
2. Book: Khalil, Hassan K. . 1996 . Second . 123 . Nonlinear Systems . Prentice Hall.