Hurwitz Matrix

In mathematics, a square matrix A is called a Hurwitz matrix if all eigenvalues of A have strictly negative real part, that is,
\Re\lambda_i < 0\,
for each eigenvalue \lambda_i. A is also called a stability matrix, because then the differential equation
\dot x = A x
is stable, that is, x(t)\to 0 as t\to\infty. 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
\dot x(t)=A x(t) + B u(t)
y(t)=C x(t) + D u(t)\,
has a Hurwitz transfer function.

References

  • Hassan K. Khalil (2002). Nonlinear Systems. Prentice Hall.

 

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