  The classical Routh-Hurwitz criterion is one of the most popular methods to
study the stability of polynomials with real coefficients, given its simplicity
and ductility. However, when moving to polynomials with complex coefficients, a
generalization exists but it is rather cumbersome and not as easy to apply. In
this paper, we make such generalization clear and understandable for a wider
public. To this purpose, we have broken down the procedure in an algorithmic
form, so that the method is easily accessible and ready to be applied. After
having explained the method, we demonstrate its use to determine the external
stability of a system consisting of the interconnection between a rotating
shaft and a PI-regulator. The extended Routh-Hurwitz criterion gives then
necessary and sufficient conditions on the gains of the PI-regulator to achieve
stabilization of the system together with regulation of the output. This
illustrative example makes our formulation of the extended Routh-Hurwitz
criterion ready to be used in several other applications.
