A generalized Routh–Hurwitz criterion for the stability analysis of polynomials with complex coefficients: Application to the PI-control of vibrating structures

The 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, some generalization exist but are either incorrect or inapplicable to most practical cases. To fill this gap, we hereby propose a directed generalization of the criterion to the case of complex polynomials, broken down in an algorithmic form, so that the method is now easily accessible and ready to be applied. Then, we demonstrate its use to determine the external stability of a system consisting of the interconnection between a rotating shaft and a PI-regulator, obtaining the necessary and sufficient conditions to achieve stabilization of the system.