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

TL;DR

This paper extends the classical Routh-Hurwitz stability criterion to polynomials with complex coefficients, providing an accessible algorithmic approach and demonstrating its application to stabilize a vibrating structure with a PI-controller.

Contribution

It presents a clear, algorithmic generalization of the Routh-Hurwitz criterion for complex polynomials, facilitating broader application in control system stability analysis.

Findings

Extended Routh-Hurwitz criterion for complex polynomials

Necessary and sufficient conditions for PI-regulator stability

Application to vibrating structure stabilization

Abstract

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.