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

Anthony Hastir, Riccardo Muolo

TL;DR

This work extends the classical Routh-Hurwitz stability test to polynomials with complex coefficients by presenting a pedagogical, algorithmic generalization grounded in Wall’s continued-fraction theory and Frank’s framework. It introduces an auxiliary polynomial $\mathfrak{p}(s)$ and a continued-fraction representation of $\mathfrak{p}(s)/p(s)$ with positive real coefficients, yielding necessary-and-sufficient stability conditions that are computable for arbitrary degree and parity. The authors explicitly derive the $n=4$ case, obtaining concrete inequalities that reduce to the classical RH criteria when all imaginary parts vanish, and demonstrate an application to PI control in rotordynamics where a cubic characteristic yields practical stability regions. The approach is intended to be implementable across domains featuring complex-coefficient dynamics, including networks and hypergraphs, and provides a direct alternative to degree-doubling methods.

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.

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 work extends the classical Routh-Hurwitz stability test to polynomials with complex coefficients by presenting a pedagogical, algorithmic generalization grounded in Wall’s continued-fraction theory and Frank’s framework. It introduces an auxiliary polynomial and a continued-fraction representation of with positive real coefficients, yielding necessary-and-sufficient stability conditions that are computable for arbitrary degree and parity. The authors explicitly derive the case, obtaining concrete inequalities that reduce to the classical RH criteria when all imaginary parts vanish, and demonstrate an application to PI control in rotordynamics where a cubic characteristic yields practical stability regions. The approach is intended to be implementable across domains featuring complex-coefficient dynamics, including networks and hypergraphs, and provides a direct alternative to degree-doubling methods.

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.
Paper Structure (9 sections, 2 theorems, 43 equations, 6 figures, 4 tables)

This paper contains 9 sections, 2 theorems, 43 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. General description of the method
  3. Example for a $4$-th degree polynomial
  4. Application to the PI-regulation of a rotating shaft
  5. Conclusion
  6. Explicit derivation of the Generalized Routh-Hurwitz conditions
  7. Attainment of Classical Routh-Hurwitz criterion in case of real coefficient
  8. Comparison with an existing method
  9. Sketch of the proof of Theorem \ref{['Thm_Cont_Frac']}

Key Result

Theorem 2.1

Let $p(s)$ and $\mathfrak{p}(s)$ be the polynomials given in (1) and Aux_Poly, respectively. The polynomial $p(s)$ is stable if and only if the ratio $\frac{\mathfrak{p}(s)}{p(s)}$ may be written as the following continued fraction where $c_i, i=0, \dots, n-1$ are real and positive and $d_i, i=1, \dots, n$ are pure imaginary or zero.

Figures (6)

  • Figure 1: Coefficients of the generalized Routh-Hurwitz criterion
  • Figure 1: A grid $(k_I,k_p)$ at which each point is either $1$ or $0$, based on conditions \ref{['cond_1_Appli']}--\ref{['cond_3_Appli']}.
  • Figure 2: Contour plot of the function $\lambda_s(k_I,k_p) := \max\{\mathfrak{Re}(\lambda), \lambda\in\sigma(\tilde{\mathbf{A}})\}$
  • Figure 3: Trajectory $x(t)$ for $x_r = 2, k_I = -1.18, k_p = -3.59$.
  • Figure 4: Trajectory $\dot{x}(t)$ for $x_r = 2, k_I = -1.18, k_p = -3.59$.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Remark 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Remark 4.1