Proving Routh’s Theorem using the Euclidean Algorithm and Cauchy’s Theorem

Abstract

This paper presents a proof of Routh’s theorem for polynomials with real coefficients, determining the number of roots in the right half plane (RHP). The proof exploits the relationship of the Routh array to the Euclidean algorithm and applies Cauchy’s theorem in an analogous way to that of applying the Nyquist criterion to investigate the stability of a control system. While a number of papers have been published over the years with different proofs of Routh’s stability criterion or Routh’s theorem, the aim in this paper is to present a proof that may offer most insight to undergraduate students of engineering. Routh’s theorem and his array are introduced without any proof in most undergraduate texts on control theory, whereas the Nyquist criterion is typically treated quite extensively in such texts. As well as presenting a proof for the regular case when all the coefficients in the first column of the Routh array are non-zero, analogous proofs are given for the singular cases when some of the leading coefficients in a row, or the coefficients of the entire row, become zero. In the first case, these result in a statement on the number of roots in the RHP, more explicit than those typically presented in papers on Routh’s theorem. In the second case, the only case where there may be roots on the imaginary axis, use is made of the modified array introduced by Routh, often referred to as the Q-method, to determine the number of such roots, differentiating between simple and multiple roots. One can thus distinguish between exponential stability, marginal stability and polynomial instability, when there are no roots in the RHP, with these results.