Explaining the Routh–Hurwitz Criterion: A Tutorial Presentation [Focus on Education]

Routh's treatise [1] was a landmark in the analysis of the stability of dynamic systems and became a core foundation of control theory. The remarkable simplicity of the result was in stark contrast to the challenge of the proof. Many researchers devoted much effort to extend the result to singular cases, with some of the earlier techniques shown to be inadequate [2]. Together with the extensions to singular cases, shorter proofs were also proposed. The proof of [3] is noteworthy, which followed the root locus arguments of [4]. A key feature of the proof is a continuity argument used in an earlier derivation [5]. In [6], the more conventional approach using Cauchy?s principle of the argument is followed. A relatively simple proof is proposed, considering the extension to complex polynomials and singular cases.