An elementary derivation of the Routh-Hurwitz criterion

Ming Tzu Ho, Aniruddha Datta, S. P. Bhattacharyya

Research output: Contribution to journalArticlepeer-review

34 Citations (Scopus)


In most undergraduate texts on control systems, the Routh-Hurwitz criterion is usually introduced as a mechanical algorithm for determining the Hurwitz stability of a real polynomial. Unlike many other stability criteria such as the Nyquist criterion, root locus, etc., no attempt whatsoever is made to even allude to a proof of the Routh-Hurwitz criterion. Recent results using the Hermite-Biehler theorem have, however, succeeded in providing a simple derivation of Routh's algorithm for determining the Hurwitz stability or otherwise of a given real polynomial. However, this derivation fails to capture the fact that Routh's algorithm can also be used to count the number of open right half-plane roots of a given polynomial. This paper shows that by using appropriately generalized versions of the Hermite-Biehler theorem, it is possible to provide a simple derivation of the Routh-Hurwitz criterion which also captures its unstable root counting capability.

Original languageEnglish
Pages (from-to)405-409
Number of pages5
JournalIEEE Transactions on Automatic Control
Issue number3
Publication statusPublished - 1998

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering


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