An elementary derivation of the Routh-Hurwitz criterion

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.