Derivation of the Routh array

The Routh array is a tabular method permitting one to establish the stability of a system using only the coefficients of the characteristic polynomial. Central to the field of control systems design, the Routh–Hurwitz theorem and Routh array emerge by using the Euclidean algorithm and Sturm's theorem in evaluating Cauchy indices.

Given the system:

Assuming no roots of ${\displaystyle f(x)=0\,}$ lie on the imaginary axis, and letting

then we have

Expressing ${\displaystyle f(x)\,}$ in polar form, we have

where

and

from (2) note that

where

Now if the i^th root of ${\displaystyle f(x)=0\,}$ has a positive real part, then (using the notation y=(RE[y],IM[y]))

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