Derivation of The Routh Array - The Routh Criterion

The Routh Criterion

To derive Routh's criterion, first we'll use a different notation to differentiate between the even and odd terms of :

Now we have:

$\begin{align} f(j\omega) & = a_0(j\omega)^n+b_0(j\omega)^{n-1}+a_1(j\omega)^{n-2}+b_1(j\omega)^{n-3}+\cdots & {} \quad (20)\\ & = a_0(j\omega)^n+a_1(j\omega)^{n-2}+a_2(j\omega)^{n-4}+\cdots & {} \quad (21)\\ & + b_0(j\omega)^{n-1}+b_1(j\omega)^{n-3}+b_2(j\omega)^{n-5}+\cdots \\ \end{align}$

Therefore, if is even,

$\begin{align} f(j\omega) & = (-1)^{n/2}\big & {} \quad (22)\\ & + j(-1)^{(n/2)-1}\big & {} \\ \end{align}$

and if is odd:

$\begin{align} f(j\omega) & = j(-1)^{(n-1)/2}\big & {} \quad (23)\\ & + (-1)^{(n-1)/2}\big & {}\\ \end{align}$

Now observe that if is an odd integer, then by (3) is odd. If is an odd integer, then is odd as well. Similarly, this same argument shows that when is even, will be even. Equation (13) shows that if is even, is an integer multiple of . Therefore, is defined for even, and is thus the proper index to use when n is even, and similarly is defined for odd, making it the proper index in this latter case.

Thus, from (6) and (22), for even:

and from (18) and (23), for odd:

Lo and behold we are evaluating the same Cauchy index for both:

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