According to Routh's stability criteria when will be pole on imaginary axis?

Question

I know this is easy for you. But I want to know. According to Routh's stability criteria when will be pole on imaginary axis?

Answer 1

As you may already know, when we have a polynomial $$p(s)=a_ns^n+a_{n-1}s^{n-1}+\cdots+a_0$$ in Routh's test, the coefficients are arranged in a table in such a way that the first row contains $a_n,a_{n-2},a_{n-4},...$ and the second row contains $a_{n-1},a_{n-3},...$ The other rows are calculated iteratively using Routh-Hurwitz algorithm.

When the table is completed, the number of sign changes in the first column will be the number of non-negative poles. But sometimes the coefficients of the Routh table in a whole row become zero and thus further calculation of the elements of the array is not possible (i.e. table cannot be completed). This happens when there exist conjugate poles on the imaginary axis. In this case, we must use the auxiliary polynomial, which is built from the coefficients of the last non-zero row, and then differentiate it.

The details can be found, for example, in here.