I was doing some research and found that when a first column 0 appears but everything else is not necessarily 0 in the row, then there exists poles with nonnegative real parts (or positive real parts if there are no imaginary roots or “opposite” roots in general). From what I saw, the proof mentions the existence of non-LHP poles, but doesn’t provide proof of number of non-LHP poles based on the first column 0 (actually, I saw one, but it doesn’t give a “hard” proof).
Based on my observations, if a first zero appears in row $s^m$, at least 2 Routh rows can be skipped including $s^m$, depending on how many consecutive zeros appear after the first zero. For example, if there’s a nonzero value after the first zero, then we go to $s^{m-2}$ and consider the sign change between $s^{m+1}$ and $s^{m-2}$. Likewise, if there are two consecutive zeros including the first row zero, then we consider the sign change between $s^{m+1}$ and $s^{m-4}$ (the rows in between are skipped). We see that the total number of rows skipped due to the first column zeros is an even number (let’s denote this as $2b$). Is there a way to prove that there is a relationship between the number of skipped rows due to first column zeros and the total number of non-LHP poles (or RHP poles if there aren’t any imaginary poles)? In other words, is there a way to prove the relationship between $P$ (total number of RHP poles), $b$, and $V\left(\infty\right)$, where $V\left(\infty\right)$ is the number of sign changes of the Sturm polynomial sequence at $\infty$ (or the number of sign changes in the Routh table)? Also, can one provide a reference to the source like an article or a journal? I have a “proof”, but it’s incorrect, because I am somehow mixing up $P$ as the total number of RHP poles and the number of RHP poles captured from the shortened Routh table.
I ask because the epsilon method can get one into a very hairy situation like this or even provide misleading results like the one mentioned by F.R. Gantmacher in The Theory of Matrices, Volume 2, page 184.
Worked “example”, using the notation from Routh and Gantmacher: $$F\left(s\right) = s^4 + s^3 + 2s^2 + 2s + 1.$$
Using the small parameter method, we see that there are two RHP poles. Without it, I get
$$V\left(\infty\right) + V\left(-\infty\right) = n - 2b = N + P - 2b = 2,$$
where $n$ is the polynomial degree, $N$ the number of LHP poles, and $V\left(-\infty\right)$ the number of sign changes of the Sturm polynomial sequence at $-\infty$. Here, $b = 1$. By the definition of the Cauchy index,
$$V\left(-\infty\right) - V\left(\infty\right) = N - P.$$
Using my previous worked relationship between the $V$’s and $N$, $P$, and $b$, I get
$$V\left(-\infty\right) = N - b = 2 \rightarrow N = 3,$$
and
$$ V\left(\infty\right) = P - b = 0 \rightarrow P = 1,$$
which contradicts with the correct answer.