# Routh's stability criterion: zeros of the auxiliary polynomial and of its derivative

When using RH criterion and using the auxiliary equation special case, which of the following is true?

1. The auxiliary equation $$A(s)=0$$ gives some(or all) of the symmetrical poles.

2. The differentiated auxiliary equation $$\displaystyle \frac{\mathrm{d}A(s)}{\mathrm{d}s}=0$$ gives some(or all) of the symmetrical poles.

I have seen both of them mentioned either separately or together in different books. I am really confused on which one is actually correct.

After solving a few problems I have observed that $$1$$ generally holds, it might be a coincidence. If someone could explain if I am missing something or if either of them(aforementioned cases) is wrong, it would be really helpful.

## 1 Answer

The zeros of the auxiliary polynomial give you the poles of equal magnitude and opposite signs (either on the real axis or on the imaginary axis). The derivative of the auxiliary polynomial gives you the coefficients that are used in the row with all zero coefficients. It's been a while, but I learned that from reading Modern Control Engineering by K. Ogata (and I had to refresh my memory for answering this question).

• Thank you for your answer. This fits my observation. I can now safely say that some authors who use the 2nd case are wrong. – paulplusx Nov 19 '18 at 4:21