The Transfer Function of a Closed-Loop system is: $T(s)=\frac{Z(s)}{1+G(s)Z(s)}$.

The BIBO stability of $T(s)$ is guaranteed if all the poles of $T(s)$ have negative real value.

In many textbooks I've read, it's written without further explanation that the zeros of the characteristic polynomial $1+G(s)Z(s)=0$ are the poles of $T(s)$, without any conditions on $Z(s)$.

In one textbook, it was written that $Z(s)$ should be bounded. If $Z(s)$ is bounded, i.e. has no poles, it's obvious that the characteristic polynomial contains all the poles of a closed-loop transfer function.

But in many physical systems, $Z(s)$ has poles. I have a theoretical example in which $Z(s)$ has a pole at $s_0$, while $G(s)$ has a zero at $s_0$, such that their product is finite and different from 1 $Z(s_0)G(s_0)\rightarrow A_0 \neq -1$. Therefore, $T(s)$ will have a pole at $s_0$, while the characteristic polynomial will not have a root at $s_0$. Therefore, we found an example for characteristic polynomial that doesn't contains all the poles of a closed-loop transfer function.

Therefore, it seems as if a necessary condition for working only with the characteristic polynomial is that the $Z(s)$'s poles won't coincide with the $G(s)$'s zeros. But is there a necessary and sufficient condition?

Why is this question not addressed in control-theory textbooks? If it does, please refer me to those textbooks.


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