Given a transfer function: $$f = \frac{\sigma(\sigma+1)}{s - \sigma + 1}$$ where $\sigma \in \mathbb{R}$. I want to check the stability of the above function.
Attempt:
Pole(s) of the function: $s = \sigma -1$
Question: Are the zero(s) of the function: $z_1 = 0$ and $z_2 = -1$? $s$ does not appear in the numerator, so this is confusing to me
Continuing, I know the system will be stable if the pole(s) of $f$ have a negative real part, but I need to be careful of pole-zero cancellation.
That is, if $\sigma = 1$, would the pole cancel with $z_1$? and $f$ reduce to: $$f = \frac{2}{s}$$ which is not stable since there are no zeros and $s = 0$
Similarly, if $\sigma = -1$ $$f = \frac{-1(-1 + 1)}{s-(-1)+1} = \frac{0}{s+2}$$ which is $0$, so how do I handle this case?
The other cases would not have any cancellations, so $f$ should be stable as long as $\sigma < 1$ exlcuding $\sigma = -1$
I am incredibly rusty with transfer functions, so any help/guidance would be greatly appreciated!
Thanks!