Can the inverse system of a stable system be unstable?
For the class of LTI systems, the criteria for stability of a system with impulse response $h(t)$ and systems function $H(s)$ are:
- $h(t)$ be absolutely integrable.
or in s-domain:
- $H(s)$ have the $j\omega$ axis (or $s=0$) in its convergence region, and do not have any derivatives of impulse in it (which means for rational system functions like $P(s)/Q(s)$ the order of the nominator be less than or equal to the denominator).
So I thought that if we consider the following system $$H(s)=\frac{s-2}{s^2-1}$$ with $ROC:s>1$, it is stable, but its inverse $H_i(s)=1/H(s)=\frac{s^2-1}{s-2}$ with $ROC:s>2$ has the first derivative of delta in it and thus is not stable.
Is my reasoning correct? and is there any way to answer the question for all of the systems (not only LTI)?