According to me, I can get an asymptotically stable system even if cancellations exist for the unstable poles. my example, Then Why does this book definition state "no hidden cancellations of unstable modes" ? It is implying necessity I understand
Asymptotic stability and BIBO stability are entirely different. System response depends on both zero state and zero input conditions. Thus we have two forms of stability criterions, one that concerns with input and other concerns only with characteristic modes of a system.When a system is observable and controllable, its external and internal descriptions are same. So BIBO and asymptotic stabilities are same.
The example that you have provided has impulse response,
which is absolutely integrable and hence BIBO stable.
But the system 1 has characteristic root at 2 while system 2 has at 2 and -3. Since these two systems do not load each other, the characteristic modes are also independent. The characteristic mode of system 1 does not converge thus making entire system asymptotically unstable.