# Understanding the role of Pole zeros cancellation from the definition of stability

I have encountered the following theorem in a book.

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

• Theoretically your diagram is stable but in practise, this wouldn't necessarily be regarded as a fuilly stable system due to delays between blocks and temperature effects making the first block s-1.99999. – Andy aka Mar 27 '18 at 9:02
• Andy's spot on. Also a step input (which is bounded) to the (s-2) block would give rise to an infinite output at the output of the (s-2) block at t=0, due to the differentiator. – Chu Mar 27 '18 at 11:34

$$h(t)=e^{-3t}u(t)$$