Why , when we have pole in RHS of any system , it is unstable ?

We will plot the graph for different pole positions , also try to understand the reason for the nature of response.

1. Pole in LHS - Stable system.

For the above attached image we can observe the Y(s) and y(t).
As the poles are there in LHS , then y(t) will converge as we  proceed in time domain .So it will give finite output.

2. Pole in RHS - Unstable System

 

As we can see y(t) is going to infinity as we proceed to infinity.
That is why we always say whenever we have pole in RHS system will be unstable.

3. Single Pole at origin -Stable system

As y(t) will be always finite .so we will conclude the system as stable one.

4. Poles on imaginary axis - Marginally stable.

Y(t) will be continuously oscillating signals so , it will never stabilizes it selfto some finite value,
nor it will go infinity. Hence marginally stable.

5. Repetitive Poles on imaginary Axis - Unstable system

 As y(t) will again grow with time . so unstable.

6. Repetitive Poles on origin - Unstable System

As t goes to infinity , then y(t) will also go to  infinity . Hence unstable.