Closed-loop control Error

Hello guys!
I'm trying to investigate the evolution of the error function in a closed loop.

What I am basically trying to find is a way to understand, checking only the error, if the control loop is working well.

My idea is that if the error is in his bounds then everything is fine and since these bounds decrease in amplitude as time goes by, in the end, I should be able to find out if the loop "goes crazy"

From what I understood:
1) The majority (>60%) of the closed loops used in the world are linear
2) In the linear controls error upper and lower bounds can be described using a mathematical functions that I can compute using the loop parameters

Are these statements correct? And, if so, what can I use in non linear systems?
Thank you, :)