We define system to be stable if for all bounded inputs, it will give bounded output at all times.
But does unstable system means it will have unbounded output for any input?
This question is motivated from a different problem I came across which was to find dc gain of a system having closed loop poles on right – hand side of complex plane. The answer states that since poles are in RH s-plane, dc gain can't be defined. My question is it valid to deduce that unstable system would be always unstable or are there inputs to which 'unstable' system might be stable?
Best Answer
Yes. There are inputs for which the output of an unstable system is bounded. For example, we can stabilise some unstable systems with feedback. In this case, someone who sees only the input and output of the unstable system (without seeing the feedback mechanism) would see bounded outputs for the unstable system.
However, you may not be able to find the DC gain if the DC input is not in the list of inputs for which the (unstable) system has bounded outputs.
Edit
Consider as an example, the first order system A $$ \frac{dx}{dt} = x+u\\ x(0) = 1 $$ The state is bounded for the input \$u(t) = -1\$. The output grows for all other constant inputs.