I am studying about PID control in engineering and i am thinking about the case when error e(t) is zero as discussed in below SE EE question https://electronics.stackexchange.com/questions/462968/confusion-in-pid-loop/462973#462973
Obiviously 1st term "P" will get to zero when it is multiplied by 0 but What will be the output of integral and derivative blocks if we have error e(t)=0 ?
It depends on past history, particularly the integral block. You might well have some "momentum" carrying you past the setpoint due to the integral term. In addition, you might not approach the setpoint at a smaller and smaller velocity. If you cross the setpoint with a distinct download slope, then the derivative term will not be zero.
In conclusion, the best way to think about $P,I,$ and $D$ is that they are independent variables.