I have encountered with the following control loop in a hydraulic system:
"command_signal+=P error+D d_error" Where d_error=error-previous_error.
The sampling time of the system is constant 0.0025 sec. P=1 and D=20.
Mathematically I would write:
$$
command(t)=\int_0^t{\left(P\cdot e(t)+D\cdot \frac{de(t)}{dt}\right)dt}
$$
Is it a common control method? If yes how is it called?
After a little more investigation and reading the comments it turned out for me
$$
\int_0^t{\left(P\cdot e(t)+D\cdot \frac{de(t)}{dt}\right)dt}=
\int_0^t{\left(P\cdot e(t)\right)dt}+\int_0^t{\left(D\cdot \frac{de(t)}{dt}\right)dt}=\\command(t) =\int_0^t{\left(P\cdot e(t)\right)dt}+D\cdot e(t)
$$
Which one really is a PI control without any derivative term. However testing these in a simulation where the actual controlling signal is restricted between -10 and +10 they acted differently due to this restriction. This limitation is actually implemented like this:
If command(t)>upper_limit then command(t)=upper_limit
If command(t)<lower_limit then command(t)=lower_limit
Else command(t)=command(t)
Could this be the reason why the first idea (the original one) was implemented in the real system? Or is there a general idea when designing the control method how to take account the domain of the controlling signal?