adelphysi:
there is no need to calculate the difference which will be multiplied with gains!
It's all about equivalent systems. Your system still involves PID control. It is not new.
See the signal "S" in the two diagrams ..... those signals will be exactly the same (for both diagrams). That signal "S" will be the input to the PID controller. This means..... it looks like the system you ended up is just the usual PID system.
Adding the signal (input) in the way you did is absolutely the same as setting a setpoint. The same.
