How to derive the PID parameters mathematically?
I saw in one of the vacuum control, proportional gain value as negative value (-300). Can the proportional value be negative? What is the reason for the value being negative?

 

Among other methods PID values may be estimated mathematically using LAPLAS Transform.

Practically most of the control engineers establish these values through trial & error practice.

Depending upon value input range in a specific controller, PLC, DCS PID Module value may be negative.

Usually "P & I" (Proportional & Integral) values are dealt with while "D" (Damping) values are rarely dealt with.

 

A negative proportional gain will provide you inverse control.

Under "normal" control, you would want to REDUCE the control variable (CV) output when the process variable (PV) is ABOVE the setpoint. Imagine a compressor filling a closed vessel with pressure for your process variable.

The situation is reversed for "inverse" control. When attempting to maintain a vacuum in a vessel, you want to INCREASE the vacuum pump speed (CV) when the pressure (PV) is ABOVE the setpoint.

Ron Davis