Converting proportional and derivative gain in PID to meaningful stiffness and damping units

[Moris]

Hi all,

I appreciate if someone could give some insight on how to convert the proportional gain (Kc) and derivative gain (Kc*Td/Ts) into meaningful units in terms of stiffness(Nm/rad) and damping (Nm/rad.s)

I am doing a position control using PID (FPGA).vi... I only perform PD control.

My process variable is the position from encoder in counts.

Thanks!

P.S: To make things more confusing (for myself) these gains are represented by a fixed-point number in the express PID (FPGA).vi and they are normalized too.

[NateBraska]

Hi Moris,

I am not familiar with converting the gain coefficients to actual meaningful units. From my experience, they are mainly just the factors you are multiplying the respective calculated errors in the system, which is then how you calculate the corresponding control forces; so I know of them as being unitless and your measurable force/torque changes as the system responds.

I really like this link. Although, it does not directly address your question, it does help with tuning a PID system.

I hope this helps.

Best Regards,

Nathan B

Applications Engineer

National Instruments

[Moris · Accepted solution]

Hi Nathan,

Thanks for your reply. I gave it some more thought and this is what I found.

My process variable is the position info from the encoder which has the unit of counts. My controller, multiplies the error in position by P, plus the rate of change in position times D and sends this to the motor as a voltage command. That is:

output of the PD controller = voltage command to the motor amplifier = P * (Pos(setpoint)- Pos (process variable)) + D * d (Pos)/dt

Therefore my P gain has the unit of volt/count. In order to convert this this to N/m or (Nm/rad) I need to know what is the relationship between 1 volt given to to the motor and the torque generated (1 volt = 1*K1 amp= 1*K1*K2 Nm). I also from the calibration, know the relationshio between 1 count from the encoder and the end effector displacement. (1 count= K3 * rad)

Knowing these K1, K2, and K3 values, I can convert my P gain to a meaningful stiffness parameter in terms of N/m or Nm/rad for torsional stiffness. Similarly, the D can be converted to a damping unit.

This is my understanding. If anyone has a comment about this approach, I will be glad to hear it.

Cheers!