Control Theorem: Gains for PD Controller of Acceleration

[G Man]

I'm trying to understand the derivation of what appears to be a basic concept in control. This is for a PD controller of acceleration. I believe there would be similar derivation for other controllers, but I do not understand its origin let alone other examples?

x^.. = K_s.x + K_v.x^.
also written as:
acceleration = (gain x displacement) + (gain x velocity)

 

[timthereaper]

AFAIK, I believe the technique for PD (and PID controllers for that matter) was based on people doing studies on ship helmsmen and how they steered large ships. They noticed that the helmsmen wouldn't simply just correct for an error in position (proportional control), but also take the rate of change into account (derivative control). Sometimes, to correct for "droop" or steady-state error, they would correct it by adding up the error in the position and use that to correct for the difference (integral control). This I believe is the origin of PID control.

The idea is that you can tune the gains based on what you observe about the system. Often, mathematical techniques are used to determine the gains (to get in the ballpark) based on properties of the system you are trying to control, but often the models are insufficient so these numbers are "tuned" to get the desired results.

 

[FactChecker]

acceleration = (gain1 x displacement) + (gain2 x velocity)

In addition to what @timethereaper has said, I think that your equation here is misleading. This is not necessarily the acceleration, it is the control feedback signal. So if you want to control something, you need to know where it is (displacement) and how it is moving (velocity). Then your control feedback signal will be some weighted (gain1, gain2) combination of those two. Other than that simple thinking, I don't think there is any derivation. It is just an idea of how to control something and applies to anything you want to control. It often will end up in some way affecting acceleration. It is usually just a starting point and the final working controller is much more complicated.