I'm not concerned about high-frequency behaviour, so I am modeling the transmission line as a first order low-pass filter fed by a variable voltage source. I'm also assuming that the voltage source will respond instantly to commands that I give it (again, at the frequencies of interest this should be a safe assumption). The power factor of the load is unity, so I can model it as a pure resistance in parallel with the capacitance of the line. So, I have a simple series RC circuit composed of Rt & Ct connected to a source Vs, with a load resistance R in parallel with Ct. Simple enough.

I would like the controller to command the source voltage Vs to vary so that the voltage across the capacitor Vc (which is the same as the voltage across the load) is held constant. At this point I'm not overly concerned with exact parameters such as steady state error, etc., I'm just having a hard time getting started (again, my control theory is rusty!).

The load resistance R is not fixed resistance. If it were, I could derive the following transfer function:

G(s) = Vc(s)/Vs(s) = (1/RtCt)/(s+(Rt+R)/(RtRCt)).

However, as the load current increases, the load "resistance" will decrease. If I replace R with a load current I(s), here is where I get stuck:

(s + 1/RtCt)Vc(s) = (1/RtCt)Vs(s) - I(s)

Clearly I(s) is a disturbance on the system, but I am not sure how to proceed beyond this point.

If it is not clear, my input is the source voltage Vs, and my output is the load voltage, which is the same as the voltage across the capacitor, Vc. I can measure the source current Is(c) directly. Although I am looking for a general solution, here are the expected values for the circuit parameters: Rt = 88 ohms, Ct = 44uF, target Vc = 300VDC. Output current will vary from 0 to about 3.5A.

Any advice on how to proceed would be greatly appreciated.

PS, I know that I could just slap a proportional controller on this, measure the response, call it a day and probably have it work out fine, but I want to do this properly with some rigour.