I think I have a good understanding of differential equations, so I would like to first try to understand my system in terms of differential equations before transforming my system into the more standard Laplace-transform, looking-for-poles-type analysis that's standard in control theory.
Specifically I'm interested in finding the differential equation that governs what is known as a "Pound-Derver-Hall (PDH) lock" in optics.
In this locking scheme, an optical system is controlled with a piezo, and the goal is to have to use the piezo to correct for changes in the optical system. The system is set up like this:
A voltage is sent to the piezo in the mechanical-optical system. An error voltage is generated from the output of this system, and is sent to the PID controller. The error voltage looks something like this:
So, if you asked me, my differential equation describing this would be the form:
$\frac{dVin}{dt} = k_p V_{error} + k_i \int V_{error} + k_d\frac{d}{dt}V_{error}$
so in the area that my error function is strictly linear:
$V_{error} = g V_{in}$
then this becomes:
$\frac{dVin}{dt} = g k_p V_{in} + g k_i \int V_{in} + g k_d\frac{d}{dt}V_{in}$
Now what's confusing to me is that when I consider only proportional gain (ie $k_i = k_d = 0$)
Then I can solve my diff.eq. exactly:
$V_{in} = V_0 e^{g k_p t}$
which has a "sink" at zero as t approaches infinity.
Now I know from experience that "position only" controllers typically have lots of oscillations and don't necessarily converge to the solution but oscillate around it. Now if I had an "inertia" term to my differential equation that added a second derivative to the differential equation, then I would get a sinusoidal solution that I recognize as the typical "oscillations" in position-only control.
But generally speaking, is there a way to infer what this "inertia term" is?
Am I making some kind of mistake in my model?
