# Differential Equation for: Locking a laser's frequency with a PI controller

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?

Regarding that part of the question, (although you already are outlining it in your question) I would suggest that the highest derivative is usually assosiated with the inertia term. So in your case the $$k_d$$ which is conviniently set to zero for the P-controller.
• mass damper spring (Harmonic oscillator): the mass in the system which is associated with $$\frac{d^2 }{dt^2}x(t)$$
• RLC system: coil inductance L which is associated with $$\frac{d^2 }{dt^2}i(t)$$
• Exactly. If you think about it, inertia is about a system's resistance to change its condition. The $k_d$ term is resisting changes... Sep 20 '20 at 19:22