Block Diagram --> System Equation question

Let me be clear this is a HW problem. I'm looking for help on the start of it, not the solution!

The system dynamics appear to be $$\ddot \theta = u$$ where u is equal to (p+a) or -(p+a). I'm not sure what p and a stand for. I kind of assume p act like an integrator like s but not sure if this makes sense.

I would probably write this system as $$\ddot \theta = u$$ where

u is

$$\begin{cases} -U(\theta + \dot \theta)\ if\ \theta > 0 \\ U(\theta + \dot \theta)\ if \ \theta < 0 \end{cases}$$

So, before I take it farther, does this make sense or am I off my rocker?

• Although I have mainly seen $s$ being used as the Laplace variable, I do have heard that sometimes $p$ is used instead. So when assuming that $a$ is just a gain then $p+a$ would just be a PD controller. I am not sure about the block in the middle, it looks like the sign function. Does your book define it anywhere? – fibonatic Jan 25 at 0:32