So I have my equation of motion for an inverted pendulum on a cart: (linearised about the upright position and X denoting the position of the cart):
$$ \ddot{\theta}-\theta = \ddot{X} $$
Rather than dealing with a 4x4 system I’d like to set $\ddot{X} = a \theta +b \dot{\theta}$ but now I’m struggling to write my system in the form:
$$\dot{x} = Ax + Bu$$
as I can’t determine what my control Bu is now that I’m not applying a direct force to the cart?
You are only using one half of the equations of motion. The other half contains an external force, which will be your input $u$.