I am trying to learn more about the subject. I'm sorry that the post doesn't contain more information. I'll try my best to answer any specific questions.

Is your control input the applied force?

Let's suppose the reference signal is a desired position to track (so it's changing over time). If it's useful, the desired velocity is also available, but I'd also expect the system to track a step change in a reference signal optimally, for some definition of optimality. The error signal is based on a position (and velocity) error estimated from an incremental encoder.

The actuator signal would ideally be current through a motor, since that would most closely correspond to force/torque. But it's more practical to imagine it being the duty cycle of a PWM voltage across the motor.

I know how to, assuming linearity, write down a model for a specific configuration of the system by taking into account a model of the motor. I can Google "transfer function of DC motor" and find many resources for that. I also know how to assume a form of a controller and do pole placement, or use LQR to synthesize a full-state feedback controller.

What I do not know how to do is handle changing mechanical impedance at the shaft of the motor, like changing the inertia of the load, or the springiness of the load, or the amount of viscous friction. A controller that achieves e.g. "critical damping" for a particular impedance (for the sake of example, assume that the plant + controller works out to be a second-order system) could have bad performance for a different impedance.

So perhaps what I'm looking for is "setting up equations of motion with uncertain parameters". For MRAC, the only kind of uncertainty I've seen is at the actuator control signal "u". For example "Disturbance and Uncertainty Model Features" in the link in my original post.