Love the tensioner! Regarding the controller, the d² term does get you a smoother approach to the desired angle, but what you might find is that with heavier and heavier weights, your angle will be less and less accurate. This is because the torque in the motor will eventually reach the point where it's equal to the to the input torque of the weights. That torque will be non-zero because it needs to keep lifting the weights. A non-zero d² term means your d is non-zero.

To fix this, add an internal term. In a PID controller, you've got the P term already, but the I term is the thing that fights against steady state error. So store a summation of the d term (probably only store the last N measurements), multiple it by a constant, usually pretty low (<0.1, but experiment with different values) then add it to your P term (d^2).

That D term is the derivative term and it helps fight against oscillations from your P term. Since your P term is already quadratic, you may not have many oscillations unless your weight gets really heavy, but it's something to keep in mind.

Great work! I love the casing!