Fundamental questions on controllability and state feedback control of a Furuta pendulum model

Hello Matlab community,

I´m trying to create a state feedback controller for a Furuta pendulum, but I´m having a few issues and hope someone can help.

In the project file attached to this post there is a model with a simple state feedback controller and one, where I use an observer to observe the states that are not measured.

When I linearize my model and check for controllability, it only returns rank(ctrb(A, B)) = 1, therefore I would assume that the model is not (fully) controllable. However, when I calculate a gain vector K and run the model, it actually manages to balance the pendulum. Obviously I´m glad that it runs, but I actually do not understand why, as I thought that controllability is a requirement for state feedback control.

Also I was surprised, that one of my eigenvalues of the A matrix is 10^4 smaller than all the other eigenvalues. Is that something I should be concerned about (maybe something wrong with the lineariation) or is it reasonable that this model has these eigenvalues.

The second model in the attached project file uses a state observer. This model is not stable, although it should be regarding the negative eigenvalues. I can´t really tell, why my first model without the observer runs, but the second one with an observer doesn´t...

Furthermore I wonder if there´s a way to choose the state variables of the state space model created in the Model Linearizer. For instance I would like to have the arm and pendulum position and velocity as states and not the current.

I´d appretiate if someone could help me clear my mind! Thanks in advance!

Jooo