Control of inverted pendulum

Hello everyone,

I'm playing with the Inverted Pendulum example : openExample('simulink_general/penddemoExample') in Command Window. I'm trying to understand which control theory stays behind the state space estimator and the choice of LQR parameters.

I don't find anything similar in literature, can anyone help me, please?

Answer 1

[M]

I don't find anything similar in literature

LQR algorithm with state estimator is very common and you can find plenty of examples in the literature.

https://scholar.google.com/scholar?hl=fr&as_sdt=0%2C5&q=state+estimator&btnG=

https://scholar.google.com/scholar?hl=fr&as_sdt=0%2C5&q=LQR+controller&btnG=

In the simulink example, the state space estimator is implemented with a discrete time state space equation:

https://mathworks.com/help/simulink/slref/discretestatespace.html

And the LQR block is simply a matrix gain. You can find details about lqr in matlab here:

https://mathworks.com/help/control/ref/lqr.html

[AG]

Dear M,

thank you for your reply and please excuse me, I don't explained well my problem. The meaning of "I don't find anything similar in literature" is referred on the Simulink example. In particular I can't find an answer to these questions:

Why A matrix is 0? Why B matrix is 1/Ts? Why D matrix is not set to 0?

Thank yoy very much!

[M]

I did not look into details but it looks like the matrices inside the estimator block come from a linearization of the functions that model the cart + pendulum. I guess the linearization is made around the vertical position.

[AG]

Its look like we use the information of cart position and angle sensor "as is", the meaning of A=0 is that we don't consider the previous state but I don't understand why. If I linearize the generic model of a inverted pendulum on a cart I must use the general model where A is a 4x4 matrix and not 2x2.

Answer 2

[AG]

No one could help me?

Thank you very much!