# Experiments with Feedback Linearization and StateTransformLinearization in Mathematica

I have some nonlinear system, and i have three big question: Non-linear ODE from closed-loop system and Response 1. how correctly use such terms, like "Feedback Linearization and StateTransform Linearization" for this case, but in Mathematica they use for affine systems. There exist tranform from clearly nonlinear system to affine? 2. is there terms relate to differential geometry control? 3. is it possible to do in Mathematica a transformation to the Brunovsky form?

• See math.stackexchange.com/questions/1376921/… for several text suggetions. as to your question 2. Sep 18, 2019 at 17:25
• Morbo, thank you. These book i have read already. Not all of them are useful, in fact.
– dtn
Sep 22, 2019 at 15:58

1. The example in the question you linked to is an affine system. It turns out to be not completely linearizable by state-transformation alone, but it is feedback linearizable. These techniques are for affine systems only.

FeedbackLinearize[AffineStateSpaceModel[x2'[t] == u[t] x2[t], x2[t], u[t], x2[t], t],
Automatic, "LinearSystem"] 3. FeedbackLinearizeby default gives the Burnovsky form if possible. There is an example, and see the last item in the details section of the FeedbackLinearize ref page.
• Yes, but because there is no systematic way to do it, as a user you have to manually do that, if possible. For example, if the system is $x'=-x + x u^2$, you can choose a new input $v=u^2$, to make the system affine $x'=-x + x v$. Sep 19, 2019 at 17:58