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?
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$\begingroup$ See math.stackexchange.com/questions/1376921/… for several text suggetions. as to your question 2. $\endgroup$– DrMrstheMonarchCommented Sep 18, 2019 at 17:25
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$\begingroup$ Morbo, thank you. These book i have read already. Not all of them are useful, in fact. $\endgroup$– ayrCommented Sep 22, 2019 at 15:58
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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"]
I don't follow your question.
FeedbackLinearizeby default gives the Burnovsky form if possible. There is an example, and see the last item in the details section of theFeedbackLinearizeref page.
answered Sep 18, 2019 at 13:48
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$\begingroup$ The OP means Nonlinear control schemes where Diff geom. plays a roll, Namely Geometric control. $\endgroup$ Commented Sep 18, 2019 at 17:26
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$\begingroup$ I do not quite understand, is this only possible for affine systems? $\endgroup$– ayrCommented Sep 19, 2019 at 7:45
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$\begingroup$ If the system is not initially affine, can it be converted to affine? $\endgroup$– ayrCommented Sep 19, 2019 at 16:40
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$\begingroup$ 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 $x'=-x + x v$. $\endgroup$ Commented Sep 19, 2019 at 17:58