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I am impressed by StateTransformationLinearize, and I feel terribly bad for not having noticed this function before.

How does it work?

Does it attempt to find a $C^r$-conjugate of the system? If so, how does it do this?

Why can't it handle Affine State Space Models with symbolic parameters?

I would like to know more about this witchcraft.

For instance, slight changes in the coefficients of the featured examples, lead to failures in finding the transformation. Perhaps if I would figure out how this straightening is done, I could grasp something more about its applicability.

Example

    sys=AffineStateSpaceModel[{{4*Subscript[x, 1], 7*Subscript[x, 1]^2 + Subscript[x, 2]}, {{1}, {-1 + 2*Subscript[x, 1]}}, {Subscript[x, 2]}, {{0}}}, {Subscript[x, 1], Subscript[x, 2]}, 
 {{Subscript[\[FormalU], 1], 0}}, {Automatic}, Automatic, SamplingPeriod -> None];

This is an example taken from the documentation. If you change the first coefficient, into a different number, i.e.:

sys=AffineStateSpaceModel[{{3*Subscript[x, 1], 7*Subscript[x, 1]^2 + Subscript[x, 2]}, {{1}, {-1 + 2*Subscript[x, 1]}}, {Subscript[x, 2]}, {{0}}}, {Subscript[x, 1], Subscript[x, 2]}, 
     {{Subscript[\[FormalU], 1], 0}}, {Automatic}, Automatic, SamplingPeriod -> None];

It already fails to find the transformation:

test = StateTransformationLinearize[ sys, {{Subscript[z, 1], Subscript[z, 2]}, "InputState"}];

Outputting the following error msg:

ControlStateAndFeedbackLinearizationsDumpStateSpaceLinarize::nclin: -- Message text not found -- (input-state)

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    $\begingroup$ It would be helpful to have examples that work or fail, respectively, to find the transformation in question. $\endgroup$
    – Daniel Lichtblau
    Mar 31, 2019 at 13:43
  • $\begingroup$ OK. I'll add an example right now. $\endgroup$
    – Mirko Aveta
    Mar 31, 2019 at 13:55
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    $\begingroup$ Linearize is misspelled in the message -- odd since it is the message head. $\endgroup$
    – Michael E2
    Mar 31, 2019 at 17:21
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    $\begingroup$ You can inspect some methods via ?*`*Linearize*. The misspelling comes from Control`StateAndFeedbackLinearizationsDump`ssLinearize0 $\endgroup$
    – Michael E2
    Mar 31, 2019 at 17:40

1 Answer 1

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The theory behind this is in Chapter 5 of the book Nonlinear Dynamical Control Systems by Henk Nijmeijer and Arjan van der Schaft.

The more powerful linearization is FeedbackLinearize which uses feedback, in addition to state transformation, to linearize a system.

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  • $\begingroup$ Thank you very much Suba! I was actually interested only in straightening because I was willing to see the "linear" properties of a nonlinear control applied to a linear system. $\endgroup$
    – Mirko Aveta
    Apr 1, 2019 at 17:05

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