Feedback linearization: Difference between revisions

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== Feedback Linearization of SISO Systems ==

Here, we consider the case of feedback linearization of a single-input single-output (SISO) system. Similar results can be extended to multiple-input multiple-output (MIMO) systems. In this case, <math>u \in \mathbb{R}</math> and <math>y \in \mathbb{R}</math>. WeThe wishobjective is to find a coordinate transformation <math>z = T(x)</math> that transforms ourthe system (1) into the so-called [[Normal form (abstract rewriting)|normal form]] which will reveal a feedback law of the form

:<math>u = a(x) + b(x)v\,</math>

that will render a linear input&ndash;output map from the new input <math>v \in \mathbb{R}</math> to the output <math>y</math>. To ensure that the transformed system is an equivalent representation of the original system, the transformation must be a [[diffeomorphism]]. That is, the transformation must not only be invertible (i.e., bijective), but both the transformation and its inverse must be [[smooth function|smooth]] so that differentiability in the original coordinate system is preserved in the new coordinate system. In practice, the transformation can be only locally diffeomorphic, but the linearization results only hold in this smaller region.

We require severalSeveral tools beforeare werequired canto solve this problem.

=== Lie derivative ===