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, and . We wish to find a coordinate transformation that transforms our system (1) into the so-called normal form which will reveal a feedback law of the form
that will render a linear input–output map from the new input to the output . 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 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 several tools before we can solve this problem.
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