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Mark viking (talk | contribs) Adding local short description: "Approach used in controlling nonlinear systems", overriding Wikidata description "common approach used in controlling nonlinear systems; involves coming up with a transformation of the nonlinear system into an equivalent linear system through a change of variables and a suitable control input" (Shortdesc helper) |
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Here, 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>. The objective is to find a coordinate transformation <math>z = T(x)</math> that transforms the 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–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
Several tools are required to solve this problem.
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