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Concept
Feedback linearization
Summary
Feedback linearization is a common strategy employed in nonlinear control to control nonlinear systems. Feedback linearization techniques may be applied to nonlinear control systems of the form
where is the state, are the inputs. The approach involves transforming a nonlinear control system into an equivalent linear control system through a change of variables and a suitable control input. In particular, one seeks a change of coordinates and control input so that the dynamics of in the coordinates take the form of a linear, controllable control system,
An outer-loop control strategy for the resulting linear control system can then be applied to achieve the control objective.
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, and . The objective is to find a coordinate transformation that transforms the 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 and the linearization results only hold in this smaller region.
Several tools are required to solve this problem.
The goal of feedback linearization is to produce a transformed system whose states are the output and its first derivatives. To understand the structure of this target system, we use the Lie derivative.
Official source
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