Feedback linearization

Feedback linearization is a common approach used in controlling nonlinear systems. The approach 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. Feedback linearization may be applied to nonlinear systems of the form

<math>begindot &= f(x) + g(x)u qquad &(1)\

y &= h(x) qquad qquad qquad &(2)end</math>

where <math>x in mathbb^n</math> is the state vector, <math>u in mathbb^p</math> is the vector of inputs, and <math>y in mathbb^m</math> is the vector of outputs. The goal is to develop a control input

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

that renders a linear input&ndash;output map between the new input <math>v</math> and the output. An outer-loop control strategy for the resulting linear control system can then be applied.

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</math> and <math>y in mathbb</math>. We wish to find a coordinate transformation <math>z = T(x)</math> that transforms our system (1) into the so-called normal form which will reveal a feedback law of the form

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

that...
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