Feedback linearization

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

${\displaystyle {\dot {x}}(t)=f(x(t))+\sum _{i=1}^{m}\,g_{i}(x(t))\,u_{i}(t)}$

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where ${\displaystyle x(t)\in \mathbb {R} ^{n}}$ is the state, ${\displaystyle u_{1}(t),\ldots ,u_{m}(t)\in \mathbb {R} }$ 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 ${\displaystyle z=\Phi (x)}$ and control input ${\displaystyle u=a(x)+b(x)\,v,}$ so that the dynamics of ${\displaystyle x(t)}$ in the coordinates ${\displaystyle z(t)}$ take the form of a linear, controllable control system,

${\displaystyle {\dot {z}}(t)=A\,z(t)+\sum _{i=1}^{m}b_{i}\,v(t).}$

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An outer-loop control strategy for the resulting linear control system can then be applied to achieve the control objective.