Feedback linearization: Difference between revisions

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[[File:Feedback linearization.svg|thumb|Block diagram illustrating the feedback linearization of a nonlinear system]]

'''Feedback linearization''' is a common approachstrategy usedemployed in controlling [[nonlinear systemscontrol]]. Theto approach involves coming up with a transformation of thecontrol [[nonlinear system into an equivalent linear system through a change of variables and a suitable control inputsystems]]. Feedback linearization techniques may be applied to nonlinear control systems of the form

:<math>\begin{align}\dot{x}(t) &= f(x(t)) + g\sum_{i=1}^{m}\,g_i(x(t))u \qquad &,u_i(1t)\\</math>

y &= h(x) \qquad \qquad \qquad &(2)\end{align}</math>

where <math>x(t) \in \mathbb{R}^n</math> is the state vector, <math>uu_1(t), \ldots, u_m(t) \in \mathbb{R}^p</math> isare the vectorinputs. The approach involves transforming a nonlinear control system into an equivalent linear control system through a change of inputs,variables and a suitable control input. In particular, one seeks a change of coordinates <math>yz = \inPhi(x)</math> and control input <math>u = a(x) + b(x)\mathbb{R}^m,v,</math> isso that the vectordynamics of outputs.<math>x(t)</math> Thein goalthe iscoordinates to<math>z(t)</math> develop take the form of a linear, controllable control inputsystem,

:<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.

:<math>\dot{z}(t) = A\,z(t) + \sum_{i=1}^{m} b_i\,v(t).</math><ref>H. Nijmeijer and A. van der Shaft 2016, p. 163</ref><ref>A. Isidori 1995, p. 152</ref>

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 to achieve the control objective.

== Feedback Linearization of SISO Systems ==