Feedback linearization: Difference between revisions

Revision as of 17:02, 3 May 2009

Revision as of 17:09, 3 May 2009

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'''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 following form:

:<math>\begin{align}\dot{x} &= f(x) + g(x)u \qquad &(1)\\

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

Wherewhere <math>x \in \mathbb{R}^n</math> is the state vector, <math>u \in \mathbb{R}^p</math> is the vector of inputs, and <math>y \in \mathbb{R}^m</math> is the vector of outputs. The goal, then, is to develop a control input <math>u</math> that renders either the input-output map linear, or results in a linearization of the full state of the system.

:<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 ==

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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{R}</math> and <math>y \in \mathbb{R}</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 will render a linear input--&ndash;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, but the linearization results only hold in this smaller region.

We require several tools before we can solve this problem.

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:<math>u = \frac{1}{L_{g}L_{f}^{n-1}h(x)}(-L_{f}^{n}h(x) + v)</math>

renders a linear input--&ndash;output map from <math>v</math> to <math>z_1 = y</math>. The resulting linearized system

:\begin{cases}\dot{z}_1 &= z_2\\