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 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 where x ∈ R n {displaystyle xin mathbb {R} ^{n}} is the state vector, u ∈ R p {displaystyle uin mathbb {R} ^{p}} is the vector of inputs, and y ∈ R m {displaystyle yin mathbb {R} ^{m}} is the vector of outputs. The goal is to develop a control input that renders a linear input–output map between the new input v {displaystyle v} and the output. An outer-loop control strategy for the resulting linear control system can then be applied. 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, u ∈ R {displaystyle uin mathbb {R} } and y ∈ R {displaystyle yin mathbb {R} } . The objective is to find a coordinate transformation z = T ( x ) {displaystyle z=T(x)} 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 v ∈ R {displaystyle vin mathbb {R} } to the output y {displaystyle y} . 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, but the linearization results only hold in this smaller region.