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 where

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.

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

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 and the linearization results only hold in this smaller region.

The goal of feedback linearization is to produce a transformed system whose states are the output

To understand the structure of this target system, we use the Lie derivative.

as, Note that the notation of Lie derivatives is convenient when we take multiple derivatives with respect to either the same vector field, or a different one.

For example, and In our feedback linearized system made up of a state vector of the output

To do this, we introduce the notion of relative degree.

if, Considering this definition of relative degree in light of the expression of the time derivative of the output

, we can consider the relative degree of our system (1) and (2) to be the number of times we have to differentiate the output

In an LTI system, the relative degree is the difference between the degree of the transfer function's denominator polynomial (i.e., number of poles) and the degree of its numerator polynomial (i.e., number of zeros).

For the discussion that follows, we will assume that the relative degree of the system is

So long as this transformation is a diffeomorphism, smooth trajectories in the original coordinate system will have unique counterparts in the

trajectories will be described by the new system, Hence, the feedback control law renders a linear input–output map from

may be chosen using standard linear system methodology.

In particular, a state-feedback control law of where the state vector

derivatives, results in the LTI system with, So, with the appropriate choice of

, we can arbitrarily place the closed-loop poles of the linearized system.

Feedback linearization can be accomplished with systems that have relative degree less than

However, the normal form of the system will include zero dynamics (i.e., states that are not observable from the output of the system) that may be unstable.

In practice, unstable dynamics may have deleterious effects on the system (e.g., it may be dangerous for internal states of the system to grow unbounded).

Minimum phase systems provide some insight on zero dynamics.

Although NDI is not necessarily restricted to this type of system, lets consider a nonlinear MIMO system that is affine in input

To use a similar derivation as for SISO, the system from Eq.

Working this out the same way as SISO, one finds that defining a virtual input

equations of the form shown in Eq.