Feedback linearization

Approach used in controlling nonlinear systems

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 is the state, 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 and control input so that the dynamics of in the coordinates take the form of a linear, controllable control system,

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

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, and The objective is to find a coordinate transformation 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 to the output 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 and the linearization results only hold in this smaller region. Several tools are required to solve this problem.

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). These unobservable states may be controllable or at least stable, and so measures can be taken to ensure these states do not cause problems in practice. Minimum phase systems provide some insight on zero dynamics.