Algebraic necessary and sufficient conditions of input-output linearization

This paper exposes two different versions of the input-output linearization problem by dynamic feedback for nonlinear control systems. The first one (weak version) considers only the input-output behavior while the second one (strong version) assures linear state equations for the input-output subsystem. It also gives the corresponding necessary and sufficient conditions for their solvability in terms of intrinsic conditions. For the first version, the necessary and sufficient condition is a rank condition, which, roughly speaking, expresses the fact that the differential output rank is equal to the rank that can be calculated when considering only the linear differential relations among the output components. The condition for the second version of the problem is the isomorphism of two algebraic structures constructed from the output components. It is shown that the structure algorithm is a convenient tool for verifying the fulfillment of these conditions, and to construct the solution when this problem is solvable. It is also established that quasi-static state feedback is sufficiently general to solve the input-output linearization for classical control systems.