Relative structure at infinity and nonlinear semi-implicit DAE´S

A notion of relative structure at infinity and the concept of relative output rank with respect to a subsystem are introduced. We introduce the problem of relative decoupling, showing that this problem is solvable if and only if the relative output rank ρ(z) coincides with dimz. The Relative Dynamic Extension Algorithm (RDEA) is presented and a geometric interpretation is also given, showing that it computes the relative structure at infinity. We develop necessary and sufficient conditions for testing if a given output z of system is relatively flat with respect to Y. As a byproduct, we obtain conditions for the decoupling problem and for flatness of a class of Differential Algebraic Equations (DAE´s). It is shown the the RDEA may be used for constructing dynamic linearizing feedback laws and/or the solution of the dynamic input-output decoupling problem for implicit systems.