Relative Structure at Infinity and Nonlinear Semi-implicit DAEs *

This paper considers explicit systems with an output η=(z,y). One may be interested in controlling the output z independently of the behavior of y, introducing problem of relative-decoupling. The Relative Dynamic Extension Algorithm (RDEA) is presented, and it is shown that it computes some geometric invariants, namely, the relative structure at infinity of z, which governs the solvability of the relative-decoupling problem. Age neralization of the notion of zero dynamics arises and, when this zero dynamics is absent, the output z is said to be a relatively flat output. The dimension of the state of the generalized zero dynamics can be computed from the relative structure at infinity, furnishing a test for verifying if z is a relatively flat output. When one adds the constraint y ≡ 0, then the system becomes a Differential Algebraic System (DAE). In this context, relative-decoupling and relative-flatness of the explicit system implies respectively, decoupling and flatness of the corresponding DAE. Hence, the RDEA may be used for computing dynamic feedback for decoupling and/or linearizing implicit systems.