A regularity condition for asymptotic tracking in discrete-time nonlinear systems

A well-defined relative degree, which is one of the basic assumptions in adaptive control or nonlinear synthesis problems, is addressed. It is shown that this is essentially a necessary condition for asymptotic tracking in discrete-time nonlinear systems. To show this, tracking problems are defined, and a local linear input-output behavior of a discrete-time system is introduced in relation to a well-defined relative degree. It is then shown that if a plant is invertible and accessible from the origin and a compensator solves the local asymptotic tracking problem, then the plant necessarily has a well-defined relative degree at the origin.