Necessary conditions for asymptotic tracking in nonlinear systems

In the literature, it has been shown that if a single-input single-output analytic nonlinear plant 1) has a well-defined relative degree and 2) is minimum-phase, it is possible to achieve asymptotic tracking for an open set of output trajectories containing the origin in C^N [0, ∞), the space of N-times continuously differentiable functions taking values in R. When either of these sufficient conditions is not met, various authors have investigated approximate analytic solutions, discontinuous solutions and solutions for restricted sets of trajectories. In this paper, it is shown that conditions 1) and 2) are necessary for the existence of an analytic compensator which yields asymptotic tracking for an open set of output trajectories. Analogous results are established for multi-input multi-output systems