  Trajectory tracking of nonlinear dynamical systems with affine open-loop
controls is investigated. The control task is to enforce the system state to
follow a prescribed desired trajectory as closely as possible. We introduce
exactly realizable desired trajectories as these trajectories which can be
tracked exactly by an appropriate control. Exactly realizable trajectories are
characterized mathematically by means of Moore-Penrose projectors constructed
from the input matrix. The approach leads to differential-algebraic systems of
equations and is considerably simpler than the related concept of system
inversion. Furthermore, we identify a particularly simple class of nonlinear
affine control systems. Systems in this class satisfy the so-called linearizing
assumption and share many properties with linear control systems. For example,
conditions for controllability can be formulated in terms of a rank condition
for a controllability matrix analogously to the Kalman rank condition for
linear time-invariant systems.
