Exactly realizable desired trajectories

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.