Local factorization of trajectory lifting morphisms for single-input affine control systems

Trajectory preserving and lifting maps have been implicitly used in many recursive or hierarchical control design techniques. Well known systems theoretic concepts such as differential flatness or more recent ones such as simulation and bisimulation can be also understood through the trajectory lifting maps they define. In this paper we initiate a study of trajectory preserving and lifting maps between affine control systems. Our main result shows that any trajectory lifting map between two single-input control affine systems can be locally factored as the composition of two special trajectory lifting maps: a projection onto a quotient system followed by a differentially flat output with respect to another control system. We use this decomposition result to show that under mild regularity conditions, trajectory preserving maps between single-input affine control systems also lift trajectories. As an additional application of the main result, we also show how the hierarchical stabilization method known as back-stepping can be used based on the existence of a trajectory preserving and lifting map having a feedback stabilizable control system as codomain.