Consistent abstractions of affine control systems

In this paper, we consider the problem of constructing abstractions of affine control systems that preserve reachability properties, and, in particular, local accessibility. In this framework, showing local accessibility of the higher level, abstracted model is equivalent to showing local accessibility of the, more detailed, lower level model. Given an affine control system and a smooth surjective map, we present a canonical construction for extracting an affine control system describing the trajectories of the abstracted variables. We then obtain conditions on the abstraction maps that render the original and abstracted system equivalent from a local accessibility point of view. Such consistent hierarchies of accessibility preserving abstractions of nonlinear control systems are then considered for various classes of affine control systems including linear, bilinear, drift free, and strict feedback systems.