  Sufficiently accurate finite state models, also called symbolic models or
discrete abstractions, allow one to apply fully automated methods, originally
developed for purely discrete systems, to formally reason about continuous and
hybrid systems, and to design finite state controllers that provably enforce
predefined specifications. We present a novel algorithm to compute such finite
state models for nonlinear discrete-time and sampled systems which depends on
quantizing the state space using polyhedral cells, embedding these cells into
suitable supersets whose attainable sets are convex, and over-approximating
attainable sets by intersections of supporting half-spaces. We prove a novel
recursive description of these half-spaces and propose an iterative procedure
to compute them efficiently. We also provide new sufficient conditions for the
convexity of attainable sets which imply the existence of the aforementioned
embeddings of quantizer cells. Our method yields highly accurate abstractions
and applies to nonlinear systems under mild assumptions, which reduce to
sufficient smoothness in the case of sampled systems. Its practicability in the
design of discrete controllers for nonlinear continuous plants under state and
control constraints is demonstrated by an example.
