Compositional abstraction refinement for control synthesis under lasso-shaped specifications

This paper presents a compositional approach to specification-guided abstraction refinement for control synthesis of a nonlinear system associated with a method to over-approximate its reachable sets. The control specification consists in following a lasso-shaped sequence of regions of the state space. The dynamics are decomposed into subsystems with partial control, partial state observation and possible overlaps between their respective observed state spaces. A finite abstraction is created for each subsystem through a refinement procedure, which starts from a coarse partition of the state space and then proceeds backwards on the lasso sequence to iteratively split the elements of the partition whose coarseness prevents the satisfaction of the specification. The composition of the local controllers obtained for each subsystem is proved to enforce the desired specification on the original system. This approach is illustrated in a nonlinear numerical example.