Complete Abstractions of Monotone Control Systems: From Model-based to Data-Driven Systems

In this paper, we introduce the approximate strong upper alternating simulation (ASUAS), a new behavioral relation for transition systems. Building on this relation, we construct upper- and lower-sparse abstractions for monotone systems that together form a complete abstraction pair: any controller synthesized for the upper-sparse abstraction can be refined into a controller for the original system, and the absence of a controller for the lower-sparse abstraction implies the absence of a controller for the original system. A key feature of our approach is the ability to provably tune the conservativeness gap between the two abstractions by tuning the space-discretization parameter. We further extend these results, beyond the model-based setting, to data-driven systems, where the abstraction is constructed directly from finite sampled data, without requiring an explicit system model. The theoretical results are illustrated through simulations.