6 research outputs found
Robustly Complete Finite-State Abstractions for Control Synthesis of Stochastic Systems
The essential step of abstraction-based control synthesis for nonlinear
systems to satisfy a given specification is to obtain a finite-state
abstraction of the original systems. The complexity of the abstraction is
usually the dominating factor that determines the efficiency of the algorithm.
For the control synthesis of discrete-time nonlinear stochastic systems
modelled by nonlinear stochastic difference equations, recent literature has
demonstrated the soundness of abstractions in preserving robust probabilistic
satisfaction of {\omega}-regular lineartime properties. However, unnecessary
transitions exist within the abstractions, which are difficult to quantify, and
the completeness of abstraction-based control synthesis in the stochastic
setting remains an open theoretical question. In this paper, we address this
fundamental question from the topological view of metrizable space of
probability measures, and propose constructive finite-state abstractions for
control synthesis of probabilistic linear temporal specifications. Such
abstractions are both sound and approximately complete. That is, given a
concrete discrete-time stochastic system and an arbitrarily small
L1-perturbation of this system, there exists a family of finite-state
controlled Markov chains that both abstracts the concrete system and is
abstracted by the slightly perturbed system. In other words, given an
arbitrarily small prescribed precision, an abstraction always exists to decide
whether a control strategy exists for the concrete system to satisfy the
probabilistic specification