Approximate Markovian abstractions for linear stochastic systems

In this paper, we present a method to generate a finite Markovian abstraction for a discrete time linear stochastic system evolving in a full dimensional polytope. Our approach involves an adaptation of an existing approximate abstraction procedure combined with a bisimulation-like refinement algorithm. It proceeds by approximating the transition probabilities from one region to another by calculating the probability from a single representative point in the first region. We derive the exact bound of the approximation error and an explicit expression for its growth over time. To achieve a desired error value, we employ an adaptive refinement algorithm that takes advantage of the dynamics of the system. We demonstrate the performance of our method through simulations.