A Stability-Based Abstraction Framework for Reach-Avoid Control of Stochastic Dynamical Systems with Unknown Noise Distributions

Finite-state abstractions are widely studied for the automated synthesis of correct-by-construction controllers for stochastic dynamical systems. However, existing abstraction methods often lead to prohibitively large finite-state models. To address this issue, we propose a novel abstraction scheme for stochastic linear systems that exploits the system's stability to obtain significantly smaller abstract models. As a unique feature, we first stabilize the open-loop dynamics using a linear feedback gain. We then use a model-based approach to abstract a known part of the stabilized dynamics while using a data-driven method to account for the stochastic uncertainty. We formalize abstractions as Markov decision processes (MDPs) with intervals of transition probabilities. By stabilizing the dynamics, we can further constrain the control input modeled in the abstraction, which leads to smaller abstract models while retaining the correctness of controllers. Moreover, when the stabilizing feedback controller is aligned with the property of interest, then a good tradeoff is achieved between the reduction in the abstraction size and the performance loss. The experiments show that our approach can reduce the size of the graph of abstractions by up to 90% with negligible performance loss.