  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 trade-off 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.
