  Designing controllers to satisfy temporal requirements has proven to be
challenging for dynamical systems that are affected by uncertainty. This is
mainly due to the states evolving in a continuous uncountable space, the
stochastic evolution of the states, and infinite-horizon temporal requirements
on the system evolution, all of which makes closed-form solutions generally
inaccessible. A promising approach for designing provably correct controllers
on such systems is to utilize the concept of abstraction, which is based on
building simplified abstract models that can be used to approximate optimal
controllers with provable closeness guarantees. The available abstraction-based
methods are further divided into discretization-based approaches that build a
finite abstract model by discretizing the continuous space of the system, and
discretization-free approaches that work directly on the continuous state space
without the need for building a finite space. To reduce the conservatism in the
sub-optimality of the designed controller originating from the abstraction
step, this paper develops an approach that naturally has the flexibility to
combine different abstraction techniques from the aforementioned classes and to
combine the same abstraction technique with different parameters. First, we
develop a multi-layered discretization-based approach with variable precision
by combining abstraction layers with different precision parameters. Then, we
exploit the advantages of both classes of abstraction-based methods by
extending this multi-layered approach guided by the specification to
combinations of layers with respectively discretization-based and
discretization-free abstractions. We achieve an efficient implementation that
is less conservative and improves the computation time and memory usage. We
illustrate the benefits of the proposed multi-layered approach on several case
studies.
