Title: Compositional (In)Finite Abstractions for Large-Scale Interconnected Stochastic Systems,
Abstract: This paper is concerned with a compositional approach for constructing both
infinite (reduced-order models) and finite abstractions (a.k.a. finite Markov
decision processes) of large-scale interconnected discrete-time stochastic
control systems. The proposed framework is based on the notion of stochastic
simulation functions enabling us to use an abstract system as a substitution of
the original one in the controller design process with guaranteed error bounds.
In the first part of the paper, we derive sufficient small-gain type conditions
for the compositional quantification of the probabilistic distance between the
interconnection of stochastic control subsystems and that of their infinite
abstractions. We then construct infinite abstractions together with their
corresponding stochastic simulation functions for a class of discrete-time
nonlinear stochastic control systems. In the second part of the paper, we
leverage small-gain type conditions for the compositional construction of
finite abstractions. We propose an approach to construct finite Markov decision
processes (MDPs) of the concrete models (or their reduced-order versions)
satisfying an incremental input-to-state stability property. We also show that
for a particular class of nonlinear stochastic control systems, the
aforementioned property can be readily checked by matrix inequalities. We
demonstrate the effectiveness of the proposed results by applying our
approaches to the temperature regulation in a circular building and
constructing compositionally a finite abstraction of a network containing 1000
rooms. We also apply our proposed techniques to a fully connected network of 20
nonlinear subsystems (totally 100 dimensions) and construct finite MDPs from
their reduced-order versions (together 20 dimensions) with guaranteed error
bounds on their output trajectories.