  In this paper, we introduce a compositional scheme for the construction of
finite abstractions (a.k.a. symbolic models) of interconnected discrete-time
control systems. The compositional scheme is based on small-gain type
reasoning. In particular, we use a notion of so-called alternating simulation
functions as a relation between each subsystem and its symbolic model. Assuming
some small-gain type conditions, we construct compositionally an overall
alternating simulation function as a relation between an interconnection of
symbolic models and that of original control subsystems. In such
compositionality reasoning, the gains associated with the alternating
simulation functions of the subsystems satisfy a certain "small-gain"
condition. In addition, we introduce a technique to construct symbolic models
together with their corresponding alternating simulation functions for
discrete-time control subsystems under some stability property. Finally, we
apply our results to the temperature regulation in a circular building by
constructing compositionally a finite abstraction of a network containing $N$
rooms for any $N\geq3$. We use the constructed symbolic models as substitutes
to synthesize controllers compositionally maintaining room temperatures in a
comfort zone. We choose $N=1000$ for the sake of illustrating the results. We
also apply our proposed techniques to a nonlinear example of fully connected
network in which the compositionality condition still holds for any number of
components. In these case studies, we show the effectiveness of the proposed
results in comparison with the existing compositionality technique in the
literature using a dissipativity-type reasoning.
