  In this paper, we develop a compositional scheme for the construction of
continuous approximations for interconnections of infinitely many discrete-time
switched systems. An approximation (also known as abstraction) is itself a
continuous-space system, which can be used as a replacement of the original
(also known as concrete) system in a controller design process. Having designed
a controller for the abstract system, it is refined to a more detailed one for
the concrete system. We use the notion of so-called simulation functions to
quantify the mismatch between the original system and its approximation. In
particular, each subsystem in the concrete network and its corresponding one in
the abstract network are related through a notion of local simulation
functions. We show that if the local simulation functions satisfy certain
small-gain type conditions developed for a network containing infinitely many
subsystems, then the aggregation of the individual simulation functions
provides an overall simulation function quantifying the error between the
overall abstraction network and the concrete one. In addition, we show that our
methodology results in a scale-free compositional approach for any
finite-but-arbitrarily large networks obtained from truncation of an infinite
network. We provide a systematic approach to construct local abstractions and
simulation functions for networks of linear switched systems. The required
conditions are expressed in terms of linear matrix inequalities that can be
efficiently computed. We illustrate the effectiveness of our approach through
an application to AC islanded microgirds.
