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In this paper, we introduce a compositional scheme for the construction offinite abstractions (a.k.a. symbolic models) of interconnected discrete-timecontrol systems. The compositional scheme is based on small-gain typereasoning. In particular, we use a notion of so-called alternating simulationfunctions as a relation between each subsystem and its symbolic model. Assumingsome small-gain type conditions, we construct compositionally an overallalternating simulation function as a relation between an interconnection ofsymbolic models and that of original control subsystems. In suchcompositionality reasoning, the gains associated with the alternatingsimulation functions of the subsystems satisfy a certain "small-gain"condition. In addition, we introduce a technique to construct symbolic modelstogether with their corresponding alternating simulation functions fordiscrete-time control subsystems under some stability property. Finally, weapply our results to the temperature regulation in a circular building byconstructing compositionally a finite abstraction of a network containing $N$rooms for any $N\geq3$. We use the constructed symbolic models as substitutesto synthesize controllers compositionally maintaining room temperatures in acomfort zone. We choose $N=1000$ for the sake of illustrating the results. Wealso apply our proposed techniques to a nonlinear example of fully connectednetwork in which the compositionality condition still holds for any number ofcomponents. In these case studies, we show the effectiveness of the proposedresults in comparison with the existing compositionality technique in theliterature using a dissipativity-type reasoning.

Compositional Synthesis of Finite Abstractions for Networks of Systems: A Small-Gain Approach