Compositional Synthesis of Safety Barrier Certificates for Infinite Networks

This paper offers a formal compositional frame-work for the construction of control barrier certificates (eBe) for an interconnected network comprised of a countably infinite number of discrete-time control subsystems. Inspired by small-gain type conditions designed inherently for the stability analysis of large-scale interconnected systems, our proposed approach aims to synthesize control strategies that ensure safety properties across infinite networks using local certificates of their individual subsystems. Specifically, this is achieved by employing a notion of control sub-barrier certificates (eSBe) for individual subsystems, using which one can ensure that the interconnected network avoids entering unsafe regions over an infinite time horizon under specific small-gain type conditions. We utilize a sum-of-squares (SOS) optimization approach to systematically search for eSBe and their associated control strategies that align with the desired safety criteria. We show case the efficacy of our compositional approach through its application to a vehicle platooning scenario, which involves a countably infinite number of vehicles with a single leader and an unlimited number of followers.