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Modern control synthesis methods that are capable of delivering safety guarantees typically rely on finding invariant sets. Computing and/or representing such sets becomes intractable for high-dimensional systems and often constitutes the main bottleneck of computational procedures. In this paper we instead analytically study a particular high-dimensional system and propose a control strategy that we prove renders a set invariant whenever it is possible to do so. The control problem---the mode-counting problem with two modes in one dimension---is inspired by scheduling of thermostatically controlled loads (TCLs) and exhibits a trade-off between local safety constraints and a global counting constraint. We improve upon a control strategy from the literature to handle heterogeneity and derive sufficient conditions for the strategy to solve the problem at hand. In addition, we show that the conditions are also necessary for the problem to have a solution, which implies a type of optimality of the proposed control strategy. We outline more general problem instances where the same control strategy can be implemented and we give sufficient (but not necessary) conditions for the closed-loop system to satisfy its specification. We illustrate our results on a TCL scheduling example.

On a Class of Maximal Invariance Inducing Control Strategies for Large Collections of Switched Systems