Compositional Construction of Abstractions for Infinite Networks of Switched Systems

Abstract

We construct compositional continuous approximations for an interconnection of infinitely many discrete-time switched systems. An approximation (known as abstraction) is itself a continuous-space system, which can be used as a replacement of the original (known as concrete) system in a controller design process. Having synthesized a controller for the abstract system, the controller is refined to a more detailed controller for the concrete system. To quantify the mismatch between the output trajectory of the approximation and of that the original system, we use the notion of so-called simulation functions. In particular, each subsystem in the concrete network and its corresponding one in the abstract network is related through a local simulation function. We show that if the local simulation functions satisfy a certain small-gain type condition developed for a network of infinitely many subsystems, then the aggregation of the individual simulation functions provides an overall simulation function between the overall abstraction and the concrete network. For a network of linear switched systems, we systematically construct local abstractions and local simulation functions, where the required conditions are expressed in terms of linear matrix inequalities and can be efficiently computed. We illustrate the effectiveness of our approach through an application to frequency control in a power gird with a switched (i.e. time-varying) topology.