Compositional construction of abstractions for infinite networks of discrete-time switched systems. (May 2022) - Sharifi, Maryam; Swikir, Abdalla; Noroozi, Navid; Zamani, Majid
Abstract: In this paper, we develop a compositional scheme for the construction of continuous abstractions for networks of infinitely many discrete-time switched systems. In particular, the constructed abstractions are themselves also continuous-space systems with potentially lower dimensions, which can be used as replacements of the original (also known as concrete) systems in the 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. 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 a spectral small-gain condition, then the aggregation of the individual simulation functions provides an overall simulation function quantifying the error between the overall abstract 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. In this case, the 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 microgrids.