From ideal to packet-based communication for spatially invariant systems with various interconnection structures

In this paper we consider systems consisting of a number of spatially invariant, i.e., identical, subsystems that use packet-based communication networks for the exchange of information. Recent literature has shown that for an infinite number of such interconnected networked subsystems, the overall system can be modeled as an infinite interconnection of identical hybrid (sub)systems. Based on this hybrid modeling perspective, conditions were derived guaranteeing uniform global exponentially stability (UGES) or Lp_p-gain performance. These conditions were formulated locally in the sense that only the local dynamics of a single hybrid subsystem in the interconnection is needed in order to obtain a maximally allowable transmission interval (MATI) for all of the individual communication networks such that these global stability properties are guaranteed for the complete infinite-dimensional system. In this work we will connect these results to other known results in the literature concerning infinite-dimensional spatially invariant systems and extend them in various directions, thereby showing the generic nature of the obtained hybrid modeling framework. In particular, it is shown that the stability and performance conditions as derived for the perfect communication case of the infinite spatially invariant interconnection guarantee robustness of the stability/performance property in the packet-based communication case. In addition, extensions to periodic interconnections and finite interconnections with boundary conditions will be discussed explicitly, bringing the theory closer to practical applications, which will be epitomized in a two-sided vehicular platooning example concerning Lp₂-stability.