Path-Complete Barrier Functions for Safety of Switched Linear Systems

In this paper, we address the safety verification problem of switched linear dynamical systems under arbitrary switching via barrier functions. Our approach is based on a notion of path-complete barrier functions, which utilizes a collection of barrier functions associated with a directed labeled graph that can encode all the possible switching sequences. We show that path-complete barrier functions effectively generalize notions of common and multiple barrier functions studied in existing literature, and can potentially provide less conservative conditions for safety verification. We demonstrate that, for switched linear systems, the inequalities imposed via path-complete barrier functions can be easily encoded into simple linear matrix inequalities under some assumptions on the regions of interest and appropriately chosen templates for the barrier functions. We also study the relationship between path-complete barrier functions and common barrier functions, and show that for any path-complete graph with an admissible path-complete barrier function, one can derive a suitable (possibly non-smooth) common barrier function by utilizing the path-complete barrier function. Finally, we utilize several examples to illustrate the effectiveness of our approach, and briefly discuss the challenges that lay foundations for future research.