Safety for Weakly-Hard Control Systems via Graph-Based Barrier Functions

Despite significant advancement in technology, communication and computational failures are still prevalent in safety-critical engineering applications. Often, networked control systems experience packet dropouts, leading to open-loop behavior that significantly affects the behavior of the system. Similarly, in real-time control applications, control tasks frequently experience computational overruns and thus occasionally no new actuator command is issued. This article addresses the safety verification and controller synthesis problem for a class of control systems subject to weakly-hard constraints, i.e., a set of window-based constraints where the number of failures are bounded within a given time horizon. The results are based on a new notion of graph-based barrier functions that are specifically tailored to the considered system class, offering a set of constraints whose satisfaction leads to safety guarantees despite communication failures. Subsequent reformulations of the safety constraints are proposed to alleviate conservatism and improve computational tractability, and the resulting trade-offs are discussed. Finally, several numerical case studies demonstrate the effectiveness of the proposed approach.