The Impact of Packet Dropouts on the Reachability Energy

We consider control systems where the input signal is transferred over a network and therefore, it is subject to packet losses. In this setting, the closed-loop behavior can be described as a constrained switching system. We investigate whether there exists a switching signal that prevents reachability of some target state, or alternatively, how much additional input energy is required to reach a target state in comparison to the dropout-free case. Mathematically, we formulate a reachability problem defined on a hybrid automaton and tackle an optimization problem, whose feasibility variants, the controllability and reachability properties, have been recently shown to be decidable. To do so, we provide automata-theoretic algorithms to study the properties of an appropriate generalization of the Controllability Gramian matrix. Additionally, we provide polynomial time heuristics for computations for a specific family of automata and show numerical evidence that they work well in practice. Last, we extend our observations to the analogous observability energy problem.