To wait or to drop: On the optimal number of retransmissions in wireless control

The dimensioning of wireless communication protocols for networked control involves a non-trivial trade-off between reliability and delay. Due to the lossy nature of wireless communications, there is a risk that sensor messages will be dropped. The end-to-end reliability can be improved by retransmitting dropped messages, but this comes at the expense of additional delays. In this work, we determine the number of retransmissions that strikes the optimal balance between communication reliability and delay, in the sense that it achieves the minimal expected linear-quadratic loss of the closed-loop system. An important feature of our setup is that it accounts for the random delays and possible losses that occur when unreliable communication is combatted with retransmissions. The resulting controller dynamically switches among a set of infinite-horizon linear-quadratic regulators, and is simple to implement. Numerical simulations are carried out to highlight the trade-off between reliability and delay.