Approximation methods for optimal network coding in a multi-hop control network with packet losses

A Multi-hop Control Network consists of a plant where the communication between sensors, actuators and computational units is supported by a (wireless) multi-hop communication network, and data flow is performed using scheduling and routing of sensing and actuation data. With the aim of rendering the system robust with respect to packet losses, we exploit network coding and redundancy in data communication (i.e. sending multiple copies of sensing and actuation data via multiple routing paths associated to possibly different delays) and assume that such data is re-combined as a weighed linear combination. The main contribution of this paper is to provide efficient computational methods to find the optimal choice of such weights either to maximize or to constrain a robustness metric based on the notion of Asymptotic Mean Square Stability. Since we observe that such metric induces an objective or constraint function that is highly non-linear, we propose an approximation that takes into account both the network parameters and the system dynamics: it is based on the assumption that packet loss probabilities are much smaller than correct transmission probabilities and leverages the well known Gerschgorin disks. We then compare our approximation with the optimal choice from the communication designer point of view, which neglects the system dynamics and is based on the idea of minimizing the quadratic error induced by the network on the actuation signal. The comparison of the above approximations quantitatively shows that the optimal choices from the points of view of control and communication theory are often at odds.