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

Abstract

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.

Codesign of controller, routing and scheduling in WirelessHART networked control systems

“…We assumed that a time‐invariant controller gain is designed via assigning the eigenvalues of the closed‐loop system in the nominal case (ie, neglecting the effect of packet losses) and that the actuator computes a linear combination of the data incoming from different routing paths and provided a suboptimal algorithm to compute the optimal weights γ i that maximize a metric induced by the notion of mean square stability. This paper extends and improves the results in the work of Smarra et al in several ways. First, we introduce the communication scheduling in our model and consider the effect of scheduling on the communication delay of each path.…”

“…In the work of Smarra et al, a state‐feedback control loop was considered as in Figure where multiple copies of the same actuation data are sent from the controller to the plant via r routing paths, each characterized by a delay equal to the number of hops n i (ie, scheduling was not considered, as instead we do in this paper) and a packet losses probability p i . We assumed that a time‐invariant controller gain is designed via assigning the eigenvalues of the closed‐loop system in the nominal case (ie, neglecting the effect of packet losses) and that the actuator computes a linear combination of the data incoming from different routing paths and provided a suboptimal algorithm to compute the optimal weights γ i that maximize a metric induced by the notion of mean square stability.…”

“…Second, we assume that the actuation packets u 1 ( k ),…, u r ( k ) are not necessarily equal and we optimally design a time‐varying gain matrix for each path in the network, ie, , taking into account the effect of packet losses and of the delays induced by the scheduling. Note that the problem formulation in the work of Smarra et al is a special case of the above definition where . Furthermore, while in the work of Smarra et al, the optimization is made on a metric based on the notion of mean square stability (ie, only taking into account the steady‐state behavior), here, we only focus on the transient behavior setting up a finite‐horizon LQR problem.…”

“…Note that the problem formulation in the work of Smarra et al is a special case of the above definition where . Furthermore, while in the work of Smarra et al, the optimization is made on a metric based on the notion of mean square stability (ie, only taking into account the steady‐state behavior), here, we only focus on the transient behavior setting up a finite‐horizon LQR problem.…”

Efficient routing redundancy design over lossy networks

“…In these works, NCSs with missing packets are modeled as time‐homogeneous MJLSs, which are an important family of stochastic hybrid systems. In particular, it is shown (eg, in other works()) that discrete‐time MJLSs (MJLSs) represent a promising mathematical model to jointly take into account the dynamics of a physical plant and the nonidealities of wireless communication such as packet losses. This is the main reason for adopting such a model in this paper.…”