Coordination Control in a Cyberphysical Environment

An important class of cyberphysical systems are flow networks. Motivated by the node load balancing problem for dynamical flow networks, we propose a self-triggered gossiping control algorithm. This is an edge-based control algorithm in which the inter-node flow is decided in a pair-wise fashion by the two nodes that exchange the flow. The times at which the flow is updated is designed on-line on the basis of the measurements relative to the quantity of goods stored at the nodes. To the best of our knowledge, this is the first time that a gossiping algorithm is devised for deterministic continuous-time systems without relying on an a priori schedule of the measurement and control times. 1 A motivating example: flow networks Flow networks are graphs where flows are associated to edges and stored quantities to the nodes. Flow networks arise in many disciplines and model a variety of transportation phenomena ranging from traffic to supply chains and water networks. In its simplest form, a flow network is an undirected graph G = (I, E), where I is the set of n nodes and E the set of m edges. Associated to the graph G is the n×m incidence matrix B. For each node i ∈ V , let xi denote the state variable representing the amount of stored material at the node. Similarly, for each edge k, let uk be the flow of material through the edge. Then the system ẋ = Bu models the evolution of the stored variables at the nodes as a function of the amount of flow exchanged among the nodes. In this system, the inter-nodes flow is viewed as a control variable at the edge. For each edge, the corresponding control variable has access to local information, namely to the difference in quantities stored at the two nodes connected by the edge. This information is compactly collected in the output vector z = BTx. Load balancing. The basic node load balancing problem for flow networks is as follows. Given the flow network ẋ = Bu find an output feedback control law u = Kz such that for any initial condition the solution of the closed-loop system converges asymptotically to the bisector {x ∈ Rn : x1 = x2 = . . . = xn}. It is well-known that if the graph G is connected, the matrix K = −Im provides a solution to the load balancing problem and the solution of the closed-loop system globally converges to a specific point of the bisector, namely to the average 1nx(0)/n of the initial conditions. Hence, the initial (possibly uneven) initial distribution of quantities stored ad the nodes is evenly distributed as time elapses. Although the setting is rather simplistic (no complex dynamics at the edges or at the nodes, no inflow and outflow, no capacity constraints, etc.) the example above highlights the importance of cooperative control algorithms in the control of dynamical flow networks. ∗Joint work with Paolo Frasca, Department of Mathematical Sciences, Politecnico di Torino, Italy. The work is partially supported by the Dutch Organization for Scientific Resaerch under the auspices of the project QUICK.