A graphic condition for the stability of dynamical distribution networks with flow constraints

TL;DR

This paper establishes a graphical condition for the stability and load balancing of dynamical distribution networks with flow constraints, extending previous work with a Lyapunov-based analysis and controller modifications.

Contribution

It introduces a necessary and sufficient graphical condition for load balancing under flow constraints and proposes a modified PI controller to steer storage variables within admissible sets.

Findings

Derived a graphic condition for load balancing with flow constraints.

Proved the condition's necessity and sufficiency using Lyapunov functions.

Showed how to steer storage variables to desired points within constraints.

Abstract

We consider a basic model of a dynamical distribution network, modeled as a directed graph with storage variables corresponding to every vertex and flow inputs corresponding to every edge, subject to unknown but constant inflows and outflows. In [1] we showed how a distributed proportionalintegral controller structure, associating with every edge of the graph a controller state, regulates the state variables of the vertices, irrespective of the unknown constant inflows and outflows, in the sense that the storage variables converge to the same value (load balancing or consensus). In many practical cases, the flows on the edges are constrained. The main result of [1] is a sufficient and necessary condition, which only depend on the structure of the network, for load balancing for arbitrary constraint intervals of which the intersection has nonempty interior. In this paper, we will consider the question about how to decide the steady states of the same model as in [1] with given network structure and constraint intervals. We will derive a graphic condition, which is sufficient and necessary, for load balancing. This will be proved by a Lyapunov function and the analysis the kernel of incidence matrix of the network. Furthermore, we will show that by modified PI controller, the storage variable on the nodes can be driven to an arbitrary point of admissible set.