The Strong Integral Input-to-State Stability Property in Dynamical Flow Networks

TL;DR

This paper introduces the Strong integral Input-to-State Stability property for dynamical flow networks, providing a unified framework to analyze stability under non-linear flow constraints and exogenous inflows, applicable to complex network types.

Contribution

It establishes sufficient and necessary conditions for input-to-state stability in various complex flow networks using the Strong iISS property, extending previous partial analyses.

Findings

Strong iISS quantifies exogenous inflow effects on flow dynamics.

Unified stability analysis for networks with cycles and non-monotone flows.

Conditions for maximum inflow to ensure network stability.

Abstract

Dynamical flow networks serve as macroscopic models for, e.g., transportation networks, queuing networks, and distribution networks. While the flow dynamics in such networks follow the conservation of mass on the links, the outflow from each link is often non-linear due to, e.g., flow capacity constraints and simultaneous service rate constraints. Such non-linear constraints imply a limit on the magnitude of exogenous inflow that is able to be accommodated by the network before it becomes overloaded and its state trajectory diverges. This paper shows how the Strong integral Input-to-State Stability (Strong iISS) property allows for quantifying the effects of the exogenous inflow on the flow dynamics. The Strong iISS property enables a unified stability analysis of classes of dynamical flow networks that were only partly analyzable before, such as networks with cycles, multi-commodity flow networks and networks with non-monotone flow dynamics. We present sufficient conditions on the maximum magnitude of exogenous inflow to guarantee input-to-state stability for a dynamical flow network, and we also present cases when this sufficient condition is necessary. The conditions are exemplified on a few existing dynamical flow network models, specifically, fluid queuing models with time-varying exogenous inflows and multi-commodity flow models.