Incremental Exponential Stability of the Unidirectional Flow Model

Incremental stability is assessed for the Unidirectional Flow Model, which describes the flow of a conserved quantity in a cascade of compartments. The model is governed by a cooperative system of nonlinear ordinary differential equations with a right-hand side that depends on an input function. Suitable conditions are given on the input function such that the solutions to the model converge to each other at an exponential rate. Milder conditions, compared with the latter, are given such that the solutions to the model converge to each other asymptotically. These conditions are related to inflow-connected and outflow-connected compartmental systems and are easy to check. Computational means to estimate the exponential rate are given and tested against numerical solutions of the model. A natural application of incremental stability is state estimation, demonstrated in the numerical experiments, put into a traffic flow context.