Contraction-based Observers using non-Euclidean Norms with an Application to Traffic Networks

TL;DR

This paper develops contraction-based observers for nonlinear systems using non-Euclidean norms, demonstrating exponential convergence of state estimates, with an application to traffic network density estimation.

Contribution

It introduces contraction conditions with respect to non-Euclidean norms for observer design, expanding applicability beyond Euclidean-based methods.

Findings

Observer estimates converge exponentially under the proposed conditions.

Contraction with respect to the one-norm is established for traffic density estimation.

The approach is demonstrated on a traffic network model.

Abstract

In this note, we study Luenberger-type full-state observers for nonlinear systems using contraction theory. We show that if the matrix measure of a suitably defined Jacobian matrix constructed from the dynamics of the system-observer interconnection is uniformly negative, then the state estimate converges exponentially to the actual state. This sufficient condition for convergence establishes that the distance between the estimate and state is infinitesimally contracting with respect to some norm on the state-space. In contrast to existing results for contraction-based observer design, we allow for contraction with respect to non-Euclidean norms. Such norms have proven useful in applications. To demonstrate our results, we study the problem of observing vehicular traffic density along a freeway modeled as interconnected, spatially homogenous compartments, and our approach relies on establishing contraction of the system-observer interconnection with respect to the one-norm.