Contraction-based Observers using non-Euclidean Norms with an Application to Traffic Networks
For control theorists and engineers, it extends contraction-based observer design to non-Euclidean norms, enabling applications where such norms are beneficial.
The paper proposes a contraction-based observer design for nonlinear systems using non-Euclidean norms, proving exponential convergence if the matrix measure of a Jacobian is uniformly negative. Applied to traffic density estimation, it demonstrates convergence under the one-norm.
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.