  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.
