  Designing estimation algorithms for systems governed by partial differential
equations (PDEs) such as fluid flows is challenging due to the high-dimensional
and oftentimes nonlinear nature of the dynamics, as well as their dependence on
unobserved physical parameters. In this paper, we propose two different
lightweight and effective methodologies for real-time state estimation of PDEs
in the presence of parametric uncertainties. Both approaches combine a Kalman
filter with a data-driven polytopic linear reduced-order model obtained by
dynamic mode decomposition (DMD). Using examples involving the nonlinear
Burgers and Navier-Stokes equations, we demonstrate accurate estimation of both
the state and the unknown physical parameter along system trajectories
corresponding to various physical parameter values.
