  Observers for PDEs are themselves PDEs. Therefore, producing real time
estimates with such observers is computationally burdensome. For both
finite-dimensional and ODE systems, moving-horizon estimators (MHE) are
operators whose output is the state estimate, while their inputs are the
initial state estimate at the beginning of the horizon as well as the measured
output and input signals over the moving time horizon. In this paper we
introduce MHEs for PDEs which remove the need for a numerical solution of an
observer PDE in real time. We accomplish this using the PDE backstepping method
which, for certain classes of both hyperbolic and parabolic PDEs, produces
moving-horizon state estimates explicitly. Precisely, to explicitly produce the
state estimates, we employ a backstepping transformation of a hard-to-solve
observer PDE into a target observer PDE, which is explicitly solvable. The MHEs
we propose are not new observer designs but simply the explicit MHE
realizations, over a moving horizon of arbitrary length, of the existing
backstepping observers. Our PDE MHEs lack the optimality of the MHEs that arose
as duals of MPC, but they are given explicitly, even for PDEs. In the paper we
provide explicit formulae for MHEs for both hyperbolic and parabolic PDEs, as
well as simulation results that illustrate theoretically guaranteed convergence
of the MHEs.
