State estimation for parabolic PDEs with reactive-convective non-linearities

An extended Luenberger observer is proposed for the solution of the state estimation problem for semi-linear parabolic PDEs with reactive-convective non-linearities. Here, the backstepping method is applied to the linearised observer error dynamics to determine the observer gains. This, however, requires a successive evaluation of the so-called Hopf-Cole transformation allowing to transform the PDE of the linearised observer error into a normal form, for which backstepping can be directly used. Moreover, the computational efficiency of determination of the gains is improved by combining the direct numerical solution approach with the sample-and-hold implementation. Finally, the observer error convergence is analysed both theoretically and by means of numerical simulations.