Modulating function-based fast convergent observer for the Coupled Tanks system

In this research, we apply the observer approach introduced by Djennoune et al. \cite{djennoune2019modulating} to estimate water levels in a coupled tanks system. Central to this approach is the use of a remarkable modulating function-based transformation $T_n$, which employs a time/output-dependent coordinate transformation. This transformation converts the original system into a form where the effects of initial conditions are effectively nullified. The primary advantage of utilizing the $T_n$ transformation is its ability to achieve instantaneous convergence, ensuring both rapid and accurate state estimation. The observer's finite-time convergence is assured, with the estimation error remaining bounded within a finite period. Numerical simulations further validate the effectiveness of this method for the Coupled Tanks system, demonstrating the robustness of the $T_n$ transformation in practical applications.