State and disturbance source estimation in Fisher–Kolmogorov equation

The problem of state and disturbance estimation based on a limited number of measurements is addressed for processes that can be modeled by Fisher–Kolmogorov-type Partial Differential Equations (PDEs). The Petrov–Galerkin approximation is employed to derive an Ordinary Differential Equation (ODE) model suitable for observer design. A nonlinear state observer is introduced to estimate the state (solution) of the Fisher–Kolmogorov PDE based on this model. The observer can efficiently reconstruct spatially distributed biological, chemical, or ecological invasion-like processes by applying only a limited number of measurements. Using Lyapunov techniques, it is demonstrated that the proposed observer ensures the convergence of the estimated state to the true state, although the system’s nonlinearity does not satisfy globally the Lipschitz condition. In cases where the system’s dynamics are influenced by an unknown disturbance source, a spatial disturbance localization method is introduced, leveraging the same model. Furthermore, a technique for estimating the magnitude of the unknown disturbance is presented using disturbance observer methods. Simulation results are provided to demonstrate the efficacy of the proposed state and source estimation methods.