Soft-Constrained Distributed Cascaded Cooperative Kalman Filter for Mobile Robots in Unknown Advection-Diffusion Field

Real-time estimation and modeling of dynamic phenomena governed by partial differential equations play a key role in applications of environmental monitoring, disaster response, and industrial control. Accurately capturing the spatiotemporal dynamics of these processes is essential for rapid analysis and effective management. Motivated by this need, we introduce a novel distributed cascaded cooperative Kalman filter for mobile robots to estimate and model a scalar field governed by an unknown advection-diffusion equation. Our approach decouples the spatiotemporal field estimation from the PDE parameter inference and incorporates soft physical constraints to ensure physically meaningful estimates while mitigating the adverse effects of feedback contamination and limited observability. We prove that the filter is convergent, fully controllable and observable, and our simulations demonstrate that the proposed method outperforms baseline techniques in both state and parameter estimation across diverse network configurations.