Boundary moving horizon estimator for approximate models of parabolic PDEs

In this work, we focus on the state estimation of the parabolic stochastic partial differential equations (PDEs) with boundary observation. The standard Kalman filter as the optimal estimator with assumption of stochastic process features and known variances on state and output disturbances can not account for the naturally present constraints on the estimated states and state disturbances. Therefore, a motivation to explore the moving horizon estimator (MHE) in the distributed parameter system setting, comes from the idea to synthesize an estimator that provides the best state estimate in a deterministic sense when process and measurement disturbances are with unknown statistics and when process constraints on states and disturbances are present. We explore the parabolic PDEs model with boundary observation, and the spectral decomposition approach is employed to yield a finite dimensional system, which incorporates low dimensional approximation of the original infinite-dimensional system. The boundary moving horizon estimator (MHE) combined with Kalman filter is built to reconstruct accurately the low dimensional approximation of the PDE state based on the noise corrupted boundary observations and estimated bounds arising from the infinite-dimensional parabolic PDEs state representation. The issue of parabolic PDEs state constraints inclusion in the MHE with Kalman filter is demonstrated by relevant simulation study of reaction-diffusion parabolic PDEs process with disturbance constraints and demonstration of accurate PDE state reconstruction.