Extended Kalman Filter Design for Semilinear Distributed Parameter Systems with Application to Anaerobic Digestion

Abstract

This paper is concerned with the online state estimation of a class of one-dimensional semilinear partial differential equation (PDE) systems considering piecewise measurements over the spatial domain. In the context of infinite-dimensional linear systems, it is well-known that the Kalman filter minimizes the mean square estimation error. For semilinear infinite-dimensional systems, the extended Kalman filter (EKF) is a widely used extension relying on successive linearizations of the estimation error dynamics. In this paper, we propose a computationally tractable implementation of the EKF using a sample-and-hold approach for which the optimal output injection operator associated to the proposed estimator is computed at each sampling time via the approximate solution of the infinite-dimensional Riccati equation. The performance of the observer is exemplified through numerical experiments which demonstrate the efficiency of the proposed approach.