A Tutorial to Multirate Extended Kalman Filter Design for Monitoring of Agricultural Anaerobic Digestion Plants

In many applications of biotechnology, measurements are available at different sampling rates, e.g., due to online sensors and offline lab analysis. Offline measurements typically involve time delays that may be unknown a priori due to the underlying laboratory procedures. This multirate (MR) setting poses a challenge to Kalman filtering, where conventionally measurement data is assumed to be available on an equidistant time grid and without delays. The present study derives the MR version of an extended Kalman filter (EKF) based on sample state augmentation, and applies it to the anaerobic digestion (AD) process in a simulative agricultural setting. The performance of the MR-EKF is investigated for various scenarios, i.e., varying delay lengths, measurement noise levels, plant-model mismatch (PMM), and initial state error. Provided with an adequate tuning, the MR-EKF could be demonstrated to reliably estimate the process state, to appropriately fuse delayed offline measurements, and to smooth noisy online measurements well. Because of the sample state augmentation approach, the delay length of offline measurements does not critically impair state estimation performance, provided observability is not lost during the delays. Poor state initialization and PMM affect convergence more than measurement noise levels. Further, selecting an appropriate tuning was found to be critically important for successful application of the MR-EKF, for which a systematic approach is presented. This study provides implementation guidance for practitioners aiming at successfully applying state estimation for multirate systems. It thereby contributes to develop demand-driven operation of biogas plants, which may aid in stabilizing a renewable electricity grid.