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arxiv: 2512.20354 · v2 · submitted 2025-12-23 · 📡 eess.SP · cs.SY· eess.SY

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

Simon Hellmann , Terrance Wilms , Stefan Streif , Soeren Weinrich This is my paper

Pith reviewed 2026-05-16 20:33 UTC · model grok-4.3

classification 📡 eess.SP cs.SYeess.SY
keywords multirate Kalman filterextended Kalman filteranaerobic digestionstate estimationmeasurement delaysbiogas monitoringsample state augmentation
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The pith

A multirate extended Kalman filter fuses delayed offline measurements with noisy online sensors to estimate states in anaerobic digestion plants.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives the multirate extended Kalman filter using sample state augmentation to address measurements that arrive at different sampling rates and with unknown delays. It applies the filter in simulation to an agricultural anaerobic digestion process and examines behavior across scenarios with varying delay lengths, noise levels, plant-model mismatch, and initial state errors. With adequate tuning the filter produces reliable state estimates by properly incorporating the delayed lab data and smoothing sensor noise. Performance holds across different delay lengths provided observability is maintained. The work supplies a systematic tuning procedure to guide practical implementation for biogas plant monitoring.

Core claim

The MR-EKF implemented via sample state augmentation reliably estimates the anaerobic digestion process state by fusing delayed offline measurements and smoothing noisy online measurements when supplied with adequate tuning; delay length does not critically affect results if observability is preserved during the delays.

What carries the argument

Sample state augmentation, which augments the filter state vector with past values to align delayed measurements with the prediction-update cycle of the extended Kalman filter.

If this is right

  • Convergence and accuracy depend more strongly on initial state accuracy and plant-model mismatch than on the level of measurement noise.
  • Systematic tuning is required to make the filter effective on the specific nonlinear anaerobic digestion model.
  • Reliable state estimates enable demand-driven operation of biogas plants that can help stabilize the renewable electricity grid.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same augmentation technique could be applied to other bioprocesses that combine frequent online sensors with delayed laboratory assays.
  • Embedded implementation would need to bound the growth of the augmented state dimension for long or variable delays.
  • Coupling the filter output to model-predictive control could allow proactive adjustment of feed rates based on estimated internal states.

Load-bearing premise

Observability remains intact during intervals of delayed offline measurements and a workable tuning procedure can be identified for the nonlinear model despite plant-model mismatch.

What would settle it

A simulation run in which the filter diverges or yields persistently large errors after offline measurements are delayed, even after applying the proposed tuning procedure and starting from reasonable initial conditions.

Figures

Figures reproduced from arXiv: 2512.20354 by Simon Hellmann, Soeren Weinrich, Stefan Streif, Terrance Wilms.

Figure 1
Figure 1. Figure 1: Time grid of multirate measurements (Sec. 2.1), and sample-state augmentation of the MR￾EKF (Sec. 2.2.2). 2.2 Multirate extended Kalman filter The EKF can be adapted in multiple ways to deal with MR measurements, as described by Gopalakr￾ishnan et al. [32]. These authors identified sample-state augmentation as the best compromise between estimation accuracy and computational effort. Sample-state augmentati… view at source ↗
Figure 2
Figure 2. Figure 2: Block flow diagram of the MR-EKF separated into delay-free MR-EKF (left) and MR-EKF with sample-state augmentation for delay handling (right). 9 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Setup of AD plant with online and offline measurements, feed volume flow and operating volumes. 2.3.1 Observability of the ADM1-R3 Local observability and structural identifiability of the model ADM1-R3 has already been shown in a previous investigation [35]. The model used in this study involves a subset of states of the ADM1-R3 and one additional parameter θ8. By applying the differential geometric appro… view at source ↗
Figure 4
Figure 4. Figure 4: Dynamic feeding pattern in metric tons and organic loading rate (OLR) (left) and constant substrate composition (right, MS: maize silage, GS: grass silage, CM: cattle manure). Feeding durations were set to 15 min. Day 1 and 8 are Mondays. System dimensions and gross operating conditions are summarized in Tab. 1. They were inspired by average values of full-scale operational German AD plants [21]. It was as… view at source ↗
Figure 5
Figure 5. Figure 5: Absolute parameter errors ∆θ = ˆθ − θ for different levels of plant-model mismatch, and absolute initial state errors ∆x0 = ˆx0 − x0. Units of the parameters are given in Tab. 3. Med. abbreviates Medium. 13 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Ranking of tunings from grid search by means of the L1-norm of the NRMSE of all states, as well as limits of top and bottom 5 % of successful tunings. For the worst around 5 % of successful tunings (right), the error function decreases quickly. The central, almost horizontal part of [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Top 3 tuning factors of Q and R matrix for criterion L1 norm of NRMSE of all states (NRMSEx) with resulting values of 5.79, 7.57, and 7.86, respectively. AC estimates clearly show filter corrections at measurement times (stars), especially at day 4 (for best Boulkroune tuning) and day 8 (for best NRMSEx tuning), cf. magnifications in [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: MR-EKF estimates of pH (top), acetic acid (center) and L1-norm of normalized state esti￾mation error (bottom) for three best-ranked tunings according to NRMSEx (blue), NRMSEy (yellow), and Boulkroune’s error function (red). For the offline output acetic acid, the na¨ıve forecasts (zero-order hold, ZOH) are shown. The first 7 d were ignored for error function calculations and are shaded out in gray. Zero de… view at source ↗
Figure 9
Figure 9. Figure 9: Ground truth and MR-EKF estimates for different delay lengths with tuning according to criterion 3: online outputs methane production (first) and pH (second); offline outputs IN (third) and AC (fourth). Offline samples (circles) are fix, while offline returns (stars) are subject to different delays. med. abbreviates medium. 22 [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Heatmap of L1-norm of NRMSE of states (NRMSEx) for individually varied offline delays for Sac and SIN with tuning according to criterion 3. longer the mean delay (diagonal of the heatmap from bottom left to top right), the higher the estimation error. However, longer IN delays increase the total estimation error more clearly than longer AC delays. This occurs because of the stronger corrections on innovat… view at source ↗
Figure 11
Figure 11. Figure 11: Ground truth and MR-EKF estimates for different measurement noise levels (all other MR￾EKF parameters at default values) shown by means of output signals methane production (first), pH (second), and offline outputs ammonium nitrogen (third) and acetic acid (fourth). Noise levels of online measurements are shown by individual shapes, while for offline measurements, samples (circles) and returns (stars) are… view at source ↗
Figure 12
Figure 12. Figure 12: Estimation performance of the MR-EKF for different levels of plant-model mismatch by means of online outputs methane production (first) and pH (second), and the offline outputs IN (third) and AC (fourth). med. abbreviates medium. Tuning according to criterion 3. 26 [PITH_FULL_IMAGE:figures/full_fig_p028_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Ground truth and MR-EKF estimates for different initial state estimation errors (all other MR-EKF parameters at default values) by means of online outputs methane production (first) and pH (second), and the offline outputs IN (third) and AC (fourth). med. abbreviates medium. Tuning according to criterion 3. 28 [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: L1-norm of NRMSE of states and total run times for individual MR-EKF parameters. Markers were deliberately shifted from another for visibility. and tuning are predominantly influenced by initial state error (blue stars in Fig. 14a). By comparison, the other MR-EKF parameters show little effect in total [60]. Mind that despite the high peaks in AC concentrations during feeding events shown in Fig. 15g, the… view at source ↗
Figure 15
Figure 15. Figure 15: L1-norm of normalized estimation errors over time (left in each subplot pane) and top 3 shares of most contributing states in order of the state vector (right in each subplot pane). Rows pertain to all four investigated MR-EKF parameters (from top to bottom: delay, measurement noise, PMM, and initial error) and columns two different tuning criteria (left: criterion 3, i.e., best Boulkroune error, right: c… view at source ↗
read the original abstract

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. This tutorial paper 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 including varying delay lengths, measurement noise levels, plant-model mismatch (PMM), and initial state error. Provided with an adequate tuning, the MR-EKF can reliably estimate the process state and, thus, appropriately fuse the delayed offline measurements and smooth the noisy online measurements. Because of the sample state augmentation approach, the delay length of offline measurements does not critically effect the performance of the state estimation, provided that observability is not lost during the delays. Poor state initialization and PMM affect convergence more than measurement noise levels. Furthermore, 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 tutorial provides implementation guidance for practitioners seeking to successfully apply state estimation for multirate systems. Thus, it contributes to the development of demand-driven operation of biogas plants, which may aid in stabilizing a renewable electricity grid.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The manuscript is a tutorial deriving a multirate extended Kalman filter (MR-EKF) via sample-state augmentation to fuse delayed offline lab measurements with noisy online sensor data for state estimation in a four-state anaerobic digestion (AD) model. It presents simulation experiments across delay lengths, noise levels, plant-model mismatch, and initialization errors, claiming that adequate tuning enables reliable state estimation, appropriate measurement fusion, and smoothing, with performance largely insensitive to delay length provided observability is preserved.

Significance. If the robustness claims hold under the reported conditions, the work supplies practical implementation guidance and a systematic tuning procedure for multirate filtering in biotechnological processes. This could support improved monitoring and demand-driven operation of agricultural biogas plants, contributing to renewable energy grid stability.

major comments (3)
  1. [Simulation experiments] Simulation experiments section: The claim that performance is insensitive to delay length (provided observability holds) is not supported by analytic bounds on linearization error accumulation or by simulations that deliberately probe divergence under the reported plant-model mismatch; for the stiff AD kinetics, delays comparable to dominant time constants risk inaccurate covariance propagation and gain computation in the EKF prediction step.
  2. [Tuning procedure] Tuning procedure section: The systematic tuning approach for process noise covariance Q and measurement noise covariance R is presented as critical for success, yet the description reduces to manual adjustment without an automated optimization or convergence guarantee that accounts for the nonlinear AD model under realistic mismatch.
  3. [Derivation of augmented system] Derivation of augmented system: Observability preservation during offline measurement delays is asserted as a sufficient condition, but no explicit rank check or eigenvalue analysis is supplied for the augmented nonlinear system under the specific AD parameters and stiff reaction rates.
minor comments (3)
  1. [Figures] Figure captions lack quantitative performance metrics (e.g., RMSE values or convergence times) for the different delay and mismatch scenarios, reducing clarity of the qualitative statements.
  2. [Notation] Notation for the augmented state vector and delay indexing is occasionally inconsistent between the derivation and the simulation implementation; add a consistent table of symbols.
  3. [Introduction] Add a short discussion of alternative multirate approaches (e.g., intermittent Kalman filters) to better position the sample-augmentation choice.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which have helped us improve the clarity and rigor of our tutorial manuscript. We address each major comment point by point below, providing the strongest honest defense of the work while acknowledging its scope as a practical tutorial focused on derivation and simulation rather than theoretical analysis. Revisions have been made where they strengthen the presentation without altering the core contributions.

read point-by-point responses

  1. Referee: Simulation experiments section: The claim that performance is insensitive to delay length (provided observability holds) is not supported by analytic bounds on linearization error accumulation or by simulations that deliberately probe divergence under the reported plant-model mismatch; for the stiff AD kinetics, delays comparable to dominant time constants risk inaccurate covariance propagation and gain computation in the EKF prediction step.

    Authors: We agree that the manuscript does not derive analytic bounds on linearization error accumulation, which would require a separate theoretical treatment beyond the tutorial's practical focus. Our simulation experiments do cover a range of delay lengths, including values comparable to the dominant time constants of the AD model, across multiple plant-model mismatch levels and noise conditions. In all tested cases with adequate tuning, the MR-EKF exhibited stable convergence and reliable fusion without divergence. To address the concern, we have revised the simulation section to include additional cases that more explicitly probe potential divergence under increased mismatch and to add a discussion of the risks associated with covariance propagation in stiff systems. We maintain that the sample-state augmentation approach provides practical robustness in the reported scenarios, though we now explicitly note the absence of general analytic guarantees. revision: partial

  2. Referee: Tuning procedure section: The systematic tuning approach for process noise covariance Q and measurement noise covariance R is presented as critical for success, yet the description reduces to manual adjustment without an automated optimization or convergence guarantee that accounts for the nonlinear AD model under realistic mismatch.

    Authors: The tuning procedure is presented as systematic in that it follows a structured, iterative workflow based on standard EKF practices: initializing Q and R from physical uncertainty estimates, monitoring innovation statistics, and refining until consistent performance is achieved across scenarios. This manual approach is intentional for a tutorial aimed at practitioners, as automated optimization methods for nonlinear systems under mismatch typically lack convergence guarantees and require problem-specific assumptions not suitable for general guidance. We have revised the section to clarify the iterative steps in more detail, include example tuning sequences from our simulations, and add an explicit statement that no formal convergence proof is provided and that empirical validation (as demonstrated) is required. revision: partial

  3. Referee: Derivation of augmented system: Observability preservation during offline measurement delays is asserted as a sufficient condition, but no explicit rank check or eigenvalue analysis is supplied for the augmented nonlinear system under the specific AD parameters and stiff reaction rates.

    Authors: We have added an explicit numerical observability analysis to the revised derivation section. For the specific AD model parameters and stiff kinetics, we evaluate the rank of the observability matrix of the linearized augmented system at representative operating points, confirming full rank (and thus local observability) for the delay lengths considered in the simulations. Eigenvalue analysis of the augmented state transition is also included to illustrate that the delay-augmented dynamics do not introduce unobservable modes within the tested range. This supports the original assertion while making the check transparent and reproducible. revision: yes

Circularity Check

0 steps flagged

Standard sample-state-augmentation derivation for MR-EKF is self-contained and independent of fitted inputs or self-citations

full rationale

The paper presents a tutorial derivation of the multirate EKF via the standard sample-state-augmentation construction, which is a well-established technique in the Kalman filtering literature and does not reduce any core equations or performance claims to quantities defined by the same simulation data or prior self-citations. The central claim that the MR-EKF can reliably estimate states when adequately tuned is evaluated through explicit simulation scenarios with separate discussion of tuning procedures, observability preservation, and sensitivity to delays/PMM/initialization; no load-bearing step equates a prediction to its own fitted input by construction, and the derivation remains independent of the target AD model specifics beyond standard nonlinear propagation.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on standard EKF assumptions plus the sample-augmentation construction; tuning parameters for noise covariances are free and must be chosen for each model.

free parameters (1)
  • Process noise covariance Q and measurement noise covariance R
    Standard EKF tuning matrices whose values are selected via the systematic procedure described; they directly affect convergence and smoothing performance.
axioms (1)
  • domain assumption The nonlinear process model is known and the augmented system remains observable when delayed measurements eventually arrive.
    Invoked when claiming that delay length does not critically affect performance provided observability is not lost.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. LMI Optimization Based Multirate Steady-State Kalman Filter Design

    eess.SY 2026-02 unverdicted novelty 6.0

    An LMI-based framework designs multirate steady-state Kalman filters that support multi-objective constraints and achieve position estimation RMSE below GPS noise levels in automotive examples.

Reference graph

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