A Moving Window Formulation for Recursive Bayesian State Estimation of Systems with Irregularly Sampled and Variable Delays in Measurements

Abstract

The time delay involved between sampling and obtaining measurements of certain quality variables is a common scenario in various process applications. Further, this delay is not fixed and can vary for various reasons. Moreover, certain measurements may be sampled at irregular time intervals. The state estimation algorithms available in the literature have been developed for the scenario where the measurements are sampled regularly or are available after a fixed time delay. In this work, a recursive moving window Bayesian state estimator formulation is proposed to utilize such measurements with variable time delays to compute the state estimates. The length of the moving window ensures that the algorithm utilizes all the available measurements (delayed or otherwise) for computing the state estimates. In practice, it may also become necessary to account for the physical bounds on the states. A constrained version of the moving window recursive state estimator is also developed to yield state estimates that are consistent with their respective bounds and constraints. The efficacy of the unconstrained moving window state estimator is demonstrated by application on the benchmark Tennessee Eastman simulation case study and an experimental two-tank heater−mixer setup, while the efficacy of the constrained moving window state estimator is demonstrated by simulation of a benchmark gas-phase batch reactor system.

Robust Trajectory Tracking in a Reactive Batch Distillation Process using Multirate Nonlinear Internal Model Control

“…As indicated in the Introduction, derivative-free filters such as UKF or EnKF are better suited for dealing with the state estimation problem associated with the systems with discontinuities in the state dynamics. When measurements are available at different sampling rates, the UKF is required to be modified to account for the time-varying measurement model. , In this section, similar to the multirate DAE-EKF developed by Purohit et al, a multirate version of DAE-UKF is developed. Similar modifications can be made to the DAE-EnKF algorithm proposed by Puranik et al to arrive at a multirate version of DAE-EnKF and are not discussed separately.…”

“…When measurements are available at different sampling rates, the UKF is required to be modified to account for the time-varying measurement model. 34,29 In this section, similar to the multirate DAE-EKF developed by Purohit et al, 34 a multirate version of DAE-UKF is developed. Similar modifications can be made to the DAE-EnKF algorithm proposed by Puranik et al 38 to arrive at a multirate version of DAE-EnKF and are not discussed separately.…”

“…The majority of the state estimation algorithms available in the literature are developed under the assumption that all the measurements are available at a fast and single sampling rate. Because concentration measurements are typically available at much slower rates and with variable time delays, deterministic as well as Bayesian state estimation approaches for asynchronously sampled linear/nonlinear multirate sampled data systems have attracted significant attention in the literature in recent years. − However, these techniques have been developed for systems modeled as ordinary differential equations. These techniques, when used for systems modeled as differential algebraic equations without any modifications, can handle measurement models that are functions of the differential states alone.…”