In the first step, the initial values for the state and covariance estimation have to be
set. Following this, the recursive estimation is performed by the prediction and
correction steps. Within the prediction step, a priori state and covariance estimation
utilizing the process model is performed. Using the unscented transformation, a set
of sigma points are chosen. These sigma points characterize the current probability
density function. Each point from the sigma matrix is propagated through the
process model to calculate the estimations of state variables and the error covariance.
Following this, a correction step is preformed when a measurement is received. This
leads to the estimations of the filtered state variables and the filtered error covariance
by calculating the Kalman gain.
The UKF has been used in various fields for non-linear sate estimations. However
a couple of alternative approaches have emerged over the last few years, namely, the
ensemble Kalman filter (EnKF) and the cubature Kalman filter (CKF) which are
widely used when the process model is of extremely high order and non-linear, the
initial states are highly uncertain and a large number of measurements are available
[16, 17].
Similar to the UKF, the EnKF and CKF select a set of sample points (sigma
points) in order to deal with the non-linearity of the system. In high-dimension
systems, the weights of the sigma points in the UKF are prone to be negative, leading
to low estimation accuracy.
In EnKF the error covariances are estimated approximately using an ensemble of
model forecasts. The main concept behind the formulation of the EnKF is that if the
dynamical model is expressed as a stochastic differential equation, the prediction
error statistics, which are described by the Fokker–Plank equation, can be estimated
using ensemble integrations, and the error covariance matrices can be calculated by
integrating the ensemble of model states [16].
The cubature Kalman filter uses the spherical–radial cubature rule to generate
some weighted sampling points to approximate integral in Bayesian estimation. A
brief overview of the unscented Kalman filtering and sigma point filtering in general
are given by van der Merwe [18].
3 Application of Kalman Filters in Bioprocess Monitoring
Here 41 recent published articles [19–60] in the period of 1991–2020 on application
of the Kalman filter and its extensions for state and parameter estimation in
bioprocesses are discussed. Due to space limitation, only some of the reported
articles are presented in Table 1. The table is organized by classifying the articles
into different categories, which include the type of the Kalman filter and the applied
process model, the type of microorganism and the cultivation process mode, the
measured process variable(s) and the objective of the filtering algorithm. This table
would help understanding how the Kalman filter was explored chronologically to
date. It should be mentioned that in some works more than one Kalman filter
The Kalman Filter for the Supervision of Cultivation Processes
103
set. Following this, the recursive estimation is performed by the prediction and
correction steps. Within the prediction step, a priori state and covariance estimation
utilizing the process model is performed. Using the unscented transformation, a set
of sigma points are chosen. These sigma points characterize the current probability
density function. Each point from the sigma matrix is propagated through the
process model to calculate the estimations of state variables and the error covariance.
Following this, a correction step is preformed when a measurement is received. This
leads to the estimations of the filtered state variables and the filtered error covariance
by calculating the Kalman gain.
The UKF has been used in various fields for non-linear sate estimations. However
a couple of alternative approaches have emerged over the last few years, namely, the
ensemble Kalman filter (EnKF) and the cubature Kalman filter (CKF) which are
widely used when the process model is of extremely high order and non-linear, the
initial states are highly uncertain and a large number of measurements are available
[16, 17].
Similar to the UKF, the EnKF and CKF select a set of sample points (sigma
points) in order to deal with the non-linearity of the system. In high-dimension
systems, the weights of the sigma points in the UKF are prone to be negative, leading
to low estimation accuracy.
In EnKF the error covariances are estimated approximately using an ensemble of
model forecasts. The main concept behind the formulation of the EnKF is that if the
dynamical model is expressed as a stochastic differential equation, the prediction
error statistics, which are described by the Fokker–Plank equation, can be estimated
using ensemble integrations, and the error covariance matrices can be calculated by
integrating the ensemble of model states [16].
The cubature Kalman filter uses the spherical–radial cubature rule to generate
some weighted sampling points to approximate integral in Bayesian estimation. A
brief overview of the unscented Kalman filtering and sigma point filtering in general
are given by van der Merwe [18].
3 Application of Kalman Filters in Bioprocess Monitoring
Here 41 recent published articles [19–60] in the period of 1991–2020 on application
of the Kalman filter and its extensions for state and parameter estimation in
bioprocesses are discussed. Due to space limitation, only some of the reported
articles are presented in Table 1. The table is organized by classifying the articles
into different categories, which include the type of the Kalman filter and the applied
process model, the type of microorganism and the cultivation process mode, the
measured process variable(s) and the objective of the filtering algorithm. This table
would help understanding how the Kalman filter was explored chronologically to
date. It should be mentioned that in some works more than one Kalman filter
The Kalman Filter for the Supervision of Cultivation Processes
103