Digital twins mainly consist of a mathematical model which describes the
dynamic behaviour observed in a biochemical reactor and a prediction or selflearning algorithm which estimates the cellular component concentrations and the
process parameters that cannot be described mechanistically [2, 3].
Bioprocess mathematical models may generally be categorized into algebraic
equations and dynamic models. Algebraic equations are developed from mass and
component balances, from mass or heat transfer laws or even from elemental
balances. Dynamic models usually consist of dynamic balances of conserved quantities in combination with kinetics to describe rate expressions as functions of the
state variables. Detailed description of mathematical modelling of bioprocesses is
covered by previous authors in greater details than space allows here [4–7]. The goal
of this chapter is to highlight state estimation methods with a specific focus on the
Kalman filter and its non-linear extensions.
For linear systems, the Luenberger observer and the Kalman filter, whose 60th
anniversary occurred in 2020 [8], are the most applied methods for estimating
parameters and process variables that cannot be measured directly. In the area of
non-linear systems, particle filtering (PF), high gain observers, non-linear extensions
of the Kalman filter such as the extended Kalman filter (EKF) and the unscented
Kalman filter (UKF) and many others have been proposed. However, due to the
simple structure and low computational effort of non-linear extensions of the
Kalman filter, these methods have gained more interest, and many research studies
have been dedicated to the implementation of such filters for state and parameter
estimation in bioprocess technologies. The main objective of this chapter is to
discuss the applications of different Kalman filter algorithms in bioprocess technologies. Therefore, this chapter is organized as follows: in the next section, a brief
overview of the Kalman filtering theory and its non-linear extensions will be
discussed. Applications of the Kalman filter for the supervision of cultivation
processes will be given in the third section, followed by a case study evaluating
the implementation of an extended Kalman filter for developing a digital twin of the
backer’s yeast batch cultivation process. In the last section, a conclusion is
presented.
2 Kalman Filtering Theory and Its Non-linear Extensions
The Kalman filter is a set of mathematical equations that provides an efficient
computational solution of the least-squares method when the considered system is
linear and the uncertainties are modelled by Gaussian random variables. When the
system state dynamics is non-linear, then certain linearization methods are applied.
The most prominent of these algorithms are the extended Kalman filter (EKF) and
the unscented Kalman filter (UKF), invented independently by several research
groups. Different extensions of the Kalman filters differ in the way the estimation
error is calculated. A brief overview of these methods are as follows.
98
A. Yousefi-Darani et al.
dynamic behaviour observed in a biochemical reactor and a prediction or selflearning algorithm which estimates the cellular component concentrations and the
process parameters that cannot be described mechanistically [2, 3].
Bioprocess mathematical models may generally be categorized into algebraic
equations and dynamic models. Algebraic equations are developed from mass and
component balances, from mass or heat transfer laws or even from elemental
balances. Dynamic models usually consist of dynamic balances of conserved quantities in combination with kinetics to describe rate expressions as functions of the
state variables. Detailed description of mathematical modelling of bioprocesses is
covered by previous authors in greater details than space allows here [4–7]. The goal
of this chapter is to highlight state estimation methods with a specific focus on the
Kalman filter and its non-linear extensions.
For linear systems, the Luenberger observer and the Kalman filter, whose 60th
anniversary occurred in 2020 [8], are the most applied methods for estimating
parameters and process variables that cannot be measured directly. In the area of
non-linear systems, particle filtering (PF), high gain observers, non-linear extensions
of the Kalman filter such as the extended Kalman filter (EKF) and the unscented
Kalman filter (UKF) and many others have been proposed. However, due to the
simple structure and low computational effort of non-linear extensions of the
Kalman filter, these methods have gained more interest, and many research studies
have been dedicated to the implementation of such filters for state and parameter
estimation in bioprocess technologies. The main objective of this chapter is to
discuss the applications of different Kalman filter algorithms in bioprocess technologies. Therefore, this chapter is organized as follows: in the next section, a brief
overview of the Kalman filtering theory and its non-linear extensions will be
discussed. Applications of the Kalman filter for the supervision of cultivation
processes will be given in the third section, followed by a case study evaluating
the implementation of an extended Kalman filter for developing a digital twin of the
backer’s yeast batch cultivation process. In the last section, a conclusion is
presented.
2 Kalman Filtering Theory and Its Non-linear Extensions
The Kalman filter is a set of mathematical equations that provides an efficient
computational solution of the least-squares method when the considered system is
linear and the uncertainties are modelled by Gaussian random variables. When the
system state dynamics is non-linear, then certain linearization methods are applied.
The most prominent of these algorithms are the extended Kalman filter (EKF) and
the unscented Kalman filter (UKF), invented independently by several research
groups. Different extensions of the Kalman filters differ in the way the estimation
error is calculated. A brief overview of these methods are as follows.
98
A. Yousefi-Darani et al.