2. 6. Model - based calibration procedure
In order to predict ethanol concentrations from the data of the gas
sensor array, the following principle component regression model (the
chemometric model) was applied.
\(\ c_{E}=\ p_{0}\ +\ \left(p_{1}\ \times\text{PC}_{1}\right)\ +\ (p_{2}\ \times\ {\text{PC}_{1}}^{2})\)
Where \(c_{E}\) is the predicted ethanol concentration,\(\text{PC}_{1}\) is the first principle component of the gas sensor
array data and \(p_{0}\), \(p_{1}\) and \(p_{2}\) are the parameters of
the model.
The simulated ethanol concentrations calculated from the process model
were used as reference data for calibrating the response of the gas
sensor array. In order to calculate the simulated ethanol
concentrations, the values of the specific growth rates were required.
For obtaining these values the following procedure was applied:
During the first step, roughly estimated starting values of the specific
growth rates that are used and the simulated ethanol concentration is
calculated. During the calibration procedure, the evaluation of the
simulated ethanol concentration is compared with the predicted ethanol
concentration and the sum of squared differences is calculated. In the
next step, the error of prediction is minimized by implementing an
optimization algorithm. The algorithm changes the process model
parameters (\(\mu_{G0}\) and \(\mu_{E0}\)) as well as the parameters of
the chemometric model (\(p_{0}\), \(p_{1}\) and \(p_{2}\)). All the
steps are processed in a cycle until the minimum of the sum of squared
differences is obtained. The flowchart of the model-based calibration
procedure is presented in Fig. 4.
The optimization method which was used to minimize the error of
prediction is a particle swarm optimization algorithm. This algorithm
works by improving a population of candidate solutions called particles,
which are the parameters of the mathematical models (here the specific
growth rates as well as the parameters of the chemometric model). The
particles are flying through the search space and the velocity of each
particle is determined by the position of its best-known performance as
well as the position of the overall swarm’s best known performance. The
swarm iteratively moves to the best solution. A more detailed
description can be found in the literature (Wang, Gandomi et al. 2014).
By applying this model-based calibration method, appropriate values for
the parameters of the theoretical process model (\(\mu_{G0}\) and\(\mu_{E0}\)) can be estimated. Furthermore, the optimal parameters of
the calibration model are calculated which are used for predicting
ethanol concentration.