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.