## 3 Results and Discussion  
A dSpace 1103 controller board was utilised for the prototyping of the proposed adaptive control algorithms. An analogue output module of a basic device, SevenMulti (Metler Toledo), was used for the transfer of the measured signal of the CO\({}_{2}\) concentration from the bioreactor system to the control device. The control algorithm (i.e., reference model, control law, and adaptation mechanism) was programmed and uploaded into the controller board. The calculated control signal (i.e., reference speed) was routed to the stirrer drive via the controller's analogue output. An additional electronic interface was required in the stirrer drive, which allowed the speed reference value to be changed. The proposed adaptive control was used for the described laboratory batch bioreactor and milk fermentation with kefir grains. The time response of the dissolved CO\({}_{2}\) concentration during the fermentation process without a control system is shown in Figure 5. In this case, the stirrer's speed (\(n=80\) min\({}^{-1}\)) stayed constant during the entire fermentation process. The CO\({}_{2}\) time response obtained was similar to the response of the first order differential equations, which is consistent with the known mathematical model of non-controlled fermentation [26]. Deviations from the response of the first order differential equations are a consequence of the nature of the biochemical process itself. Microorganisms' activity often varies because they are highly sensitive to process condition changes (temperature, concentration, pH), viability changes and mutations. Disturbances are also seen clearly in the time course of the fermentation product. Therefore, the repetition of the fermentation, despite the same initial concentrations and unchanged bioreactor, does not yield exactly the same time responses. On the basis of systematic tests, it is possible to determine empirically what is the maximum possible achieved product concentration, and what can be the shortest duration of fermentation. Operators of batch bioreactors possess empirical experience on a trajectory of output quantity that ensures a desired yield of a quality product in a fermentation process. On the basis of systematic tests, it is possible to determine empirically what is the desired final product concentration, and what can be the shortest duration of a fermentation process. Operators of batch bioreactors possess empirical experience on the reference profile of output quantity that ensures high yield of a quality product in a short time. The presented study does not deal with the methods of determining this optimal reference profile. The primary purpose of this study is the development of a control system that will ensure that the CO\({}_{2}\) response will follow the arbitrarily set reference profile. To estimate the reference CO\({}_{2}\) profile for the purposes of our study, we executed the fermentation process with the same mixture systematically, but with different stirrer speeds. Because too high a gradient affects the fermentation process negatively, we tried to estimate the maximum allowable gradient that led to the reference final yield. We chose a similar shape of the reference profile as the CO\({}_{2}\) response in the fermentation process, only with a higher maximum gradient and final value. By the measurements of various bioreactors and microorganisms, we found that the CO\({}_{2}\) transient in the fermentation process is very similar to the step response of the first order differential equation [26]. Based on the estimated values about maximum allowable gradient 0.57 g/L/h and reference final product concentration 0.8 g/L, we defined the reference profile \(\gamma_{r}(t)\) with Equation (30):  
\[\gamma_{r}(t)=0.8\Big{(}1-\mathrm{e}^{-\frac{t}{L_{\mathrm{0}}}}\Big{)} \tag{30}\]  
Figure 5: Time course of dissolved CO\({}_{2}\) concentration \(\gamma\) of a non-controlled laboratory batch bioreactor by constant stirrer’s speed \(n=80\) min\({}^{-1}\).