## 3 Results and Discussion
The time course of the reference profile is shown in Figure 6. As can be seen in the Figures 5 and 6, the reference trajectory has a negligible lag phase compared to the non-controlled course, and the acceleration part of the growth phase has a maximum allowed gradient. Therefore, the stationary phase is reached faster, which results in a shortening of the fermentation time. Additionally, the steady state value of the reference trajectory is set higher than the stationary value of the non-controlled fermentation process. A duty of an adaptive controller is to ensure the time course of the output quantity will follow the prescribed reference trajectory. The time course of the reference trajectory differs from the response of the uncontrolled batch bioreactor. The difference is in the time course of the transitive phenomenon and in the stationary value. From the comparison of the non-controlled and reference CO\({}_{2}\) release, we conclude that the task of the adaptive controller will be to increase the gradient of the CO\({}_{2}\) release at the beginning of the fermentation process, to achieve higher steady-state value, and to hold the CO\({}_{2}\) value in the batch reactor almost constant (the same as prescribed with the reference trajectory) during the last phase of the fermentation process. A parallel AMFC system was used with signal-synthesis adaptation. Its goal was to achieve that the time response of the CO\({}_{2}\) concentration would be as close as possible to the reference profile. It follows that the dynamics of the fermentation process would be similar to the dynamics of the reference model. The selected reference model was determined based on the reference profile defined with Equation (30). The parameters of the reference model were chosen in a way that the step response of the model was equal to the empirically obtained trajectory. The following reference model was obtained:  
\[\dot{x}_{\text{m}}(t)=-0.71\;x_{\text{m}}(t)+0.57\;u_{\text{m}}(t) \tag{31}\]  
The parameters in the adaptation mechanism (i.e., values of **D**, **F**, **M**, **G**, **N**, **F**', and M\({}^{\prime}\)) were estimated with support of a simulation based on a typical enhanced model. The goal was to find such parameters of the adaptation mechanism which ensured stable operation and good dynamic properties in the case of non-linear models of different batch bioreactors. We did not look intentionally for parameters that would ensure optimum operation of the selected batch bioreactor, but we tried to determine as many universal parameters that would ensure good control of various batch bioreactors in the entire range of operation. The selected parameters of the reference model, control law, and adaptation mechanism are presented in Table 4. Since the reference model was of the first degree, all elements are scalar. Figure 6: Time course of dissolved CO\({}_{2}\) concentration \(\gamma\) of the chosen reference model for the laboratory batch bioreactor. The time response of the CO\({}_{2}\) release obtained with the AMFC system is shown in Figure 7. The trajectory is almost the same as prescribed with the reference model. The output of the digital adaptive controller represents the reference signal for the stirrer's drive system. The reference signal is shown in Figure 8. The oscillations are also a consequence of insufficiently filtered measurement noise. \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{A_{m}}=-0.71\) & \(\mathbf{B_{m}}=0.57\) & \\ \(\mathbf{K_{p}}=0\) & \(\mathbf{K_{m}}=0\) & \(\mathbf{K_{u}}=0\) \\ \(\mathbf{D}=25\) & \(\mathbf{G}=10\) & \(\mathbf{N}=10\) \\ \(\mathbf{F}=10\) & \(\mathbf{F^{\prime}}=10\) & \(\mathbf{M}=10\) & \(\mathbf{M^{\prime}}=10\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of the reference model, control law, and adaptation mechanism for the control of the fermentation process in the laboratory batch bioreactor.