## Materials and Methods
### Controlled Plant
#### 2.1.4 Mathematical Model for a Batch Bioreactor with a Changeable Stirrer Speed  
Almost all known mathematical models of batch bioreactors (also this one, presented with Equations (1)-(4)) represent a bioreactor as an autonomous system. They describe only the time responses of the bioreactor's media concentration on different initial conditions. These models are not convenient for the design of the bioreactor's control system, because they do not express the influence of the possible control variables (the stirrer's drive speed or the heater's temperature) on the bioreactor's controlled variables (concentrations of biomass, substrate, and product). Until now, there have only been a few publications available wherein the influence of the heat (temperature) and mechanical work (stirrer's speed) were studied on the bioprocess's dynamics. An original approach based on experimental analysis and theoretical insight used for modelling the bioreactor's product release has been published in [26; 33]. In these papers, the bioreactor's response to different heater temperatures and stirrer speed variations was investigated in different phases of the fermentation process. The observed bioreactor's output variable was CO\({}_{2}\). The milk fermentation process studied was considered as a two-input (heater's temperature, stirrer's speed), one-output (CO \begin{table}
\begin{tabular}{c c c c} \hline \hline \(t\) **(h)** & \(A_{\mathbf{1}}\) & \(A_{\mathbf{2}}\) & \(A_{\mathbf{3}}\) \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 2: Eigenvalues of the linearised mathematical model of a non-linear model of the analysed fermentation process in a batch bioreactor. concentration) dynamic system. The main conclusion of the comprehensive laboratory work was that the batch bioreactor's mathematical model could be considered as composed of two parts [33]:  
* An autonomous mathematical model, which describes the dynamics of the bioreactor as a response to initial states and assumes constant media temperature and stirrer's rotation speed. * A supplementary input/output mathematical model, which describes the influence of heat or mixing on the CO\({}_{2}\) release. Reference [33] shows that we get a good match between the results of laboratory experiments and simulations if we use the described non-linear fundamental model (1)-(4) for an autonomous model, and we apply a linear input-output model of appropriate level as a supplementary input/output model. The enhanced mathematical model was developed using a two-stage procedure. Firstly, the determination of the autonomous non-linear third order mathematical model was performed, which describes CO\({}_{2}\) release behaviour in response to initial conditions by constant process inputs. Secondly, transfer functions (or Differential Equations) were developed, describing CO\({}_{2}\) release according to two changeable process inputs. For the design and synthesis of the control system, a supplementary input/output mathematical model is crucial. In [33], the order of the supplementary input/output model was obtained by means of systematic research with a least-square identification method for different batch bioreactors and different fermentation media in different phases of fermentation. It was concluded that the heater's temperature or the stirrer's speed impact on the CO\({}_{2}\) release can be modelled with a dynamical system built of two parallel-connected transfer functions:  
* A first-order term with gain \(k_{1}\) and time constant \(T_{1}\). * A second-order term with gain \(k_{2}\), the time constant \(T_{2}\), and damping \(z\), which is in series with a differentiator with the time constant \(T_{\text{d}}\). The first-order term denotes a major (dominant) connection between the bioreactor's inputs and the dynamics of the CO\({}_{2}\) release. The second part (the second order term with the differentiator) has an impact during a transient response, and is important for fine tuning of the controller.