## Materials and Methods
### Controlled Plant
#### 2.1.3 Linearisation and Eigenvalue Analysis  
The majority of adaptive control approaches are derived theoretically for controlled plants, which are represented by linear models with unknown parameters. The stability of the entire control system is proved theoreticaly only in the case of a controlled plant described with a linear model with (unknown) constant parameters. It was found out by numerical calculation that the majority of adaptive controllers are also successful at controlling processes in which parameters change relatively slowly, or are even not distinctly non-linear. For this purpose, before laboratory tests, we checked thoroughly by simulation the characteristics and the behaviour of the analysed non-linear model of the batch bioreactor. The analysis of the dynamic characteristics of the fermentation process during the operation was made by means of the linearisation of the non-linear model (1)-(4) and eigenvalue analysis of the obtained linearised model. During usual working circumstances, the analysed non-linear \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline The substrate inhibition constant & \(K_{i}=74.58\) g/L \\ The substrate saturation constant & \(K_{\text{m}}=0.50\) g/L \\ The product inhibition constant & \(P_{\text{m}}=12.60\) g/L \\ The maximum microorganism’s growth rate & \(\mu_{\text{m}}=0.45\) h\({}^{-1}\) \\ The product yield related to microorganism’s growth & \(\alpha=0.90\) \\ The product yield independent of the microorganism’s growth & \(\beta=0.001\) h\({}^{-1}\) \\ The initial value of the concentration of the biomass & \(X(0)=2.9\) g/L \\ The initial value of the concentration of the substrate & \(S(0)=9.0\) g/L \\ The initial value of the concentration of the product & \(P(0)=0.1\) g/L \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters and initial values of the non-linear (autonomous) mathematical model of the analysed fermentation process in the batch bioreactor [31]. Figure 1: Time courses of concentrations of microorganisms, substrate, and product of analysed fermentation process in batch bioreactor with constant stirrer speed \(n=80\) min\({}^{-1}\), obtained with a non-linear mathematical model (1)–(4) with the parameters in Table 1.  
model operates along the trajectory, which depends on model parameters and initial concentrations of substances. Small non-zero disturbances lead to small deviations in model state variables. The behaviour within a certain operating range of a trajectory can be approximated reasonably by that of a linearised model [32]. The trajectory which follows the presented model parameters and initial values (Table 1) is denoted with \(X^{\star}(t)\), \(S^{\star}(t)\), and \(P^{\star}(t)\), the actual state-space variables are denoted with \(X(t)\), \(S(t)\), and \(P(t)\), and the deviations between actual state-space variables and the variables of the linearisation trajectory are marked with \(X_{\Delta}(t)\), \(S_{\Delta}(t)\), and \(P_{\Delta}(t)\), as defined with Equations (5):  
\[ \begin{array}{l}X_{\Delta}(t)=X(t)-X^{\star}(t)\\ S_{\Delta}(t)=S(t)-S^{\star}(t)\\ P_{\Delta}(t)=P(t)-P^{\star}(t)\end{array}  \tag{5}\]  
The non-linear model in Equations (1)-(4) can be rewritten in the form of Equations (6)-(8):  
\[\dot{X}(t)=\frac{\mu_{m}\!\!\left(\!1-\frac{P(t)}{P_{m}}\!\right)\!S(t)}{K_{m} +S(t)+\frac{(S(t))^{2}}{K_{i}}}X(t)=f_{1}(X(t),S(t),P(t)) \tag{6}\]  
\[\dot{S}(t)=-\frac{\mu_{m}\!\!\left(\!1-\frac{P(t)}{P_{m}}\!\right)\!S(t)}{K_{m }+S(t)+\frac{(S(t))^{2}}{K_{i}}}X(t)=f_{2}(X(t),S(t),P(t)) \tag{7}\]  
\[\dot{P}(t)=\!\!\left[\alpha\,\frac{\mu_{m}\!\!\left(\!1-\frac{P(t)}{P_{m}}\! \right)\!S(t)}{K_{m}+S(t)+\frac{(S(t))^{2}}{K_{i}}}+\beta\right]\!\!X(t)=f_{3} (X(t),S(t),P(t)) \tag{8}\]  
The linearised model is described with the state-space Equation (9):  
\[\left[ \begin{array}{c}\dot{X}_{\Delta}(t)\\ \dot{S}_{\Delta}(t)\\ \dot{P}_{\Delta}(t)\end{array} \right]=\mathbf{A}(t)\!\left[ \begin{array}{c} X_{\Delta}(t)\\ S_{\Delta}(t)\\ P_{\Delta}(t)\end{array}