## 5 Numerical Simulation  
To assess the effectiveness of the proposed FOSMC controller, a numerical simulation of the controlled photobioreactor was carried out in Matlab/Simulink using the FOMCON toolbox [37]. The simulation considered the transition from batch operation to continuous operation. In this transition, where the photobioreactor operates in fed-batch mode, the controller should ensure that the biomass concentration follows a predefined path as close as possible despite unmeasured disturbances. This means that the system in Equation (7) should follow the dynamics of the reference model until the maximum volume, \(x_{3}(t)\), is reached. The reference model is given by the following:  
\[\dot{x}_{1r} = \mu_{m}x_{1}(t)-u(t)x_{1}(t),\] \[\dot{x}_{2r} = -\alpha_{1}\mu_{m}x_{1r}(t)+u(t)(\alpha_{2}-x_{2r}(t)),\] \[\mu_{mr} = \frac{a_{1,nom}x_{2r}(t)}{a_{2,nom}+x_{2r}(t)}, \tag{27}\]  
where \(a_{1,nom}\) and \(a_{2,nom}\) are the nominal model parameters (used in the design of the controller). The non-modelled disturbance function \(\Sigma\) in Equation (7) represents in our case model parameter changes due to unmeasured variations in light incidence, pH, and substrate concentration gradients. We consider that all these disturbances impact only the maximum specific growth rate as an additive term, i.e., \(a_{1}(t)=a_{1,nom}+\delta(t)\). It is assumed that this additive term suffers step changes at arbitrary times. Therefore, the \(\Sigma\) function has the following structure:  
\[\dot{x}_{1} = \mu_{m}x_{1}-\frac{x_{1}}{x_{3}}u+\Sigma x_{1},\] \[\mu_{m} = \frac{a_{1,nom}x_{2}}{a_{2}+x_{2}},\] \[\Sigma = \frac{\delta(t)x_{2}}{a_{2}+x_{2}}. \tag{28}\]  
Since the maximum norm \(\delta(t)\) and the \(a_{2}\) constant are positive, the maximum \(\Sigma\) disturbance value will always be limited by the measurable state at the output, so it is possible to expand the FOSMC shown in Equation (25) for the reduced system in Equation (28), such that \(a_{2},\delta(t)\
eq 0\rightarrow\left\|\Sigma\right\|\leq\left\|x_{2} \right\|\leq\alpha_{1}^{-1}\left\|x_{1}\right\|,\)  
Therefore, the trajectory tracking error is \(e=x_{r1}-x_{1}\). The numerical simulation parameters are given in the following Table 2. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(k\) & \(-\)30,119 \\ \(\theta\) & 1 \\ \(\tau\) & 60.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the Internal Model Control (IMC) applied to the photosynthetic plant. A robust FOSMC is proposed for the system in Equation (28) as follows:  
\[u = -(\mu_{m}x_{1}+\eta(\cdot))\frac{x_{3}}{x_{1}},\] \[\eta(\cdot) = -k_{2}I^{\beta}s_{\beta}n(s)-k_{3}s-\mu_{m}x_{r1}-x_{r1}u_{r}-k_{1}e\] \[s = e+k_{1}\int_{0}^{t}\!e(\tau)\,d\tau,\] \[\mu_{m} = \frac{a_{1,nom}x_{2}}{a_{2}+x_{2}}. \tag{29}\]  
The process is simulated in Matlab/Simulink with a variable step Dormand-Prince numerical method. The simulation experiment considers 500 h of cultivation of the microalgae _Spirulina Maxima_, according to real-time experiments taken from the literature, see to Table 3.  
The complete nonlinear dynamics of the photobioreactor in Equation (7) is simulated using a robust FOSMC control law (Equation (29)) in the presence of \(\delta(t)\) disturbances on parameter \(a_{1}\) (see Figure 3). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Plant** & **Reference plant** & **Units** \\ \hline \(a_{1,nom}\) & 0.027 & 0.027 & h\({}^{-1}\) \\ \(a_{2}\) & 9.25 & 9.25 & mg/L \\ \(\alpha_{1}\) & 3.45 & 3.45 &... \\ \(\alpha_{2}\) & 205 & 205 & mg/L \\ \(x_{1}(0)\) & 600 & 540 & mg/L \\ \(x_{2}(0)\) & 80 & 60 & mg/L \\ \(x_{3}(0)\) & 0.001 & NA & L \\ \(x_{r1}(0)\) & NA & 540 & mg/L \\ \(x_{r1}(0)\) & NA & 60 & mg/L \\ \(x_{3,\max}\) & 3 & 3 & L \\ \(u_{r}\) & NA & 0.001 & L/h \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters of the reference plant and the simulated fed-batch plant, where \(u_{r}\) is an optimal open loop feed flow for the continuous photobioreactor model.