## 6 A Simulation Study: Optimization of Micro-Algae Lipid Productivity
### Asymptotic Observer Design  
In contrast to \(P\), \(P_{l}\) is not directly measurable as \(Q\) is an intracellular variable. However, \(X\) and \(S\) can be measured online using NIR and UV probes combined with appropriated spectrometers [41], and a software sensor can be developed for \(\tilde{Q}\). To not lose the advantage of ESC and require prior model knowledge, a robust estimator has to be designed notwithstanding the lack of knowledge regarding the kinetic structures and model parameter values. Such a robust estimator of the internal quota has been developed by [42] using super-twisting observers. Here, for the sake of simplicity, an asymptotic observer (AO) is proposed as follows:  
\[\dot{Q}=\hat{\rho}-\hat{\mu}\dot{Q} \tag{34}\]  
whose rate of convergence is linked to the growth rate. Indeed, defining the estimation error \(\tilde{Q}=Q-\hat{Q}\):  
\[\dot{Q} =\rho-\mu Q-\hat{\rho}+\hat{\mu}\dot{Q}\] \[=(\rho-\hat{\rho})-(\hat{\mu}+\hat{\mu})(\hat{Q}+\hat{Q})+\hat{\mu} \dot{Q}\] \[=\hat{\rho}-\hat{\mu}\dot{Q}-\hat{\mu}\dot{Q}-\hat{\mu}\dot{Q}+\hat {\mu}\dot{Q}\] \[=\hat{\rho}-\hat{\mu}\dot{Q}-\hat{\mu}\dot{Q}\] \[\dot{Q} \rightarrow-\mu\dot{Q}\quad\text{if}\quad\hat{\rho}\to 0 \quad\text{and}\quad\bar{\mu}\to 0\quad\text{fast enough} \tag{35}\]  
It is therefore necessary to develop estimators of the reaction rates \(\rho\) and \(\mu\). This can be achieved in a simple way using forward discretization (with sampling period \(T_{s}\)) of the biomass Equation (31a):  
\[X_{k} =X_{k-1}+(\mu\,X_{k-1}-D\,X_{k-1})\,T_{s}\] \[=\mu\,T_{s}\,X_{k-1}+(1-D\,T_{s})\,X_{k-1}\] \[=\psi_{\rho}\,\theta+(1-D\,T_{e})\,X_{k-1} \tag{36}\]  
with \(\psi_{\rho}=T_{s}\,X_{k-1}\), \(\theta_{\mu}=\mu\). In the same way, a forward discretization of the substrate Equation (31b) yields:  
\[S_{k} =S_{k-1}+\big{[}-\rho X_{k-1}+D(S_{in}-S_{k-1})\big{]}T_{s}\] \[=-\rho\,T_{s}\,X_{k-1})+(1-D\,T_{s})\,S_{k-1}+D\,S_{in}T_{s}\] \[=\psi_{\mu}\,\theta+(1-D\,T_{s})\,S_{k-1}+D\,S_{in}\,T_{s} \tag{37}\]  
\(\psi_{\rho}=-T_{s}\,X_{k-1}\), \(\theta_{\rho}=\rho\).  
RELS with noise model and constant trace algorithm (Equations (6)-(11)) can be used to estimate online the reaction rates considered as slowly varying parameters. This pragmatic approach is less rigorous than the solution proposed in [42] but works quite well in practice and is preferred here for its great simplicity.  
Moreover, even if a new  
\begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Value Using \(P\)** & **Value Using \(P_{l}\)** \\ \hline \(Q^{opt}\) (gN\(\cdot\)gX\({}^{-1}\)) & 0.097 & 0.0747 \\ \hline \(X^{opt}\) (gX\(\cdot\)L\({}^{-1}\)) & 1.02 & 1.3355 \\ \hline \(D^{opt}\) (d\({}^{-1}\)) & 0.85 & 0.58 \\ \hline \(l^{opt}\) (\(\mu\)E/m\({}^{2}\)\(\cdot\)s) & 210 & 210 \\ \hline \(p^{opt}\) (gX\(\cdot\)L\(\cdot\)d\({}^{-1}\)) & 0.87 & 0.7746 \\ \hline \(l^{opt}_{l}\) (gX\({}^{2}\).gN\({}^{-1}\).L\(\cdot\)d\({}^{-1}\)) & 8.97 & 10.37 \\ \hline \end{tabular}
\end{table}
Table 3: Optimal values following bifurcation analysis.  
online measurement is required, it can be achieved using an available (at reasonable costs) substrate probe. This approach is preferred to a dynamic model, which, in essence, would be uncertain and require the same types of adaptation schemes.