## 6 A Simulation Study: Optimization of Micro-Algae Lipid Productivity
### Numerical Results  
Initial conditions are set as \(X(0)=\)1 gX\(\cdot\)L\({}^{-1}\), \(S(0)=\) 0.1 gN\(\cdot\)L\({}^{-1}\), \(Q(0)=\) 0.05 gN\(\cdot\)gX\({}^{-1}\), \(D(0)=\) 0.4 d\({}^{-1}\), \(I(0)=\) 600 \(\upmu\)mol/(m\({}^{2}\cdot\)s), and \(S_{in}=\) 0.1 gN\(\cdot\)L\({}^{-1}\). The initial estimates for the AO and RELS estimating \(\rho\) and \(\mu\) are, respectively, \(\widehat{Q}(0)=\) 0.07 gN\(\cdot\)gX\({}^{-1}\), \(\widehat{\rho}(0)=\) 0.1 gN\(\cdot\)gX\({}^{-1}\).d\({}^{-1}\) and \(\widehat{\mu}(0)=\) 0.2 d\({}^{-1}\).  
The AO is first run together with the two RELS for \(\widehat{\rho}\) and \(\widehat{\mu}\) during the first 10 days in open loop. After this initialization period, the adaptive slope-seeking strategy is activated. SOP is first performed following a step sequence \([t_{switch},P_{l}]=\{(10,8)(15,6)(20,7)(25,8)\}\). Then, ESC is performed during the rest of the simulation starting from \(t=30\) d. The controller settling time is set to \(t_{s}=2\) d, and the second eigenvalue of each pole placement controllers \(\lambda_{c}=-100\) d\({}^{-1}\). The constant trace algorithm is defined with \(k_{0}=1000\) for the RELS of the adaptive slope seeking algorithm and the two RELS estimating \(\rho\) and \(\mu\). Measurement noises on \(X\) and \(S\) are defined as white noises with \(\sigma_{X}^{2}=0.007^{2}\) and \(\sigma_{S}^{2}=0.001^{2}\) (this corresponds to about 1 % of the variable ranges, which is realistic). The excitation signals are provided by MPRS obtained by adding 3 PRBS as suggested in Section 3.3. For the incident light intensity, \(T_{prbs_{i}}=p_{i}\,T_{s}\), with \(p_{1}=2\) (amplitude = 10 \(\upmu\)mol/(m\({}^{2}\cdot\)s)), \(p_{2}=5\) (amplitude = 25 \(\upmu\)mol/(m\({}^{2}\cdot\)s)), \(p_{3}=11\) (amplitude = 15 \(\upmu\)mol/(m\({}^{2}\cdot\)s)), whereas for the dilution rate \(p_{1}=3\) (amplitude = 0.02 d\({}^{-1}\)), \(p_{2}=8\) (amplitude = 0.01 d\({}^{-1}\)), \(p_{3}=13\) (amplitude = 0.02 d\({}^{-1}\)).  
Figure 3 shows that the AO provides a fast and accurate estimation of the internal quota despite noisy estimates computed by the 2 RELS. It should also be highlighted that the convergence of the AO is ensured by the good cell growth rate maintained all along the culture.  
Figure 4 shows that the productivity reference trajectory (i.e., a succession of step changes) is well followed by the controller (see the evolution after 10 days of open-loop operation). The incident light intensity is set at the optimum, and its value quickly settles, whereas the dilution rate is varied so as to achieve the different productivity levels. In this figure, the blue dashed curves correspond to the suboptimal open-loop signals determined by the slope generator, while the black continuous lines represent the actual signals generated by the controller, which also include the PBRS dither signals. Figure 5 shows the phase plane plot evolution, while Figure 6 shows the gradient evolution.  
Figure 3: Time evolution of \(Q\), \(\rho\), and \(\mu\) in black and their estimates provided by the AO and the 2 RELS in blue.  
Figure 7 confirms that PPC with integral action behaves as expected, the latter showing good performance during the transient phases despite the dependence on the estimate convergence and accuracy.  
Figure 4: Time evolution of productivity \(P_{l}\), incident light \(l\), and dilution rate \(D\) following SOP and ESC (red: optimal values; blue: estimated references; black: trajectory; magenta: \(\bar{l}\))  
Figure 5: Level lines of the performance index in the plane \((D,I)\) and corresponding productivity \(P_{l}\).  
Figure 7: PPC controller performance: Time evolution of productivity \(P_{l}\), incident light \(I\), and dilution rate \(D\) following SOP and ESC (red: optimal values; solid blue: PPC; The SOC step changes are indicated by the vertical dashed lines.