## 5 Experimental Validation  
The PI controller and the IMC were implemented and validated in some experiments at the available test rig. An activation of the sliding-mode control and the state-feedback control in the experiment confirms the unacceptably large control amplitudes, leading to actuator saturations, which make the employed linear plant models invalid. For the experiments, the desired trajectory was slightly adjusted compared to the simulation. There, the challenge was to find a trajectory which remains in the validity domain of the model. The hydrogen mass flow has a lower limitation, which marks the minimum mass flow necessary to guarantee the desired current, here \(I_{\mathrm{d}}=3\mathrm{A}\). By undershooting this limitation, the load would consume the whole hydrogen inside the fuel cell due to excessive fuel utilization, where the approximated model obviously turns invalid as soon as the current and electric power break down. The designed desired trajectory has to prevent such kind of invalidity.  
The results of the experiments are listed in Table 1 and are further shown in Figures 10 and 11-13, which validates the simulation results. Both the PI and IMC controllers have a good trajectory tracking accuracy with average errors of the electric power \(\mu_{\mathrm{Pel}}<3\) W. Because of a smaller standard deviation obtained by the IMC, it is, as for the simulation, the best controller, which is confirmed by the reduced tracking errors in transient phases. Moreover, the repetition of the experimental validation of the IMC for Experiments 2-4 shows both the reproducibility and robustness of the control accuracy for various initial power levels and, hence, confirms the robustness of the controller for a quite wide range of operating conditions despite the actual plant nonlinearities.  
Besides the measured data of the electric power and its errors, the temporal variation of the gain  
\[K_{\mathrm{Pel}}=\frac{P_{\mathrm{el,m}}}{\mu_{\mathrm{H_{2},in}}} \tag{34}\]  
is illustrated. It is clearly visible that a large variation of the gain of several tens of percent along the time axis occurs, which was not yet modeled. This gain variation results from unavoidable changes of the internal temperature distribution in the SOFC stack module and from variations of the cathode gas temperature which is employed to keep the stack at a constant thermal operating point despite exothermal reaction enthalpies [12]. Despite this model uncertainty resulting from actually nonlinear dynamic phenomena, the PI controller and the IMC are able to follow the desired trajectory. These model uncertainties could be analyzed in future work. Therefore, an alternative concept is to estimate this gain online and to make use of it in a generalization of the proposed PI controller and IMC towards an adaptive control approach.  
**Remark 6**: _Furthermore, experiments showed a huge dependency between the electric power and the temperature distribution inside the stack. Due to an inhomogeneous temperature distribution with a low core temperature, the maximum electric power achieved by the cold combustion decreases. The temperature dependency has not yet been considered._  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Controller** & **Experiment no.** & \(\theta_{\mathrm{AG,in}}\) & \(\theta_{\mathrm{CG,in}}\) & \(\theta_{\mathrm{stack,min}}\) & \(\theta_{\mathrm{stack,max}}\) & \(\mu_{\mathrm{Pel}}\) & \(\sigma_{\mathrm{Pel}}\) \\ \hline PI & 1 & 902.92 K & 942.92 K & 936.36 K & 969.57 K & \(-0.12\) W & 4.03 \\ \hline IMC & 2 & 910.2 K & 969.42 K & 937.7 K & 971.04 K & 1.4 W & 0.92  
Hence, a further point for future work will be to interface the presented power controls with currently independent temperature controllers [12].  
Figure 11: Experiment 2: IMC.  
Figure 10: Experiment 1: PI controller.  
Figure 12: Experiment 3: IMC.  
Figure 13: Experiment 4: IMC.