## 4 Simulation Case Study  
To forecast the control performance in experiments at the test rig, the four controllers of Section 3 were tested in a simulation regarding trajectory tracking and robustness against measurement noise and  
Figure 8: Block diagram of the Luenberger observer.  
Figure 7: Block diagram of state-feedback control with the static gain compensation.  
model uncertainties. Therefore, in the simulation, some external inputs were added with white sensor noise with a variance similar to the real measurement noise. Furthermore, the controllers were tested with 10% gain and time constant deviations between the approximated system models for plant representation and control synthesis. To simulate both the transient and static behavior of each controller, the desired trajectory was chosen in such a way that it rises to a new piece-wise constant value whereby a Bernstein polynomial of order seven determines the transient phase (see Figure 9).  
As a result, the IMC produces the smallest tracking error, i.e., the smallest difference between desired and simulated power, closely followed by the PI controller. The control law of the IMC is based on the inverted system model, which means that it is robust against input errors. Furthermore, the integral term inside the PI controller ensures steady-state accuracy. Here, the time constants of the PI controller are intentionally chosen very large to prevent the wind-up effects that may otherwise occur on the test rig.  
Although the sliding-mode controller and the linear state-feedback controller were improved with a gain compensation, the expected gain error occurs during steady-state phases of the desired trajectory. A possible reason for this is the model mismatch between the approximated third-order lag system used for designing the sliding-mode controller and the simulated SOFC system given by a sixth-order transfer function. Note that this type of mismatch is a simple possibility to check the expected robustness of each of the presented controllers in simulations before they are applied to the test rig, where gain and time constant mismatches are further caused by neglected nonlinearities.  
In the presented simulation results of the PI and IMC approaches, the manipulated variables were used more efficiently, by reducing the control amplitudes and variations in contrast to the linear state-feedback and sliding-mode controller and thus help to avoid actuator saturations at the test rig. Overall, despite the dynamic and static gain error compensation used for the sliding-mode controller and the state-feedback controller, the trajectory tracking is not as good as with the other two controllers. The design of the sliding-mode controller and the state-feedback controller could be improved with the help of an LQR design instead of pole placement. Due to the structurally implemented optimality criterion, the controller gain would then be equally optimized as opposed to the pole placement. Alternatively, an H\({}_{\infty}\) norm-based design can be used for control synthesis, simultaneously improving the controller gain in terms of accounting for input limitations and maximizing the impact of the controller, while additionally accounting for process noise [25].  
Figure 9: Simulation results of the controllers under consideration of a measurement noise with a standard deviation of \(\sigma_{\text{in}_{\text{in}_{\text{q}}}}=1\cdot 10^{-7}\frac{\text{kg}}{ \text{s}}\) and \(\sigma_{\text{P}_{\text{el}}}=\sqrt{3}\text{W}\).