# Adaptive Feedback Control for Four-Phase Interleaved Boost Converter Used with PEM Fuel Cell +
## 4 Testing the Robustness of Control Systems Against Uncertainty  
To assess the efficacy and resilience of the suggested controller, it is essential to account for two categories of uncertainty. Firstly, variations in converter operating conditions, such as fluctuating input voltages and varying load power levels, introduce one form of uncertainty. Secondly, discrepancies in circuit components arising from manufacturing variations and aging effects in inductors or capacitors constitute another source of uncertainty. For comparison purposes, three controllers are selected:  
* The first controller selected for this comparison is the double-loop PI controller. The initial step required for this controller involves deriving the small-signal model of the studied converter. Subsequently, the transfer functions \((H_{eu})\) from the duty cycle to the output voltage and \((H_{iu})\) from the duty cycle to the current in the coil can be obtained using Laplace transformation. Finally, the synthesis of the controller parameters can be determined using pole placement methods. * The second one is a double-loop super-twisting sliding mode controller (DL-STSM). Its principle consists of simultaneously controlling the voltage and current loops using two distinct STSM controllers. Two reasons explain why the DL-STSM is chosen. Firstly, it can offer good robustness with a reasonable limitation of input current ripple for the FC application; secondly, it demonstrates the capacity to inhibit overshooting the FC current [11]. For a fair comparison, the synthesis of the DL-STSM controller parameters is based on the PSO algorithm for better optimization. * The third controller implemented, referred to as the backstepping and super-twisting sliding mode (BS + STSM) controller, also features a dual-loop structure. The outer loop, based on the STSM algorithm, ensures precise tracking of the output voltage while simultaneously generating the appropriate current reference. The inner loop, utilizing the backstepping control algorithm, enhances resilience to disturbances and indirectly controls the fuel cell stack current. Stability analyses of both loops are conducted using the Lyapunov approach. Finally, the selection and optimization of parameters for this controller are also achieved through the PSO algorithm. In order to verify the robustness of the proposed controller, it is necessary to consider the uncertainties in the circuit parameters with different operating points of the converter. For this, five different representative operations were selected, as shown in Figure 3. The performance of the output voltage with the proposed controller and other controllers such as the DL-STSM controller and the backstepping controller coupled with STSM are presented in Figure 3 case (a), case (b), case (c), case (d), and case (e), respectively. It can be clearly observed that the proposed controller exhibits almost the same output performance with a lower overshoot and faster settling time than the best controller chosen for comparison. The maximum overshoot of the DL-STSM controller is 4 V and the settling time is 0.15 s with ripples, while the BS + STSM controller and the PI controller exhibit lesser robustness when the circuit parameters are changed; in comparison, a reduction of 62.5% in output voltage overshoot and a superior reduction of 40% in settling time (0.149 s) are achieved with the proposed controller.