# Adaptive Feedback Control for Four-Phase Interleaved Boost Converter Used with PEM Fuel Cell +
## 1 Introduction
Control algorithms based on an adapted observer to cancel disturbances and ensure compliance with imposed specifications are currently the most recent research topics, such as flatness control with an extended state observer (ESO) in [13] and active disturbance rejection control with ESO in [21]. Since the general form of the disturbance is unknown, this necessitates an additional observer to enhance the system's robustness. Although the aforementioned controller demonstrates the expected performance with experimental results for fuel cell applications, the drawback is that the experimental validation of the fuel cell is conducted through a fuel cell emulator [1]. A double-loop PI controller has also been introduced and verified with an IBC for fuel cells in several studies [11; 16]. In the literature, a state-feedback control has been developed to stabilize linear and nonlinear dynamical systems. This involves a procedure for designing a control law that connects the choice of internal states of the system to the design of a closed-loop controller, ensuring the asymptotic global stability of the systems. The number of controller gains to be synthesized depends on the number of state equations of the designed system. In a real application, the system's differential equations would be transformed into state equations of the inductor current and output voltage, and the control technique (LQR) is established to complete the stability proof. Some research has reported that state-feedback control is widely used in many fields, such as photovoltaic systems [22]. The following reasons prompted us to select a state-feedback controller for regulating both the voltage and current loops:State-feedback control ensures global output regulation and disturbance attenuation for dynamic systems, whether linear or nonlinear. This method also offers great flexibility in designing control laws, allowing adaptation to various dynamic system configurations [23]. * The design of this controller is linked to the internal and external states of the system in closed-loop, thus allowing precise and efficient control of the internal states and the system output. * Algorithmic simplicity: Although this type of control may add complexity, it also helps reduce algorithmic complexity by simplifying implementation, focusing on regulating relevant internal states of the system. In our research, the corresponding double-loop controller is proposed. The error of the outer loop is defined as the difference between the output voltage and the reference voltage, and the control law is the reference current that would be used to regulate the error of the inner loop to constitute the duty cycle in this loop. Furthermore, this article aims to design an innovative and robust double-loop controller for IBC with FC. The proposed controller structure consists of two parts: an inner loop and an outer loop. Firstly, a state-feedback control for the inner loop is used to ensure the tracking of the inductor current reference. Then, another state-feedback control for the outer loop is used to increase the fuel cell stack voltage to the desired voltage level. The synthesis of the parameters of both controllers is carried out using the linear quadratic regulation (LQR) method, also utilizing the static error cancellation condition. This synthesis is performed in two stages: the first stage involves determining the gains of the inner loop to ensure its stability, and then, using these parameters, the inner loop acts as a direct chain for the outer loop. Similarly, the parameters of the outer controller are synthesized. The robustness of the proposed control system is validated by variations in circuit parameters and the uncertainty in operating conditions. Although numerous controllers have been proposed and tested for fuel cell (FC) system converters, establishing a fair method for evaluating each controller, especially those that are nonlinear, presents a challenge.