Backstepping active disturbance rejection control for nonlinear fractional-order systems with a fixed-time synergetic extended state observer

Active disturbance rejection control (ADRC) has received considerable attention from researchers in recent years. It provides an effective control strategy, especially in the presence of uncertainties and disturbances, and is particularly valuable when a precise system model is not available. Despite its simple control structure, ADRC exhibits high robustness to plant uncertainties and external disturbances. Numerous studies have used this technique in the case of integer-order systems, so this technique has given good results. However, only a few works have been realised in the case of fractional-order systems. Fractional systems have become important for modelling many physical phenomena. Some research has taken advantage of the fractional calculus to design ADRC for both linear and nonlinear fractional-order systems. This paper presents a novel fractional-order backstepping active disturbance rejection control (ADRC) approach that integrates the backstepping control technique with a fixed-time synergetic extended state observer (ESO). This approach, which employs a fractional macro-variable to ensure fixed-time convergence in nonlinear systems, represents a new direction in ADRC design for fractional-order systems. The method begins with the development of a fixed-time synergetic ESO for estimating the extended state, which includes uncertainties and external disturbances, the fixed-time convergence of this observer is demonstrated. A backstepping controller is then designed based on the estimated states and total disturbance to achieve system stabilisation and reference tracking, while effectively compensating for the total disturbance. The proposed backstepping ADRC is compared with existing methods, including backstepping ADRC with a fractional high-order sliding mode ESO (FHOSMESO), sliding mode ADRC with FHOSMESO, and sliding mode ADRC with a fixed-time synergetic ESO (FTSESO). Superior performance in disturbance rejection and reference tracking is demonstrated by simulation results of the proposed approach.