Fractional controller based on a robust PIα observer for uncertain fractional systems

In this paper, we present a fractional sliding mode controller based on a robust $ PI^{\alpha } $ PIα observer for fractional systems with bounded disturbances. This controller is introduced to achieve closed-loop Mittag–Leffler stability for uncertain fractional systems; meanwhile, the robust $ PI^{\alpha } $ PIα observer is designed such that a $ \mathcal {L}_{2} $ L2 stability is obtained via quadratic Lyapunov functions and the tuning is achieved with linear matrix inequality approach. The performance and stability of the proposed controller are analysed and demonstrated through a generalised $ n\alpha $ nα-differentiator. Finally, to illustrate the effectiveness of the proposed approach a comparative analysis against the fractional Super-Twisting is carried out.