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In this article, a novel adaptive super-twisting nonlinear Fractional-order PID sliding mode control (ASTNLFOPIDSMC) strategy using extended state observer (ESO) for the speed operation of permanent magnet synchronous motor (PMSM) is proposed. Firstly, this paper proposes a novel nonlinear Fractional-order PID (NLFOPID) sliding surface with nonlinear proportion term, nonlinear integral term and nonlinear differential term. Secondly, the novel NLFOPID switching manifold and an adaptive supertwisting reaching law (ASTRL) are applied to obtain excellent control performance in the sliding mode phase and the reaching phase, respectively. The novel ASTNLFOPIDSMC strategy is constructed by the ASTRL and the NLFOPID sliding surface. Due to the utilization of NLFOPID switching manifold, the characteristics of fast convergence, good robustness and small steady state error can be ensured in the sliding mode phase. Due to the utilization of ASTRL, the chattering phenomenon can be weakened, and the characteristics of high accuracy and strong robustness can be obtained in the reaching phase. Further, an ESO is designed to achieve dynamic feedback compensation for external disturbance. Furthermore, Lyapunov stability theorem and Fractional calculus are used to prove the stability of the system. Finally, comparison results under different controllers demonstrate that the proposed control strategy not only achieves good stability and dynamic properties, but also is robust to external disturbance. INDEX TERMS Adaptive super-twisting nonlinear Fractional-order PID sliding mode control (ASTNL-FOPIDSMC) strategy, extended state observer (ESO), permanent magnet synchronous motor (PMSM), nonlinear Fractional-order PID (NLFOPID) sliding surface, adaptive super-twisting reaching law (ASTRL).

Overhead cranes with double-pendulum effect seem more practical than those with singlependulum effect. However, in this case, the dynamic performance analysis and the controller design become more difficult. Moreover, achieving both high-precision trajectory tracking and load sway rejection is a more significant issue for crane systems. For these purposes, the nonlinear dynamics of an overhead crane with double-pendulum effect is derived for the controller design. Then, a novel adaptive hierarchical sliding mode controller is presented. Unlike a traditional approach, the proposed one can make fixed sliding mode surface active to search the state trajectories. Such an adaptive design can make the states of the system to enter the desired sliding surface as soon as possible and, at the same time, can also improve the cart tracking precision. The Lyapunov technique and LaSalle's principle are employed to confirm the stability of the whole system. The experimental results validate that the proposed method has superior control performance and robustness with respect to parameter variations. INDEX TERMS Overhead crane, double-pendulum effect, load sway rejection, tracking control, adaptive hierarchical sliding mode control.

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