In this paper, the design of a fractional-order hyperbolic adaptive neuro-fuzzy backstepping sliding mode controller (HANFBSMC) has been addressed for a class of fractional-order chaotic systems with time-varying delays in their states, control inputs, disturbances and uncertainties. In the proposed controller, adaptive rules are used both in a neuro-fuzzy estimator to estimate the unknown system dynamics, and in updating uncertainty bounds of system. The robust part of the proposed hybrid controller includes backstepping sliding mode controller, in which, the hyperbolic tangential fractional-order sliding surfaces are employed to prevent large tracking errors. Employing backstepping control strategy also extends the flexibility of controller to deal with higher order systems and under more extensive design issues. Adaptive rules based on Lyapunov stability analysis are also employed for tuning of relating robust control parameters according to the estimated upper bounds of the uncertainties. Analysis of stability of this controller has been performed via Lyapunov–Krasovskii theorem and Barbalat's lemma. Besides, finite time reaching to sliding surfaces has been proved. Finally, out performance of the proposed controller has been reflected via stabilization of a fractional-order hyper-chaotic system with time varying delays in its states as well as fractional-order Chen system with time delay in its inputs and states, where both systems experience unknown uncertainties and disturbances.

在本文中，针对一类分数阶混沌系统，其状态，控制输入，扰动和不确定性具有时滞，提出了分数阶双曲自适应神经模糊反推滑模控制器（HANFBSMC）的设计。 。在提出的控制器中，自适应规则既可用于神经模糊估计器中以估计未知的系统动力学，也可用于更新系统的不确定性范围。提出的混合控制器的鲁棒性部分包括反步滑模控制器，其中采用双曲正切分数阶滑动面来防止较大的跟踪误差。采用反步控制策略还可以扩展控制器的灵活性，以应对更高阶的系统和更广泛的设计问题。根据估计的不确定性上限，还使用基于Lyapunov稳定性分析的自适应规则来调整相关的鲁棒控制参数。该控制器的稳定性已通过Lyapunov–Krasovskii定理和Barbalat引理进行了分析。此外，已经证明了有限时间到达滑动表面。最后，该控制器的性能已通过其状态具有时变延迟的分数阶超混沌系统以及其输入和状态具有时滞的分数阶Chen系统的稳定性得以反映。未知的不确定性和干扰。该控制器的稳定性已通过Lyapunov–Krasovskii定理和Barbalat引理进行了分析。此外，已经证明了有限时间到达滑动表面。最后，该控制器的性能已通过其状态具有时变延迟的分数阶超混沌系统以及其输入和状态具有时滞的分数阶Chen系统的稳定性得以反映。未知的不确定性和干扰。该控制器的稳定性已通过Lyapunov–Krasovskii定理和Barbalat引理进行了分析。此外，已经证明了有限时间到达滑动表面。最后，该控制器的性能已通过其状态具有时变延迟的分数阶超混沌系统以及其输入和状态具有时滞的分数阶Chen系统的稳定性得以反映。未知的不确定性和干扰。