In this paper, the problem of projective synchronization between time-delayed fractional-order nonlinear systems with mismatched fractional orders, non-identical time-delays, unknown parameters and external disturbances is investigated. To solve this problem, a time-delayed fractional-order integral sliding surface is firstly introduced. Then, based on the fractional-order Lyapunov stability analysis and sliding mode control strategy, a novel flexible robust control scheme which includes the compensation controller, the sliding mode controller and the adaptive controller is derived that ensures the projective synchronization error dynamical system operates in the sliding mode. Furthermore, the necessary conditions for the error dynamics to be globally asymptotically stable in the sliding mode are determined. Finally, the validity of the achieved theoretical results is verified by a numerical example. The advantages of the proposed control scheme in comparison with previous approaches are also shown.

中文翻译：

---

具有未知参数和扰动的失配时滞分数阶非线性系统的投影同步的新型灵活滑模控制

本文研究了分数阶不匹配、时滞不相同、参数未知和外部扰动的时滞分数阶非线性系统之间的射影同步问题。为了解决这个问题，首先引入了时滞分数阶积分滑动面。然后，基于分数阶Lyapunov稳定性分析和滑模控制策略，推导出了一种包括补偿控制器、滑模控制器和自适应控制器的新型柔性鲁棒控制方案，以确保投影同步误差动态系统在滑动模式。此外，确定了在滑动模式下误差动力学全局渐近稳定的必要条件。最后，通过数值算例验证了所得理论结果的有效性。还显示了所提出的控制方案与以前的方法相比的优点。