In this paper, a novel optimal synchronization control scheme for fractional–integer hybrid–order chaotic systems is formulated. To deal with the fractional–order (FO) constraint, a transformation programme is developed and then the master system considered as an FO chaotic system is transformed into an integer–order one. A fuzzy echo state network (FESN) with the advantages of both fuzzy logic system and echo state network is introduced to approximate system uncertainty. Simultaneously, to alleviate the resource pressure, an optimal synchronization control is proposed in the light of the reinforcement learning mechanism, in which critic–actor update laws are constructed through the negative gradient of a positive function with regard to Bellman residual. Compared to typical fuzzy synchronization control, the devised FESN synchronization scheme with the same computational complexity possess superior approximation ability and synchronization performance. Ultimately, three simulation cases are exhibited to check the validity of the proposed approach.

中文翻译：

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基于强化学习的混合阶混沌系统自适应模糊回波状态网络最优同步控制

在本文中，提出了一种新颖的分数整数混合阶混沌系统的最优同步控制方案。为了处理分数阶（FO）约束，开发了一个转换程序，然后将被视为 FO 混沌系统的主系统转换为整数阶系统。结合模糊逻辑系统和回波状态网络的优点，引入模糊回波状态网络（FESN）来近似系统的不确定性。同时，为了缓解资源压力，根据强化学习机制，提出了一种最优同步控制，其中通过关于贝尔曼残差的正函数的负梯度构造批评者-行动者更新律。与典型的模糊同步控制相比，所设计的FESN同步方案在相同计算复杂度下具有优越的逼近能力和同步性能。最后，展示了三个模拟案例来检验所提出方法的有效性。