Optimal Synchronization of Unidirectionally Coupled FO Chaotic Electromechanical Devices with the Hierarchical Neural Network

Abstract

This article solves the problem of optimal synchronization, which is important but challenging for coupled fractional-order (FO) chaotic electromechanical devices composed of mechanical and electrical oscillators and electromagnetic filed by using a hierarchical neural network structure. The synchronization model of the FO electromechanical devices with capacitive and resistive couplings is built, and the phase diagrams reveal that the dynamic properties are closely related to sets of physical parameters, coupling coefficients, and FOs. To force the slave system to move from its original orbits to the orbits of the master system, an optimal synchronization policy, which includes an adaptive neural feedforward policy and an optimal neural feedback policy, is proposed. The feedforward controller is developed in the framework of FO backstepping integrated with the hierarchical neural network to estimate unknown functions of dynamic system in which the mentioned network has the formula transformation and hierarchical form to reduce the numbers of weights and membership functions. Also, an adaptive dynamic programming (ADP) policy is proposed to address the zero-sum differential game issue in the optimal neural feedback controller in which the hierarchical neural network is designed to yield solutions of the constrained Hamilton-Jacobi-Isaacs (HJI) equation online. The presented scheme not only ensures uniform ultimate boundedness of closed-loop coupled FO chaotic electromechanical devices and realizes optimal synchronization but also achieves a minimum value of cost function. Simulation results further show the validity of the presented scheme.