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Conference Proceeding

Publication Date



In this paper, optimal tracking control is found for an inputaffine nonlinear quadcopter using Single Network Adaptive Critics (SNAC). The quadcopter dynamics consists of twelve states and four controls. The states are defined using two related reference frames: the earth frame, which describes the position and angles, and the body frame, which describes the linear and angular velocities. The quadcopter has six outputs and four controls, so it is an underactuated nonlinear system. The optimal control for the system is derived by solving a discrete-time recursive Hamilton-Jacobi-Bellman equation using a linear in-parameter neural network. The neural network is trained to find a mapping between a target costate vector and the current states. The network’s weights are iteratively trained using the least-squares approximation method until the maximum number of iterations or convergence is reached, and training begins at the final time and proceeds backward to the initial time. The trained neural controller applies online optimal feedback control that tracks a trajectory, minimizes control effort, and satisfies the optimality condition. The SNAC method provides a controller that can handle all initial conditions within the domain of training and all times less than the training’s final time.


Copyright © 2022 by ASME. Original published version available at

Publication Title

Proceedings of the ASME 2022 International Mechanical Engineering Congress and Exposition





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