Deep neural networks (DNN) are increasingly being used to learn controllers due to their excellent approximation capabilities. However, their black-box nature poses significant challenges to closed-loop stability guarantees and performance analysis. In this paper, we introduce a structured DNN-based controller for the trajectory tracking control of Lagrangian systems using backing techniques. By properly designing neural network structures, the proposed controller can ensure closed-loop stability for any compatible neural network parameters. In addition, improved control performance can be achieved by further optimizing neural network parameters. Besides, we provide explicit upper bounds on tracking errors in terms of controller parameters, which allows us to achieve the desired tracking performance by properly selecting the controller parameters. Furthermore, when system models are unknown, we propose an improved Lagrangian neural network (LNN) structure to learn the system dynamics and design the controller. We show that in the presence of model approximation errors and external disturbances, the closed-loop stability and tracking control performance can still be guaranteed. The effectiveness of the proposed approach is demonstrated through simulations.

翻译：深度神经网络（DNN）因其优异的逼近能力，正越来越多地被用于学习控制器。然而，其黑箱特性给闭环稳定性保证和性能分析带来了重大挑战。本文针对拉格朗日系统的轨迹跟踪控制问题，利用反步技术，提出了一种基于结构化DNN的控制器。通过合理设计神经网络结构，所提出的控制器能够确保在任何兼容的神经网络参数下闭环系统稳定。此外，通过进一步优化神经网络参数，可以实现改进的控制性能。同时，我们给出了跟踪误差关于控制器参数的显式上界，这使得我们可以通过适当选择控制器参数来实现期望的跟踪性能。进一步地，当系统模型未知时，我们提出了一种改进的拉格朗日神经网络（LNN）结构来学习系统动力学并设计控制器。我们证明，在存在模型逼近误差和外部干扰的情况下，闭环稳定性和跟踪控制性能仍然可以得到保证。仿真结果验证了所提方法的有效性。