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.

本研究提出了一种基于深度神经网络的控制器，用于拉格朗日系统的轨迹跟踪控制，并通过适当设计神经网络结构来确保任何兼容的神经网络参数下的闭环稳定性，进一步优化神经网络参数可实现改进的控制性能。同时，在系统模型未知的情况下，通过改进的拉格朗日神经网络结构学习系统动力学并设计控制器，在模型逼近误差和外部干扰存在时仍可保证闭环稳定性和跟踪控制性能，并通过仿真验证了该方法的有效性。

基于结构化深度神经网络的拉格朗日系统反馈轨迹跟踪控制

Structured Deep Neural Networks-Based Backstepping Trajectory Tracking  Control for Lagrangian Systems

This paper presents CART, an analytical method to augment a learning-based,
distributed motion planning policy of a nonlinear multi-agent system with
real-time collision avoidance and robust tracking guarantees, independently of
learning errors. We first derive an analytical form of an optimal safety filter
for Lagrangian systems, which formally ensures a collision-free operation in a
multi-agent setting in a disturbance-free environment, while allowing for its
distributed implementation with minimal deviation from the learned policy. We
then propose an analytical form of an optimal robust filter for Lagrangian
systems to be used hierarchically with the learned collision-free target
trajectory, which also enables distributed implementation and guarantees
exponential boundedness of the trajectory tracking error for safety, even under
the presence of deterministic and stochastic disturbance. These results are
shown to extend further to general control-affine nonlinear systems using
contraction theory. Our key contribution is to enhance the performance of the
learned motion planning policy with collision avoidance and tracking-based
robustness guarantees, independently of its original performance such as
approximation errors and regret bounds in machine learning. We demonstrate the
effectiveness of CART in motion planning and control of several examples of
nonlinear systems, including spacecraft formation flying and rotor-failed UAV
swarms.

本文介绍了 CART，一种分析方法，旨在增强一个非线性多智能体系统的基于学习的分布式动态规划策略，独立于学习误差，具有实时的避碰和强鲁棒性保证，并正式推导出安全滤波器和强鲁棒滤波器的解析形式，证明了这些结果进一步扩展到使用收缩理论的一般控制仿形非线性系统。

基于学习的多智能体系统运动规划中的 CART: 碰撞避免和鲁棒跟踪增强

CART: Collision Avoidance and Robust Tracking Augmentation in  Learning-based Motion Planning for Multi-Agent Systems

High performance tracking control can only be achieved if a good model of the
dynamics is available. However, such a model is often difficult to obtain from
first order physics only. In this paper, we develop a data-driven control law
that ensures closed loop stability of Lagrangian systems. For this purpose, we
use Gaussian Process regression for the feed-forward compensation of the
unknown dynamics of the system. The gains of the feedback part are adapted
based on the uncertainty of the learned model. Thus, the feedback gains are
kept low as long as the learned model describes the true system sufficiently
precisely. We show how to select a suitable gain adaption law that incorporates
the uncertainty of the model to guarantee a globally bounded tracking error. A
simulation with a robot manipulator demonstrates the efficacy of the proposed
control law.

使用基于数据的控制法律，通过高斯过程回归来补偿系统未知的动力学，并基于学习的模型的不确定性来自适应反馈部分的增益，以保证全局有界跟踪误差。