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

Jiajun Qian, Liang Xu, Xiaoqiang Ren, Xiaofan Wang

TL;DR

A structured DNN-based controller for the trajectory tracking control of Lagrangian systems using backing techniques and an improved Lagrangian neural network (LNN) structure is proposed to learn the system dynamics and design the controller.

Abstract

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.

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

TL;DR

A structured DNN-based controller for the trajectory tracking control of Lagrangian systems using backing techniques and an improved Lagrangian neural network (LNN) structure is proposed to learn the system dynamics and design the controller.

Abstract

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.
Paper Structure (17 sections, 4 theorems, 30 equations, 6 figures, 1 table)

This paper contains 17 sections, 4 theorems, 30 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Euler-Lagrange Equation
  4. Fully Connected Neural Network
  5. Lagrangian Neural Network
  6. Input Convex Neural Network
  7. Main Results
  8. Neural Backstepping Tracking Control Design
  9. Stability of the NBS Controller
  10. Learning Optimization Formulation
  11. LNNs based Model Approximation and Controller Design
  12. Modified LNN for Learning Dynamics
  13. Controller Design and Performance Analysis
  14. Simulations
  15. Two-Link Planar Robot Arm with Known Model Information
  16. ...and 2 more sections

Key Result

Theorem 1

Consider the system eq.manipulator without disturbance $\bm{\tau^d}$, if the controller is designed as eq.trcaking controller, where $\bm{\Phi}(\bm{z_1})$ is strongly convex in $\bm{z_1}$ with a unique minimum at $\bm{z_1}=\bm{0}$ satisfying $\bm{\Phi}(0)=0$, and $\bm{D}(\bm{z_2})$ is positive defin

Figures (6)

  • Figure 1: The structure for proposed NBS controller.
  • Figure 2: The model of a 2-link planar robot arm
  • Figure 3: The time evolution of angles of the two-link planar robot arm using the NBS tracking controller without training. Different initial states and state variables are marked by different line colors and styles respectively.
  • Figure 4: The time evolution of angles of the two-link planar robot arm (a) using the NBS tracking controller after training, (b) using the PID controller.
  • Figure 5: The steady state tracking error under different $\alpha$ and the corresponding upper bounds in Theorem \ref{['Th. disturbance_regularizer']}
  • ...and 1 more figures

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof