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Robust Taylor-Lagrange Control for Safety-Critical Systems

Wei Xiao, Christos Cassandras, Anni Li

Abstract

Solving safety-critical control problem has widely adopted the Control Barrier Function (CBF) method. However, the existence of a CBF is only a sufficient condition for system safety. The recently proposed Taylor-Lagrange Control (TLC) method addresses this limitation, but is vulnerable to the feasibility preservation problem (e.g., inter-sampling effect). In this paper, we propose a robust TLC (rTLC) method to address the feasibility preservation problem. Specifically, the rTLC method expands the safety function at an order higher than the relative degree of the function using Taylor's expansion with Lagrange remainder, which allows the control to explicitly show up at the current time instead of the future time in the TLC method. The rTLC method naturally addresses the feasibility preservation problem with only one hyper-parameter (the discretization time interval size during implementation), which is much less than its counterparts. Finally, we illustrate the effectiveness of the proposed rTLC method through an adaptive cruise control problem, and compare it with existing safety-critical control methods.

Robust Taylor-Lagrange Control for Safety-Critical Systems

Abstract

Solving safety-critical control problem has widely adopted the Control Barrier Function (CBF) method. However, the existence of a CBF is only a sufficient condition for system safety. The recently proposed Taylor-Lagrange Control (TLC) method addresses this limitation, but is vulnerable to the feasibility preservation problem (e.g., inter-sampling effect). In this paper, we propose a robust TLC (rTLC) method to address the feasibility preservation problem. Specifically, the rTLC method expands the safety function at an order higher than the relative degree of the function using Taylor's expansion with Lagrange remainder, which allows the control to explicitly show up at the current time instead of the future time in the TLC method. The rTLC method naturally addresses the feasibility preservation problem with only one hyper-parameter (the discretization time interval size during implementation), which is much less than its counterparts. Finally, we illustrate the effectiveness of the proposed rTLC method through an adaptive cruise control problem, and compare it with existing safety-critical control methods.
Paper Structure (7 sections, 2 theorems, 37 equations, 1 figure, 2 tables)

This paper contains 7 sections, 2 theorems, 37 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. PRELIMINARIES
  3. Robust Taylor-Lagrange Control
  4. Robust Taylor-Lagrange Control
  5. Optimal Control with rTLC
  6. CASE STUDIES AND RESULTS
  7. Conclusion

Key Result

Theorem 1

Given a TLC function $h(\bm x)$ from Def. def:tlc with the associated safe set defined as in (eqn:C-1), if $h(\bm x(t_0))\geq 0$, then any Lipschitz continuous controller $\bm u(\xi)\in K_{tlc}(\bm x(t_0), \bm x(\xi)), \forall t_0\in[0,\infty), \xi\in(t_0,t), t>t_0$ renders the set C forward invaria

Figures (1)

  • Figure 1: Comparison of speed, control and safety function $h(\bm x)$ profiles between between time-driven HOCBF, time-driven TLC, event-driven TLC, and the proposed rTLC ($\Delta t = 0.85, dt = 0.1$).

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4: Taylor-Lagrange Control xiao2025taylor
  • Theorem 1: xiao2025taylor
  • Definition 5: Robust Taylor-Lagrange Control
  • Theorem 2
  • Remark 1: rTLC addresses the inter-sampling effect