Robust Taylor-Lagrange Control for Safety-Critical Systems

Wei Xiao; Christos Cassandras; Anni Li

eess.SYFeb 23, 2026

Robust Taylor-Lagrange Control for Safety-Critical Systems

Authors: Wei Xiao, Christos Cassandras, Anni Li

Organizations: Robotics Engineering Department at Worcester Polytechnic Institute · MIT CSAIL, USA · the division of Systems Engineering at Boston University, USA · the Electrical and Computer Engineering Department at The University of North Carolina at Charlotte, Charlotte, NC, 28223, USA

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.

Figures & tables

Parameter	Value	Units	Parameter	Value	Units
$v(0)$	24	$m/s$	$z(0)$	90	$m$
$v_{0}$	13.89	$m/s$	$v_{d}$	24	$m/s$
$M$	1650	$kg$	g	9.81	$m/s^{2}$
$f_{0}$	0.1	$N$	$f_{1}$	5	$Ns/m$
$f_{2}$	0.25	$Ns^{2}/m$	$c$	10	$m$
$\Delta t$	0.1	$s$	$dt$	0.1	$s$

TABLE I: Simulation parameters for ACC

Fig. 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

Method	min $h(\bm{x})$	Compute time (s)
Time-driven HOCBF	1.8250e-7	0.0013 $\pm$ 1.8714e-4
Time-driven TLC	-0.6414	0.0013 $\pm$ 1.2619e-4
Event-driven TLC	0.6511	0.0017 $\pm$ 6.8994e-4
rTLC $\Delta t=0.85,dt=0.1$	6.2487	0.0014 $\pm$ 1.9303e-4
rTLC $\Delta t=0.5,dt=0.5$	1.1867	0.0013 $\pm$ 2.0533e-4
rTLC $\Delta t=0.1,dt=0.1$	0.0092	0.0014 $\pm$ 3.8113e-4

TABLE II: Comparison of minimum CBF values and computation time between time-driven HOCBF, time-driven TLC, event-driven TLC, and the proposed rTLC. Safety in time-driven is violated due to the inter-sampling effect.

Safety-Critical Control

Safety-critical control designs controllers that keep physical systems within hard safety constraints despite uncertainty, disturbances, and limited actuation.

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Merely pursuing performance may adversely affect safety, while a conservative policy for safe exploration will degrade the performance. How to guarantee both safety and performance in learning-based control problems is an interesting yet challenging issue. This paper aims to enhance system performance with a safety guarantee by solving reinforcement learning (RL)-based optimal control problems for nonlinear systems subject to high-relative-degree state constraints and unknown time-varying disturbance/actuator faults. A new type of control barrier functions (CBFs), termed high-order reciprocal-based control barrier function, is proposed to handle high-relative-degree constraints, which extends the design of CBFs to enforce robust safety without knowing the disturbance bound. The concept of gradient similarity is proposed to quantify the relationship between safety and performance. Finally, gradient manipulation and adaptive mechanisms are introduced in the model-based safe RL framework to enhance the performance with a safety guarantee. Two simulation examples illustrate the efficacy of the proposed algorithms.

Abstract

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