Discrete implementations of sliding-mode controllers with barrier-function adaptations require a revised framework

TL;DR

This paper identifies fundamental limitations in the discrete implementation of sliding-mode controllers with barrier functions, proposes a revised framework incorporating actuator and sampling constraints, and introduces a modified controller ensuring finite-time safety guarantees.

Contribution

It presents a new control framework that explicitly accounts for sampling and actuator constraints, enabling reliable digital implementation of barrier-function-based sliding-mode controllers.

Findings

Established a relation between actuator capacity, sampling rate, and barrier width.

Demonstrated the revised framework resolves design issues in digital SMC implementations.

Introduced a modified BFASMC ensuring finite-time convergence to safety sets.

Abstract

Challenges in the discrete implementation of sliding-mode controllers (SMC) with barrier-function-based adaptations are analyzed, revealing fundamental limitations in conventional design frameworks. It is shown that under uniform sampling, the original continuous-time problem motivating these controllers becomes theoretically unsolvable under standard assumptions. To address this incompatibility, a revised control framework is proposed, explicitly incorporating actuator capacity constraints and sampled-data dynamics. Within this structure, the behavior of barrier function-based adaptive controllers (BFASMC) is rigorously examined, explaining their empirical success in digital implementations. A key theoretical result establishes an explicit relation between the actuator capacity, the sampling rate, and the width of the barrier function, providing a principled means to tune these controllers for different application requirements. This relation enables the resolution of various design problems with direct practical implications. A modified BFASMC is then introduced, systematically leveraging sampling effects to ensure finite-time convergence to a positively invariant predefined set, a key advancement for guaranteeing predictable safety margins.