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

Luis Ovalle, Andrés González, Leonid Fridman, Hernan Haimovich

TL;DR

This work addresses the challenge of implementing barrier-function-based adaptive sliding-mode controllers in discrete-time systems. It shows that the traditional predefined performance problem (PPP) cannot be guaranteed under uniform sampling and finite actuator authority, and thus introduces a revised sampling-based framework (PPPS) that accounts for actuator limits and sampling effects. A formal relationship among the maximum disturbance, sampling period, and barrier width is derived, enabling principled tuning of BFASMC in practice. The authors then propose a modified BFASMC with a region-dependent reaching law that ensures finite-time convergence to a positively invariant barrier set, supported by simulations that illustrate reduced chattering, bounded control, and improved robustness to sampling. Collectively, the results bridge continuous-time theory and practical digital implementations, offering a principled methodology for barrier-function-based sliding-mode control in embedded systems.

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.

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

TL;DR

This work addresses the challenge of implementing barrier-function-based adaptive sliding-mode controllers in discrete-time systems. It shows that the traditional predefined performance problem (PPP) cannot be guaranteed under uniform sampling and finite actuator authority, and thus introduces a revised sampling-based framework (PPPS) that accounts for actuator limits and sampling effects. A formal relationship among the maximum disturbance, sampling period, and barrier width is derived, enabling principled tuning of BFASMC in practice. The authors then propose a modified BFASMC with a region-dependent reaching law that ensures finite-time convergence to a positively invariant barrier set, supported by simulations that illustrate reduced chattering, bounded control, and improved robustness to sampling. Collectively, the results bridge continuous-time theory and practical digital implementations, offering a principled methodology for barrier-function-based sliding-mode control in embedded systems.

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.

Paper Structure

This paper contains 13 sections, 2 theorems, 7 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Motivation Example
  3. Considered System
  4. Barrier-Function Adaptive Control
  5. Unsolvability of the PPP under Sampling
  6. Predefined Performance Problem under Sampling
  7. PPPS and BFASMC
  8. Modifying the BFA
  9. Simulation results
  10. Unsolvability of the PPP under sampling
  11. Behavior of sampled BFA
  12. Finite-time convergence to the barrier set
  13. Conclusions

Key Result

Theorem 1

Let system eq:sys be controlled by eq:BFA, and Assumptions ass:sys and ass:icc hold. Then for all $t\geq t_0$, $|x(t)|\leq\max\left\{\beta,\frac{\varphi}{\varphi+1}\right\}\varepsilon<\varepsilon$ with $\varphi = \bar{\delta} + \theta$ and some $\theta\in(0,1)$. Moreover, there exists a $T \geq 0$ s

Figures (7)

  • Figure 1: Graphical representation of the BFASMC
  • Figure 2: Divisions of the state-space: [Div. 1] unstable region [Div. 2] saturation region [Div. 3] region where $|u|>|\delta(t,x)|$ [Div. 4] region where nothing can be ensured but might serve as a final set
  • Figure 3: Simulations for a perturbed integrator with $\delta(t)$ as in Proposition \ref{['prop:del']}. (left) behavior of the system for 1 second (right) behavior of the system for two sampling intervals; the solution leaves the predefined set before the first sampling interval
  • Figure 4: Response of a first order system under sampling. (left) The trajectories of the system are kept within Div. 4 (right) The controller achieves the task without crossing the limits for the control effort
  • Figure 5: Comparison between a saturated linear constant gain controller and the BFASMC. (left) Control effort. (right) Trajectories of the system
  • ...and 2 more figures

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • proof
  • Theorem 2
  • proof