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Integral Control Barrier Functions with Input Delay: Prediction, Feasibility, and Robustness

Adam K. Kiss, Ersin Das, Tamas G. Molnar, Aaron D. Ames

Abstract

Time delays in feedback control loops can cause controllers to respond too late, and with excessively large corrective actions, leading to unsafe behavior (violation of state constraints) and controller infeasibility (violation of input constraints). To address this problem, we develop a safety-critical control framework for nonlinear systems with input delay using dynamically defined (integral) controllers. Building on the concept of Integral Control Barrier Functions (ICBFs), we concurrently address two fundamental challenges: compensating the effect of delays, while ensuring feasibility when state and input constraints are imposed jointly. To this end, we embed predictor feedback into a dynamically defined control law to compensate for delays, with the predicted state evolving according to delay-free dynamics. Then, utilizing ICBFs, we formulate a quadratic program for safe control design. For systems subject to simultaneous state and input constraints, we derive a closed-form feasibility condition for the resulting controller, yielding a compatible ICBF pair that guarantees forward invariance under delay. We also address robustness to prediction errors (e.g., caused by delay uncertainty) using tunable robust ICBFs. Our approach is validated on an adaptive cruise control example with actuation delay.

Integral Control Barrier Functions with Input Delay: Prediction, Feasibility, and Robustness

Abstract

Time delays in feedback control loops can cause controllers to respond too late, and with excessively large corrective actions, leading to unsafe behavior (violation of state constraints) and controller infeasibility (violation of input constraints). To address this problem, we develop a safety-critical control framework for nonlinear systems with input delay using dynamically defined (integral) controllers. Building on the concept of Integral Control Barrier Functions (ICBFs), we concurrently address two fundamental challenges: compensating the effect of delays, while ensuring feasibility when state and input constraints are imposed jointly. To this end, we embed predictor feedback into a dynamically defined control law to compensate for delays, with the predicted state evolving according to delay-free dynamics. Then, utilizing ICBFs, we formulate a quadratic program for safe control design. For systems subject to simultaneous state and input constraints, we derive a closed-form feasibility condition for the resulting controller, yielding a compatible ICBF pair that guarantees forward invariance under delay. We also address robustness to prediction errors (e.g., caused by delay uncertainty) using tunable robust ICBFs. Our approach is validated on an adaptive cruise control example with actuation delay.

Paper Structure

This paper contains 10 sections, 4 theorems, 48 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Integral Control Barrier Functions
  4. State and Input Constraints
  5. Integral CBFs with Input Delay
  6. Predictor Feedback
  7. Feasibility
  8. Robust Safety
  9. Adaptive Cruise Control with Delay
  10. Conclusion

Key Result

Theorem 1

If $h$ is an ICBF for eq:plant_nodelay with the controller eq:ctrl_nodelay, then any locally Lipschitz controller ${\mathbf{v}=\mathbf{q}(\mathbf{x},\mathbf{u})}$, with ${\mathbf{q} : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^m}$, satisfying: for all ${(\mathbf{x},\mathbf{u})\in\mathcal{S}}$, renders $\mathcal{S}$ forward invariant (safe), i.e., ${(\mathbf{x}(0),\mathbf{u}(0)) \in \mathcal{

Figures (2)

  • Figure 1: Simulation of ACC, showing the effect of input delay and predictor compensation: (a) velocity; (b) acceleration; (c) CBF $h_x$; (d) phase portrait. Blue: delay-free reference; yellow: naive controller unaware of input delay; purple: the proposed predictor feedback.
  • Figure 2: Simulation of ACC, demonstrating robustness to prediction error (with $50\%$ delay underestimation): (a) velocity; (b) acceleration; (c) CBF $h_x$; (d) phase portrait. Blue: perfect prediction; yellow: uncertain prediction and no robustification; purple: uncertain prediction with robustification.

Theorems & Definitions (8)

  • Definition 1: ames2020integral
  • Theorem 1: ames2020integral
  • Lemma 1: molnar2023composingcohen2025compatibility
  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Remark 1: Special Cases