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Robust Safety-Critical Control of Integrator Chains with Mismatched Perturbations via Linear Time-Varying Feedback

Imtiaz Ur Rehman Moussa Labbadi, Amine Abadi, Lew Lew Yan Voon

TL;DR

This work addresses safety-critical control for chains of integrators in the presence of matched and mismatched disturbances by introducing a linear time-varying feedback design that couples backstepping with control barrier function (CBF) theory. A key contribution is a backstepping-based construction of a relative-degree-one CBF, augmented with a QP-based safety filter, and guided by a time-varying gain Υ(t) to enforce safety while preserving performance. The framework is extended from the double integrator to generalized n-th order chains, including smooth robust CBF formulations (SRCBF) to handle disturbances without disturbance observers, and explicit KKT-based solutions for the safety filter. The approach yields persistent safety (no singularities) and robust performance under bounded disturbances, demonstrated through simulations that confirm obstacle avoidance and reduced control effort relative to conventional SBCBF methods. The results offer a scalable, practical foundation for robust safety-critical control in robotics and autonomous systems where disturbances are unavoidable and safety must be maintained indefinitely.

Abstract

In this paper, we propose a novel safety-critical control framework for a chain of integrators subject to both matched and mismatched perturbations. The core of our approach is a linear, time-varying state-feedback design that simultaneously enforces stability and safety constraints. By integrating backstepping techniques with a quadratic programming (QP) formulation, we develop a systematic procedure to guarantee safety under time-varying gains. We provide rigorous theoretical guarantees for the double integrator case, both in the presence and absence of perturbations, and outline general proofs for extending the methodology to higher-order chains of integrators. This proposed framework thus bridges robustness and safety-critical performance, while overcoming the limitations of existing prescribed-time approaches.

Robust Safety-Critical Control of Integrator Chains with Mismatched Perturbations via Linear Time-Varying Feedback

TL;DR

This work addresses safety-critical control for chains of integrators in the presence of matched and mismatched disturbances by introducing a linear time-varying feedback design that couples backstepping with control barrier function (CBF) theory. A key contribution is a backstepping-based construction of a relative-degree-one CBF, augmented with a QP-based safety filter, and guided by a time-varying gain Υ(t) to enforce safety while preserving performance. The framework is extended from the double integrator to generalized n-th order chains, including smooth robust CBF formulations (SRCBF) to handle disturbances without disturbance observers, and explicit KKT-based solutions for the safety filter. The approach yields persistent safety (no singularities) and robust performance under bounded disturbances, demonstrated through simulations that confirm obstacle avoidance and reduced control effort relative to conventional SBCBF methods. The results offer a scalable, practical foundation for robust safety-critical control in robotics and autonomous systems where disturbances are unavoidable and safety must be maintained indefinitely.

Abstract

In this paper, we propose a novel safety-critical control framework for a chain of integrators subject to both matched and mismatched perturbations. The core of our approach is a linear, time-varying state-feedback design that simultaneously enforces stability and safety constraints. By integrating backstepping techniques with a quadratic programming (QP) formulation, we develop a systematic procedure to guarantee safety under time-varying gains. We provide rigorous theoretical guarantees for the double integrator case, both in the presence and absence of perturbations, and outline general proofs for extending the methodology to higher-order chains of integrators. This proposed framework thus bridges robustness and safety-critical performance, while overcoming the limitations of existing prescribed-time approaches.

Paper Structure

This paper contains 8 sections, 4 theorems, 61 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES AND PROBLEM FORMULATION
  3. Main results
  4. Double integrator case without disturbance
  5. Double integrator case with disturbance
  6. Generalized n-th order chain of integrators
  7. Simulations
  8. Conclusions

Key Result

Proposition 1

Consider the states of the unperturbed system (eq:1) for $(n=2)$, which are initially safe, (i.e., $h_1(x_0) > 0$), and Assumption eq:ass_2 hold. Then, the control law (eq:18_a), designed from the CBFs (eq:11) with the initial gain (eq:10), ensures that $h_1(x(t)\geq 0)$$\forall t \geq t_0$.

Figures (2)

  • Figure 1: Double integrator system under nominal law, SBCBF, and the proposed safety filter. The nominal controller reaches the desired goal but intersects the obstacle. The SBCBF fails to prevent collision under perturbations, as indicated by the trajectory portion inside the obstacle. In contrast, the proposed safety filter accomplishes the objective by effectively guiding the system to evade the obstacle, thus preventing a collision.
  • Figure 2: The evolution of the desired candidate CBF $h_1(p)$ wrt time for each controller is presented. This illustrates that, in contrast to the nominal controller and SBCBF, our proposed method guarantees $h_1(p)>0$, thereby maintaining safety $\forall t \geq 0$. Although the nominal law requires less effort than our method, it does not consider safety. Moreover, the SBCBF spends more control effort in comparison to our proposed technique, yet still fails to ensure safety.

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 1
  • proof
  • Remark 4
  • Theorem 1
  • ...and 5 more