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Explicit Solution of Tunable Input-to-State Safe-Based Controller Under High-Relative-Degree Constraints

Yan Wei, Yu Feng, Linlin Ou, Yueying Wang, Xinyi Yu

TL;DR

This work addresses safety verification for affine nonlinear systems with high-relative-degree constraints under disturbances by deriving explicit, closed-form solutions for $HOCLF$-$HOCBF$ based controllers, with and without a nominal controller, to avoid online quadratic programs. It introduces tunable input-to-state safety ($TISSf$) through a tunable function to balance safety margins under bounded disturbances, reducing conservatism while maintaining safety. The key contributions include explicit QP solutions for both CLF-HOCBF and ISSf-HOCBF formulations, a tunable safety mechanism via $\\varrho$ and $\\epsilon$, and simulation studies demonstrating safety guarantees and improved performance over traditional robust HOCBF approaches. The results offer a practical, analyzable framework for safety-critical control under disturbances in high-relative-degree settings, with potential extension to discrete-time systems in future work.

Abstract

This paper investigates the safety analysis and verification of nonlinear systems subject to high-relative-degree constraints and unknown disturbance. The closed-form solution of the high-order control barrier functions (HOCBF) optimization problem with and without a nominal controller is first provided, making it unnecessary to solve the quadratic program problem online and facilitating the analysis. Further, we introduce the concept of tunable input-to-state safety(ISSf), and a new tunable function in conjunction with HOCBF is provided. When combined with the existing ISSf theorem, produces controllers for constrained nonlinear systems with external disturbances. The theoretical results are proven and supported by numerical simulations.

Explicit Solution of Tunable Input-to-State Safe-Based Controller Under High-Relative-Degree Constraints

TL;DR

This work addresses safety verification for affine nonlinear systems with high-relative-degree constraints under disturbances by deriving explicit, closed-form solutions for - based controllers, with and without a nominal controller, to avoid online quadratic programs. It introduces tunable input-to-state safety () through a tunable function to balance safety margins under bounded disturbances, reducing conservatism while maintaining safety. The key contributions include explicit QP solutions for both CLF-HOCBF and ISSf-HOCBF formulations, a tunable safety mechanism via and , and simulation studies demonstrating safety guarantees and improved performance over traditional robust HOCBF approaches. The results offer a practical, analyzable framework for safety-critical control under disturbances in high-relative-degree settings, with potential extension to discrete-time systems in future work.

Abstract

This paper investigates the safety analysis and verification of nonlinear systems subject to high-relative-degree constraints and unknown disturbance. The closed-form solution of the high-order control barrier functions (HOCBF) optimization problem with and without a nominal controller is first provided, making it unnecessary to solve the quadratic program problem online and facilitating the analysis. Further, we introduce the concept of tunable input-to-state safety(ISSf), and a new tunable function in conjunction with HOCBF is provided. When combined with the existing ISSf theorem, produces controllers for constrained nonlinear systems with external disturbances. The theoretical results are proven and supported by numerical simulations.

Paper Structure

This paper contains 15 sections, 8 theorems, 102 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. System Distribution
  4. High Order Control Barrier Functions
  5. High Order Control Lyapunov Functions
  6. Explicit solution of tunable ISSf HOCBF
  7. Tunable Input to State Safe HOCBF
  8. Explicit Solution of HOCBF-Based Controller Without Disturbance
  9. Tunable ISSf HOCBF-Based Controller
  10. Simulation Studies
  11. Conclusion
  12. Proof of Lemma \ref{['FI-Th-HOCLF']}
  13. Proof of Theorem \ref{['Th-CLF-HOCBF-1']}
  14. Proof of Theorem \ref{['Th-HOCBF-1']}
  15. Proof of Theorem \ref{['Th-TISSf-HOCBF-ForwardInvariant']}

Key Result

Lemma 1

Consider an HOCBF $B(\mathbf{x})$, ${\mathcal{\varphi}}_{i-1}(\mathbf{x})$, $1\le i \le r$ defined in sequence-HOCBF. If $\mathbf{x}(t_0)\in \mathcal{S}(t_0)$, then any locally Lipschitz continuous controller $\mathbf{u}: \mathcal{R}^n\to \mathcal{R}^m$ such that $u\in \mathcal{U}_{cbf}$, $\forall t

Figures (4)

  • Figure 1: Trajectories comparison of $x_1$ under constraints (Case 1).
  • Figure 2: Trajectories of CBF comparison (Case 1).
  • Figure 3: Trajectories comparison of $x_1$ under constraints (Case 2).
  • Figure 4: Trajectories of CBF comparison (Case 2).

Theorems & Definitions (21)

  • Definition 1: Forward Invariant7782377
  • Definition 2: HOCBFXiaoBelta-282
  • Lemma 1: TanShaw-Cortez-296
  • Definition 3: HOCLFWOS:001196709200041
  • Lemma 2
  • proof
  • Definition 4
  • Definition 5: TISSf-HOBF
  • Lemma 3: LyuXu-356
  • Definition 6: TISSf-HOCBF
  • ...and 11 more