Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Synthesis and verification of robust-adaptive safe controllers

Simin Liu, Kai S. Yun, John M. Dolan, Changliu Liu

TL;DR

The paper addresses safety guarantees for systems with constant unknown parameters by introducing robust-adaptive CBFs (raCBFs) and solving synthesis and verification problems via sum-of-squares programming (SOSP). It presents a multi-phase, bilinear alternating SOSP framework that certifies raCBFs and enlarges their invariant safe sets while respecting safety constraints. Empirical results on a 2D toy system, a cart-pole, and a 7D planar quadrotor show 100% safety and substantial performance improvements over a robust baseline, with clear scalability to higher dimensions. The approach offers provable safety for uncertain systems and paves the way for extensions to time-varying unknowns and broader controller families.

Abstract

Safe control with guarantees generally requires the system model to be known. It is far more challenging to handle systems with uncertain parameters. In this paper, we propose a generic algorithm that can synthesize and verify safe controllers for systems with constant, unknown parameters. In particular, we use robust-adaptive control barrier functions (raCBFs) to achieve safety. We develop new theories and techniques using sum-of-squares that enable us to pose synthesis and verification as a series of convex optimization problems. In our experiments, we show that our algorithms are general and scalable, applying them to three different polynomial systems of up to moderate size (7D). Our raCBFs are currently the most effective way to guarantee safety for uncertain systems, achieving 100% safety and up to 55% performance improvement over a robust baseline.

Synthesis and verification of robust-adaptive safe controllers

TL;DR

The paper addresses safety guarantees for systems with constant unknown parameters by introducing robust-adaptive CBFs (raCBFs) and solving synthesis and verification problems via sum-of-squares programming (SOSP). It presents a multi-phase, bilinear alternating SOSP framework that certifies raCBFs and enlarges their invariant safe sets while respecting safety constraints. Empirical results on a 2D toy system, a cart-pole, and a 7D planar quadrotor show 100% safety and substantial performance improvements over a robust baseline, with clear scalability to higher dimensions. The approach offers provable safety for uncertain systems and paves the way for extensions to time-varying unknowns and broader controller families.

Abstract

Safe control with guarantees generally requires the system model to be known. It is far more challenging to handle systems with uncertain parameters. In this paper, we propose a generic algorithm that can synthesize and verify safe controllers for systems with constant, unknown parameters. In particular, we use robust-adaptive control barrier functions (raCBFs) to achieve safety. We develop new theories and techniques using sum-of-squares that enable us to pose synthesis and verification as a series of convex optimization problems. In our experiments, we show that our algorithms are general and scalable, applying them to three different polynomial systems of up to moderate size (7D). Our raCBFs are currently the most effective way to guarantee safety for uncertain systems, achieving 100% safety and up to 55% performance improvement over a robust baseline.
Paper Structure (17 sections, 3 theorems, 18 equations, 7 figures, 3 tables, 1 algorithm)

This paper contains 17 sections, 3 theorems, 18 equations, 7 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related works
  3. Problem formulation
  4. Safe control for uncertain systems
  5. Robust-adaptive CBFs
  6. Preliminaries: SOSP
  7. Methodology
  8. raCBF verification
  9. raCBF synthesis
  10. Results
  11. Toy 2D
  12. Cartpole
  13. Quadrotor
  14. Conclusion
  15. Appendix
  16. ...and 2 more sections

Key Result

Theorem 1

Define the set $\mathcal{I}^{r}_{\hat{\theta}} = \{x \in \mathcal{X}, \hat{\theta}\in\Theta \;|\; \phi(x, \hat{\theta}) \geq \frac{1}{2 \gamma} \tilde{\theta}^{\top} \tilde{\theta} \}$ where $\tilde{\theta}$ is the maximum possible estimation error. $\tilde{\theta} = \bar{\theta} - \underaccent{\bar

Figures (7)

  • Figure 1: Two of our three test systems: cartpole with unknown joint friction and quadrotor with unknown drag coefficients.
  • Figure 2: Random trajectories generated by the safety test. Observe that all stay within the invariant set (in red).
  • Figure 3: Note the significant increase in size from the initial to final invariant sets. Also, observe that the final raCBF depends on $\hat{\theta}$ in an intuitive way.
  • Figure 4: Notice that raCBF's invariant set is much larger than the baseline's, which accounts for the difference in the performance test.
  • Figure 5: Detailed analysis of one trial from cartpole's performance test.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Definition 1: Set invariance
  • Definition 2: Safety specification
  • Theorem 1
  • proof
  • Definition 3: raCBF quadratic program (raCBF-QP)
  • Definition 4: Sum-of-squares polynomial
  • Definition 5: Sum-of-squares program
  • Theorem 2: Positivstellensatz stengle1974nullstellensatz
  • Definition 6: S-procedure parrilo2000structured
  • Theorem 3: raCBF verification
  • ...and 1 more