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A Duality-Based Optimization Formulation of Safe Control Design with State Uncertainties

Xiao Tan, Rahal Nanayakkara, Paulo Tabuada, Aaron D. Ames

Abstract

State estimation uncertainty is prevalent in real-world applications, hindering the application of safety-critical control. Existing methods address this by strengthening a Control Barrier Function (CBF) condition either to handle actuation errors induced by state uncertainty, or to enforce stricter, more conservative sufficient conditions. In this work, we take a more direct approach and formulate a robust safety filter by analyzing the image of the set of all possible states under the CBF dynamics. We first prove that convexifying this image set does not change the set of possible inputs. Then, by leveraging duality, we propose an equivalent and tractable reformulation for cases where this convex hull can be expressed as a polytope or ellipsoid. Simulation results show the approach in this paper to be less conservative than existing alternatives.

A Duality-Based Optimization Formulation of Safe Control Design with State Uncertainties

Abstract

State estimation uncertainty is prevalent in real-world applications, hindering the application of safety-critical control. Existing methods address this by strengthening a Control Barrier Function (CBF) condition either to handle actuation errors induced by state uncertainty, or to enforce stricter, more conservative sufficient conditions. In this work, we take a more direct approach and formulate a robust safety filter by analyzing the image of the set of all possible states under the CBF dynamics. We first prove that convexifying this image set does not change the set of possible inputs. Then, by leveraging duality, we propose an equivalent and tractable reformulation for cases where this convex hull can be expressed as a polytope or ellipsoid. Simulation results show the approach in this paper to be less conservative than existing alternatives.

Paper Structure

This paper contains 11 sections, 5 theorems, 41 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Control Barrier Functions
  4. A Dynamic Uncertainty Perspective
  5. Problem Formulation
  6. A duality-based optimization reformulation
  7. Exactness of the Convex Hull Relaxation
  8. Polytopic Uncertainty
  9. Ellipsoidal Uncertainty
  10. Simulations
  11. Conclusion

Key Result

Theorem 1

Consider the nonlinear system in eq:system with state estimate $\hat{x}$ and state uncertainty set $\mathcal{B}$ in eq:state uncertainty. Suppose that $k_{\rm ro}(\hat{x},\mathcal{B})$ is well defined and locally Lipschitz for any $x\in \mathcal{D}\supset \mathcal{C}$. Then, for the closed-loop syst

Figures (3)

  • Figure 1: State uncertainty and its image under $(a(x), b(x))$
  • Figure 2: Comparison of standard CBFs, R-CBFs, MR-CBFs and the proposed duality based method.
  • Figure 3: State trajectories for Segway

Theorems & Definitions (14)

  • Definition 1: Control Barrier Functions Ames2017
  • Example 1
  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Proposition 1
  • ...and 4 more