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Synthesis of safety certificates for discrete-time uncertain systems via convex optimization

Marta Fochesato, Han Wang, Antonis Papachristodoulou, Paul Goulart

Abstract

We study the problem of co-designing control barrier functions and linear state feedback controllers for discrete-time linear systems affected by additive disturbances. For disturbances of bounded magnitude, we provide a semi-definite program whose feasibility implies the existence of a control law and a certificate ensuring safety in the infinite horizon with respect to the worst-case disturbance realization in the uncertainty set. For disturbances with unbounded support, we rely on martingale theory to derive a second semi-definite program whose feasibility provides probabilistic safety guarantees holding joint-in-time over a finite time horizon. We examine several extensions, including (i) encoding of different types of input constraints, (ii) robustification against distributional ambiguity around the true distribution, (iii) design of safety filters, and (iv) extension to general safety specifications such as obstacle avoidance.

Synthesis of safety certificates for discrete-time uncertain systems via convex optimization

Abstract

We study the problem of co-designing control barrier functions and linear state feedback controllers for discrete-time linear systems affected by additive disturbances. For disturbances of bounded magnitude, we provide a semi-definite program whose feasibility implies the existence of a control law and a certificate ensuring safety in the infinite horizon with respect to the worst-case disturbance realization in the uncertainty set. For disturbances with unbounded support, we rely on martingale theory to derive a second semi-definite program whose feasibility provides probabilistic safety guarantees holding joint-in-time over a finite time horizon. We examine several extensions, including (i) encoding of different types of input constraints, (ii) robustification against distributional ambiguity around the true distribution, (iii) design of safety filters, and (iv) extension to general safety specifications such as obstacle avoidance.
Paper Structure (23 sections, 11 theorems, 89 equations, 4 figures, 1 algorithm)

This paper contains 23 sections, 11 theorems, 89 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related works
  3. Contributions and Outline
  4. Notation and Basic Preliminaries
  5. Problem setup
  6. Safety description
  7. Problem formulation
  8. Co-design of CBFs and state feedback controllers
  9. Design for infinite-horizon safety
  10. Design for finite-horizon safety
  11. Extensions
  12. Dealing with input constraints
  13. Dealing with distributional ambiguity
  14. Safety filters
  15. General safety specifications
  16. ...and 8 more sections

Key Result

Lemma 1.2

Let $X_0, X_1,\ldots$ be a non-negative supermartingale. Then, for any real $a > 0$,

Figures (4)

  • Figure 1: Illustration of the safety filter concept.
  • Figure 2: Empirical bound.
  • Figure 3: Theoretical bound.
  • Figure 4: 100 one-second-long trajectories.

Theorems & Definitions (19)

  • Definition 1.1: Supermartingale
  • Lemma 1.2: Ville's inequality, ville1939etude
  • Lemma 1.3: S-Lemma
  • Definition 2.2: Semi-algebraic set
  • Definition 2.3: Infinite-horizon safety
  • Definition 2.4: $(1-\alpha)$-safety in probability
  • Theorem 3.3
  • Remark 3.4: On optimizing over $\lambda$ and $\beta$
  • Lemma 3.5
  • Proposition 3.6
  • ...and 9 more