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Control Barrier Functions for Stochastic Systems and Safety-critical Control Designs

Yuki Nishimura, Kenta Hoshino

TL;DR

This paper addresses safety guarantees for stochastic control systems by extending control barrier function (CBF) methods to stochastic dynamics. It introduces three main constructs: almost-sure reciprocal CBFs (AS-RCBF), almost-sure zeroing CBFs (AS-ZCBF), and a new stochastic ZCBF that directly incorporates diffusion, enabling explicit probabilistic safety bounds. The authors prove forward-invariance-in-probability (FIiP) results, design safety-critical controls that diverge toward safe-set boundaries when appropriate, and validate the approach via simple numerical examples including constrained inputs. The framework provides a principled way to quantify and enforce safety in stochastic environments, with potential applicability to safety-critical robotics and human-robot interaction where disturbances are present and guarantees are required.

Abstract

In recent years, the analysis of a control barrier function has received considerable attention because it is helpful for the safety-critical control required in many control application problems. While the extension of the analysis to a stochastic system studied by many researchers, it remains a challenging issue. In this paper, we consider sufficient conditions for reciprocal and zeroing control barrier functions ensuring safety with probability one and design a control law using the functions. Then, we propose another version of a stochastic zeroing control barrier function to evaluate a probability of a sample path staying in a safe set and confirm the convergence of a specific expectation related to the attractiveness of a safe set. We also show a way of deisgning a safety-critical control law based on our stochastic zeroing control barrier function. Finally, we confirm the validity of the proposed control design and the analysis using the control barrier functions via simple examples with their numerical simulation.

Control Barrier Functions for Stochastic Systems and Safety-critical Control Designs

TL;DR

This paper addresses safety guarantees for stochastic control systems by extending control barrier function (CBF) methods to stochastic dynamics. It introduces three main constructs: almost-sure reciprocal CBFs (AS-RCBF), almost-sure zeroing CBFs (AS-ZCBF), and a new stochastic ZCBF that directly incorporates diffusion, enabling explicit probabilistic safety bounds. The authors prove forward-invariance-in-probability (FIiP) results, design safety-critical controls that diverge toward safe-set boundaries when appropriate, and validate the approach via simple numerical examples including constrained inputs. The framework provides a principled way to quantify and enforce safety in stochastic environments, with potential applicability to safety-critical robotics and human-robot interaction where disturbances are present and guarantees are required.

Abstract

In recent years, the analysis of a control barrier function has received considerable attention because it is helpful for the safety-critical control required in many control application problems. While the extension of the analysis to a stochastic system studied by many researchers, it remains a challenging issue. In this paper, we consider sufficient conditions for reciprocal and zeroing control barrier functions ensuring safety with probability one and design a control law using the functions. Then, we propose another version of a stochastic zeroing control barrier function to evaluate a probability of a sample path staying in a safe set and confirm the convergence of a specific expectation related to the attractiveness of a safe set. We also show a way of deisgning a safety-critical control law based on our stochastic zeroing control barrier function. Finally, we confirm the validity of the proposed control design and the analysis using the control barrier functions via simple examples with their numerical simulation.
Paper Structure (23 sections, 7 theorems, 68 equations, 4 figures)

This paper contains 23 sections, 7 theorems, 68 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Notations
  4. Target system, the related functions, and a global solution
  5. Motivating Example
  6. An example of safety-critical control for a deterministic system
  7. Trying extension of safety-critical control to a stochastic system
  8. Main Claim
  9. Definitions of a safe set and safety for a stochastic system
  10. CBFs ensuring almost sure safety
  11. A Stochastic ZCBF and Safety-critical Control Design
  12. Examples
  13. Revisit to the motivating example
  14. Confinement in a bounded subset
  15. Numerical simulation
  16. ...and 8 more sections

Key Result

Theorem 1

(A slight modification of Proposition 17 in nishimura2018automatica): Let us consider the system eq:sys-sto-gen, an open subset $M \subset {\mathbb{R}}^n$, a continuous mapping $\phi: M \to {\mathbb{R}}^n$ and an initial condition $x_0 \in M$. If there exists a proper and $C^2$ mapping $Y: M \to [0, is satisfied for all $x \in M$ and for some $c_1 \in [0,\infty)$ and $c_2 \in [0,\infty)$, then the

Figures (4)

  • Figure 1: The shape of $h_1(x)$ with $\alpha=1$ and $N=7$.
  • Figure 2: The shape of $h_2(x)$ for $x_2=0$ with $M=1$.
  • Figure 4: Simulation results of Ex. \ref{['ex:ex1sim']}. The grey colored lines denote 10 sample paths, the red colored lines denote the average of the paths, and the blue lines denote the results for the deterministic system (i.e., $\sigma'=0$ ). Fig. \ref{['fig:ex1-zcbf']} shows that the safety condition $h_1(x)>0$ is satisfied in all 10 trials while the probability of the safety is $0.96$.
  • Figure 9: Time responses of $u_1$.

Theorems & Definitions (16)

  • Definition 1: FIiP and FCiP; a slight modification of (C2) in nishimura2018automatica
  • Theorem 1
  • Definition 2: AS-RCBF
  • Theorem 2
  • Definition 3: AS-ZCBF
  • Theorem 3
  • Corollary 1
  • Remark 1
  • Remark 2
  • Definition 4: Stochastic ZCBF
  • ...and 6 more