Safety-Critical Control for Discrete-time Stochastic Systems with Flexible Safe Bounds using Affine and Quadratic Control Barrier Functions

TL;DR

This paper tackles safety in discrete-time stochastic systems under Gaussian disturbances by developing a flexible Control Barrier Function (CBF) framework that yields probabilistic, finite-horizon safety guarantees. It introduces an auxiliary function $\Phi$ to bound the $K$-step exit probability via Ville's inequality and derives concrete conditions for both bounded and unbounded $h$, including affine and quadratic CBFs, with corresponding convexity guarantees for the safety-filter optimization. The approach supports tight probability bounds through polynomial and exponential constructions and a scaling technique to tighten bounds further, while also enabling safety under unbounded safe sets. Numerical examples in affine, inverted-pendulum, and obstacle-avoidance settings demonstrate tighter, practically relevant safety bounds and the effectiveness of safety-filter modifications to nominal controllers.

Abstract

This paper presents a safe controller synthesis of discrete-time stochastic systems using Control Barrier Functions (CBFs). The proposed condition allows the design of a safe controller synthesis that ensures system safety while avoiding the conservative bounds of safe probabilities. In particular, this study focuses on the design of CBFs that provide flexibility in the choice of functions to obtain tighter bounds on the safe probabilities. Numerical examples demonstrate the effectiveness of the approach.

Figures (4)

• Figure 1: Safe control with an affine CBF over 200 trials. The color map shows the bounds of $P(x, 150)$ for each state $x$.
• Figure 2: Safe control of an inverted pendulum using different conditions for 500 trials. The dotted lines show the boundary of the safe set $h(x)=0$, and the color map shows the bounds of $P(x, 100)$ for each state $x$.
• Figure 3: Safe control of a single integrator model using Theorem \ref{['thm ville']} for 200 trials. The dotted line shows the boundary of the safe set $h(x)=0$, and the color map shows the bounds of $P(x, 300)$ for each state $x$.
• Figure 4: Safe control of a single integrator model using Theorem \ref{['thm ville']} for 200 trials. The dotted line shows the boundary of the safe set $h(x)=0$, and the color map shows the bounds of $P(x, 300)$ for each state $x$.