Source-linked AI summary
Control Barrier Functions for Stochastic Systems
Andrew Clark
TL;DR
The paper addresses how to provide computationally tractable safety guarantees for control systems with stochastic dynamics and noisy state measurements. It constructs reciprocal and zero stochastic CBFs for complete and incomplete state information, combines them with CLFs for safety and stability, and validates the approach in multi-agent collision avoidance. The reported guarantees are probability 1 in the complete-information case and depend on estimation accuracy with incomplete information.
Problem
Existing CBF techniques target deterministic systems with exact state observations, motivating stochastic CBFs for systems affected by process and measurement noise.
Method
The paper constructs stochastic reciprocal and zero CBFs for complete and incomplete information, extends them to high relative degree systems, and combines them with CLFs in optimization-based controllers.
Results
The constructions guarantee safety with probability 1 under complete state information, while incomplete-information guarantees depend on estimation accuracy; numerical collision avoidance shows safety alongside reaching desired final states.
Takeaways & Limitations
Stochastic CBFs provide a framework for safe control under complete or noisy state information, with control policies that also address stability.
Takeaways & Limitations
The incomplete-information results can become inapplicable for highly nonlinear systems when the EKF accuracy constant δ is small, and some systems may not admit probability-1 safety policies.
Abstract
from arXiv · showhide
Control Barrier Functions (CBFs) aim to ensure safety by constraining the control input at each time step so that the system state remains within a desired safe region. This paper presents a framework for CBFs in stochastic systems in the presence of Gaussian process and measurement noise. We first consider the case where the system state is known at each time step, and present reciprocal and zero CBF constructions that guarantee safety with probability 1. We extend our results to high relative degree systems with linear dynamics and affine safety constraints. We then develop CBFs for incomplete state information environments, in which the state must be estimated using sensors that are corrupted by Gaussian noise. We prove that our proposed CBF ensures safety with probability 1 when the state estimate is within a given bound of the true state, which can be achieved using an Extended Kalman Filter when the system is linear or the process and measurement noise are sufficiently small. We propose control policies that combine these CBFs with Control Lyapunov Functions in order to jointly ensure safety and stochastic stability. Our results are validated via numerical study on a multi-agent collision avoidance example.
1 Introduction
The paper motivates extending computationally tractable Control Barrier Functions from deterministic, fully observed systems to stochastic systems with process and measurement noise. It develops stochastic reciprocal and zero CBFs, including complete- and incomplete-information settings, and evaluates them for multi-agent collision avoidance.
- Safety requires keeping a control-system state within a desired region and is important in transportation, medicine, and energy.
- Existing CBFs support tractable safety control but assume deterministic dynamics and exact state observations.CBFs use reciprocal or zero constructions whose control constraints keep the barrier finite or positive near the safe-region boundary.
- The paper generalizes CBF methods to stochastic systems with both complete state information and noisy state measurements.For both settings, it formulates stochastic reciprocal and zero CBFs and derives linear control constraints with provable safety guarantees.
- The contributions include probability-1 safety conditions, EKF-based incomplete-information analysis, high-relative-degree constructions, and CBF-CLF controllers implemented through quadratic programs.
- Numerical multi-agent collision-avoidance results show that the proposed ZCBF guarantees safety while allowing agents to reach desired final states.
2 Related Work
The related work positions CBFs among barrier-based safety methods and distinguishes controller synthesis from verification. This paper extends an earlier stochastic-CBF conference version with zero CBFs, high-relative-degree methods, and broader incomplete-information results.
- CBFs have been extended to settings including input constraints, delays, self-triggering, linearization, and signal temporal logic.
- Barrier certificates provide provable safety verification for a given controller but do not synthesize controllers with safety guarantees.
- The present paper adds stochastic zero CBFs, high-relative-degree CBF computation, and incomplete-information results for outputs nonlinear in the input beyond the preliminary conference version.
3 Background
The background introduces probability-space and stochastic-process concepts needed for the paper’s analysis. It covers martingales, stopping times, semimartingales, Itô calculus, and stochastic differential equations.
- Martingales, submartingales, and supermartingales are classified by conditional expectations relative to the current value.
- Stopping times are random times whose threshold events belong to the filtration available by that time.
- Doob’s martingale inequality and its supermartingale corollary provide bounds on process maxima over time intervals.
- A continuous semimartingale decomposes into an initial value, a martingale, and the difference of two continuous nondecreasing adapted processes.
- Itô’s lemma composes twice-differentiable functions with semimartingales, while an Itô SDE uses drift, diffusion, and Brownian-motion terms.
4 Complete-Information CBFs
The paper constructs stochastic reciprocal and zero Control Barrier Functions for complete-information systems, using control constraints to guarantee safety with probability 1. It also extends zero-CBF reasoning to high relative degree systems, while identifying conditions under which safety guarantees may fail.
- 4.1 Problem Statement: The complete-information problem seeks a policy mapping the observed state history to controls that keep the state in the safe region for all time with probability 1.The safe region is defined by C={x:h(x)≥0}, with its boundary given by h(x)=0.
- 4.1 Problem Statement: For some systems, probability-1 safety may be impossible, leaving policies with safety probability ϵ∈(0,1) as a future-work direction.The paper does not construct these lower-probability policies in the presented framework.
- 4.2 Reciprocal Control Barrier Function Construction: Choosing the control input to satisfy the reciprocal-CBF constraint guarantees that the state remains in the safe region with probability 1 when initialized safely.The reciprocal construction uses a barrier that becomes unbounded at the safe-region boundary, and boundedness arguments establish almost-sure safety.
- 4.3 Zero Control Barrier Function Construction: Choosing the control input to satisfy the stochastic zero-CBF constraint guarantees safety with probability 1 when the initial state lies in the safe region.The stochastic construction replaces the deterministic Lie derivative with an Itô derivative to account for noise.
- 4.4 High Relative Degree Systems: High relative degree can prevent the standard zero-CBF condition from being satisfied, so the paper constructs a sequence of functions h_i to address such systems.The issue arises when the control does not directly affect the safety function derivative, with ∂h/∂x g(x)=0 at some states.
5 Incomplete Information CBFs
The paper extends stochastic CBFs to systems with noisy state measurements by combining EKF-based state estimation with safety margins and linear control constraints. Under bounded estimation error, the resulting reciprocal and zero CBF constructions provide safety with probability at least 1−ϵ.
- 5 Incomplete Information CBFs: EKF-based estimation converts incomplete-information safety into a bounded-error problem, enabling CBF design on a safe region for the estimated state.The approach first bounds estimation error, then applies complete-information stochastic CBF methods to the estimated-state dynamics.
- 5 Incomplete Information CBFs: Under the stated assumptions, EKF accuracy can be bounded when process noise is sufficiently small, while highly nonlinear systems may require a small noise threshold δ.The guarantees do not depend on control magnitude, but their applicability can shrink for highly nonlinear dynamics.
- 5 Incomplete Information CBFs: If the estimation error remains within γ and the estimated barrier satisfies the margin hγ, then the true state remains safe for all time.Lemma 4 establishes the deterministic implication from estimation accuracy and the shifted barrier condition to safety.
- 5 Incomplete Information CBFs: The incomplete-information RCBF imposes a linear constraint on each control input and guarantees Pr(x_t ∈ C ∀t) ≥ 1−ϵ when the initial shifted barrier is positive.The other parameters can be evaluated from the noise characteristics and Kalman-filter matrices.
- 5 Incomplete Information CBFs: The incomplete-information ZCBF provides the same probability guarantee when the initial shifted barrier is positive and each control input satisfies its constraint.The proof maintains positivity of the estimated barrier whenever the estimation error stays within γ.
6 CBF-Based Control Policies
The paper combines stochastic CBF constraints with stochastic CLF constraints in quadratic programs for control. When both constraints are jointly feasible, the controller guarantees safety while asymptotically approaching the desired operating point.
- 6 CBF-Based Control Policies: The controller minimizes a positive-definite quadratic objective subject to an affine CBF-feasible set, using the state estimate in the incomplete-information case.With noisy measurements, the EKF supplies the estimate used by the quadratic program.
- 6 CBF-Based Control Policies: Multiple safety constraints can be represented by one linear constraint per condition, but the resulting quadratic program is not guaranteed to have a feasible control input.This feasibility limitation applies when the safe region is an intersection of multiple constraint sets.
- 6 CBF-Based Control Policies: The stochastic CLF constraint alone yields stochastic asymptotic stability, providing the stability component of the joint controller.The joint guarantee follows when the CLF and appropriate CBF constraints can be satisfied simultaneously.
- 6 CBF-Based Control Policies: A feasible control input satisfying both the appropriate stochastic CBF constraint and the CLF constraint guarantees safety and asymptotic approach to the desired operating point.The CBF and CLF requirements are incorporated as linear constraints in the optimization-based controller.
7 Numerical Study
A multi-agent collision-avoidance simulation evaluates stochastic CBF policies under noisy dynamics and measurements. Both CBF approaches avoid safety violations, while the stochastic ZCBF more closely tracks the desired control input than the simplified CBF.
- 7 Numerical Study: Agents are placed on a circle and travel toward opposite destinations while noisy position and velocity dynamics, sensing, and high-relative-degree CBF constraints are used.The simulation minimizes deviation from a nominal linear control law.
- 7 Numerical Study: The stochastic ZCBF produces less deviation from the desired control input than the simplified CBF in the multi-agent simulation.The comparison uses the gap between desired and actual control inputs over time.
- 7 Numerical Study: Both CBF-based policies avoid safety violations, whereas the linear control law violates safety while agents move toward their destinations.The stochastic ZCBF approaches the unsafe region before recovering a safe distance and still reaches the desired final position.
8 Conclusion
The paper develops stochastic reciprocal and zero CBFs for complete and incomplete state information, with probability-one guarantees in the complete case and estimation-dependent guarantees in the incomplete case. It combines these barriers with stochastic CLFs and validates them in multi-agent collision avoidance.
- 8 Conclusion: The framework constructs reciprocal and zero CBFs for complete and incomplete information, with safety guarantees determined by exact state knowledge or estimation accuracy.Complete-information guarantees hold with probability 1, while incomplete-information guarantees depend on bounded estimation error.
- 8 Conclusion: Quadratic programs combining stochastic CBFs and CLFs provide joint safety and stability guarantees at each time step.The approach was evaluated through a multi-agent collision-avoidance simulation.
- 8 Conclusion: Future work targets more general high-relative-degree systems and systems that are not affine in the control input.
(a) (b) (c)
Figure 2 evaluates stochastic CBFs in a noisy multi-agent collision-avoidance task, comparing stochastic ZCBF, simplified CBF, and an unsafe linear controller.
- (a): Ten agents move toward opposite destinations on a circle while avoiding the center under process and measurement noise.
- (b): The simplified CBF produces greater deviation from the desired control action than the proposed stochastic ZCBF.
- (c): The linear controller violates safety, whereas both CBF-based approaches avoid collisions.