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arxiv: 2604.12956 · v1 · submitted 2026-04-14 · 📡 eess.SY · cs.SY

Output-Feedback Safe Control of Discrete-Time Stochastic Systems with Chance Constraints

Jianing Zhao , Zhuoting Cai , Xiang Yin This is my paper

Pith reviewed 2026-05-10 14:34 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords output-feedback controlcontrol barrier functionsstochastic systemschance constraintssafety-critical controldiscrete-time systemsestimation uncertainty
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The pith

An output-feedback control barrier function framework keeps discrete-time stochastic systems safe under estimation uncertainty by converting chance constraints into tractable deterministic conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a method for enforcing safety in stochastic systems when only noisy measurements are available instead of full state information. It introduces an output-feedback version of control barrier functions that works with an expectation-based barrier condition defined over the evolving belief state. To make the method practical for real-time use, it derives conservative deterministic bounds on that expectation using Jensen inequalities applied to belief statistics. The result is a safety filter formulated as a simple optimization problem that can be added to existing controllers while respecting probabilistic safety requirements.

Core claim

The central discovery is an output-feedback CBF framework for discrete-time stochastic systems with chance constraints. It rests on an expectation-based discrete-time barrier condition that explicitly accounts for estimation uncertainty through the belief distribution over the state. Deterministic sufficient conditions are obtained by bounding the expectation with computable functions of the belief statistics via Jensen inequalities, yielding a tractable quadratic program for the safety filter that remains compatible with standard online controllers.

What carries the argument

The expectation-based discrete-time barrier condition, enforced via Jensen-inequality bounds on belief statistics to produce deterministic sufficient conditions for the output-feedback CBF.

If this is right

  • Standard controllers can be augmented with a real-time safety filter that uses only output measurements and belief statistics.
  • The approach handles both process noise and measurement uncertainty while keeping the online optimization tractable.
  • Numerical examples confirm fast computation and reliable safety performance when the derived bounds are applied.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same bounding technique could be tested on systems with non-Gaussian noise to check whether Jensen bounds remain practical.
  • Tighter moment-based or sampling approximations might reduce conservatism while preserving the deterministic structure.
  • The framework naturally pairs with recursive estimators that improve the belief over time, potentially allowing less conservative safety margins as uncertainty decreases.

Load-bearing premise

The Jensen inequalities produce bounds that are tight enough to enforce the original chance constraints reliably without excessive conservatism or practical violations.

What would settle it

Simulate the closed-loop system with deliberately high measurement noise; if the observed frequency of safety-constraint violations exceeds the allowed probability bound over repeated trials, the deterministic sufficient conditions fail to guarantee the claimed safety.

Figures

Figures reproduced from arXiv: 2604.12956 by Jianing Zhao, Xiang Yin, Zhuoting Cai.

Figure 1
Figure 1. Figure 1: One realization of the closed-loop trajectories by the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Closed-loop state trajectories and evolution of the safety function [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Closed-loop state trajectories and evolution of the safety function [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: One realization of the closed-loop trajectories by the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison between the state-feedback controller and [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

In this paper, we investigate safety-critical control problem of discrete-time stochastic systems with incomplete information, where safety constraints must be enforced using state estimates obtained from noisy measurements. We develop an output-feedback control barrier function (CBF) framework based on an expectation-based discrete-time barrier condition that explicitly incorporates estimation uncertainty through the evolving belief over the state. To enable real-time implementation, we derive deterministic sufficient conditions that conservatively enforce the expectation-based CBF by bounding the expectation with computable functions of the belief statistics using Jensen inequalities. The resulting safety filter is formulated as a tractable optimization problem compatible with standard online controllers. Numerical simulations demonstrate that the proposed output-feedback approach achieves fast online computation while providing reliable safety performance in the presence of process noise and measurement uncertainty.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops an output-feedback control barrier function (CBF) framework for discrete-time stochastic systems subject to chance constraints under incomplete state information. It introduces an expectation-based discrete-time barrier condition that incorporates estimation uncertainty via the evolving belief distribution over the state, then derives deterministic sufficient conditions by applying Jensen inequalities to bound the expectation using computable belief statistics. The resulting safety filter is cast as a tractable optimization problem suitable for online use with standard controllers, with numerical simulations demonstrating real-time performance and safety under process and measurement noise.

Significance. If the Jensen-derived bounds are verifiably sufficient (i.e., their satisfaction implies the original expectation-based condition), the work offers a practical bridge between stochastic control, output-feedback estimation, and CBF techniques for safety-critical systems. The emphasis on deterministic, online-solvable surrogates and the provision of numerical validation are strengths that could support deployment in robotics or autonomous systems with partial observations.

major comments (2)
  1. [§4] §4 (derivation of deterministic sufficient conditions): The central step applies Jensen inequalities to obtain a deterministic surrogate for the expectation-based CBF condition. Because Jensen supplies a one-sided bound whose direction depends on convexity/concavity of the composed barrier function, the manuscript must explicitly state the barrier form, the sign of the CBF inequality, and prove that satisfaction of the deterministic condition implies the original expectation condition (rather than merely the converse). Without this, the claim that the bounds 'conservatively enforce' the chance constraints remains unestablished.
  2. [§5] §5 (numerical simulations): The examples illustrate online computation and safety behavior, but do not report quantitative metrics such as empirical violation rates of the chance constraints, measured conservatism (e.g., distance to the boundary of the true expectation condition), or performance under varying noise intensities. These data are needed to assess whether the bounds are tight enough for the central safety claim.
minor comments (2)
  1. [Throughout] Notation for belief statistics (mean, covariance, etc.) should be introduced once and used uniformly; occasional redefinition in later sections reduces readability.
  2. [Abstract and §1] The abstract and introduction would benefit from a single sentence clarifying that the Jensen bounds are claimed to be sufficient (not necessary) for the stochastic safety condition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below and will revise the paper to incorporate the suggested clarifications and additional evaluations.

read point-by-point responses

  1. Referee: [§4] §4 (derivation of deterministic sufficient conditions): The central step applies Jensen inequalities to obtain a deterministic surrogate for the expectation-based CBF condition. Because Jensen supplies a one-sided bound whose direction depends on convexity/concavity of the composed barrier function, the manuscript must explicitly state the barrier form, the sign of the CBF inequality, and prove that satisfaction of the deterministic condition implies the original expectation condition (rather than merely the converse). Without this, the claim that the bounds 'conservatively enforce' the chance constraints remains unestablished.

    Authors: We agree that the directionality of the Jensen bound must be rigorously established to substantiate the conservative enforcement claim. In the revised manuscript, we will explicitly state the barrier function form (a concave function of the state, consistent with standard discrete-time CBF constructions), clarify the sign of the CBF inequality (ensuring the expected barrier change satisfies an upper bound that is strictly negative when the barrier is positive), and add a formal proof in §4 showing that satisfaction of the deterministic surrogate implies the original expectation-based condition holds. This direction follows from the concavity of the composed barrier and the one-sided nature of Jensen's inequality applied to the belief distribution, thereby confirming that our bounds conservatively enforce the chance constraints. revision: yes

  2. Referee: [§5] §5 (numerical simulations): The examples illustrate online computation and safety behavior, but do not report quantitative metrics such as empirical violation rates of the chance constraints, measured conservatism (e.g., distance to the boundary of the true expectation condition), or performance under varying noise intensities. These data are needed to assess whether the bounds are tight enough for the central safety claim.

    Authors: We appreciate this recommendation to strengthen the empirical support. In the revised version of §5, we will include additional quantitative results: empirical violation rates of the chance constraints computed via Monte Carlo simulations (e.g., over 1000 runs), quantitative conservatism metrics comparing the surrogate control to the boundary of the true expectation-based condition (where tractable), and performance evaluations across multiple noise intensity levels. These additions will better demonstrate the tightness and reliability of the proposed bounds while preserving the real-time computation focus. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives an output-feedback CBF framework from an expectation-based discrete-time barrier condition that incorporates belief over the state, then applies standard Jensen inequalities to produce deterministic sufficient conditions for real-time optimization. This chain relies on external mathematical tools (Jensen's inequality for bounding expectations) and established stochastic control concepts rather than any self-definitional reduction, fitted-input prediction, or load-bearing self-citation. No equations or steps reduce the central safety filter to its own inputs by construction; the derivation remains self-contained with independent grounding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The approach rests on standard assumptions of discrete-time stochastic dynamics and noisy measurements; no free parameters or invented entities are explicitly fitted or postulated beyond the new framework itself.

axioms (2)
  • domain assumption The system evolves as a discrete-time stochastic process with additive process noise and linear or known measurement model.
    Invoked implicitly to define the belief evolution and expectation-based barrier condition.
  • standard math Jensen's inequality can be applied to produce conservative deterministic bounds on the probabilistic safety constraint.
    Used to convert the expectation-based CBF into a tractable optimization problem.
invented entities (1)
  • Output-feedback CBF framework with expectation-based barrier condition no independent evidence
    purpose: To enforce safety constraints using only state estimates and belief statistics in stochastic systems.
    The central new construct introduced to handle incomplete information.

pith-pipeline@v0.9.0 · 5422 in / 1406 out tokens · 35997 ms · 2026-05-10T14:34:48.907354+00:00 · methodology

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Reference graph

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