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

Published 14 Apr 2026 in eess.SY | (2604.12956v1)

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.

Abstract PDF Upgrade to Chat

Authors (3)

Summary

The paper introduces an expectation-based output-feedback control barrier function framework using Kalman filtering and Jensen-type inequalities to manage chance constraints.
It reformulates safety constraints into a convex optimization problem, enabling tractable real-time control with explicit finite-horizon probabilistic guarantees.
Empirical results demonstrate significant computational gains and robust safety performance in sensor-limited, high-noise stochastic environments.

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

Introduction and Motivation

This work addresses the safe control of discrete-time stochastic systems under incomplete state information, a scenario prevalent in practical cyber-physical applications where sensor noise and process disturbances are inevitable. Traditional CBF-based approaches either presume perfect state feedback or are constrained to deterministic settings. Existing stochastic CBF methods emphasize continuous-time systems and rarely address the output-feedback case in discrete time. The authors' main innovation is an output-feedback control barrier function framework leveraging expectation-based discrete-time conditions that accommodate the presence of estimation uncertainty. This is achieved by integrating Kalman filtering with a statistically rigorous deterministic reformulation of the CBF constraint using Jensen-type inequalities.

Problem Formalization and Methodological Advancement

The paper considers discrete-time linear stochastic systems:

$x_{k+1} = A_k x_k + B_k u_k + w_k,\quad y_k = C_k x_k + v_k,$

where $w_k$ and $v_k$ denote mutually independent, zero-mean Gaussian noise components. Safety is encoded via a function $h(x) \geq 0$ , demarcating forward invariance of a designated safe set. The primary objective is to design an output-feedback controller that tracks a nominal policy while ensuring the probability of violating the safety constraint over a finite horizon $T$ remains below a specified threshold $\epsilon$ .

The critical limitation with expectation-based stochastic CBFs is their intractability in presence of nonconvex domains or complex safety sets, especially when the true system state is unavailable for feedback. The authors address this by:

Introducing an expectation-based CBF condition over the Kalman filter estimate, thus transferring safety properties from the unknown state to the estimated state;
Employing a second-order Taylor expansion and a sharp Jensen-type inequality to deterministically upper bound the expectation in terms of mean and covariance statistics of the belief state;
Reformulating the output-feedback CBF condition to a tractable optimization constraint, compatible with established online control and MPC solvers.

This machinery enables converting safety enforcement, previously reliant on sampling or min-max risk estimation (as required by CVaR-based methods such as [kishida2024risk]), into a deterministic program with explicit finite-horizon probabilistic guarantees.

Output-Feedback CBF Design and Theoretical Guarantees

The output-feedback CBF synthesis proceeds as follows:

State Estimation: A time-varying Kalman filter generates the belief mean $\hat{x}_k$ and error covariance $P_k$ . The key insight is that the estimation error covariance is completely characterized by $P_k$ for linear-Gaussian models.
Safety Function Transformation: The CBF is rendered over the estimated state, and the barrier function is offset by the worst-case error across the confidence set, ensuring nonnegativity of the true state implies that of the estimate with high probability.
Jensen-Based Reformulation: For a twice continuously differentiable $h$ with bounded Hessian, the following deterministic constraint is enforced:

$w_k$ 0

where $w_k$ 1 is a tunable slack capturing second-order uncertainty, parameterized in terms of the Kalman gain, noise covariances, and the Hessian bound.

Probabilistic Guarantee: By leveraging $w_k$ 2-step exit probability bounds and relating true and estimated safety functions, the closed-loop system is guaranteed to satisfy:

$w_k$ 3

where $w_k$ 4 is the probability that the estimate violates the offset safety condition and $w_k$ 5 controls the confidence level over the estimation error.

This entire procedure is formalized as a convex optimization problem (for concave $w_k$ 6), thus ensuring computational tractability. The framework generalizes beyond half-space and ellipsoidal sets to general nonlinear safety specifications, provided regularity and boundedness conditions on $w_k$ 7.

Empirical Evaluation and Numerical Results

The framework's efficacy is assessed via simulation studies paralleling and extending [kishida2024risk]. Specifically:

Linear quadratic systems with half-plane and ellipsoidal safety constraints are examined.
The proposed method matches the empirical safety probabilities of the worst-case CVaR-CBF method under identical settings, but with reduced computational burden in ellipsoidal cases due to avoidance of auxiliary variables and min-max operations.
Figure 1: One realization of the closed-loop trajectories by the two methods under the half-space safety constraint. Both methods achieve the same empirical safety probability $w_k$ 8.

Critically, the Jensen-based output-feedback CBF filter achieves the same empirical safety probability ( $w_k$ 9 under half-space; $v_k$ 0 under the ellipsoidal constraint) as the Kalman-CVaR-CBF, while the average runtime decreases by up to two orders of magnitude in complex sets (e.g., ~46 ms for the proposed method vs. ~2800 ms for CVaR-CBF in ellipsoidal constraints).

Figure 2: One realization of the closed-loop trajectories by the two methods under the ellipsoidal safety constraint. Both methods achieve the same empirical safety probability $v_k$ 1.

Robustness across Monte Carlo trials and generalization to alternative initial conditions and nominal policies validate that theoretical probabilistic guarantees are not overly conservative and align tightly with measured performance.

Practical and Theoretical Implications

On the practical side, the presented methodology enables real-time enforcement of finite-horizon chance constraints in safety-critical stochastic control, directly applicable to robotics, autonomous systems, and digital control implementations. The ability to handle general nonlinear safety specifications (beyond ellipsoids or half-spaces) enhances applicability to complex, real-world safe set geometries. The deterministic reformulation also facilitates deployment on embedded platforms without the computation overhead of scenario-sampling or robust counterpart derivations.

Theoretically, the results extend the class of tractable stochastic CBF designs for discrete-time, output-feedback settings. Unlike previous approaches limited to state feedback or bounding convex sets, this framework systematically accounts for estimation error and statistical covariance propagation, providing explicit, computable finite-horizon guarantees. The connection to convex programming for concave $v_k$ 2 broadens the spectrum of safety verifiable systems within existing optimization-based control toolchains.

Future Directions

Potential future avenues include:

Extension to output-feedback stochastic CBF design within model predictive control architectures, leveraging longer receding horizons for reduced conservatism and anticipatory safety enforcement.
Application to nonlinear systems via extended or unscented Kalman filtering, or leveraging distributional reinforcement learning for belief propagation.
Integration with multi-agent safety assurance in the presence of networked measurement and correlated disturbances.
Exploring less conservative higher-order truncations or non-Gaussian uncertainty propagation for robustness enhancement.

Conclusion

This paper contributes a principled, expectation-based output-feedback CBF framework for discrete-time stochastic systems, integrating Kalman filtering and Jensen-type deterministic reformulation for tractable enforcement of chance constraints. Empirical studies demonstrate reliability and strong computational advantages, particularly for complex safety sets, while providing less conservative and rigorously grounded theoretical safety guarantees for real-world, sensor-limited, stochastic control problems.

Markdown Report Issue