A Bayesian Framework for Safe Optimal Control of Unknown Systems from Input-Output Data

Robert Lefringhausen; Annalena Daniels; Michael Fink; Sandra Hirche

eess.SYNov 2, 2026

A Bayesian Framework for Safe Optimal Control of Unknown Systems from Input-Output Data

Authors: Robert Lefringhausen, Annalena Daniels, Michael Fink, Sandra Hirche

Organizations: School of Computation, Information and Technology, Technical University of Munich, Germany · NTFE GmbH, Munich, Germany

Abstract

As control engineering is applied to increasingly complex systems, learning-based control, which uses empirical data to reduce the reliance on first-principles models, has emerged as a promising paradigm. However, such approaches must cope with nonlinear and stochastic dynamics, the absence of state measurements, and scarce data, while ensuring safe operation under constraints, which requires rigorous uncertainty quantification. Whereas existing methods treat these challenges in isolation, this article proposes a Bayesian framework that jointly addresses them. A prior in state-space form is updated from input-output data using particle Markov chain Monte Carlo methods, yielding a representation of joint uncertainty over dynamics and latent states. Building on this representation, a validation procedure is developed that provides probabilistic guarantees for constraint satisfaction under arbitrary causal control laws. For control synthesis, a formulation penalizing the conditional value-at-risk of the constraint function is proposed, together with an adaptive calibration mechanism that adjusts the performance-robustness trade-off to achieve a desired robustness level. The probability of successful certification is characterized theoretically. The framework is extended to a receding-horizon setting, where posterior updates from new measurements allow the controller to progressively reduce uncertainty and improve performance. The approach is demonstrated on the optimal control of a vertical farm.

Figures & tables

Panel of Figure 1 — Figure 1: Overview of the proposed approach. Training and validation scenarios are generated from PMCMC posterior samples. In attempt $j$ , a candidate input sequence is synthesized from the training scenarios with risk weight $\omega_{j}$ and evaluated on a fresh validation set, yielding a high-confidence bound $\delta_{j}$ on its violation probability. The risk weight is updated until a candidate is certified within the acceptance band or the attempt budget is exhausted. Colors indicate scenario generation (blue, Section 3 and Algorithm 1 ), validation (green, Section 4 ), and synthesis and calibration (orange, Section 5 ).

Panel of Figure 2 — Figure 2: Schematic of the vertical farm. The inputs (temperature, drought level, lighting, and CO 2 concentration) act on the crop, whose state $\bm{x}_{t}$ evolves according to the SIMPLE model with uncertain parameters $\bm{\theta}$ . Cameras provide noisy measurements $y_{t}$ of the biomass, while the remaining states are latent (AI-assisted, Claude Opus 5.5).

Panel of Figure 3 — Figure 3: Biomass trajectories for a representative Monte Carlo run. Solid lines show the trajectories realized on the true system, dashed lines the predictions, and the shaded area the $5$ – $95\,\%$ interval of the posterior predictive distribution under the certified plan. The dotted line shows the predicted mean of the initial attempt. The triangle marks the terminal biomass required to meet the yield target. The certified plan reaches the target, whereas the certainty-equivalent (CE) plan misses it despite predicting otherwise.

Method	Plan found	Violations	$J/J_{\mathrm{oracle}}$
Proposed	$155/200$	$0/155$	$1.24$
Prior-based	$0/200$	–	–
CE	$191/200$	$119/191$	$1.00$
Oracle	$192/200$	–	$1$

Table 1: Open-loop results over

200

Monte Carlo runs. Plans are counted if certified (proposed and prior-based) or feasible (CE and oracle). Violations are reported among runs with a plan, and costs are medians of the per-run cost ratios to the oracle over the

155

runs in which the proposed, CE, and oracle planners all find a plan.

Panel of Figure 4 — Figure 4: Biomass trajectories for the run of Figure 3 under shrinking-horizon planning. Solid lines show the trajectories realized on the true system. The shaded area shows the $5$ – $95\,\%$ interval of the posterior predictive biomass under the certified plan at $t=75$ , while the error bar shows the corresponding interval of the terminal biomass under the certified plan at $t=95$ . The triangle marks the terminal biomass required to meet the yield target. Replanning brings the terminal biomass closer to the target than the open-loop plan, whereas the certainty-equivalent (CE) planner misses the target despite replanning.

Method	Plan in all cycles	Violations	$J/J_{\mathrm{oracle}}$
Proposed	$153/155$	$0/155$	$1.14$
CE	$143/155$	$113/155$	$1.00$

Table 2: Shrinking-horizon results over the

155

runs certified at

t_{0}

. A plan is counted in each cycle if certified (proposed) or feasible (CE). Violations are reported over all runs, including those in which no new plan is found in a later cycle. Costs are medians of the per-run cost ratios to the oracle over all

155

runs.

Optimal Control

Optimal control studies how to choose actions over time to optimize performance while satisfying system dynamics and constraints.

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Uncertainty Quantification Safety-Critical Control Bayesian Optimization Safe Bayesian Optimization

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Uncertainty quantification is essential when deploying learning-based control methods in safety-critical systems. This is commonly realized by constructing uncertainty tubes that enclose the unknown function of interest, e.g., the reward and constraint functions or the underlying dynamics model, with high probability. However, existing approaches for uncertainty quantification typically rely on restrictive assumptions that encode smoothness properties of the unknown function, such as a known norm in a function space. Moreover, these methods usually struggle with discontinuities. In this paper, we model the unknown function as a random function from which independent and identically distributed realizations can be generated. We then construct uncertainty tubes via the scenario approach that hold with high probability. Our uncertainty tubes rely solely on sampled realizations and can therefore accommodate discontinuities represented by the sampling model. We integrate these uncertainty tubes into a safe Bayesian optimization algorithm with which we safely tune control parameters on a real Furuta pendulum.

Abstract

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