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
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 ωj and evaluated on a fresh validation set, yielding a high-confidence bound δj 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 ).
Figure 2: Schematic of the vertical farm. The inputs (temperature, drought level, lighting, and CO 2 concentration) act on the crop, whose state xt evolves according to the SIMPLE model with uncertain parameters θ . Cameras provide noisy measurements yt of the biomass, while the remaining states are latent (AI-assisted, Claude Opus 5.5).
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/Joracle
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.
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/Joracle
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 t0 . 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.
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.
Abdullah Tokmak, Toni Karvonen, Thomas B. Schön +1
Cyber-physical Systems Group, Aalto University, Espoo, Finland · School of Engineering Sciences, Lappeenranta–Lahti University of Technology LUT, Lappeenranta, Finland · Department of Information Technology, Uppsala University, Uppsala, Sweden
We present a distributionally robust PAC-Bayesian framework for certifying the performance of learning-based finite-horizon controllers. While existing PAC-Bayes control literature typically assumes bounded losses and matching training and deployment distributions, we explicitly address unbounded losses and environmental distribution shifts (the sim-to-real gap). We achieve this by drawing on two modern lines of research, namely the PAC-Bayes generalization theory and distributionally robust optimization via the type-1 Wasserstein distance. By leveraging the System Level Synthesis (SLS) reparametrization, we derive a sub-Gaussian loss proxy and a bound on the performance loss due to distribution shift. Both are tied directly to the operator norm of the closed-loop map. For linear time-invariant systems, this yields a computationally tractable optimization-based framework together with high-probability safety certificates for deployment in real-world environments that differ from those used in training.
Domagoj Herceg, Duarte Antunes
The authors are with the Department of Mechanical Engineering, Eindhoven university of Technology, Eindhoven, The Netherlands.
This paper proposes a safe data-driven control framework for nonlinear systems with partially known dynamics. The method ensures stability and constraint satisfaction during online learning, assuming only a stabilizable linear approximation of the process is available. Unmodeled nonlinear dynamics are captured by a Gaussian process residual learned in real time. Safety is enforced through a probabilistic control-invariant set derived from Lyapunov theory, guaranteeing high-probability stability. A convex quadratic program computes control inputs that maximize information gain while respecting probabilistic safety constraints. The framework provides finite-sample safety guarantees and allows adaptive expansion of the invariant set as uncertainty decreases. Numerical results validate the approach, demonstrating safe and informative exploration under model uncertainty: the safe set expands by about 30% while the Gaussian process root-mean-square error drops from 1.11 to 0.03.