A new control theory paper offers a way to certify learning based controllers from limited data, something that could eventually matter to advanced 3D printers.
Domagoj Herceg of Eindhoven University of Technology published the June 2026 preprint “PAC-Bayesian Certificates for Quadratic Closed-Loop Control.” The work is generic control theory rather than an additive manufacturing study, but the problem it addresses is increasingly relevant to machines that use sensors and adaptive control — like 3D printers.
The issue is generalization. A controller may look excellent on the disturbance data used to tune it, yet behave poorly when the real machine encounters new vibration, friction, thermal drift or other variation.
PAC Bayes methods try to put a finite sample bound around that uncertainty. Instead of considering only one optimized controller, the approach examines a distribution of possible controller responses and penalizes solutions that move too far from a predefined prior.
The difficulty is that control systems commonly use quadratic trajectory costs. Those costs are unbounded and do not fit neatly into many learning theory tools.
Herceg uses System Level Synthesis, or SLS, to change the coordinates of the problem. Rather than directly optimizing a controller law, SLS parameterizes feasible closed loop responses. In this method, the loss becomes quadratic in both the disturbance trajectory and the free response coordinates, allowing explicit PAC Bayes certificates to be derived.
For Gaussian disturbance trajectories, including correlated disturbances, the paper derives an exact one sided transform and a tractable sensitivity based upper bound. The interesting twist is deployment: the learning process keeps a probability distribution around candidate responses, but the actual controller can be the deterministic mean response.
Where 3D Printing Might Fit
There are several obvious places this idea could eventually intersect with additive manufacturing. Motion control, extrusion pressure, melt flow, thermal regulation and even powder handling can all be seen as systems responding to disturbances.
A learning based controller might be trained from a relatively small number of print runs, particularly on an experimental machine or a newly introduced material. In that situation, a method that explicitly discourages controllers that are highly sensitive to unseen disturbances could be valuable.
Their test used a double integrator with 110 free SLS coordinates. With only ten training trajectories, the data driven PAC Bayes mean response achieved a held out cost of 3.22, compared with 4.96 for ridge regularization. Its measured sensitivity was 0.85 versus 4.19 for ridge. As their dataset grew, the methods converged, suggesting the main benefit to the approach is when data is scarce.
But there is still a large jump from this experiment to a real printer. The strongest results assume finite horizon linear systems and Gaussian or bounded disturbances, without hard state or input constraints. The paper itself identifies constrained model predictive control and model uncertainty as future work.
This is an interesting approach that at some point in the future might find its way into the motion systems of tomorrow’s 3D printers.
Via arXiv
