The paper studies the certification of a fixed candidate trajectory on a finite certification grid under parametric uncertainty. For every pair of constraint and time, a scalar measure of the violation of the constraint is defined, and the resulting pointwise chance constraints are aggregated into a worst case margin of the value at risk. The aim is not to generate a new trajectory, but to assess online whether a trajectory produced by a planner or by a predictive controller is sufficiently safe on the certification grid.
Direct evaluation requires the uncertainty to be propagated repeatedly and is often too expensive for models that are computationally demanding. The authors therefore adopt a scheme that separates offline and online work. Offline, a surrogate of the map of constraint violations along the candidate trajectory is constructed with a polynomial chaos expansion when the law of the uncertainty is known, or with kernel regression when only sampled data of inputs and outputs are available. Online, the surrogate is sampled in order to evaluate conservative bounds of the value at risk at low computational cost.
On the theoretical side, a finite sample upper bound for the margin on the grid is derived from empirical quantiles, the inequality of Dvoretzky, Kiefer and Wolfowitz, and a union bound over all pairs of constraint and time, without assuming a parametric family for the underlying distribution of the violations. It is also shown how a uniform bound on the error of the surrogate transfers to the certified margin.
The approach is illustrated on a population balance model of a crystallisation process. The risk estimates based on the surrogate follow the results of direct Monte Carlo simulations, while the time needed for the evaluation during operation is reduced substantially.
The study was carried out at the Institute of Control Systems at Hamburg University of Technology within the Collaborative Research Centre CRC 1615 (SMART Reactors).
Srimanta Santra, Oleksii Molodchyk, Matti Noack, Timm Faulwasser (2026). Fast Risk Certification of Candidate Trajectories under Uncertain Time-Varying Constraints. arXiv:2606.31566.
