Probabilistic feasibility guarantees for solution sets to uncertain variational inequalities

F. Fabiani, K. Margellos and P. J. Goulart

@article{FMG:2022,
  author = {F. Fabiani and K. Margellos and P. J. Goulart},
  title = {Probabilistic feasibility guarantees for solution sets to uncertain variational inequalities},
  journal = {Automatica},
  year = {2022},
  volume = {137},
  pages = {110-120},
  url = {https://www.sciencedirect.com/science/article/pii/S000510982100649X},
  doi = {10.1016/j.automatica.2021.110120}
}

We leverage on a data-driven paradigm to provide a-posteriori feasibility certificates to the set of solutions to variational inequalities affected by uncertainty. Specifically, we focus on instances with a deterministic mapping and an uncertain feasibility set, and represent uncertainty by means of scenarios. Building upon recent advances in the scenario approach literature, we quantify the robustness properties of the entire set of solutions against a new unseen realization of the uncertainty. This allows us to circumvent the necessity that the variational inequality admits a unique solution. We show that assessing the violation probability of the entire set of solutions rather than a single one requires enumeration of the support constraints that ‘‘shape’’ this set. In this context, we also propose a general procedure to enumerate the support constraints that does not need a closed form description of the solution set, which is unlikely to be available. We show that robust game theory constitutes an applications class that falls within the considered framework of uncertain variational inequalities, and illustrate our theoretical results through extensive numerical simulations on a case study involving an electric vehicle charging coordination problem.

Probabilistic stabilizability certificates for a class of black-box linear systems

Filippo Fabiani, Kostas Margellos and Paul J. Goulart

@article{FMG:2022b,
  author = {Filippo Fabiani and Kostas Margellos and Paul J. Goulart},
  title = {Probabilistic stabilizability certificates for a class of black-box linear systems},
  journal = {IEEE Control Systems Letters},
  year = {2022},
  volume = {6},
  pages = {584-589},
  url = {https://ieeexplore.ieee.org/document/9441020},
  doi = {10.1109/LCSYS.2021.3083962}
}

We provide out-of-sample certificates on the controlled invariance property of a given set with respect to a class of black-box linear systems generated by a possibly inexact quantification of some parameters in the state-space matrices. By exploiting a set of realizations of those undetermined parameters, verifying the controlled invariance property of the given set amounts to a linear program, whose feasibility allows us to establish an a-posteriori probabilistic certificate on the controlled invariance property of such a set with respect to the unknown linear time-invariant dynamics. We apply this framework to the control of a networked system with unknown weighted graph.