An optimal transport approach to data compression in distributionally robust control

We consider the problem of controlling a stochastic linear time-invariant system using a behavioural approach based on the direct optimization of controllers over input-output pairs drawn from a large dataset. In order to improve the computational efficiency of controllers implemented online, we propose a method for compressing this large data set to a smaller synthetic set of representative behaviours using techniques based on optimal transport. Specifically, we choose our synthetic data by minimizing the Wasserstein distance between atomic distributions supported on both the original data set and our synthetic one. We show that a distributionally robust optimal control computed using our synthetic dataset enjoys the same performance guarantees onto an arbitrarily larger ambiguity set relative to the original one. Finally, we illustrate the robustness and control performances over the original and compressed datasets through numerical simulations.