Data-Driven Ambiguity Sets for Linear Systems Under Disturbances and Noisy Observations

This paper studies the characterization of Wasserstein ambiguity sets for dynamic random variables when noisy partial observations are progressively collected from their evolving distribution. The ambiguity sets are accompanied by quantitative guarantees about the true distribution of the data, which renders them appropriate for the formulation of robust stochastic optimization problems. To describe the evolution of the variable, we consider a linear discrete-time dynamic model with random initial conditions, stochastic uncertainty in the dynamics, and partial noisy measurements. The probability distribution of all the involved random elements is supposed to be unknown. To make inferences about the distribution of the state vector, we collect several output samples from multiple realizations of the process. We use a classical Luenberger observer to obtain full-state estimators for the independent realizations and exploit these further to build the centers of the ambiguity sets.