Probabilistic Multi-Step Identification With Implicit State Estimation for Stochastic MPC

Stochastic Model Predictive Control (SMPC) is a promising solution for controlling multivariable systems in the presence of uncertainty. However, a core challenge lies in obtaining a probabilistic system model. Recently, multi-step system identification has been proposed as a solution. Multi-step models simultaneously predict a finite sequence of future states, which traditionally involves recursive evaluation of a state-space model. Particularly in the stochastic context, the recursive evaluation of identified state-space models has several drawbacks, making multi-step models an appealing choice. As a main novelty of this work, we propose a probabilistic multi-step identification method for a linear system with noisy state measurements and unknown process and measurement noise covariances. We show that, in expectation, evaluating the identified multi-step model is equivalent to estimating the initial state distribution and subsequently propagating this distribution using the known system dynamics. Therefore, using only recorded data of an unknown linear system, our proposed method yields a probabilistic multi-step model, including the state estimation task, that can be directly used for SMPC. As an additional novelty, our proposed SMPC formulation considers parametric uncertainties of the identified multi-step model. We demonstrate our method in two simulation studies, showcasing its effectiveness even for a nonlinear system with output feedback.