  A powerful result from behavioral systems theory known as the fundamental
lemma allows for predictive control akin to Model Predictive Control (MPC) for
linear time invariant (LTI) systems with unknown dynamics purely from data.
While most of data-driven predictive control literature focuses on robustness
with respect to measurement noise, only few works consider exploiting
probabilistic information of disturbances for performance-oriented control as
in stochastic MPC. In this work, we propose a novel data-driven stochastic
predictive control scheme for chance-constrained LTI systems subject to
measurement noise and additive stochastic disturbances. In order to render the
otherwise stochastic and intractable optimal control problem deterministic, our
approach leverages ideas from tube-based MPC by decomposing the state into a
deterministic nominal state driven by inputs and a stochastic error state
affected by disturbances. By tightening constraints probabilistically with
respect to the additive disturbance and robustly with respect to the
measurement noise, satisfaction of original chance-constraints is guaranteed.
The resulting data-driven receding horizon optimal control problem is
lightweight and recursively feasible, and renders the closed loop
input-to-state stable with respect to both additive disturbances and
measurement noise. We demonstrate the effectiveness of the proposed approach in
a simulation example.
