(33c) On Using Feedback Control to Contend with Nature's Randomness

Probability distributions are often used to characterize the randomness of nature. In stochastic model predictive control (SMPC), disturbances are described by a probability distribution that is used within a stochastic optimization problem to construct a feedback control law. While powerful, these probability distributions are themselves subject to their own type of uncertainty, often called distributional uncertainty. In this work, we establish that SMPC, under suitable assumptions, provides a nonzero margin of robustness to this distributional uncertainty. This inherent distributional robustness is afforded by feedback and careful algorithm design. Through a small example, we demonstrate the implications of this result for incorrectly modeled, out-of-sample, and even unmodeled disturbances. This result also covers scenario-based approximations of stochastic optimal control problems and unifies the description of robustness for nominal and stochastic model predictive control.