Recursive Feasibility of Nonlinear Stochastic Model Predictive Control With Gaussian Process Dynamics

ABSTRACT Data‐based learning of system dynamics allows model‐based control approaches to be applied to systems with partially unknown dynamics. Gaussian process regression is a preferred approach that outputs not only the learned system model but also the variance of the model, which can be seen as a measure of uncertainty. Stochastic model predictive control uses a stochastic system model like this to propagate the probability density functions of the predicted states and solves a stochastic optimal control problem in every time step. This paper proposes a stochastic model predictive control algorithm with guarantees on recursive feasibility and stability for nonlinear systems with unknown parts that are learned from data using Gaussian process regression. To this end, an error bound of the Gaussian process prediction is established based on an upper bound of the norm of the unknown dynamics function in the corresponding reproducing kernel Hilbert space. Unlike related work, this allows to consider state and input dependent disturbances instead of independent and identically distributed (iid) white Gaussian noise which is usually considered in the context of stochastic model predictive control. The numerical evaluation illustrates that the resulting control algorithm stabilizes nonlinear systems with partially unknown dynamics.