  This paper investigates the problem of designing data-driven stochastic Model
Predictive Control (MPC) for linear time-invariant systems under additive
stochastic disturbance, whose probability distribution is unknown but can be
partially inferred from data. We propose a novel online learning based
risk-averse stochastic MPC framework in which Conditional Value-at-Risk (CVaR)
constraints on system states are required to hold for a family of distributions
called an ambiguity set. The ambiguity set is constructed from disturbance data
by leveraging a Dirichlet process mixture model that is self-adaptive to the
underlying data structure and complexity. Specifically, the structural property
of multimodality is exploit-ed, so that the first- and second-order moment
information of each mixture component is incorporated into the ambiguity set. A
novel constraint tightening strategy is then developed based on an equivalent
reformulation of distributionally ro-bust CVaR constraints over the proposed
ambiguity set. As more data are gathered during the runtime of the controller,
the ambiguity set is updated online using real-time disturbance data, which
enables the risk-averse stochastic MPC to cope with time-varying disturbance
distributions. The online variational inference algorithm employed does not
require all collected data be learned from scratch, and therefore the proposed
MPC is endowed with the guaranteed computational complexity of online learning.
The guarantees on recursive feasibility and closed-loop stability of the
proposed MPC are established via a safe update scheme. Numerical examples are
used to illustrate the effectiveness and advantages of the proposed MPC.
