  We present a learning model predictive control (MPC) scheme for
chance-constrained Markov jump systems with unknown switching probabilities.
Using samples of the underlying Markov chain, ambiguity sets of transition
probabilities are estimated which include the true conditional probability
distributions with high probability. These sets are updated online and used to
formulate a time-varying, risk-averse optimal control problem. We prove
recursive feasibility of the resulting MPC scheme and show that the original
chance constraints remain satisfied at every time step. Furthermore, we show
that under sufficient decrease of the confidence levels, the resulting MPC
scheme renders the closed-loop system mean-square stable with respect to the
true-but-unknown distributions, while remaining less conservative than a fully
robust approach. Finally, we show that the value function of the learning MPC
converges from above to its nominal counterpart as the sample size grows to
infinity. We illustrate our approach on a numerical example.

Optimization and Control;Systems and Control;Systems and Control