  We consider the problem of designing control laws for stochastic jump linear
systems where the disturbances are drawn randomly from a finite sample space
according to an unknown distribution, which is estimated from a finite sample
of i.i.d. observations. We adopt a distributionally robust approach to compute
a mean-square stabilizing feedback gain with a given probability. The larger
the sample size, the less conservative the controller, yet our methodology
gives stability guarantees with high probability, for any number of samples.
Using tools from statistical learning theory, we estimate confidence regions
for the unknown probability distributions (ambiguity sets) which have the shape
of total variation balls centered around the empirical distribution. We use
these confidence regions in the design of appropriate distributionally robust
controllers and show that the associated stability conditions can be cast as a
tractable linear matrix inequality (LMI) by using conjugate duality. The
resulting design procedure scales gracefully with the size of the probability
space and the system dimensions. Through a numerical example, we illustrate the
superior sample complexity of the proposed methodology over the stochastic
approach.
