  Markovian jump linear systems (MJLS) are an important class of dynamical
systems that arise in many control applications. In this paper, we introduce
the problem of controlling unknown (discrete-time) MJLS as a new benchmark for
policy-based reinforcement learning of Markov decision processes (MDPs) with
mixed continuous/discrete state variables. Compared with the traditional linear
quadratic regulator (LQR), our proposed problem leads to a special hybrid MDP
(with mixed continuous and discrete variables) and poses significant new
challenges due to the appearance of an underlying Markov jump parameter
governing the mode of the system dynamics. Specifically, the state of a MJLS
does not form a Markov chain and hence one cannot study the MJLS control
problem as a MDP with solely continuous state variable. However, one can
augment the state and the jump parameter to obtain a MDP with a mixed
continuous/discrete state space. We discuss how control theory sheds light on
the policy parameterization of such hybrid MDPs. Then we modify the widely used
natural policy gradient method to directly learn the optimal state feedback
control policy for MJLS without identifying either the system dynamics or the
transition probability of the switching parameter. We implement the
(data-driven) natural policy gradient method on different MJLS examples. Our
simulation results suggest that the natural gradient method can efficiently
learn the optimal controller for MJLS with unknown dynamics.
