  In this paper, we consider Markov Decision Processes (MDPs) with error
states. Error states are those states entering which is undesirable or
dangerous. We define the risk with respect to a policy as the probability of
entering such a state when the policy is pursued. We consider the problem of
finding good policies whose risk is smaller than some user-specified threshold,
and formalize it as a constrained MDP with two criteria. The first criterion
corresponds to the value function originally given. We will show that the risk
can be formulated as a second criterion function based on a cumulative return,
whose definition is independent of the original value function. We present a
model free, heuristic reinforcement learning algorithm that aims at finding
good deterministic policies. It is based on weighting the original value
function and the risk. The weight parameter is adapted in order to find a
feasible solution for the constrained problem that has a good performance with
respect to the value function. The algorithm was successfully applied to the
control of a feed tank with stochastic inflows that lies upstream of a
distillation column. This control task was originally formulated as an optimal
control problem with chance constraints, and it was solved under certain
assumptions on the model to obtain an optimal solution. The power of our
learning algorithm is that it can be used even when some of these restrictive
assumptions are relaxed.
