Approximate Information States for Worst-case Control of Uncertain Systems

In this paper, we investigate a worst-case-scenario control problem with a partially observed state. We consider a non-stochastic formulation, where noises and disturbances in our dynamics are uncertain variables which take values in finite sets. In such problems, the optimal control strategy can be derived using a dynamic program (DP) with respect to the memory. The computational complexity of this DP can be improved using a conditional range of the state instead of the memory. We present a more general definition of an information state which is sufficient to construct a DP without loss of optimality, and show that the conditional range is an example of an information state. Next, we extend this notion to define an approximate information state and an approximate DP. We also bound the maximum loss of optimality when using an approximate DP to derive the control strategy. Finally, we illustrate our results in a numerical example.