Random-Sampling Multipath Hypothesis Propagation for Cost Approximation in Long-Horizon Optimal Control

In this paper, we develop a Monte-Carlo based heuristic approach to approximate the objective function in long horizon optimal control problems. In this approach, we evolve the system state over multiple trajectories into the future while sampling the noise disturbances at each time-step, and find the weighted average of the costs along all the trajectories. We call these methods random sampling - multipath hypothesis propagation or RS-MHP. These methods (or variants) exist in the literature; however, the literature lacks convergence results for a generic class of nonlinear systems. This paper fills this knowledge gap to a certain extent. We derive convergence results for the cost approximation error from the MHP methods and discuss their convergence (in probability) as the sample size increases. As a case study, we apply RS-MHP to approximate the cost function in a linear quadratic control problem and demonstrate the benefits of our approach against an existing and closely related approximation approach called nominal belief-state optimization.