The Mixed-Observable Constrained Linear Quadratic Regulator Problem: The Exact Solution and Practical Algorithms

This article studies the problem of steering a linear system subject to state and input constraints toward a goal location that may be inferred only through noisy partial observations. We assume mixed-observable settings, where the system's state is fully observable and the environment's state defining the goal location is only partially observed. In these settings, the planning problem is an infinite-dimensional optimization problem where the objective is to minimize the expected cost. We show how to reformulate the control problem as a finite-dimensional deterministic problem by optimizing over a trajectory tree. Leveraging this result, we demonstrate that when the environment is static, the observation model piecewise, and cost function convex, the original control problem can be reformulated as a mixed-integer convex program that can be solved to global optimality using a branch-and-bound algorithm. The effectiveness of the proposed approach is demonstrated on navigation tasks, where the goal location should be inferred through noisy measurements.