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

TL;DR

This paper presents an exact solution and practical algorithms for the mixed-observable constrained linear quadratic regulator problem, enabling optimal control in systems with partial environment observations and state constraints.

Contribution

It reformulates the complex control problem as a finite-dimensional deterministic problem and as a mixed-integer convex program for static environments, providing a globally optimal solution approach.

Findings

The approach effectively solves navigation tasks with partial observations.

The reformulation enables global optimality via branch-and-bound.

Demonstrated on systems reaching goals from partial environment data.

Abstract

This paper studies the problem of steering a linear time-invariant system subject to state and input constraints towards a goal location that may be inferred only through 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 (MICP) 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 system has to reach a goal location identified from partial observations.