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

• Published in: IEEE transactions on automatic control Vol. 68; no. 7; pp. 1 - 8
• Main Authors: Rosolia, Ugo, Chen, Yuxiao, Daftry, Shreyansh, Ono, Masahiro, Yue, Yisong, Ames, Aaron D.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.07.2023; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Constraints; Cost function; Costs; Exact solutions; Hidden Markov models; Linear quadratic regulator; measurement uncertainty; Mixed integer; Noise measurement; observability; Optimal control; Optimization; Planning; Steering; Trajectory; Uncertainty
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2022.3210871

This paper studies the problem of steering a linear system subject to state and input constraints towards 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 (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 goal location should be inferred through noisy measurements.