Analysis of the optimization landscape of Linear Quadratic Gaussian (LQG) control

• 成果类型: Article
• 署名作者: Tang, Yujie; Zheng, Yang; Li, Na
• 署名单位: Peking University; University of California System; University of California San Diego; Harvard University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-023-01938-4
• 发表日期: 2023
• 页码: 399-444
• 关键词: convergence

摘要

This paper revisits the classical Linear Quadratic Gaussian (LQG) control from a mod-ern optimization perspective. We analyze two aspects of the optimization landscape of the LQG problem: (1) Connectivity of the set of stabilizing controllers C-n; and (2) Structure of stationary points. It is known that similarity transformations do not change the input-output behavior of a dynamic controller or LQG cost. This inherent symmetry by similarity transformations makes the landscape of LQG very rich. We show that (1) The set of stabilizing controllers C-n has at most two path-connected components and they are diffeomorphic under a mapping defined by a similarity transformation; (2) There might exist many strictly suboptimal stationary points of the LQG cost function over C-n that are not controllable and not observable; (3) All controllable and observable stationary points are globally optimal and they are identical up to a sim-ilarity transformation. These results shed some light on the performance analysis of direct policy gradient methods for solving the LQG problem.