On the Optimization Landscape of Dynamic Output Feedback Linear Quadratic Control

TL;DR

This paper analyzes the complex optimization landscape of dynamic output-feedback linear quadratic control (dLQR), providing theoretical insights into policy gradient methods and establishing conditions for their optimality and equivalence with LQG control.

Contribution

It characterizes the optimization landscape of dLQR, proves the uniqueness of stationary points under observability, and links dLQR with LQG control for stochastic systems.

Findings

Uniqueness of stationary points for observable dLQR.

Conditions under which dLQR and LQG are equivalent.

Insights into policy gradient algorithm design for partially observed systems.

Abstract

The convergence of policy gradient algorithms hinges on the optimization landscape of the underlying optimal control problem. Theoretical insights into these algorithms can often be acquired from analyzing those of linear quadratic control. However, most of the existing literature only considers the optimization landscape for static full-state or output feedback policies (controllers). We investigate the more challenging case of dynamic output-feedback policies for linear quadratic regulation (abbreviated as dLQR), which is prevalent in practice but has a rather complicated optimization landscape. We first show how the dLQR cost varies with the coordinate transformation of the dynamic controller and then derive the optimal transformation for a given observable stabilizing controller. One of our core results is the uniqueness of the stationary point of dLQR when it is observable, which provides an optimality certificate for solving dynamic controllers using policy gradient methods. Moreover, we establish conditions under which dLQR and linear quadratic Gaussian control are equivalent, thus providing a unified viewpoint of optimal control of both deterministic and stochastic linear systems. These results further shed light on designing policy gradient algorithms for more general decision-making problems with partially observed information.