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

Abstract

This paper revisits the classical Linear Quadratic Gaussian (LQG) control from a modern optimization perspective. We analyze two aspects of the optimization landscape of the LQG problem: (1) Connectivity of the set of stabilizing controllers $$\mathcal {C}_n$$ ; 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 $$\mathcal {C}_n$$ 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 $$\mathcal {C}_n$$ that are not controllable and not observable; (3) All controllable and observable stationary points are globally optimal and they are identical up to a similarity transformation. These results shed some light on the performance analysis of direct policy gradient methods for solving the LQG problem.