Analysis of the Optimization Landscape of Linear Quadratic Gaussian (LQG) Control

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 Cn; and 2) structure of stationary points. It is known that similarity transformations do not change the input-output behavior of a dynamical 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 Cn 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 Cn and these stationary points are always non-minimal; 3) all minimal 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.