First Order Methods For Globally Optimal Distributed Controllers Beyond Quadratic Invariance

TL;DR

This paper demonstrates that first-order gradient methods can find globally optimal distributed controllers beyond the quadratic invariance condition, expanding the class of problems where optimal solutions are computationally feasible.

Contribution

It introduces a direct gradient descent approach for output-feedback controllers under QI constraints and characterizes a broader class of problems where global optimality is guaranteed.

Findings

Gradient descent converges to global optima despite non-convexity.

Identifies a larger class of problems (US) where first-order methods are effective.

Provides a tractable test for the US property.

Abstract

We study the distributed Linear Quadratic Gaussian (LQG) control problem in discrete-time and finite-horizon, where the controller depends linearly on the history of the outputs and it is required to lie in a given subspace, e.g. to possess a certain sparsity pattern. It is well-known that this problem can be solved with convex programming within the Youla domain if and only if a condition known as Quadratic Invariance (QI) holds. In this paper, we first show that given QI sparsity constraints, one can directly descend the gradient of the cost function within the domain of output-feedback controllers and converge to a global optimum. Note that convergence is guaranteed despite non-convexity of the cost function. Second, we characterize a class of Uniquely Stationary (US) problems, for which first-order methods are guaranteed to converge to a global optimum. We show that the class of US problems is strictly larger than that of strongly QI problems and that it is not included in that of QI problems. We refer to Figure 1 for details. Finally, we propose a tractable test for the US property.