On the Semidefinite Duality of Finite-Horizon LQG Problem

TL;DR

This paper explores the semidefinite duality in finite-horizon LQG control problems, deriving SDP formulations, duality, and Riccati equations, offering new convex relaxations and insights for structured control design.

Contribution

It introduces an SDP formulation for finite-horizon LQG, links Riccati equations to KKT conditions, and develops convex relaxations for complex control problems.

Findings

Riccati equations derived from SDP KKT conditions

Decoupling of system and gain matrices in the primal problem

Effective convex relaxations for decentralized control

Abstract

In this paper, our goal is to study fundamental foundations of linear quadratic Gaussian (LQG) control problems for stochastic linear time-invariant systems via Lagrangian duality of semidefinite programming (SDP) problems. In particular, we derive an SDP formulation of the finite-horizon LQG problem, and its Lagrangian duality. Moreover, we prove that Riccati equation for LQG can be derived the KKT optimality condition of the corresponding SDP problem. Besides, the proposed primal problem efficiently decouples the system matrices and the gain matrix. This allows us to develop new convex relaxations of non-convex structured control design problems such as the decentralized control problem. We expect that this work would provide new insights on the LQG problem and may potentially facilitate developments of new formulations of various optimal control problems. Numerical examples are given to demonstrate the effectiveness of the proposed methods.