Structural Results and Explicit Solution for Two-Player LQG Systems on a Finite Time Horizon

TL;DR

This paper extends classical LQG results to a two-controller setting with different information sets, providing explicit solutions and efficient recursions for finite-horizon systems.

Contribution

It proves structural properties of controllers, derives explicit coupled state-space solutions, and demonstrates efficient solution methods for the two-controller LQG problem.

Findings

Sufficient statistics are conditional means of the global state.

Explicit coupled recursions for optimal controllers are derived.

Efficient solution algorithms with complexity similar to centralized cases.

Abstract

It is well-known that linear dynamical systems with Gaussian noise and quadratic cost (LQG) satisfy a separation principle. Finding the optimal controller amounts to solving separate dual problems; one for control and one for estimation. For the discrete-time finite-horizon case, each problem is a simple forward or backward recursion. In this paper, we consider a generalization of the LQG problem in which there are two controllers. Each controller is responsible for one of two system inputs, but has access to different subsets of the available measurements. Our paper has three main contributions. First, we prove a fundamental structural result: sufficient statistics for the controllers can be expressed as conditional means of the global state. Second, we give explicit state-space formulae for the optimal controller. These formulae are reminiscent of the classical LQG solution with dual forward and backward recursions, but with the important difference that they are intricately coupled. Lastly, we show how these recursions can be solved efficiently, with computational complexity comparable to that of the centralized problem.