Optimal Decentralized State-Feedback Control with Sparsity and Delays

This work presents the solution to a class of decentralized linear quadratic state-feedback control problems, in which the plant and controller must satisfy the same combination of delay and sparsity constraints. Using a novel decomposition of the noise history, the control problem is split into independent subproblems that are solved using dynamic programming. The approach presented herein both unifies and generalizes many existing results

Optimal Control for LQG Systems on Graphs---Part I: Structural Results

In this two-part paper, we identify a broad class of decentralized output-feedback LQG systems for which the optimal control strategies have a simple intuitive estimation structure and can be computed efficiently. Roughly, we consider the class of systems for which the coupling of dynamics among subsystems and the inter-controller communication is characterized by the same directed graph. Furthermore, this graph is assumed to be a multitree, that is, its transitive reduction can have at most one directed path connecting each pair of nodes. In this first part, we derive sufficient statistics that may be used to aggregate each controller's growing available information. Each controller must estimate the states of the subsystems that it affects (its descendants) as well as the subsystems that it observes (its ancestors). The optimal control action for a controller is a linear function of the estimate it computes as well as the estimates computed by all of its ancestors. Moreover, these state estimates may be updated recursively, much like a Kalman filter

Optimal Control of Two-Player Systems with Output Feedback

In this article, we consider a fundamental decentralized optimal control problem, which we call the two-player problem. Two subsystems are interconnected in a nested information pattern, and output feedback controllers must be designed for each subsystem. Several special cases of this architecture have previously been solved, such as the state-feedback case or the case where the dynamics of both systems are decoupled. In this paper, we present a detailed solution to the general case. The structure of the optimal decentralized controller is reminiscent of that of the optimal centralized controller; each player must estimate the state of the system given their available information and apply static control policies to these estimates to compute the optimal controller. The previously solved cases benefit from a separation between estimation and control which allows one to compute the control and estimation gains separately. This feature is not present in general, and some of the gains must be solved for simultaneously. We show that computing the required coupled estimation and control gains amounts to solving a small system of linear equations

Exponential Convergence Bounds using Integral Quadratic Constraints

The theory of integral quadratic constraints (IQCs) allows verification of stability and gain-bound properties of systems containing nonlinear or uncertain elements. Gain bounds often imply exponential stability, but it can be challenging to compute useful numerical bounds on the exponential decay rate. In this work, we present a modification of the classical IQC results of Megretski and Rantzer that leads to a tractable computational procedure for finding exponential rate certificates