On the Optimal Control of Network LQR with Spatially-Exponential Decaying Structure

TL;DR

This paper investigates the structure of optimal controllers in network LQR problems with spatially-exponential decaying system matrices, showing they are 'quasi'-SED and enabling near-optimal decentralized control.

Contribution

It proves the 'quasi'-SED structure of optimal LQR controllers and disturbance response controls in SED systems, extending to unstable systems and providing performance bounds for local controllers.

Findings

Optimal LQR gain is 'quasi'-SED in stable systems.

Performance bounds for truncated local controllers are established.

Optimal disturbance response control also exhibits 'quasi'-SED property.

Abstract

This paper studies network LQR problems with system matrices being spatially-exponential decaying (SED) between nodes in the network. The major objective is to study whether the optimal controller also enjoys a SED structure, which is an appealing property for ensuring the optimality of decentralized control over the network. We start with studying the open-loop asymptotically stable system and show that the optimal LQR state feedback gain $K$ is `quasi'-SED in this setting, i.e. $\|[K]_{ij}\|\sim O\left(e^{-\frac{c}{\mathrm{poly}\ln(N)}}\mathrm{dist}(i,j)\right)$. The decaying rate $c$ depends on the decaying rate and norms of system matrices and the open-loop exponential stability constants. Then the result is further generalized to unstable systems under a SED stabilizability assumption. Building upon the `quasi'-SED result on $K$, we give an upper-bound on the performance of $\kappa$-truncated local controllers, suggesting that distributed controllers can achieve near-optimal performance for SED systems. We develop these results via studying the structure of another type of controller, disturbance response control, which has been studied and used in recent online control literature; thus as a side result, we also prove the `quasi'-SED property of the optimal disturbance response control, which serves as a contribution on its own merit.