Performance Improvement in Noisy Linear Consensus Networks with Time-Delay

TL;DR

This paper analyzes how time delays affect the performance of linear consensus networks and proposes methods to improve it by re-weighting, growing, or sparsifying the network topology, with efficient algorithms that approximate optimal solutions.

Contribution

It introduces a convex approximation of the performance measure and presents three novel algorithms for network optimization considering time delays.

Findings

Convexity of performance measure w.r.t. Laplacian eigenvalues and weights.

Non-monotonic performance behavior due to time-delay effects.

Algorithms achieve near-optimal performance with lower complexity.

Abstract

We analyze performance of a class of time-delay first-order consensus networks from a graph topological perspective and present methods to improve it. The performance is measured by network's square of H-2 norm and it is shown that it is a convex function of Laplacian eigenvalues and the coupling weights of the underlying graph of the network. First, we propose a tight convex, but simple, approximation of the performance measure in order to achieve lower complexity in our design problems by eliminating the need for eigen-decomposition. The effect of time-delay reincarnates itself in the form of non-monotonicity, which results in nonintuitive behaviors of the performance as a function of graph topology. Next, we present three methods to improve the performance by growing, re-weighting, or sparsifying the underlying graph of the network. It is shown that our suggested algorithms provide near-optimal solutions with lower complexity with respect to existing methods in literature.