Performance Analysis and Optimal Design of Time-Delay Directed Consensus Networks

Abstract

The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_2$</tex-math></inline-formula> performance of a class of noisy time-delay consensus networks with directed interconnection topologies is considered. For networks with normal Laplacian, we derive a closed-form expression for the performance measure and study its functional properties. The scaling properties of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_2$</tex-math></inline-formula> -norm are investigated for some families of networks. Finally, the problem of tuning feedback gains of a network with fixed topology in order to improve its performance is considered. Our proposed design method can handle networks with a thousand nodes.