Abstract
We study performance of noisy linear time-invariant (LTI) consensus networks in the presence of time-delay with directed graph topologies. The H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm squared of the network is adopted as the performance measure. One of our main results provides an explicit expression for the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm of directed networks with normal Laplacian matrices. This expression is in terms of Laplacian eigenvalues and time-delay. Furthermore, we exploit topology of some well-known graphs and show how their H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm scale with network size and time delay. It is also proven that H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm of a directed network with normal Laplacian is an increasing function of time-delay.
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