1 research outputs found
Performance of Single and Double-Integrator Networks over Directed Graphs
This paper provides a framework to evaluate the performance of single and
double integrator networks over arbitrary directed graphs. Adopting vehicular
network terminology, we consider quadratic performance metrics defined by the
L2-norm of position and velocity based response functions given impulsive
inputs to each vehicle. We exploit the spectral properties of weighted graph
Laplacians and output performance matrices to derive a novel method of
computing the closed-form solutions for this general class of performance
metrics, which include H2-norm based quantities as special cases. We then
explore the effect of the interplay between network properties (e.g. edge
directionality and connectivity) and the control strategy on the overall
network performance. More precisely, for systems whose interconnection is
described by graphs with normal Laplacian L, we characterize the role of
directionality by comparing their performance with that of their undirected
counterparts, represented by the Hermitian part of L. We show that, for
single-integrator networks, directed and undirected graphs perform identically.
However, for double-integrator networks, graph directionality -- expressed by
the eigenvalues of L with nonzero imaginary part -- can significantly degrade
performance. Interestingly in many cases, well-designed feedback can also
exploit directionality to mitigate degradation or even improve the performance
to exceed that of the undirected case. Finally we focus on a system coherence
metric -- aggregate deviation from the state average -- to investigate the
relationship between performance and degree of connectivity, leading to
somewhat surprising findings. For example increasing the number of neighbors on
a \omega-nearest neighbor directed graph does not necessarily improve
performance. Similarly, we demonstrate equivalence in performance between
all-to-one and all-to-all communication graphs.Comment: Index Terms: L2, H2 norm, directed graph, performanc