5 research outputs found
A Resistance Distance-Based Approach for Optimal Leader Selection in Noisy Consensus Networks
We study the performance of leader-follower noisy consensus networks, and in
particular, the relationship between this performance and the locations of the
leader nodes. Two types of dynamics are considered (1) noise-free leaders, in
which leaders dictate the trajectory exactly and followers are subject to
external disturbances, and (2) noise-corrupted leaders, in which both leaders
and followers are subject to external perturbations. We measure the performance
of a network by its coherence, an norm that quantifies how closely the
followers track the leaders' trajectory. For both dynamics, we show a
relationship between the coherence and resistance distances in an a electrical
network. Using this relationship, we derive closed-form expressions for
coherence as a function of the locations of the leaders. Further, we give
analytical solutions to the optimal leader selection problem for several
special classes of graphs
Submodular Optimization for Consensus Networks with Noise-Corrupted Leaders
We consider the leader selection problem in a network with consensus dynamics
where both leader and follower agents are subject to stochastic external
disturbances. The performance of the system is quantified by the total
steady-state variance of the node states, and the goal is to identify the set
of leaders that minimizes this variance. We first show that this performance
measure can be expressed as a submodular set function over the nodes in the
network. We then use this result to analyze the performance of two greedy,
polynomial-time algorithms for leader selection, showing that the leader sets
produced by the greedy algorithms are within provable bounds of optimal.Comment: 6 pages, 1 figur
On the Trade-off Between Controllability and Robustness in Networks of Diffusively Coupled Agents
In this paper, we demonstrate a conflicting relationship between two crucial
properties---controllability and robustness---in linear dynamical networks of
diffusively coupled agents. In particular, for any given number of nodes
and diameter , we identify networks that are maximally robust using the
notion of Kirchhoff index and then analyze their strong structural
controllability. For this, we compute the minimum number of leaders, which are
the nodes directly receiving external control inputs, needed to make such
networks controllable under all feasible coupling weights between agents. Then,
for any and , we obtain a sharp upper bound on the minimum number of
leaders needed to design strong structurally controllable networks with
nodes and diameter . We also discuss that the bound is best possible for
arbitrary and . Moreover, we construct a family of graphs for any
and such that the graphs have maximal edge sets (maximal robustness) while
being strong structurally controllable with the number of leaders in the
proposed sharp bound. We then analyze the robustness of this graph family. The
results suggest that optimizing robustness increases the number of leaders
needed for strong structural controllability. Our analysis is based on
graph-theoretic methods and can be applied to exploit network structure to
co-optimize robustness and controllability in networks.Comment: IEEE Transactions on Control of Network System
Submodularity in Systems with Higher Order Consensus with Absolute Information
We investigate the performance of m-th order consensus systems with
stochastic external perturbations, where a subset of leader nodes incorporates
absolute information into their control laws. The system performance is
measured by its coherence, an norm that quantifies the total steady-state
variance of the deviation from the desired trajectory. We first give conditions
under which such systems are stable, and we derive expressions for coherence in
stable second, third, and fourth order systems. We next study the problem of
how to identify a set of leaders that optimizes coherence. To address this
problem, we define set functions that quantify each system's coherence and
prove that these functions are submodular. This allows the use of an efficient
greedy algorithm that to find a leader set with which coherence is within a
constant bound of optimal. We demonstrate the performance of the greedy
algorithm empirically, and further, we show that the optimal leader sets for
the different orders of consensus dynamics do not necessarily coincide.Comment: 13 pages, 3 figure
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