12,004 research outputs found
Graph Laplacians and Stabilization of Vehicle Formations
Control of vehicle formations has emerged as a topic of significant interest to the controls community. In this paper, we merge tools from graph theory and control theory to derive stability criteria for formation stabilization. The interconnection between vehicles (i.e., which vehicles are sensed by other vehicles) is modeled as a graph, and the eigenvalues of the Laplacian matrix of the graph are used in stating a Nyquist-like stability criterion for vehicle formations. The location of the Laplacian eigenvalues can be correlated to the graph structure, and therefore used to identify desirable and undesirable formation interconnection topologies
Information flow and cooperative control of vehicle formations
We consider the problem of cooperation among a collection of vehicles performing a shared task using intervehicle communication to coordinate their actions. Tools from algebraic graph theory prove useful in modeling the communication network and relating its topology to formation stability. We prove a Nyquist criterion that uses the eigenvalues of the graph Laplacian matrix to determine the effect of the communication topology on formation stability. We also propose a method for decentralized information exchange between vehicles. This approach realizes a dynamical system that supplies each vehicle with a common reference to be used for cooperative motion. We prove a separation principle that decomposes formation stability into two components: Stability of this is achieved information flow for the given graph and stability of an individual vehicle for the given controller. The information flow can thus be rendered highly robust to changes in the graph, enabling tight formation control despite limitations in intervehicle communication capability
Stability and Equilibrium Analysis of Laneless Traffic with Local Control Laws
In this paper, a new model for traffic on roads with multiple lanes is
developed, where the vehicles do not adhere to a lane discipline. Assuming
identical vehicles, the dynamics is split along two independent directions: the
Y-axis representing the direction of motion and the X-axis representing the
lateral or the direction perpendicular to the direction of motion. Different
influence graphs are used to model the interaction between the vehicles in
these two directions. The instantaneous accelerations of each car, in both X
and Y directions, are functions of the measurements from the neighbouring cars
according to these influence graphs. The stability and equilibrium spacings of
the car formation is analyzed for usual traffic situations such as steady flow,
obstacles, lane changing and rogue drivers arbitrarily changing positions
inside the formation. Conditions are derived under which the formation
maintains stability and the desired intercar spacing for each of these traffic
events. Simulations for some of these scenarios are included.Comment: 8 page
Agreement Problems in Networks with Directed Graphs and Switching Topology
In this paper, we provide tools for convergence and performance analysis of an agreement protocol for a network of integrator agents with directed information flow. Moreover, we analyze algorithmic robustness of this consensus protocol for the case of a network with mobile nodes and switching topology. We establish a connection between the Fiedler eigenvalue of the graph Laplacian and the performance of this agreement protocol. We demostrate that a class of directed graphs, called balanced graphs, have a crucial role in solving average-consensus problems. Based on the properties of balanced graphs, a group disagreement function (i.e. Lyapunov function) is proposed for convergence analysis of this agreement protocol for networks with directed graphs. This group disagreement function is later used for convergence analysis for the agreement problem in networks with switching topology. We provide simulation results that are consistent with our theoretical results and demonstrate the effectiveness of the proposed analytical tools
Consensus problems in networks of agents with switching topology and time-delays
In this paper, we discuss consensus problems for networks of dynamic agents with fixed and switching topologies. We analyze three cases: 1) directed networks with fixed topology; 2) directed networks with switching topology; and 3) undirected networks with communication time-delays and fixed topology. We introduce two consensus protocols for networks with and without time-delays and provide a convergence analysis in all three cases. We establish a direct connection between the algebraic connectivity (or Fiedler eigenvalue) of the network and the performance (or negotiation speed) of a linear consensus protocol. This required the generalization of the notion of algebraic connectivity of undirected graphs to digraphs. It turns out that balanced digraphs play a key role in addressing average-consensus problems. We introduce disagreement functions for convergence analysis of consensus protocols. A disagreement function is a Lyapunov function for the disagreement network dynamics. We proposed a simple disagreement function that is a common Lyapunov function for the disagreement dynamics of a directed network with switching topology. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results
Stability Margin Scaling Laws for Distributed Formation Control as a Function of Network Structure
We consider the problem of distributed formation control of a large number of
vehicles. An individual vehicle in the formation is assumed to be a fully
actuated point mass. A distributed control law is examined: the control action
on an individual vehicle depends on (i) its own velocity and (ii) the relative
position measurements with a small subset of vehicles (neighbors) in the
formation. The neighbors are defined according to an information graph.
In this paper we describe a methodology for modeling, analysis, and
distributed control design of such vehicular formations whose information graph
is a D-dimensional lattice. The modeling relies on an approximation based on a
partial differential equation (PDE) that describes the spatio-temporal
evolution of position errors in the formation. The analysis and control design
is based on the PDE model. We deduce asymptotic formulae for the closed-loop
stability margin (absolute value of the real part of the least stable
eigenvalue) of the controlled formation. The stability margin is shown to
approach 0 as the number of vehicles N goes to infinity. The exponent on the
scaling law for the stability margin is influenced by the dimension and the
structure of the information graph. We show that the scaling law can be
improved by employing a higher dimensional information graph.
Apart from analysis, the PDE model is used for a mistuning-based design of
control gains to maximize the stability margin. Mistuning here refers to small
perturbation of control gains from their nominal symmetric values. We show that
the mistuned design can have a significantly better stability margin even with
a small amount of perturbation. The results of the analysis with the PDE model
are corroborated with numerical computation of eigenvalues with the state-space
model of the formation.Comment: This paper is the expanded version of the paper with the same name
which is accepted by the IEEE Transactions on Automatic Control. The final
version is updated on Oct. 12, 201
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