1

Cooperative Control of Vehicle FormationSanghoon Kim CDSL2007-12-26

J. Alex Fax, Richard M. Murry, Information Flow and Cooperative Control of Vehicle Formations, IEEE A.C. 20041Introduction1)CooperationGiving consent to providing ones state and following a common protocol that serves the group objectiveConsensusMeans to reach an agreement regarding a certain quantity of interest that depends on the states of all agentsDecentralized ControlDepends on only neighbors of each vehicle21) Consensus and Cooperation in Networked Multi-Agent System, IEEE A.C. 2006 Recent Research in Cooperative Control of Multi-Vehicle Systems ,2006IntroductionDecentralized Cooperative Control3

Dynamics of i-th Vehicle Task in terms of Cost Function Additively Decoupled Task (or just Decoupled) Decentralized Control Cannot decoupled Cooperative TaskDepends on neighborsRole of vehicleApplications1/2 Military SystemsFormation Flight Alignment Reduction of a drag forceCooperative Classification and Surveillance agent , agent ()Cooperative Attack and Rendezvous, Mixed Initiative SystemsHuman operator + Autonomous vehicles 4Applications2/2Mobile Sensor NetworksEnvironmental SamplingDistributed Aperture ObservingEx) Collective of microsatellites Virtual big single satelliteTransportation SystemsIntelligent HighwaysSafety , Density Air traffic controlCollision warning, Congestion Control Free Flight

5Graph Theory Definitions Directed graph G Vertex / ArcUndirectedIn(Out)-degreeCompletePath / AccessStrongly ConnectedDisconnectedCommunication / ComponentInitial / Final vertexN-cycle / k-periodic Acycle / Primitive

6

Adjacency matrixNormalized adjacency matrixLaplacian matrixStochastic matrix

Irreducible / Reducible MatrixReducible if permutation P exists such that

Positive (Nonnegative) MatrixGraph TheoryLaplacian Matrix7

8/23

EquivalentGraph TheoryPerron-Frobenius TheoremSpectral Radius of A =

9

Graph TheoryEigenvalues of Laplacians10

11/24

Kronecker ProductDefinition

Properties12

13/23 AIn =? Collection of Dynamics In A=? Manipulating scalar data from N vehicles

Formation ControlStabilization with constant referencesLeader Follower approachSimple Reference by the leaderFormation stability individual vehicles stabilityPoor disturbance rejectionHeavily on the leader / over-reliance on a single vehicleVirtual Leader approachGood disturbance rejectionHigh communication and computation Communication Topology Robustness to changes in a topology

14/23Formation Equations15

Dynamics of i-th VehicleDecentralized ControllerAll Collective System Internal state measurement External relative state measurement V is internal state Consensus Algorithm

Set of vehicles which vehicle i can sense16

To representation of L17/23

18

19

NOTE : block diagonal To Upper TriangularEquivalence Transformation To Decompose collective dynamics20

U is upper triangular with eigenvalues of L on diagonal T : Schur Transformation of L

21/23

Decompose into pieces22

23/23

Formation Stabilityver.124

Proof) Dynamics of each vehicle Eq. (13) is equivalent to eq.(11)NOTE) zero eigenvalue unobservability of absolute motion of the formation (states x)Formation Stabilityver.2via Nyquist CriterionAssumptionEach internal vehicle is stable (inner loop) PA has no eigenvalues in RHPDont use y PC1 =zero Stabilization of Relative formation dynamics25

Transfer function of x z for all iNyquist Criterionfor all i

Let

Formation Stabilityver.2 via Nyquist Criterion (2)26

Evaluating Formations via Laplacian Eigenvalues27

Complete

Acycle (Directed)Leader-Follower

Single Directed CycleNonzero

Nonzero

Perron DiskMagnitude of nonzero eigenvaluesBound on Real part of eigenvaluesPeriodicityBADExample28

K(s) = More arc not better performance Periodicity Bad 28DiscussionMeasures of Graph Periodicity to quantify stabilityWeighted GraphLatency on NetworkVehicles with Nonlinear Dynamics

Next Coming SeminarInformation FlowsRobustness to Graph TopologyAnalogous to Disturbance Observer

29