Sanghoon Kim CDSL J. Alex Fax, Richard M. Murry, Information Flow and Cooperative Control of Vehicle Formations, IEEE A.C. 2004 Cooperation Giving consent to providing ones state and following a common protocol that serves the group objective Consensus Means to reach an agreement regarding a certain quantity of interest that depends on the states of all agents Decentralized Control Depends on only neighbors of each vehicle 2 1) Consensus and Cooperation in Networked Multi-Agent System, IEEE A.C Recent Research in Cooperative Control of Multi-Vehicle Systems,2006 3 Dynamics of i-th Vehicle Task in terms of Cost Function Additively Decoupled Task (or just Decoupled) Decentralized Control Cannot decoupled Cooperative Task Depends on neighbors Role of vehicle Military Systems Formation Flight Alignment Reduction of a drag force Cooperative Classification and Surveillance agent , agent ( ) Cooperative Attack and Rendezvous , Mixed Initiative Systems Human operator + Autonomous vehicles 4 Mobile Sensor Networks Environmental Sampling Distributed Aperture Observing Ex) Collective of microsatellites Virtual big single satellite Transportation Systems Intelligent Highways Safety, Density Air traffic control Collision warning, Congestion Control Free Flight 5 Directed graph G Vertex / Arc Undirected In(Out)-degree Complete Path / Access Strongly Connected Disconnected Communication / Component Initial / Final vertex N-cycle / k-periodic Acycle / Primitive 6 Adjacency matrix Normalized adjacency matrix Laplacian matrix Stochastic matrix Irreducible / Reducible Matrix Reducible if permutation P exists such that Positive (Nonnegative) Matrix 7 8/23 Equivalent Spectral Radius of A = 9 10 11/24 Definition Properties 12 13/23 A I n =? Collection of Dynamics I n A=? Manipulating scalar data from N vehicles Stabilization with constant references Leader Follower approach Simple Reference by the leader Formation stability individual vehicles stability Poor disturbance rejection Heavily on the leader / over-reliance on a single vehicle Virtual Leader approach Good disturbance rejection High communication and computation Communication Topology Robustness to changes in a topology 14/23 15 Dynamics of i-th Vehicle Decentralized Controller All Collective System Internal state measurement External relative state measurement V is internal state Consensus Algorithm Set of vehicles which vehicle i can sense 16 To representation of L 17/23 18 19 NOTE : block diagonal To Upper Triangular 20 U is upper triangular with eigenvalues of L on diagonal T : Schur Transformation of L 21/23 22 23/23 24 Proof) Dynamics of each vehicle Eq. (13) is equivalent to eq.(11) NOTE) zero eigenvalue unobservability of absolute motion of the formation (states x) Assumption Each internal vehicle is stable (inner loop) P A has no eigenvalues in RHP Dont use y P C1 =zero Stabilization of Relative formation dynamics 25 Transfer function of x z for all i Nyquist Criterion for all i Let 26 27 Complete Acycle (Directed) Leader-Follower Single Directed Cycle Nonzero Perron Disk Magnitude of nonzero eigenvalues Bound on Real part of eigenvalues Periodicity BAD 28 K(s) = More arc not better performance Periodicity Bad Measures of Graph Periodicity to quantify stability Weighted Graph Latency on Network Vehicles with Nonlinear Dynamics Next Coming Seminar Information Flows Robustness to Graph Topology Analogous to Disturbance Observer 29