IPAM, 9 Jan 07 R. M. Murray, Caltech 1 Distributed Consensus and Cooperative Estimation Richard M. Murray Control and Dynamical Systems California Institute of Technology Domitilla Del Vecchio (U Mich) Bill Dunbar (UCSC) Alex Fax (NGC) Eric Klavins (U Wash) Reza Olfati-Saber (Dartmouth) Vijay Gupta Zhipu Jin Demetri Spanos Abhishek Tiwari Yasi Mostofi Cedric Langbort (CMI/UIUC)
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Distributed Consensus and Cooperative Estimationhelper.ipam.ucla.edu/publications/sn2007/sn2007_6707.pdfA = adjacency matrix D = diagonal matrix, weighted by outdegree Properties of
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IPAM, 9 Jan 07 R. M. Murray, Caltech 1
Distributed Consensus andCooperative Estimation
Richard M. Murray
Control and Dynamical Systems
California Institute of Technology
Domitilla Del Vecchio (U Mich) Bill Dunbar (UCSC) Alex
Fax (NGC) Eric Klavins (U Wash)
Reza Olfati-Saber (Dartmouth)
Vijay Gupta Zhipu Jin Demetri Spanos
Abhishek Tiwari Yasi Mostofi
Cedric Langbort (CMI/UIUC)
IPAM, 9 Jan 07 R. M. Murray, Caltech 2
RoboFlag Subproblems
Goal: develop systematic techniques for solving subproblems
• Cooperative control and graph Laplacians
• Distributed receding horizon control
• Verifiable protocols for consensus and control
1. Formation control
• Maintain positions toguard defense zone
2. Distributed estimation
• Fuse sensor data todetermine opponentlocation
3. Distributed consensus
• Assign individuals totag incoming vehicles
Implement and testas part of annual RoboFlag competition
IPAM, 9 Jan 07 R. M. Murray, Caltech 3
Information Flow in Vehicle Formations
Example: satellite formation
• Blue links represent sensed
information
• Green links representcommunicated information
Sensed information
• Local sensors can see some subset of nearbyvehicles
• Assume small time delays, pos’n/vel info only
Communicated information
• Point to point communications (routing OK)
• Assume limited bandwidth, some time delay
• Advantage: can send more complexinformation
Topological features
• Information flow (sensed or communicated)represents a directed graph
• Cycles in graph ! information feedback loops
Question: How does topological structure of information flow affectstability of the overall formation?
• Maintain fixed relative spacing between left andright neighbors
Can extend to more sophisticated “formations”
• Include more complex spatia-temporal constraints
( )i
i j i j ij
j N
e w y y h!
= " "#
relativeposition
weightingfactor
offset
1 2
3
45
6
1 2
3
45
6
IPAM, 9 Jan 07 R. M. Murray, Caltech 5
Graph Laplacian
Construction of (weighted) Laplacian
A = adjacency matrix
D = diagonal matrix, weighted by outdegree
Properties of Laplacian
• Row sum equal 0 (stochastic matrix)
• All eigenvalues are non-negative, with atleast one zero eigenvalue (from row sum)
• Multiplicity of 0 as an eigenvalue is equalto the number of strongly connectedcompo-nents of the graph
• All eigenvalues lie in a circle of radius onecentered at 1 + 0i (Perron Frobenius)
• For bidirectional (eg, undirected) graphs,eigenvalues are all real, in [0,2]
1 11 0 0 0
2 2
0 1 0 0 1 0
1 1 10 0 1
3 3 3
0 0 0 1 1 0
1 1 10 0 1
3 3 3
1 10 0 0 1
2 2
L
! "# #$ %
$ %#$ %
$ %# # #$ %
$ %=#$ %
$ %$ %# # #$ %$ %
# #$ %$ %& '
1L I D A
!= !
IPAM, 9 Jan 07 R. M. Murray, Caltech 6
Mathematical Framework
Analyze stability of closed
loop
• Interconnection matrix, L, isthe Laplacian of the graph
• Stability of closed looprelated to eigenstructure ofthe Laplacian
ˆ ( )K s ˆ( )P s
yL I!
y
h
e u
IPAM, 9 Jan 07 R. M. Murray, Caltech 7
Stability Condition
Theorem The closed loop system is (neutrally) stable iff the Nyquist plot ofthe open loop system does not encircle -1/"i(L), where "i(L) are the nonzeroeigenvalues of L.
Example
ˆ ( )K s ˆ( )P s
yL I!
y
h
e u
2( )
se
P ss
!"
= ( ) d pK s K s K= +
Fax and Murray
IFAC 02, TAC 04
IPAM, 9 Jan 07 R. M. Murray, Caltech 8
Spectra of Laplacians
0,1! =
Unidirectional
tree
2 ( 1) /1 i j N
i e!" #
= #
Cycle
[0,2]! "
Undirected
graph
10, 2
N! != =
Periodic
graph
IPAM, 9 Jan 07 R. M. Murray, Caltech 9
Example Revisited
Example
• Adding link increases the number of three cycles (leads to “resonances”)
• Change in control law required to avoid instability
• Q: Increasing amount of information available decreases stability (??)
• A: Control law cannot ignore the information ! add’l feedback inserted
2( )
se
P ss
!"
= ( ) d pK s K s K= +
x
x
x
x
IPAM, 9 Jan 07 R. M. Murray, Caltech 10
Improving Performance through Communication
Baseline: stability only
• Poor performance due to interconnection
Method #1: tune information flow filter
• Low pass filter to damp response
• Improves performance somewhat
Method #2: consensus + feedforward
• Agree on center of formation, then move
• Compensate for motion of vehicles byadjusting information flow
Fax and Murray
IFAC 02
IPAM, 9 Jan 07 R. M. Murray, Caltech 11
Special Case: Consensus
Consensus: agreement between agents using information flow graph
• Can prove asymptotic convergence to single value if graph is connected
• If wij = 1/(in-degree) + graph is balanced (same in-degree for all nodes) ! all
agents converge to average of initial condition
Variations and extensions (Jadbabaie, Leonard, Moreau, Morse, Olfati-