A Primer on Graph Theory
Richard M. Murray
Control and Dynamical Systems
California Institute of Technology
Check use of Laplacian versus
weighted Laplacian in
consensus protocol. Not clear
that we converge the the
average due to the weighting.
Goals
• Introduce some motivating cooperative control problems
• Describe basic concepts in graph theory (review)
• Introduce matrices associated with graphs and related properties
(spectra)
Based on CDS 270 notes by Reza Olfati-Saber (Dartmouth) and PhD thesis of
Alex Fax (Northrop Grumman).
References
• R. Diestel, Graph Theory. Springer-Verlag, 2000.
• C. Godsil and G. Royle, Algebraic Graph Theory. Springer, 2001.
• R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge Univ Press,1987.
• R. Olfati-Saber and M, “Consensus Problems in Networks of Agents”, IEEE
Transactions on Automatic Control, 2004.
Cooperative Control Applications
Transportation
Air traffic control
Intelligent transportation systems
(ala California PATH project)
•
•
Military
Distributed aperture imaging
Battlespace management
•
•
Scientific
Distributed aperture imaging
Adaptive sensor networks (eg,
Adaptive Ocean Sampling
Network)
•
•
Commercial
Building sensor networks (related)
•
Non-vehicle based applications
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Richard M. Murray, Caltech CDS
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Example: RoboFlag (D’Andrea, Cornell)
D’Andrea & M
ACC 2003
Arbiter
Humans
(2-3 per team)
computers
for each vehicle
Robot version of “Capture the Flag”
Teams try to capture flag of opposing
team without getting tagged
Mixed initiative system: two humans
controlling up to 6-10 robots
Limited BW comms + limited sensing
•
•
•
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Problem Framework
Agent dynamics
ẋi = f i (xi , ui ) xi ∈ Rn , ui ∈ Rm
y i = hi (xi )
y i ∈ SE(3),
Task
Encode as finite horizon optimal control
•
J=
!
T
L(x, α, u) dt + V (x(T ), α(T )),
0
• Assume task is coupled
Vehicle “role”
α ∈ A encodes internal state +
relationship to current task
Transition α! = r(x, α)
•
Strategy
Control action for individual agents
•
•
{gji (x, α) : rji (x, α)}
ui = γ(x, α)
!
rji (x, α)
g(x, α) = true
i!
α =
unchanged otherwise.
Communications graph G
Encodes the system information flow
Neighbor set N i (x, α)
•
•
Decentralized strategy
ui (x, α) = ui (xi , αi , x−i , α−i )
x−i = {xj1 , . . . , xjmi }
jk ∈ N i
mi = |N i |
• Similar structure for role update
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Richard M. Murray, Caltech CDS
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Information Flow in Vehicle Formations
Sensed information
• Local sensors can see some subset of nearby
vehicles
• 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 complex
information
Example: satellite formation
Blue links represent sensed
information
Green links represent
communicated information
•
•
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 affect
stability of the overall formation?
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Richard M. Murray, Caltech CDS
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Basic
Definitions (1
(1/2)
Basic
Definitions
of 2)
Definition 1. A graph is a pair G = (V, E) that consists of a set of vertices V and a
set of edges E ⊆ V × V:
• Vertices: vi ∈ V
• Edges: eij = (vi , vj ) ∈ E
Example:
V = {1, 2, 3, 4, 5, 6}
E = {(1, 6), (2, 1), (2, 3), (2, 6), (6, 2), (3, 4),
(3, 6), (4, 3), (4, 5), (5, 1), (6, 1), (6, 2), (6, 4)}
Notation:
• Order of a graph = number of nodes: |V|
• vi and vj are adjacent if there exists e = (vi , vj )
• An adjacent node vj for a node vi is called a neighbor of vi
• Ni = set of all neighbors of vi
• G is complete if all nodes are adjacent
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2
Basic Definitions (2/2)
Basic Definitions (2 of 2)
Undirected graphs
• A graph is undirected if eij ∈ E =⇒ eji ∈ E
• Degree of a node: deg(vi ) := |Ni |
• A graph is regular (or k-regular) if all vertices of a graph
have the same degree k
Directed graphs (digraph)
• Out-degree of vi : degout = number of edges eij = (vi , vj )
• In-degree of vi : degin = number of edges eki = (vk , vi )
Balanced graphs
• A graph is balanced if out-degree = in-degree at each node
1
2
6
2
5
3
4
6
8
1
1
10
2
4
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Richard
M. Murray,
Caltech
CDS
R. M.(b)
Murray,
Caltech
CDS
Figure 3: Three examples of balanced graphs.
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6
3
4
5
3
10
4
1
(c)
9
2
(a)
3
7
5
4.3. Algebraic Graph Theory
Connectedness of Graphs (1/2)
Connectedness of Graphs (1 of 2)
Paths
• A path is a subgraph π = (V, Eπ ) ⊂ G with
distinct nodes V = {v1 , v2 , . . . , vm } and
Eπ := {(v1 , v2 ), (v2 , v3 ), . . . , (vm−1 , vm )}.
1
2
6
8
3
9
7
10
• The length of π is defined as |Eπ | = m − 1.
• A cycle (or m-cycle) C = (V, EC ) is a path
(of length m) with an extra edge (vm , v1 ) ∈ E.
5
Connectivity of undirected graphs
4
1
• An undirected graph G is called connected if there
exists a path π between any two distinct nodes of G. 1
• For a connected graph G, the length of the longest
path is called the diameter of G.
6
• A tree is a connected acyclic graph
8
5
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3
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6
10
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4
12
Figure 1.4: A bipartite graph.
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Figure 4.1: Sample Graph G.
2
3
• A graph with no cycles is called acyclic
12
11
Figure 4.2: Induced Subgraphs of Components of G.
Richard M. Murray, Caltech CDS
For R.
example,
the adjacency
M. Murray,
Caltech matrix
CDS of the graph in Figure 1.1 is

0 0 0 0 0 1

4
8
(or vertices) of a graph is called the order of the graph. This is equal to |V| w
|S| denotes the number of elements of S. Figure 1.1 shows an example of a gra
the following set of vertices and edges:
V =(2/2)
{1, 2, 3, 4, 5, 6}
Connectedness of Graphs
E = {16,
23, 26,2)
62, 34, 36, 43, 45, 51, 61, 62, 64}
Connectedness of Graphs
(221,of
Connectivity of directed graphs
4.3.
1
• A digraph is called strongly connected if there
exists a directed path π between any two distinct
nodes
G.
Algebraic
GraphofTheory
51
• A digraph is called weakly connected if there
exists an undirected path between any two distinct
nodes of G.
1
2
8
6
3
5
4
7
6
3
4
Strongly connected
Figure 1.1: A graph of order 6.
9
1.2
5
2
Neighbors of a Node
10
Two nodes of the graph are called adjacent iff there exists an edge between th
node vj of the node vi is called the neighbor of vi . The set of all neighbors of
12
11
11
???
Weakly
connected
Figure
4.1: Sample
Graph G.
• Q: how do we check connectivity of a graph?
1
2
8
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Richard M. Murray, Caltech CDS
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Matrices Associated with a Graph
er 1
Matrices Associated with a Graph
Capture properties of a graph using matrices
• The adjacency matrix A = [aij ] ∈ Rn×n of a graph G of order n is given by:
h Theory

1 if (v , v ) ∈ E
i j
aij :=
0 otherwise
provides some background on graph theory based on the materials in [7, 5].
phs
• The degree matrix of a graph as a diagonal n × n (n = |V|) matrix
pair G = (V, E) that
of a set of vertices
V and a set of edges E ⊆ V × V.
∆ consists
= diag{deg
out (vi )} with diagonal elements equal to the out-degree of each node and
of the graph is denoted by vi or i where i is an index that belongs to an index set
everywhere
, n}. Each edge of zero
the graph
is denoted byelse.
(vi , vj ) with i, j ∈ I. Equivalently, for
nvenience, we sometimes denote (vi , vj ) by eij or simply ij. The number of the nodes
• The
Laplacian
matrix
L oftoa|V|graph
is any
defined
of a graph is called
the order
of the graph.
This is equal
where for
set S, as
e number of elements of S. Figure 1.1 shows an example of a graph of order 6 with
set of vertices and edges:
L=∆−A
V = {1, 2, 3, 4, 5, 6}
E = •{16,
26, 62,
34, 36,of
43, the
45, 51,Laplacian
61, 62, 64}
The
row
sums
. 21, 23,
1
2
5
2
6
3
4
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1.1: A graph of order 6.
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ghbors of
a Node
(1.1)
are all 0.
3
0
0
0
0
0
1
6
61
6
6
60
A=6
6
60
6
61
4
0
1
0
0
0
0
1
0
0
1
0
1
0
0
0
0
1
0
1
0
7
17
7
7
17
7
7
07
7
07
5
0
1
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1
6
6−1
6
6
6 0
L=6
6
6 0
6
6−1
4
−1
3
0
0
0
0
−1
3
−1
0
0
0
2
−1
0
0
−1
2
−1
0
0
0
1
−1
0
−1
0
7
−17
7
7
−17
7
7
0 7
7
0 7
5
3
10
6
Periodic Graphics and Weighted Graphs
Periodic Graphs and Weighted Graphs
Periodic and acyclic graphs
• A graph with the property that the set of all cycle
lengths has a common divisor k > 1 is called k-periodic.
• A graph without cycles is said to be acyclic.
Weighted graphs
• A weighted graph is graph (V, E) together with a map
ϕ : E → R that assigns a real number wij = ϕ(eij ) called
a weight to an edge eij = (vi , vj ) ∈ E.
• The set of all weights associated with E is denoted by
W.
• A weighted graph can be represented as a triplet G =
(V, E, W).
Weighted Laplacian

1
− 1
 2
 0

 0

 0
0
− 12
1
0
0
0
−1
0
− 12
1
−1
− 12
0
0
0
0
1
− 12
0
0
0
− 21
0
1
0
Weighted Laplacian

− 12
0 

− 12 

0 

0 
1
• In some applications it is natural to “normalize” the
Laplacian by the outdegree
• L̃ := ∆−1 L = I − P , where P = ∆−1 A (weighted adjacency matrix).
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7
Consensus
protocols
Application:
Consensus
Protocols
Consider a collection of N agents that communicate along
a set of undirected links described by a graph G. Each
agent has a state xi with initial value xi (0) and together
they wish to determine the average of the initial states
!
Ave(x(0)) = 1/N
xi (0).
The agents implement the following consensus protocol:
"
ẋi =
(xj − xi ) = −|Ni |(xi − Ave(xNi ))
j∈Ni
Sensor Network
which is equivalent to the dynamical system
ẋ = u
u = −Lx.
40
35
Proposition 1. If the graph is connected, the state of the
agents converges to x∗i = Ave(x(0)) exponentially fast.
xi
30
• Proposition 1 implies that the spectra of L controls the
stability (and convergence) of the consensus protocol.
• To (partially) prove this theorem, we need to show
that the eigenvalues of L are all positive.
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20
15
10
0
10
20
iteration
30
40
12
8
Gershgorin
Theorem
Gershgorin Disk
Disk Theorem
Theorem 2 (Gershgorin Disk Theorem). Let A = [aij ] ∈ Rn×n and define the deleted
absolute row sums of A as
n
!
|aij |
(1)
ri :=
j=1,j"=i
Then all the eigenvalues of A are located in the union of n disks
G(A) :=
n
"
Gi (A), with
Gi (A) := {z ∈ C : |z − aii | ≤ ri }
(2)
i=1
Furthermore, if a union of k of these n disks forms a connected region that is disjoint from
all the remaining n − k disks, then there are precisely k eigenvalues of A in this region.
Sketch of proof Let λ be an eigenvalue of A and let v be a corresponding eigenvector.
Choose i such that |vi | = maxj |vj > 0. Since v is an eigenvector,
!
!
λvi =
Aij vj
=⇒
(λ − aii )vi =
Aij vj
i
i"=j
Now divide by vi %= 0 and take the absolute value to obtain
!
!
|λ − aii | = |
aij vj | ≤
|aij | = ri
j"=i
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j"=i
Richard M. Murray, Caltech CDS
R. M. Murray, Caltech CDS
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9
Propertiesof
of the
the Laplacian
(1) (1)
Properties
Laplacian
Proposition 3. Let L be the Laplacian matrix of a digraph G with maximum node
out–degree of dmax > 0. Then all the eigenvalues of A = −L are located in a disk
B(G) := {s ∈ C : |s + dmax | ≤ dmax }
(3)
Remark 5. The notion of algebraic connectivity (or λ2 ) of graphs was o
that is located in the closed LHP
of s-plane
and is tangent
to the
axis atthis
s =notion
0.
M. Fiedler
for undirected
graphs
[13,imaginary
14]. We extend
to alge
digraphs by defining the mirror operation on digraphs that produces an
Proposition 4. Let L̃ be the weighted Laplacian matrix of a digraph G. Then all the
from a digraph G (See Definition 2).
eigenvalues of A = −L are located inside a disk of radius 1 that is located in the closed
The key in the stability analysis of system (8) is in the spectral
LHP of s-plane and is tangent to the imaginary axis at s = 0.
Laplacian. The following result is well-known for undirected graphs (e.g
result
digraphs
and digraph
prove itofusing
Geršgorin
Theorem 5 (Olfati-Saber). state
Let G the
= (V,
E, Wfor
) be
a weighted
order
n with disk theorem
Laplacian L. If G is strongly connected, then rank(L) = n − 1.
Im
r=d max
Remarks:
• Proof for the directed case is standard
• Proof for undirected case is available in
Olfati-Saber & M, 2004 (IEEE TAC)
• For directed graphs, need G to be strongly connected
and converse is not true.
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r
Spec(!L)
Re
Spec(L)
Richard M. Murray, Caltech CDS
14 to grap
Figure 1: A demonstration of Geršgorin Theorem applied
R. M. Murray, Caltech CDS
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Proof of Consensus Protocol
Proof of the Consensus Protocol
ẋ = −Lx
L=∆−A
Note first that the subspaced spanned by 1 = (1, 1, . . . , 1)T is an invariant subspace since
L · 1 = 0 Assume that there are no other eigenvectors with eigenvalue 0. Hence it suffices
to look at the convergence on the complementary subspace 1⊥ .
Let δ be the disagreement vector
δ = x − Ave(x(0)) 1
and take the square of the norm of δ as a Lyapunov function candidate, i.e. define
V (δ) = "δ"2 = δ T δ
(4)
Differentiating V (δ) along the solution of δ̇ = −Lδ, we obtain
V̇ (δ) = −2δ T Lδ < 0,
∀δ $= 0,
(5)
where we have used the fact that G is connected and hence has only 1 zero eigenvalue
(along 1). Thus, δ = 0 is globally asymptotically stable and δ → 0 as t → +∞, i.e.
x∗ = limt→+∞ x(t) = α0 1 because α(t) = α0 = Ave(x(0)), ∀t > 0. In other words, the
average–consensus is globally asymptotically achieved.
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Perron-Frobenius Theory
Perron-Frobenius Theory
Spectral radius:
• spec(L) = {λ1 , . . . , λn } is called the spectrum of L.
• ρ(L) = |λn | = maxk |λk | is called the spectral radius of L
Theorem 6 (Perron’s Theorem, 1907). If A ∈ Rn×n is a positive matrix (A > 0), then
1. ρ(A) > 0;
2. r = ρ(A) is an eigenvalue of A;
3. There exists a positive vector x > 0 such that Ax = ρ(A)x;
4. |λ| < ρ(A) for every eigenvalue λ "= ρ(A) of A, i.e. ρ(A) is the unique eigenvalue of
maximum modulus; and
5. [ρ(A)−1 A]m → R as m → +∞ where R = xy T , Ax = ρ(A)x, AT y = ρ(A)y, x > 0,
y > 0, and xT y = 1.
Theorem 7 (Perron’s Theorem for Non–Negative Matrices). If A ∈ Rn×n is a
non-negative matrix (A ≥ 0), then ρ(A) is an eigenvalue of A and there is a non–negative
vector x ≥ 0, x "= 0, such that Ax = ρ(A)x.
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12
51
4.3. Algebraic Graph Theory
Irreducible Graphs and Matrices
Irreducible Graphs and Matrices
Irreducibility
• A directed graph is irreducible if, given any two vertices, there exists a path from the first vertex to the
second. (Irreducible = strongly connected)
1
2
6
• A matrix is irreducible if it is not similar to a block
upper triangular matrix via a permutation.
8
3
5
• A digraph is irreducible if and only if its adjacency
matrix is irreducible.
9
7
4
10
12
11
Figure 4.1: Sample Graph G.
1
2
8
9
n×n
Theorem 8 (Frobenius). Let A ∈ R
and suppose that
A is irreducible and non-negative. Then
1.
2.
3.
4.
ρ(A) > 0;
r = ρ(A) is an eigenvalue of A;
There is a positive vector x > 0 such that Ax = ρ(A)x;
r = ρ(A) is an algebraically simple eigenvalue of A;
and
5. If A has h eigenvalues of modulus r, then these eigenvalues are all distinct roots of λh − rh = 0.
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3
5
4
7
10
12
11
Figure 4.2: Induced Subgraphs of Components of G.
{1, . . . , 6}
{7, 8, 9, 11, 12}
{10}
Figure 4.3: Graph of Components of G.
17
13
Spectra of the Laplacian
Properties of Laplacians (2)
Properties of L
• If G is strongly connected, the zero eigenvalue of L is simple.
• If G is aperiodic, all nonzero eigenvalues lie in the interior of the Gershgorin disk.
• If G is k-periodic, L has k evenly spaced eigenvalues on the boundary of the
Gershgorin disk.
Unidirectional
tree
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Undirected
graph
Cycle
Richard M. Murray, Caltech CDS
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Periodic
graph
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14
Algebraic Connectivity
Algebraic
Connectivity
Theorem 9 (Variant of Courant-Fischer). Let A ∈ Rn×n be a Hermitian matrix with
eigenvalues λ1 ≤ λ2 ≤ · · · ≤ λn and let w1 be the eigenvector of A associated with the
eigenvalue λ1 . Then
λ2 =
min
x != 0, x ∈ Cn ,
x∗ Ax
=
x∗ x
x⊥w1
min
x∗ Ax
(6)
x∗ x = 1,
x⊥w1
4.3. Algebraic Graph Theory
Remarks:
• λ2 is called the algebraic connectivity of L
• For an undirected graph with Laplacian L, the rate of convergence for the consensus
protocol is bounded by the second smallest eigenvalue λ2
1
Example
Directed graph with a bottleneck
•
Would be better to
delete some of the
extra links to make
the bottleneck link
more clear.
6
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3
5
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4
9
7
10
12
Figure 4.1: Sample Graph G.
11
19
15
CyclicallySeparable
Separable Graphs
Cyclically
Graphs
Definition (Cyclic separability). A digraph G = (V, E) is cyclically
separable if and only if there exists a partition of the set of edges
c
E = ∪n
k=1 Ek such that each partition Ek corresponds to either the
edges of a cycle of the graph, or a pair of directed edges ij and ji
that constitute an undirected edge. A graph that is not cyclically
separable is called cyclically inseparable.
Lemma 10. Let L be the Laplacian matrix of a cyclically separable
!n
digraph G and set u = −Lx, x ∈ Rn . Then i=1 ui = 0, ∀x ∈ Rn
and 1 = (1, . . . , 1)T is the left eigenvector of L.
1
6
2
5
3
1
4
(a)
Figure 3: Three
Proof. The proof follows from the fact that by definition of cyclic
separability. We have
G = (V, E, A) is called balanced if and
nc "
n
!
"
"
"
ui =
−
(xj − xi ) =
(xj − xi ) = 0
i=1
ij∈E
k=1 ij∈Ek
j
because the inner sum is zero over the edges of cycles and undiAny undirected graph is balanced.
rected edges of the graph.
balanced. Here is our first main result
• Provides a “conservation” principle for average consensus
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R. M. Murray, Caltech CDS
Theorem 4. Consider a network of in
strongly connected digraph. 20Then, pro
consensus problem if and only
16 if G is
Proof. The proof follows from Theore
Consensus on Balanced Graphs
Balanced Graphs
Let G = (V, E) be a digraph. We say G is balanced if and only if the in–degree and
out–degree of all nodes of G are equal, i.e.
degout (vi ) = degin (vi ),
∀vi ∈ V
(7)
Theorem 11. A digraph is cyclically separable if and only if it is balanced.
Corollary 11.1. Consider a network of integrators with a directed information flow G
and nodes that apply the consensus protocol. Then, α = Ave(x) is an invariant quantity if
and only if G is balanced.
Remarks
• Balanced graphs generalized undirected graphs and retain many key properties
1
1
2
6
2
5
3
4
6
8
10
2
1
3
5
7
3
9
4
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(a)
Richard M. Murray,(b)
Caltech CDS
R. M. Murray, Caltech CDS
Figure 3: Three examples of balanced graphs.
(c)
21
17
Consensus Protocols for Balanced Graphs
1
2
3
4
5
1
2
3
4
5
10
9
8
7
6
10
9
8
7
6
1
2
3
4
5
1
2
3
4
5
Algebraic Connectivity=0.191
0
!10
!20
0
10
9
8
(c)
5
7
6
10
0
!10
!20
0
15
(b)
Algebraic Connectivity=0.205
10
node value
10
node value
(a)
10
9
5
8
(d)
7
10
6
15
200
disagreement
disagreement
time(
time( sec)
Figure 4: Four examples
ofsec)balanced and strongly connected digraphs:
(a) Ga , (b) Gb , (c)
300
300
Gc , and (d) Gd satisfying.
200
node value
node value
as the 100
number of the edges of the graph increases, 100
algebraic connectivity (or λ2 ) increases,
and the0 settling time of the state trajectories decreases.
0
0
5
10
15
0
5
10
15
The case of a directed
of length 10, or Ga , has
the highest
over-shoot.
In all four
time( cycle
sec)
time(
sec)
(a)
cases, a consensus is asymptotically
reached and the performance is(b)
improved as a function
Algebraic Connectivity=0.213
Algebraic Connectivity=0.255
of λ2 (Ĝ10k ) for k ∈ {a,
b, c, d}.
10
In Figure 6(a), a finite automaton is shown with the set of states {Ga , Gb , Gc , Gd } rep0 CDS
CRM, Dec 07 0
Richard M. Murray, Caltech
resenting the discrete-states of a network
with switching
topology as a hybrid system. The
!10
hybrid!10system starts at the discrete-state Gb and switches
every T = 1 second to the next
state according to the state machine in Figure 6(a). The continuous-time state trajectories
22
Formation Operations: Graph Switching
Control questions
How do we split and rejoin teams of vehicles?
How do we specify vehicle formations and control them?
How do we reconfigure formations (shape and topology)
•
•
•
Consensus-based approach using balanced graphs
If each subgraph is balanced, disagreement vector provides common Lyapunov fcn
By separately keeping track of the flow in and out of nodes, can preserve center of
mass of of subgraphs after a split manuever
•
•
CRM, Dec 07
Richard M. Murray, Caltech CDS
23
Other Uses of Consensus Protocols
8
Reza Olfati-Saber
8
Reza
Computation
of other functions besides
the Olfati-Saber
average
Can
to compute
max,
etc
with aadopt
state (ethe
∈ R2mapproach
, input ui , and
output
withqia. This
state
filter
(eimin,
, qiis) ∈
used
R2mfor
, input ui , and output qi . This filter is used for
i , qi )basic
inverse-covariance consensus
thatcompute
calculates superidempotent
Ŝinverse-covariance
consensus
i by that calculates Ŝi column-wise for node i by
i column-wise for node
Chandy/Charpentier:
can
functions:
! −1
! −1
•
• applying the filter on columns of H R
i
matrix version of this filter can ftake
(X
applying
the of
filter
oni.columns
of Hi Ri Hi as
the inputs
node
The
i Hi as
! −1
input. of this filter can take Hi! Ri−1 Hi
Hi Ri Hmatrix
i as theversion
∪ Y ) = f (f (X) ∪ Y )
the inputs of node i. The
as the input.
Fig. 2 shows the architecture of each node ofFig.
the 2sensor
showsnetwork
the architecture
for dis- of each node of the sensor network for dis-
Basic idea: local conservation implies
global
conservation
tributed Kalman filtering. Note that consensustributed
filteringKalman
is performed
filtering.
withNote
the that consensus filtering is performed with the
same
frequency
as these
Kalmancases
filtering.
is a same
unique
frequency
feature
asrejoining
Kalman
completely
filtering.
Can
extend
toThis
handle
splitting
andthat
as wellThis is a unique feature that completely
•
distinguishes our algorithm with some relateddistinguishes
work in [30, our
33]. algorithm with some related work in [30, 33].
Distributed Kalman filtering
Maintain local estimates of global average and covariance
Need to be careful about choosing rates of convergence
•
•
Node i
Sensor
Data
Covariance
Data
CRM, Dec 07
Node i
Sensor
Data
Low-Pass
Consensus Filter
Micro
Kalman
Filter
(µKF)
Band-Pass
Consensus Filter
Low-Pass
Consensus Filter
x̂
Covariance
Data
Micro
Kalman
Filter
(µKF)
x̂
Band-Pass
Consensus Filter
Richard M. Murray, Caltech CDS
24
Summary
Graphs
Directed, undirected, connected, strongly complete
Cyclic versus acyclic; irreducible, balanced
•
•
Graph Laplacian
L=Δ-A
Spectral properties related to connectivity of graph
# zero eigenvalues = # strongly connected components
Second largest nonzero eigenvalue ~ weak links
•
•
•
•
Spectral Properties of Graphs
Gershgorin’s disk theorem
Perron-Frobenius theory
•
•
Examples
Consensus problems, distributed computing
Cooperative control (coming up next...)
•
•
CRM, Dec 07
Richard M. Murray, Caltech CDS
25