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UTA Research Institute (UTARI)The University of Texas at Arlington
F.L. Lewis, NAIMoncrief-O’Donnell Endowed Chair
Head, Controls & Sensors Group
Multi-agent Consensus andAlgebraic Graph Theory
Supported by NSF, ARO, AFOSR
Motions of biological groups
Fishschool
Birdsflock
Locustsswarm
Firefliessynchronize
Graphs and Dynamic Graphs
Communication Graph – Algebraic Graph Theory1
2
3
4
56
N nodes
[ ]ijA a=
0 ( , )ij j i
i
a if v v Eif j N
> ∈
∈
oN1
Noi ji
jd a
=
=∑ Out-neighbors of node iCol sum= out-degree
42a
Adjacency matrix
0 0 1 0 0 01 0 0 0 0 11 1 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 1 0
A
=
iN1
N
i ijj
d a=
=∑ In-neighbors of node iRow sum= in-degreei
G= (V,E)
i
1
2
3
4
56
Diameter= length of longest path between two nodes
Volume = sum of in-degrees1
N
ii
Vol d=
=∑
Spanning treeRoot node
Strongly connected if for all nodes i and j there is a path from i to j.
Tree- every node has in-degree=1Leader or root node
Followers
1
2
3
4
56
Exists a spanning tree iff there is a node having a path to all other nodes
Graph strongly connected implies exists a spanning treeStrongly connected implies Every Node is a root node
Quasi-strongly connected if for all nodes i and jthere exists a node k with a path to i and a path to j
i j
k
Exists a spanning tree iff quasi-strongly connected
The Multi-agent Consensus Problem – Cooperative Control
i ix u=Agent dynamics
Find control so that all agents reach the same steady-state valueiu
0, ,i jx x i j− → ∀
i ix u=
Dynamic Graph- the Graphical Structure of Control
Each node has an associated state i ix u=
Standard local voting protocol ( )i
i ij j ij N
u a x x∈
= −∑
[ ]1
1i i
i i ij ij j i i i iNj N j N
N
xu x a a x d x a a
x∈ ∈
= − + = − +
∑ ∑
x
1
N
uu
u
=
1
N
dD
d
=
i
j
[ ]ijA a=
First-order Consensus of Multi-agent systems
Global Vectors
1 12 11 1 1
21
1
00
0
N
N N NN N
d a au x x
au Dx Ax
u x xd a
= = − + = − +
( )i
i ij j ij N
x a x x∈
= −∑Dynamics
Dynamic Graph- the Graphical Structure of Control
( )x u Dx Ax D A x Lx= = − + = − − = − L=D-A = graph Laplacian matrix
x Lx= −
If x is an n-vector then ( )nx L I x= − ⊗
Global Closed-loop dynamics
First-order Consensus of Multi-agent systems
1 12 11 1 1 1
21
1
00
0
N
N N N NN N
d a ax u x x
au Dx Ax
x u x xd a
= = = − + = − +
( )i
i ij j ij N
x a x x∈
= −∑
Global Closed-loop dynamics depends on L
x Lx= −
Closed-loop system with local voting protocol
1 12 1
21
1
00
0
N
N N
d a aa
L D A
d a
= − = −
L has row sum of 0
10 1 1
1L L Lc = = =
1,
N
i ijj
d a=
=∑ = row sum
L= graph Laplacian matrix
( ) (0)Ltx t e x−=
1 0λ =
For any constant c
ij N∈jx ( )
i
i ij j ij N
e a x x∈
= −∑ i ix u= ixiu
i ix u=( )
i
i ij j ij N
u a x x∈
= −∑
( )i
i ij j ij N
e a x x∈
= −∑
Multi-agent Cooperative Regulator
Dynamics
Cooperative Control
Local Neighborhood Consensus Error
In the single-agent regulator problem, the state goes to zero.
In the multi-agent cooperative regulator problem, the states go to a nonzero consensus value
Multi-agent Cooperative Regulator
( )i
i ij j ij N
x a x x∈
= −∑Closed-loop system
FlockingReynolds, Computer Graphics 1987
Reynolds’ Rules:Alignment : align headings
Cohesion : steer towards average position of neighbors- towards c.g.Separation : steer to maintain separation from neighbors
( )i
i ij j ij N
aθ θ θ∈
= −∑
Distributed Adaptive Control for Multi-Agent Systems
Nodes converge to consensus heading
( )ci
i ij j ij N
aθ θ θ∈
= −∑
Consensus Control for Swarm Motions
heading angle
Convergence of headings
1
2
3
4
56
0 5 10 15 20 25 30 35 400
1
2
3
4
5
6
-350 -300 -250 -200 -150 -100 -50 0 50-300
-250
-200
-150
-100
-50
0
50
cossin
i i
i i
x Vy V
θθ
==
time x
y
Leader
Followers
0 5 10 15 20 25 30-120
-100
-80
-60
-40
-20
0
20
40
60
Time
Hea
ding
Heading Consensus using Equations (21) and (22)
Nodes converge to heading of leader
Consensus Control for Formations
heading angle
Formation- a Tree network
Convergence of headings
10 20 30 40 50 60 70 80 90 100-200
-150
-100
-50
0
50
100
150
200
Heading Update using Spanning Tree Trust Update
x
y
Leader
y
x
( )ci
i ij j ij N
aθ θ θ∈
= −∑
cossin
i i
i i
x Vy V
θθ
==
time
leader
16
Eigenstructure of Graphs
Closed-loop dynamics depends on L
x Lx= −
Closed-loop system with local voting protocol
L =D-A has row sum of 0
10 1
1L c Lc = =
L= D-A= graph Laplacian matrix
( ) (0)Ltx t e x−=
Eigenvalues of L determine time response
At steady-state 0 s sx Lx= = −
Therefore at steady state
11
1sx c c
= =
For any constant c
and consensus is reached
( )i
i ij j ij N
x a x x∈
= −∑
Eigenstructure of Graph Laplacian Matrix
1L MJM −=
In general the Jordan form matrix is
State Space Transformation1
1
i
i
i
J block diag
λλ
λ
=
We consider the case of simple L which has diagonal Jordan form1
2
N
J
λλ
λ
=
Define transformation matrix as the modal matrix Then[ ]1 2 NM v v v=
[ ] [ ]1
21 2 1 2N N
N
LM MJ L v v v v v v
λλ
λ
= = =
( ) 0i iI L vλ − =
So that
and
or
i i iLv vλ=
is said to be a right eigenvector for the eigenvalue iv iλ
Right Eigenvectors
1L MJM −=
So can write either as before
11 1
21 1 2 2
T T
T T
T TNN N
w ww w
M L JM L
w w
λλ
λ
− −
= = =
LM MJ=
Or 1 1M L JM− −=
Define inverse modal matrix as
1
1 2
T
T
TN
ww
M
w
−
=
and
or
is said to be a left eigenvector for the eigenvalue iw iλ
Then
( ) 0Ti iw I Lλ − =
T Ti i iw L wλ=
Left Eigenvectors
Connectivity for Undirected Graphs
Connected if for all nodes i and j there is a path from i to j.If there is a path from i to j, there is a path from j to i
[ ] [ ] Tij jiA a a A= = =
ij jia a=
T TL D A D A L= − = − =
Any undirected graph has Hence, all its eigenvalues are real and can be ordered as
Since L has all row sums zero :
where and c is any constant
1 10 1 ( )L c I L vλ= − = −
so 1 0λ = Is an eigenvalue and 1 1v c= is the right e-vector
1 0λ =if 1 1v c=and
1
2
3
4
56
Theorem. Graph contains a spanning tree iff e-val at is simple.
Graph strongly connected implies exists a spanning tree
Then 2 0λ >
Then L has one e-val at zero and all the rest in open right-half plane
1 0λ =
Then -L has one e-val at zero and all the rest stable
Theorem : L has rank N-1, i.e is non-repeated, if and only if graph G has a spanning tree.
1 0λ =
1( ) (0) (0)i
NtLt T
i ij
x t e x v e w xλ−−
=
= =∑Modal decomposition
[ ]1 1
21 21 2
1
T
T NT
N i i ij
TN N
ww
L MJM v v v v w
w
λλ
λ
λ
−
=
= = =
∑
Modal decomposition
x Lx= −
( ) (0)Ltx t e x−=
Closed-loop dynamics depends on eigenstructure of L1
( ) (0) (0)i
NtLt T
i ij
x t e x v e w xλ−−
=
= =∑
x Lx= −
Closed-loop system with local voting protocol
Modal decomposition
vi = right e-vectorswi = left e-vectors
1 12 1
21
1
00
0
N
N N
d a aa
L D A
d a
= − = −
L has row sum of 0
10 1
1L L = =
or 1 10 1 ( )L I L vλ= − = −
so 1 0λ = and 1 1v = is the right e-vector
1,
N
i ijj
d a=
=∑ = row sum
1L MJM −=Jordan Normal Form
Consensus: Final Consensus Value
x Lx= −
Closed-loop system with local voting protocol
Modal decomposition
[ ]1
1
1 1 11
1 (0) 1( ) (0) (0)
1 (0) 1
Nt T
N j jj
N
xx t v e w x x
x
λ γ γ γ−
=
→ = =
∑
Let be simple. Then at steady-state1 0λ =
with [ ]1 1T
Nw γ γ= the normalized left e-vector of 1 0λ =
Therefore 1
( ) (0)N
i j jj
x t xγ=
→∑ for all nodes CONSENSUS
( )1 1
( ) (0) (0) (0)i i
N Nt tLt T T
i i i ij j
x t e x v e w x w x e vλ λ− −−
= =
= = =∑ ∑
Importance of left e-vector of 1 0λ =
Consensus value depends on communication graph structure
Consensus Leaders
with [ ]1 1T
Nw γ γ= the normalized left e-vector of 1 0λ =
1( ) (0)
N
i j jj
x t xγ=
→∑ for all nodesCONSENSUS
Theorem.
1
2
3
4
56
Spanning treeRoot node
0iγ ≠ iff node i is a Root node
So the consensus value is a weighted average of the initial conditions of all the Root nodes
Corollary. Let the graph be a tree. Then all nodes converge to IC of the root node
( )1 1
( ) (0) (0) (0)i i
N Nt tLt T T
i i i ij j
x t e x v e w x w x e vλ λ− −−
= =
= = =∑ ∑
Convergence Rate
x Lx= −
Closed-loop system with local voting protocol
Modal decomposition
Let be simple. Then for large t1 0λ =
2 1 22 2 1 1 2 2
1( ) (0) (0) (0) 1 (0)
Nt t tT T T
j jj
x t v e w x v e w x v e w x xλ λ λ γ− − −
=
→ + = + ∑
2λ determines the rate of convergence and is called the FIEDLER e-value
1 0λ =
and the Fiedler e-val 2λThere is a big push to find expressions for the left e-vector for
Fiedler e-val depends on left e-vect for 1 0λ =
[ ]1 1T
Nw γ γ=
For a certain type of graph
Olshevsky and Tsitsiklis
212 2
1 11 min ( )
N N
i ij i jx S i ja x xλ γ
∈= =
= − −∑ ∑
Bounds for Fiedler e-val: Convergence RateUndirected Graphs Row sum = col sum, in-degree = out-degree, balanced
21
( )D vol Gλ ≥ D= diameter= length of longest path between 2 nodes
( ) ii
vol G d=∑
2 1N
Nλ δ≤
−δ, where is the minimum in-degree
In-degree = out-degree
Strogatz Small World Networks are faster
Directed Graphs- less is known
Star network has largest Fiedler e-val of any graph with the same number of nodes and edges
{ }1( ) min max out degree, max in degree1
na Ln
≤−
OUT degree is important
Book by Chai Wah WuWork of Fan Chung
Left e-vector for 1 0λ =
For a connected undirected graph1
2i
idL N
γ +=
+
where di is the out-degree= in-degree, N the number of nodes, L the number of edges
[ ]1 1T
Nw γ γ=
Hovareshthi & Baras
1( ) (0)
N
i j jj
x t xγ=
→∑ = weighted average of initial conditions of nodes
A graph is balanced if in-degree=out-degree
Balanced means that row sum= column sum
[ ]ijA a=
1
N
i ijj
d a=
=∑ Row sum= in-degree
1
No
i iji
d a=
=∑ Column sum= out-degree
Then L has row sum=0 and column sum=0
1 0Tw L = means that left e-vector is also 1
11
1w
= =
Consensus value is
1
1( ) (0)N
i jj
x t xN =
→ ∑ = average of initial conditions of nodes
Then Consensus value is
i
i
Consensus value depends on communication graph structure
Independent of graph structure
Undirected Graphseij is an edge if eji is an edge, and eij = eji
1
2
3
4
56
0 1 1 0 0 01 0 1 1 0 11 1 0 0 1 00 1 0 0 0 10 0 1 0 0 10 1 0 1 1 0
TA A
= =
A is symmetric
L=D-A is symmetric and positive semidefiniteA symmetric graph is balancedConnected undirected graph has average consensus value
Row sum= column sum so that in-degree= out-degree
1
1( ) (0)N
i jj
x t xN =
→ ∑ = average of initial conditions of nodes
Then Consensus value is
Independent of graph structure
Olshevsky and TsitsiklisBidirectional graph 0 0ij jia a≠ ⇒ ≠
Equal neighbor
1i in N= + Number of neighbors +1 (i.e. include node itself)
1 1,ij iii i
a an n
= =
Then left evector for 1 0λ =
[ ]1 1T
Nw γ γ=
( )i
in
Vol Gγ = ( ) i
iVol G n=∑where
Consensus value is 1 1
( ) (0) (0)( )
N Ni
i i i ii i
nx t x xVol G
γ= =
→ =∑ ∑
ij jia a≠But maybe
Graph Laplacian Eigenstructure
L D A= −Laplacian
2λ
First eigenvalue is and 1 1v c= is the right e-vector
All other eigenvalues have positive real parts Iff there exists a spanning treeA sufficient condition for this is strongly connected.
1 0λ =
x Lx= −First order consensus system
Then the Fiedler eigenvalue has positive real partAnd is a measure of the convergence rate
What about the other eigenvalues? Where are they?
[ ] n nijE e R ×= ∈
1
:n
ii ijj ii
z C z e e≠=
∈ − ≤
∑
Thm. Gerschgorin Circle Criterion. All eigenvalues of matrix
are located within the following union of n discs
0 dmax2dmax
L has row sum = 0 implies there is an eigenvalue at s=0
All e-vals are in the circles
1 20 Nλ λ λ= ≤ ≤ ≤
2λ Nλ
-2dmax
-dmax0
E-vals of -L
1λ
1 12 1
21
1
00
0
N
N N
d a aa
L D A
d a
= − = −
( , )i iC d d
[ ] n nijE e R ×= ∈
1
:n
ii ijj ii
z C z e e≠=
∈ − ≤
∑
Thm. Gerschgorin Circle Criterion. All eigenvalues of matrix
are located within the following union of n discs
Normalized Laplacian 1 1 1( )D L D D A I D A− − −= − = − has diagonal entries =1
0 1 2
L has row sum = 0 implies there is an eigenvalue at s=0
All e-vals are in the circle
1 20 Nλ λ λ= ≤ ≤ ≤
2λ Nλ
-2 -1 0E-vals of -L
1λ
0 1 2
E.G. if graph is k-cyclic (all circular paths have a GCD of k)
then e-vals of are uniformly spaced around this circle
Then consensus time plot oscillates until steady-state
e.g. k=4
12
3
4 5
6
Captures the structure of social networks and rumor passing
1D L−
1
2 3
4 5 6
12
3
4 5
6
Graph Eigenvalues for Different Communication Topologies
Directed Tree-Chain of command
Directed Ring-Gossip networkOSCILLATIONS
Graph Eigenvalues for Different Communication Topologies
Directed graph-Better conditioned
Undirected graph-More ill-conditioned
65
34
2
1
4
5
6
2
3
1
Synchronization on Good Graphs
Chris Elliott fast video
65
34
2
1
1
2 3
4 5 6
Regular 2D mesh
N
N
Synchronization on Gossip Rings
Chris Elliott weird video
12
3
4 5
6
N
N
cycle2N
Control of a Network with a Leader Node
Controlled Consensus: Cooperative Tracker
Node state i ix u=Local voting protocol with control node v
( ) ( )i
i ij j i i ij N
u a x x b v x∈
= − + −∑
( ) 1x L B x B v= − + +
i i
i i ij i ij j ij N j N
u b a x a x b v∈ ∈
= − + + +
∑ ∑
0ib ≠ If control v is in the neighborhood of node i
Control node is in some neighborhoods iN
{ }iB diag b=
Theorem. Let at least one . Then L+B is nonsingular with all e-vals positiveand -(L+B) is asymptotically stable
0ib ≠
So initial conditions of nodes in graph A go away.Consensus value depends only on vIn fact, v is now the only spanning node
Get rid of dependence on initial conditions control node vRon Chen
Strongly connected graph L
Pinning Control
ibPinninggains
1
2
3
4
A = [0 0 0 0;1 0 1 0;0 0 0 1;1 0 0 0];
Lamdas = 0 1 1 2Consensus time approx 7.5 sec
0 2 4 6 8 10 12 14 16 18 200
1
2
3
4
5
6
7
8
9
Time
Sta
tes
with
diff
. Ini
. con
d.
Controlled Consensus
Average of ICS
Original network
Controlled network L
Lamdas = 0 1 1 1 2
0 2 4 6 8 10 12 14 16 18 200
1
2
3
4
5
6
7
8
9
Time
Sta
tes
with
diff
. Ini
. con
d.
Consensus time approx 8 sec
Leader’s IC
1
2
3
4
Open Research TopicTime-varying edge weights
( )ci
i ij ij j ij N
aθ ξ θ θ∈
= −∑
ijξ
????ijξ =
Is the control input
An extension of Adaptive Control Methods
Lyapunov Design?
S. Boyd showed that if some edge weights are negative, convergence speed is faster
Second Order Consensus – Kevin Moore and Wei Ren
Let each node have an associated state i ix u=
Second-order local voting protocol
0 1( ) ( )i i
i ij j i ij j ij N j N
u a x x a x xγ γ∈ ∈
= − + −∑ ∑
x Lx= −
Consensus works because the closed-loop system is Type I.
has an integrator- simple e-val at 0.
Closed-loop system
0 1
0 Ix x
L Lγ γ
= − −
Reaches consensus in both iff graph has a spanning tree and gains are chosen for stability
Has 2 integrators- Can follow a ramp consensus input
,i ix x
PD cooperative control
Second Order Controlled Consensus for Position Offset Control – Kevin Moore and Wei Ren
node state i ix u=
Second-order controlled protocol
[ ]0 1 0 0( ) ( ) ( ) ( )i i
i ij j i ij ij j i i i ij N j N
u a x x a x x b x x x xγ γ∈ ∈
= − −∆ + − + − + − ∑ ∑
where node 0 is a leader node.
ij∆ is a desired separation vector
Good for formation offset position control Leader
Followers
ij∆
x0
v
The cloud-capped towers, the gorgeous palaces,The solemn temples, the great globe itself,Yea, all which it inherit, shall dissolve,And, like this insubstantial pageant faded,Leave not a rack behind.
We are such stuff as dreams are made on, and our little life is rounded with a sleep.
Our revels now are ended. These our actors, As I foretold you, were all spirits, and Are melted into air, into thin air.
Prospero, in The Tempest, act 4, sc. 1, l. 152-6, Shakespeare
52
Graph Matrix Theory
0A ≥ with row sum positive= di
L D A= −
Zhihua Qu 2009 book
is an M matrix
00
M+ ≤
= ≤ + Off-diagonal entries 0≤Principal minors nonnegative – singular M matrix
Nonsingular M matrix if all principal minors positive
L also has all row sums = 0
Do not confuse with stochastic matrix
0E ≥ is row stochastic if all row sums =1
LtE e−= is row stochastic with positive diagonal elements
0E ≥ be row stochastic. Then A=I-E is an M matrix with row sums zero.Let
Discrete-time voting protocol gives stochastic c.l. matrix 1k kx Ex+ =1 1( ) ( ) ( )E I D I A I I D L− −= + + = − +
F is said to be lower triangularly complete if in every row i there exists at least one
is a scalar, it is equal to 0.) is square and irreducible. (note- if
Matrix E is reducible if it can be brought by row/column permutations to the form* 0* *
Two matrices that are similar using permutation matrices are said to be cogredient.
Irreducibility
A graph G(A) is strongly connected iff its adjacency matrix A is irreducible.
11
21 22
1 2
0 00T
p p pp
FF F
F TET
F F F
= =
iiF iiF
A reducible matrix E can be brought by a permutation matrix T to the lower block triangular (LBT) Frobenius canonical form
where
j i<0ijF ≠
0,ijF j i> ∀ <
such that (i.e. it has least one nonzero entry).
.
F is said to be lower triangularly positive if
F is lower triang. Complete iff the associated graph has a spanning tree
Then is an eigenvalue of E iff all row sums are equal to 0.Moreover, let E be irreducible with all row sums equal to zero. Then
It is strictly diagonally dominant if these inequalities are all strict. E is strongly diagonally dominant if at least one of the inequalities is strict [Serre 2000]E is irreducibly diagonally dominant if it is irreducible and at least one of the inequalities is strict.
Let E be a diagonally dominant M matrix (i.e. nonpositive elements off the diagonal, nonnegative elements on the diagonal).
[ ] n nijE e R ×= ∈ ii ij
j ie e
≠
≥∑
0λ =
Matrix is diagonally dominant if, for all i,
has multiplicity of 1.
Diagonal Dominance
0λ =
Thm. Let E be strictly diagonally dominant or irreducibly diagonally dominant. Then E is nonsingular. If in addition, the diagonal elements of E are all positive real numbers, then
Re ( ) 0, 1i E i nλ > ≤ ≤
SO. Let graph be irreducible. Then the Laplacian L has simple e-value at 0.Add a positive number to any diagonal entry of L to get .
Then is nonsingular and is stable.L
L L−
Thm. Properties of Irreducible M matrices.
Let E=sI-A be an irreducible M matrix, that is,
5. exists and is nonnegative for all nonnegative diagonal matrices D with at least one positive element.
6. Matrix E has Property c.7. There exists a positive diagonal matrix P such that
Then,1. E has rank n-1.2. there exists a vector v>0 such that Ev=0.3. implies Ex=0.4. Each principal submatrix of order less that n is a nonsingular M matrix
0A ≥
0Ex ≥
1( )D E −+
TPE E P+That is, matrix E is pseudo-diagonally dominant, That is, matrix -E is Lyapunov stable.
and is irreducible.
is positive semidefinite, Used for Lyapunov Proofs
Interpret E as the Laplacian matrix L
Zhihua Qu Book 2009
Original network to be controlledHow to pick
Injection nodes?
u1
u2 u3 External control nodes added
Lei Guo – Soft controlDo not change the local protocols of the nodescan only add additional neighbors to influence existing nodes
Extension of virtual leader approach
[ ]11 12 1 11 12 1
21 21 2 ( )
1 2 1
[ ]
N m
m N N mij
N N NN N Nm
a a a b b ba b b
A A B a R
a a a b b
× +
= = ≡ ∈
1 2 1{ : 1, } { , , , , , , }i N mx i N m x x x u u= + =
Add m additional control nodes[ ] 0ijB b= ≥ 0ijb ≥ is the weight from control node uj to existing network node vi. where
Augmented state
New connectivity matrix
Row sum = id
11 12 11 11 12 1
21 221
11 2
mN
m
N NmN N N NN
b b bd a a ab ba
D A B
b ba a d a
− − −− − −
− −− − − =
− −− − −
Laplacian
L D A= −
Modified local voting protocol
( )i
i ij j ij N
a x xµ∈
= −∑
{ }iD diag d=
1,i ij
j N md a
= +
= ∑ A
modified diagonal matrix of in-degrees
with the i-th row sum of , which includes the new control nodes.
SOFT CONTROL, includes new control nodes in some nbhds
N NL D A R ×= − ∈
x Lx Bu= − +
Modified Laplacian
New closed-loop system
Row sum of [ ]L B D A B − = − − is zero
But i-th row sum of L D A= −
i i id d δ= + where iδ = i-th row sum of control matrix B.
has been increased by iδ
i ix µ= Lei Guo
1 11 12 1
21 2 22
1 2
[ ]
N
N Nij
N N N NN
L L LL L
L D A L L R
L L L
δδ
δ
×
+ + = − = ∆ + ≡ ≡ ∈ +
iδ = i-th row sum of control matrix B.
Lemma Let L have row sum zero and be irreducible.At least one 0iδ ≠
Then 1 11 12 1
21 2 22
1 2
[ ]
N
N Nij
N N N NN
L L LL L
L D A L L R
L L L
δδ
δ
×
+ + = − = ∆ + ≡ ≡ ∈ +
is irreducibly diagonally dominant and hence nonsingular.
Lemma. Then is asymp. stable.L−
RTP 1. Let { } , 0i iL diag Lδ δ= + ≥
Then relate eigenvalues of to those of L . They are shifted right by some amount.
and L be irreducible
L
Special case. ,i c iδ = ∀
Then all e-vals are shifted right by c
Conjecture. Let L be irreducible. Then L∆ + > ∆
{ }idiag δ∆ =Define
1 11 12 1
21 2 22
1 2
[ ]
N
N Nij
N N N NN
L L LL L
L D A L L R
L L L
δδ
δ
×
+ + = − = ∆ + ≡ ≡ ∈ +
iδ = i-th row sum of control matrix B.
1 11 12 1
21 2 22
1 2
[ ]
N
N Nij
N N N NN
L L LL L
L D A L L R
L L L
δδ
δ
×
+ + = − = ∆ + ≡ ≡ ∈ +
My Theorem
Let
{ }idiag δ∆ =
N NL R ×∈
( ) ( )1 2
1 2 1 21 21 1
j j js
j j j j j js ss
ki i i
i i i i i is j j j k
L L L Lδ δ δ= ≤ < < < ≤
= ∆ + = +∑ ∑
Then the determinant of is given by
Define
Minor with rows and columns struck out
( )
11 1 1 11 1 1 11 1
1 1 1
1 1 1
0
0
i N i N N
i i ii i ii i i
N Ni NN N Ni NN N NN
ii i
L L L L L L L LL L L L L
L L
L L L L L L L L
L L
δ δ
δ
+= ∆ + ≡ = +
= +
Example- one diagonal entry positiveiδ
To increase the determinant as much as possible-Add to the node with the largest OUT-degree, i.e. largest column sumiδ
We call the out-degree of node i its influence or social standing. - John Baras
Original network to be controlledwith fixed structure
Control graphwith desired structure
0 0x xL Bu u
− =
Overall Dynamics
Does not reach consensus unless matrix is irreducible.
00 00 0 0 0
aug A BL B DL
− = = −
augL irreducible iff m=1
Add control graph 000 0
augG G
A BL B DL
GL D −
= = −
0 G
x xL Bu uL
− = −
Induced Strogatz Small World StructureReduced diameter= longest path length, larger Fiedler e-val, so faster
has m e-vals at 0
Original network to be controlled
u1u2 External control nodes addedy1y2
Control outputs
Results. To make the controlled network as fast as possible,
Tap into the node with the LARGEST out-degree (highest social standing)
And take measured outputs from the nodes with the SMALLEST out-degree
- Zhihua Qu