multi-agent consensus and algebraic graph theory › utari › acs › ee5329 › lectures › 2017...

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