1. First-Order Consensus

8/13/2019 1. First-Order Consensus

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Prepared by F.L. Lewis

Updated: Saturday, February 02, 2013

1.FirstOrderConsensusEquation Chapter 1 Section 1

Add discussion. Refer to refs.

1.1 Some Graph Theory

A graphis a pair ( , )G V E with 1{ , , }NV v v a set ofNnodes or vertices andEa set of edges

or arcs. We assume the graph is simple, i.e. ( , ) ,i iv v E i no self loops, and no multiple edges

between the same pairs of nodes. Elements ofEare denoted as ( , )i jv v which is termed an edge

or arc from vito vj, and represented as an arrow with tail at viand head at vj. Edge ( , )i jv v is saidto be outgoing with respect to node vi and incoming with respect to vj; node vi is termed theparent and vjthe child. The in-degree of viis the number of edges having vias a head. The out-

degree of a node viis the number of edges having vias a tail. The set of (in-) neighbors of a node

vi is { : ( , ) }i j j iN v v v E , i.e. the set of nodes with edges incoming to vi. The number of

neighbors iN of node viis equal to its in-degree.

If ( , ) ( , ) , ,i j j iv v E v v E i j the graph is said to be bi-directional, otherwise it is

termed a directed graph or digraph. If the in-degree equals the out-degree for all nodes v V thegraph is said to be balanced.

A directed path is a sequence of nodes 0 1, , , rv v v such that1( , ) , {0,1, , 1}i iv v E i r . Node vi is said to be connected to node vj if there is a directed

path from vito vj. The distance from vito vjis the length of the shortest path from vito vj. Graph

G is said to be strongly connected if ,i j

v v are connected for all distinct nodes ,i j

v v V . For

bidirectional graphs, if there is a directed path from vito vj, then there is a directed path from vjtovi, and the qualifier strongly is omitted. A (directed) tree is a connected digraph where every

node except one, called the root, has in-degree equal to one. A spanning tree of a digraph is a

directed tree formed by graph edges that connects all the nodes of the graph. A graph is said tohave a spanning tree if a subset of the edges forms a directed tree. This is equivalent to saying

that all nodes in the graph are reachable from a single (root) node by following the edge arrows.

A graph may have multiple spanning trees. Define the root set or leader set of a graph as the setof nodes that are the roots of all spanning trees. If a graph is strongly connected it contains at

least one spanning tree. In fact, if a graph is strongly connected that all nodes are leader nodes.

Associate with each edge ,i j

v v E a weight aji. Note the order of the indices in this

definition. We assume in this chapter that the nonzero weights are strictly positive. The graph

can be represented by an adjacency or connectivity matrix [ ]ijA a with weights

0 ( , )ij j i

a if v v E and 0ij

a otherwise. Note that 0iia . Define the weighted in-degree of


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Figure 1. A directed graph.

Figure 2. A spanning tree with root

node 1.

node vias the i-th row sum ofA

1

N

i ij

j

d a

and the weighted out-degree of node vias the i-th column sum ofA

1

No

i ji

j

d a

A graph is said to be undirected if , ,ij ji

a a i j , that is, if it is bi-directional and the

weights of edges ( , )i jv v and ( , )j iv v are the same. Then the adjacency matrixAis symmetric.

A graph is said to be weight balancedif the weighted in-degree equals the weighted out-degree

for all i. If all the nonzero edge weights are all equal to 1, this is the same as the definition of

balanced graph. An undirected graph is weight balanced, since if TA A then the i-th row sumequals the i-th column sum. We may be loose at times and refer to node visimply as node i, andrefer simply to in-degree, out-degree, and the balanced property, without the qualifier weight,

even for graphs having non-unity weights on the edges.

Two global graph properties are the diameter DiamG , which is the greatest distance

between two nodes in a graph, and the (in-)volume, which is the sum of the in-degrees

i

i

Vol G d (1.1.1)

Define the diagonal in-degree matrix { }iD diag d and the (weighted) graph Laplacian

matrix L D A . Note thatLhas row sums equal to zero. Many properties of a graph may bestudied in terms of its graph Laplacian.

Example 1: Graph Matrices

Consider the digraph shown in Figure 1 with all edgeweights equal to 1. The graph is strongly connected sincethere is a path between any two pairs of nodes. A

spanning tree with root node 1 is shown in bold in Figure

2. There are other spanning trees in this graph.

The adjacency matrix is given by

0 0 1 0 0 0

1 0 0 0 0 1

1 1 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 00 0 0 1 1 0

A

and the diagonal in-degree matrix and Laplacian are


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1 0 0 0 0 0

0 2 0 0 0 0

0 0 2 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 2

D

,

1 0 1 0 0 0

1 2 0 0 0 1

1 1 2 0 0 0

0 1 0 1 0 0

0 0 1 0 1 0

0 0 0 1 1 2

L

Note that the row sums ofLare all zero.

1.2 Dynamic Systems on Graphs

Equation Section (Next)

A network may be considered as a set of nodes or agents that collaborates to achieve what each

cannot achieve alone. To capture the notion of dynamical agents, endow each node iof a graph

with a time-varying state vector ( )ix t . A graph with node dynamics [Olfati-Saber and Murray

2004] is ( , )G x with Ga graph havingNnodes and1

[ ]T T T

N

x x x a global state vector, where

the state of each node evolves according to some dynamics ( , )i i i ix f x u , with iu a control input

and (.)if some function. Given a graph ( , )G V E , we interpret ,i jv v E to mean that node vj

can obtain information from node vi for feedback control purposes. The control given by

1 2( , , , )

mii i i i iu k x x x for some function (.)ik is said to be distributed if ,im N i , that is, the

control input of each node depends on some proper subset of all the nodes. It is said to be a

protocol with topology Gif { } , {1, }ji i i i

v v N j m , that is, each node can obtain information

about the state only of itself and its (in)-neighbors.

Cooperative control, or control of distributed dynamical systems on graphs, refers to the

situation where each node can obtain information for controls design only from itself and itsneighbors. The graph might represent a communication network topology that restricts the

allowed communications between the nodes. This has also been referred to as multi-agent

control, but is not the same as the notion of multi-agent systems used by the Computer Sciencecommunity [Shoham 2009].

1.3 Consensus with Single Integrator Dynamics


We start our study of dynamical systems on graphs, or cooperative control, by considering thecase where all nodes of the graph Ghave scalar single-integrator dynamics

i i

x u (1.3.1)

with ,i ix u R . This corresponds to endowing each node or agent with a memory. Consider the

local control protocols

( )i

i ij j i

j N

u a x x

(1.3.2)

withija the graph edge weights. This is known as a local voting protocolsince the control input

of each node depends on the difference between its state and all its neighbors. Note that if these


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states are all the same, then 0i ix u . In fact, it will be seen that, under certain conditions, this

protocol drives all states to the same value. Let us therefore define the following.

Consensus Problem. Find a control protocol that drives all states to the same constant steady-

state values , ,i jx x i j . This value is known as a consensus value.

We wish to show that protocol (1.3.2) solves the consensus problem and to determine theconsensus value reached. Write the closed-loop dynamics as

( )i

i ij j i

j N

x a x x

(1.3.3)

1

1

i i

i i ij ij j i i i iN

j N j N

N

x

x x a a x d x a a

x

(1.3.4)

with id the in-degree. Define the global state vector 1[ ]T N

Nx x x R and write

{ }iD diag d so that

( )x Dx Ax D A x (1.3.5)

x Lx (1.3.6)

Note that the global control input vector1

[ ]T NN

u u u R is given by

u Lx (1.3.7)

It is seen that, using the local voting protocol, the closed-loop dynamics depends on the

graph Laplacian matrix L. We shall now see how the evolution of first-order integratordynamical systems on graphs depends on the graph properties through the Laplacian matrix. The

eigenvalues ofLare instrumental in this analysis.

Non-Scalar Node States

It was assumed that the node states and controls in the dynamics (1.3.1) are scalars ,i ix u R . If

the states and controls are vectors so that , n

i ix u R , then 1[ ]T T T nN

Nx x x R ,

1[ ]T T T nN

Nu u u R and the elements

ija and id in (1.3.4) are multiplied by the n n identity

matrix nI . Then, the global dynamics are written as

( )nu L I x (1.3.8)

( )n

x L I x (1.3.9)

with the Kronecker product (Appendix A). The following developments still hold if theKronecker product is added where appropriate.

Eigenstructure of Graph Laplacian Matrix

The eigenstructure of the graph Laplacian matrix Lplays a key role in the analysis of consensus

for (1.3.6). Define the Jordan normal form (Appendix A) of the graph Laplacian matrix by1

L MJM (1.3.10)


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with the Jordan form matrix and transformation matrix given as

1

2

N

J

, 1 2 NM v v v (1.3.11)

where the eigenvaluesi and right eigenvectors iv satisfy

( ) 0i iI L v (1.3.12)

with I the identity matrix. Let the eigenvalues be ordered so that 1 2 N . It is

assumed for ease of notation that L is simple, that is the Jordan form is diagonal. This is

guaranteed if all eigenvalues of L are distinct. The analysis here extends to the general case.

The inverse of the transformation matrix is given as

1

1 2

T

T

T

N

w

wM

w

(1.3.13)

where the left eigenvectors iw satisfy

( ) 0T

i iw I L (1.3.14)

and are normalized so that 1T

i iw v .

Any undirected graph has TL L so all its eigenvalues are real and one can order them as

1 2 N .

SinceLhas all row sums zero, one has1 0L c (1.3.15)

with 1 1 1T N

R the vector of 1s and c any constant. Therefore, 1 0 is an

eigenvalue with a right eigenvector of 1c . That is, 1 ( )c N L the nullspace of L. If the

dimension of the nullspace of L is equal to one, i.e. the rank of L is N-1, then 1 0 is

nonrepeated and 1c is the only vector inN(L). The next result states when this occurs.

Theorem 1. Lhas rank N-1, i.e. 1 0 is nonrepeated, if and only if graphGhas a spanning

tree.

Proof: Appendix 1a.

If the graph is strongly connected, then it has a spanning tree andLhas rankN-1.

The Laplacian has at least one eigenvalue at 1 0 . The remaining eigenvalues can be

localized using the following result.


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Figure 3. Gershgorin discs of L in the

complex plane.

Gerschgorin (sp. Gergorin) Circle Criterion. All eigenvalues of a matrix [ ] N NijE e R are

located within the union ofNdiscs

1

:N

ii ij

j ii

z C z e e

The i-th disc in this theorem is drawn with a

center at the diagonal element iie and with a radius

equal to the i-th absolute row sum with the diagonal

element deleted, ijj i

e

. Therefore the Gerschgorin

discs for the graph Laplacian matrix L are centered at

the in-degrees id and have radius equal to id . Let Maxd

be the maximum in-degree of G. Then, the largestGerschgorin disc of the Laplacian matrixLis given by

a circle centered at Maxd and having radius of Maxd .

This circle contains all the eigenvalues ofL. See Figure 3. This theorem ties the eigenvalues ofLrather closely to the graph structural properties in terms of the in-degrees.

We have thus discovered that if the graph has a spanning tree, there is a nonrepeated

eigenvalue at 1 0 and all other eigenvalues have positive real parts, i.e. in the open right half-

plane, and are within a circle centered at Maxd and having radius of Maxd .

When comparing eigenvalues between two graphs, it is often more useful to use the

normalized Laplacian matrix1 1 1( )L D L D D A I D A (1.3.16)

Since the normalized adjacency matrix 1A D A has row sums equal to one, 1,id i and L

has all Gerschgorin discs centered at s=1 with radius of 1.

Consensus for Single Integrator Dynamics

The dynamics given by (1.3.6) has a system matrix of L , and hence has eigenvalues in the left-half plane, specifically inside the circle in Figure 3 reflected into the left half plane. At steady-

state, according to (1.3.6) one has

0ssLx (1.3.17)

Therefore, the steady-state global state is in the nullspace of L. According to (1.3.15) one vector

inN(L)is 1c . Therefore, ifLhas rank ofN-1, one has 1ssx c for some constant c. Then onehas at steady-state

1

1

N

x c

x c

x c

(1.3.18)

and , ,i jx x c const i j . Then, consensus is reached. The next result formalizes this


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discussion and delivers the consensus value c.

Theorem 2. Consensus. The local voting protocol (1.3.2) guarantees consensus of the single-

integrator dynamics (1.3.1) if and only if the graph has a spanning tree. Then, all node states

come to the same steady-state values , ,i jx x c i j . The consensus value is given by

1

(0)N

i i

i

c p x

(1.3.19)

where 1 1T

Nw p p is the normalized left eigenvector of the Laplacian L for

10 .

Finally, consensus is reached with a time constant given by

21/ (1.3.20)

with2

the second eigenvalue ofL.

Proof: The proof is given for case thatLis simple, i.e. has all Jordan blocks of order one. Thegeneral case follows similarly. Using modal decomposition (Appendix A) write the solution of

(1.3.6) in terms of the Jordan form ofLas

11 1

( ) (0) (0) (0) (0)i iN N

t tLt Jt T T

i i i i

j j

x t e x Me M x v e w x w x e v

(1.3.21)

Here, the left and right eigenvectors are normalized so that 1Ti i

w v . If the graph has a spanning

tree, then Theorem 1 shows that1

0 is simple and Gerschgorin shows that all other

eigenvalues of L are in the open right half of the complex plane. Then system (1.3.6) ismarginally stable with one pole at the origin and the rest in the open left half plane. Therefore,

in the limit as t one has2 1

2 2 1 1( ) (0) (0)t tT Tx t v e w x v e w x (1.3.22)

with2

the smallest magnitude nonzero eigenvalue. One has1

0 , and take1

1v . Define

the left eigenvector 1 1T

Nw p p , normalized so that 1Ti iw v , that is, 1i

i

p . Then

2

2 2

1

( ) (0) 1 (0)N

t T

i i

i

x t v e w x p x

(1.3.23)

The last term in this equation is the steady-state value 1ss

x c , with c given in (1.3.19). The

first term on the right verifies that the consensus value is reached with a time constant given by

2

1/ .

If the graph does not have a spanning tree, then the dimension of nullspace ofLis greater

than one and there is a ramp term in (1.3.21) that increases with time. Then, consensus is not

reached.

If the graph is strongly connected, then it has a spanning tree and consensus is reached.

The theorem shows that the consensus value is in the convex hull of the initial node


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states, and is the normalized linear combination of the initial states weighted by the elements of

the left eigenvector 1w for 1 0 .

Due to their importance in the analysis of dynamics and consensus on graphs, we call 1

the first right eigenvector ofLand 1w its first left eigenvector.

If the graph has a spanning tree, there is a simple pole at zero and the system is of Type I.Then, it reaches a steady-state value. Otherwise, the system is of Type 2 or higher and a constant

steady-state value is not reached. Thus, we have the curious situation that the system dynamical

behavior, notably the system type, depends on the graph topology, that is, the manner in whichthe nodes communicate.

This development has made clear the importance for networked dynamical systems of the

communication graph topology as given in terms of the eigenstructure of the Laplacian matrix L,

including its first eigenvalue 1 0 , its right eigenvector 1 1v and left eigenvector 1w , and the

second (Fiedler) eigenvector2

. In some cases, especially in undirected graphs, these quantities

can be more closely tied to topological properties of the graph.

Motion Invariants for First-Order Consensus

Let 1 1T

Nw p p be a (not necessarily normalized) left eigenvector ofLfor 1 0 , then

1 1 1( ) 0T T Td

w x w x w Lxdt

(1.3.24)

so that the quantity

1

1 1

T

N i i

i

N

x

x w x p p p x

x

(1.3.25)

is invariant. That is, regardless of the values of the node states ( )ix t , the quantity ( )i ii

x p x t

is a constant of the motion. This means that the global velocity vector ( ) N

x t R is always

orthogonal to the vector 1N

w R .

Accordingly, (0) ( ) ,i i i ii i

p x p x t t . Therefore, if the graph has a spanning tree, at

steady-state one has consensus so that , ,i j

x x c i j where the consensus value is given by

(0)i ii

i

i

p x

c p

(1.3.26)

This provides another proof for (1.3.19) (note that in that equation, the left eigenvector is

normalized).

According to (1.3.7) the global control input is u Lx so that

1 1 0T T

i i

i

w u p u w Lx (1.3.27)


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with ( )iu t the node control inputs. This states that the linear combination of control inputs

weighted by the elements of the left eigenvector for 1 0 is equal to zero. This can be

interpreted as a statement that the internal forces of the graph do no work.

Center of Gravity Dynamics and Shape DynamicsThe quantity x in (1.3.25) is a weighted centroid, or center of gravity, of the group, which

remains stationary under the local voting protocol. Define the state-space coordinate

transformation [Lee and Spong ] z Mx with

1 2 3

1 1

1 1

1 1

Np p p p

M

(1.3.28)

with 1 1 TNw p p the normalized first left eigenvector of L, that is, 1 0Tw L , 1ii

p .Then

1 2

1N N

x

x x xz

x

x x

(1.3.29)

with 1( ) Nx t R an error vector that shows how far the node states are from consensus. It can

easily be seen that

2 3

2 3

1

2 3

32

1

1 1

1 1 1

1 111

N

N

N

N

P P P

P P P

M P P

P

P PP

(1.3.30)

whereN

k ii kP p

.

Now, one has1

z MLM z (1.3.31)

where

10 0

0MLM

L

(1.3.32)

(Note that the first row ofMis the first left eigenvector ofLand the first column of 1M the first

right eigenvector ofL.) Therefore,


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0x

x Lx

(1.3.33)

Since a state-space transformation does not change the eigenvalues, the eigenvalues of L are the

eigenvalues 2 , , N ofL. If the graph has a spanning tree, therefore, L has all eigenvalues

in the left-half plane and so is asymptotically stable. This means that the weighted centroidremains stationary, while, if there is a spanning tree, the error vector ( )x t converges to zero.

Vector ( )x t is called the shape vector; it indicates the spread of the error vectors about the

centroid.

The Fiedler Eigenvalue of the Graph Laplacian Matrix

According to (1.3.23), the second eigenvalue 2 of the graph Laplacian matrixLis important in

determining the speed of consensus. Graph topologies that have a large value of 2 are better

for achieving fast consensus. The second eigenvalue ofL is known as the Fiedler eigenvalue

and it has important and subtle connections to the topology of the graph. It is also known as the

graph algebraic connectivity. We discuss 2 further in Chapter **.

For undirected graphs there are many useful bounds on the Fiedler eigenvalue, including[Wu 2007]

2 min1

Nd

N

(1.3.34)

where mind is the minimum in-degree, and for connected undirected graphs

2

1

DiamG Vol G

. (1.3.35)

The distance between two nodes in an undirected graph is the length of the shortest pathconnecting them, and if they are not connected the distance between them is infinity. The

diameter of a graph DiamG is the greatest distance between any two nodes in Gand the volume

VolG is given by (1.1.1).

According to (1.3.35) convergence to consensus is faster in graphs that are more fully

connected, that is, have shorter distances between any pair of nodes, and for graphs whose nodes

have fewer neighbors. Among the fastest graph topologies is the star graph [Wu 2007], whereone root node is connected to all other nodes. The star is a tree graph with depth equal to one.

Example 2: Fiedler Eigenvalues for Various Graph Types

Example 3: Consensus Simulation

Example 4: Consensus of Headings in Swarm Motion Control

The local voting protocol (1.3.2) can be used in many applications to yield consensus. An

example is the consensus of headings in a formation or in animal groups. Simple motions of a

group of N agents in the ( , )x y plane as shown in Figure * can be described by the node


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Fi ure *. Consensus of headin s.

Figure *. Motion consensus in (x,y)plane

Simulation of Cooperative Systems with Vector States

If the states in (1.3.1) are vectors so that , n

i ix u R , then 1[ ]T T T nN

Nx x x R ,

1[ ]T T T nN

Nu u u R and the global dynamics are written as (1.3.9). This can cause numerical

integration problems in simulations since the dimension nN of the global state becomes large

quickly as the node state dimension n and the number of nodes N increase. The matrix

( ) nN nN nL I R R is a large sparse matrix with many zero entries. For simulation purposes, it

is better to define the n N matrix of states

1

2

T

T

T

N

x

xX

x

(1.3.38)

ThenX LX (1.3.39)

which does not involve the Kronecker product and has better numerical properties for simulation.

Consensus Value for Balanced Graphs- Average Consensus

The consensus value (1.3.26) is the weighted average of the initial conditions of the states

of the nodes. It depends on the graph topology through the left eigenvector 1 1T

Nw p p

of the zero eigenvalue. It is not desirable for the consensus value to depend on the graphtopology, i.e. on the manner in which the nodes communicate. An important problem in

cooperative control is therefore the following.

Average Consensus Problem. Find a control protocol that drives all states to the same constant

steady-state values , ,i jx x c i j , where c is the average of the initial states of the nodes

1(0)i

i

c xN

(1.3.40)

This value does not depend on the graph structure. The average consensus problem is important


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is many applications. For instance, in wireless sensor networks each node measures some

quantity (e.g. temperature, salinity content, etc.) and it is desired to determine the best estimateof the measured quantity, which is the average if all sensors have identical noise characteristics.

A graph is said to be weight balanced if the weighted in-degree equals the weighted out-

degree for all nodes v V . We shall omit the qualifier weight for simplicity. The row sums of

the LaplacianL are all zero. For balanced graphs, the i-th row sum and i-th column sum of A,and henceL, are equal. Then, the column sums of the Laplacian matrix are also all equal to zero

and

1 1 0T Tw L c L (1.3.41)

so that the left eigenvector for 1 0 is 1 1w c . Then the consensus value (1.3.26) is (1.3.40)

and average consensus is reached.

Note that if a graph is undirected, it is balanced, so that average consensus is reached.

X-consensus? Cortes and Chinese Academy guys.

Max, min consensus

Consensus Leaders

A (directed) tree is a connected digraph where every node except one, called the root or leader,

has in-degree equal to one. It was seen in Example 4 that all nodes reached a consensus heading

equal to the initial heading of the leader. The consensus value is given by (1.3.26) withi

p the i-

th component of the left eigenvector1

w for the zero eigenvalue. If the graph is strongly

connected, one has 0,ip i since each node influences all other nodes along directed paths. If

the graph has a spanning tree but is not strongly connected, then some nodes do not have paths to

all other nodes. Thus, all other nodes cannot be aware of the initial states of such nodes.

Suppose a graph contains a spanning tree. Then it may contain more than one. Definethe root set or leader set of a graph as the set of nodes that are the roots of all spanning trees.

Theorem 3. Consensus Leaders. Suppose a graph ( , )G V E contains a spanning tree and

define the leader set L V . Define the left eigenvector for 1 0 as 1 1T

Nw p p .

Then 0,ip i L and 0,ip i L .

Proof: Appendix 1a.

This result shows that the consensus value (1.3.26) is actually the weighted average of the

initial conditions of the root or leader nodes in a graph, that is, of the nodes that have a path to all

other nodes in the graph. In Example 4 the graph is a tree. Therefore, all nodes reach aconsensus heading equal to the initial heading of the leader node. In a strongly connected graph,

all nodes are leader nodes so that 0,i

p i .

1.4 Consensus with First-Order Discrete-Time Dynamics



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Figure 4. Region of eigenvalues of

Perron matrix P.

Now suppose each node or agent has scalar discrete-time (DT) dynamics given by

( 1) ( ) ( )i i ix k x k u k (1.4.1)

with ,i ix u R . This corresponds to endowing each node with a memory in the form of a shift

register of order one. We discuss two ways to select the control input protocols.

Perron DT Systems

Consider the local control protocols

( ) ( ( ) ( ))i

i ij j i

j N

u k a x k x k

(1.4.2)

withija the graph edge weights and 0 . The closed-loop system becomes

( 1) ( ) ( ( ) ( )) (1 ) ( ) ( )i i

i i ij j i i i ij j

j N j N

x k x k a x k x k d x k a x k

(1.4.3)

The corresponding global input is

( ) ( )u k Lx k (1.4.4)and the global dynamics is

( 1) ( ) ( ) ( )x k I L x k Px k (1.4.5)

with1

[ ]T NNu u u R , 1[ ]T N

Nx x x R .

The matrix P I L is known as the Perronmatrix. Since Lhas all eigenvalues in the right half of

the s-plane, -Lhas all eigenvalues in the left half-plane,

and Phas all eigenvalues in the shaded region of the z-

plane shown in Figure 4. If is small enough, Phas all

eigenvalues inside the unit circle. Then, if the graph hasa spanning tree, it has a simple eigenvalue1

1 at

1z , and the rest strictly inside the unit circle. Then,(1.4.5) is marginally stable and of Type 1, and a steady-

state value is reached. A sufficient condition for P to

have all eigenvalues in the unit circle is

1/ Maxd (1.4.6)

Note that the conditions for stability of each agents dynamics (1.4.3) are 1/ id .

Since Laplacian matrix Lhas row sums of zero, Phas row sums of 1, and so is a row

stochastic matrix. That is,

1 1P (1.4.7)

( )1 0I P (1.4.8)

and 1 is the right eigenvector of 1 1 . Let 1w be a left eigenvector of L for 1 0 . Then

1 1 1( )T T T

w P w I L w so that 1w is a left eigenvector of Pfor 1 1 .

If the system (1.4.5) reaches steady state, then


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ss ssx Px (1.4.9)

If the graph has a spanning tree, then the only solution of this is 1ssx c for some 0c . Then,

consensus is reached so that , ,i jx x c i j .

Let 1 1T

Nw p p be a left eigenvector of Pfor 1 1 , not necessarily normalized.

Then

1 1 1( 1) ( ) ( )T T Tw x k w Px k w x k (1.4.10)

so that the quantity

1

1 1

T

N i i

i

N

x

x w x p p p x

x

(1.4.11)

is invariant. That is, the quantity ( )i ii

x p x k is a constant of the motion. Thus,

(0) ( ) ,i i i ii i

p x p x k k . Therefore, if the graph has a spanning tree, at steady-state one hasconsensus so that , ,

i jx x c i j where the consensus value is given by (1.3.26), the

normalized linear combination of the initial states weighted by the elements of the left

eigenvector of P for 1 1 . This depends on the graph topology and hence on how the nodes

communicate.

If the graph is balanced, then row sums ofLare equal to column sums, and this property

carries over to P. Then 1 1w c is a left eigenvector of Pfor 1 1 and the consensus value is

the average of the initial conditions (1.3.40), which is independent of the graph topology.

Normalized Protocol DT Systems

Consider the local control protocols

1( ) ( ( ) ( ))

1i

i ij j i

j Ni

u k a x k x k d

(1.4.12)

which are normalized by dividing by 1i

d , withi

d the in-degree of node i. Thus, nodes with

larger in-degrees must use less relative control effort than in the protocol (1.4.2). The closed-loop system becomes

1 1( 1) ( ) ( ( ) ( )) ( ) ( )

1 1i ii i ij j i i ij j

j N j Ni i

x k x k a x k x k x k a x k

d d

(1.4.13)

which is a weighted average of the previous states of node i and its neighbors. The

corresponding global input is1( ) ( ) ( )u k I D Lx k

(1.4.14)

with { }iD diag d . The global dynamics is

1 1( 1) ( ) ( ) ( ) ( ) ( ) ( ) ( )x k x k I D Lx k I D I A x k Fx k (1.4.15)


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Figure 4. Region of eigenvalues of

discrete-time graph matrix F.

where1 1( ) ( ) ( )F I I D L I D I A (1.4.16)

is the normalized DT graph matrix.

According to the Gershgorin Theorem the

normalized Laplacian matrix1

( )I D L

has alleigenvalues in the right half s-plane within a disc

centered at / (1 )Max Max

d d with radius equal to

/ (1 )Max Maxd d . Therefore, the F matrix has

eigenvalues inside the shaded region in Figure 5. Since

/ (1 ) 1Max Max

d d , this region is always inside the unit

circle. Therefore, if the graph has a spanning tree, F has

a simple eigenvalue at 1 1 and all other eigenvalues

strictly inside the unit circle. Then, the system (1.4.15)

is marginally stable and of Type I, and the state reaches

a steady-state value.Since Laplacian matrix Lhas row sums of zero, Fhas row sums of 1, and so is a row

stochastic matrix since

11 ( ) 1 1F I I D L (1.4.17)

Then, 1 is the right eigenvector of Ffor 1 1 .

Let 1 1T

Nw p p be a left eigenvector of Ffor 1 1 , not necessarily normalized.

Then

1 1 1( 1) ( ) ( )T T Tw x k w Fx k w x k (1.4.18)

so that the quantity (1.4.11) is invariant, but now ip are defined with respect to matrix F. Then,

the quantity ( )i ii

x p x k is a constant of the motion. If the graph has a spanning tree, the

consensus value , ,i j

x x c i j is given by (1.3.26) in terms of the elements of the left

eigenvector for 1 1 of F.

If the graph is balanced, then row sums of Lare equal to column sums and 1c is a left

eigenvector of L for 1 0 . However, in1( )I D L the i-th row is divided by 1 id .

Therefore, 1( )I D L is not balanced. Hence, F is not balanced and so even for balanced

graphs the average consensus (1.3.40) is not reached. The consensus value of (1.4.15) depends

on graph properties even for balanced graphs. This is true also for general undirected graphs.However, if all nodes have the same in-degree, then L balanced implies that 1( )I D L is

balanced. Hence Fis balanced and average consensus is reached. An undirected graph where all

nodes have the same degree dis called d-regular.

Let 1w be a left eigenvector of Lfor 1 0 . Then 1w is not a left eigenvector of Ffor

1 1 unless the graph is regular. Then, it can be checked that 1( )T T Ti i iw F w I I D L w .


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Example 5: Discrete-Time Consensus Simulation

Average Consensus Using Two Parallel Protocols at Each Node

The average consensus problem yields the average of the initial states (1.3.40), independently of

the left eigenvector of 1 1 and hence of the graph topology. This is important in a variety of

applications. It has been seen that if the graph is balanced, the continuous-time protocol (1.3.3)

and the Perron protocol (1.4.3) both yield average consensus (1.3.40) independently of the

detailed topology of the graph. The normalized DT protocol (1.4.13) does not yield average

consensus unless the graph is regular (all nodes have the same in-degree). There are severalways to obtain average consensus for this protocol [Olshevsky and Tsitsiklis 2009].

Suppose each node iknows its componenti

p of the normalized left eigenvector 1w of F

for 1 1 , and the number of nodes N in the graph. Suppose it is desired to reach average

consensus on a prescribed vector 1 2(0) (0) (0) (0) T

Nx x x x . Then, let each node run

the normalized protocol (1.4.13) with initial state selected as (0) /i ix Np . If the graph has a

spanning tree, the weighted average consensus (1.3.26) is achieved. Substituting the selected

initial conditions into (1.3.26) yields average consensus (1.3.40).

The requirement for each node to know its owni

p as well as the number of nodes Nin

the graph assumes that there is a central authority which computes the graph structure and

disseminates this global information, which is contrary to the idea of distributed cooperativecontrol. It is possible in some graph topologies for each node to estimate global graph structural

informationby using multiple protocols at each node, as follows.

Let each node be endowed with two states ( ), ( )i iy k z k , each running local protocols in

the form (1.4.13) so that

1( 1) ( ) ( )

1i

i i ij j

j Ni

y k y k a y kd

(1.4.19)

1( 1) ( ) ( )

1i

i i ij j

j Ni

z k z k a z kd

(1.4.20)

with prescribed initial states (0), (0)i iy z . The global dynamics of 1[ ]T N

Ny y y R , are then

1( 1) ( ) ( ) ( ) ( )y k I D I A y k Fy k (1.4.21)

1( 1) ( ) ( ) ( ) ( )z k I D I A z k Fz k (1.4.22)

Now suppose the graph has a special topology known as the bidirectional equal-neighbor

weight topology [Olshevsky and Tsitsiklis 2009]. In this topology, one has 0 0ij jia a and

0 1ij ij

a a . Thus, each node equally weights its own current state and its neighbors

current states by 1/ (1 )i

d in computing the next state. For this special topology, it is easy to

give explicit formulae for the componentsi

p of left eigenvector 1w . Define the (modified)


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volume of the graph as1

' (1 )N

i

i

Vol d

. Then for the bidirectional equal-neighbor weight

topologies

1

'

ii

dp

Vol

(1.4.23)

which is easily verified by verifying that 11 1 1( ) ( )T T T

w F w I D I A w for the given graph

topology and left eigenvector components. Note that (1.4.23) gives a normalized 1w since

1ii

p .

Now suppose it is desired to reach average consensus on a prescribed vector

1 2(0) (0) (0) (0) T

Nx x x x . Then, select initial states (0) 1/ (1 )

i iy d ,

(0) (0) / (1 )i i i

z x d This is local information available to each node i. If the graph has a

spanning tree, then according to (1.3.26) the two protocols (1.4.19), (1.4.20) reach respective

consensus values of

1( ) , ( ) (0)

' 'i i i

i

Ny k z k x

Vol Vol (1.4.24)

Then, the estimate of the average consensus value is given by

( ) 1( ) (0)

( )

ii i

ii

z kx k x

y k N (1.4.25)

This procedure results in average consensus independent of the graph topology. This has

been achieved by running two local voting protocols at each node, one of which, ( )iy k ,

estimates the global graph property / 'N Vol . This is used to divide out the volume in the second

protocol for ( )iz k .


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References

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A. Jadbabaie, J. Lin, and S. Morse. 2003. Coordination of Groups of Mobile Autonomous

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Lee and Spong

L. Moreau, Stability of multiagent systems with time-dependent communication links, IEEE

Trans. Automatic Control, vol. 50, no. 2, pp. 169-182, Feb. 2005.

R. Olfati-Saber and R.M. Murray, Consensus Problems in Networks of Agents with SwitchingTopology and Time-Delays, IEEE Transaction of Automatic Contro, vol.49, no. 9, pp. 1520-

1533, 2004.

R. Olfati-Saber, J.A. Fax, and R.M. Murray, Consensus and cooperation in networked multi-

agent systems, Proc. IEEE, vol. 95, no. 1, pp. 215-233, Jan. 2007.

A. Olshevskyand J.N. Tsitsiklis, Convergencespeed in distributed consensus and averaging,

SIAM Journal on Control and Optimization, vol. 48, no. 1, pp. 33-55, 2009.

W. Ren and R.W. Beard. 2005,Consensus Seeking in Multiagent Systems Under Dynamically

Changing Interaction Topologies,IEEE Trans. Automatic Control50, no. 5 (May): 655-661.

W. Ren, R. Beard, and E. Atkins, A survey of consensus problems in multi-agent coordination,

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C. Reynolds, Flocks, herds, and schools: A distributed behavior model, Proceedings of

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Y. Shoham and K. Leyton-Brown,Multiagent Systems, Cambridge Press, New York, 2009.

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Zhihua Qu

Wu 2007

Documents

1. First-Order Consensus