Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
SIAM J. CONTROL OPTIM.
c
2008 Society for Industrial and Applied Mathematics
Vol. 47, No. 2, pp. 601–623
REACHING A CONSENSUS IN A DYNAMICALLY CHANGING
ENVIRONMENT: CONVERGENCE RATES, MEASUREMENT
DELAYS, AND ASYNCHRONOUS EVENTS
∗
MING CAO
†
, A. STEPHEN MORSE
†
, AND BRIAN D. O. ANDERSON
‡
Abstract. This paper uses recently established properties of compositions of directed graphs
together with results from the theory of nonhomogeneous Markov chains to derive worst case conver-
gence rates for the headings of a group of mobile autonomous agents which arise in connection with
the widely studied Vicsek consensus problem. The paper also uses graph-theoretic constructions to
solve modified versions of the Vicsek problem in which there are measurement delays, asynchronous
events, or a group leader. In all three cases the conditions under which consensus is achieved prove
to be almost the same as the conditions under which consensus is achieved in the synchronous,
delay-free, leaderless case.
Key words. cooperative control, graph theory, switched systems, convergence rates, delays,
asynchronism
AMS subject classifications. 93C05, 05C50, 05C75, 15A51, 40A20, 68W15
DOI. 10.1137/060657029
1. Introduction. In a recent paper [6] the present authors defined the notion of
“graph composition” and established a number of basic properties of compositions of
directed graphs which are useful in explaining how a consensus might be reached by
a group of mobile autonomous agents in a dynamically changing environment. The
aim of this paper is to use the graph-theoretic findings of [6] to address several issues
related to the well-known Vicsek consensus problem [20] which have either not been
considered before or have been only partially resolved.
The paper begins with a brief review in section 2 of the basic leaderless consensus
problem treated in [6, 14, 16]. Section 3 exploits the connection between “neighbor-
shared” graphs and the elegant theory of “scrambling matrices” found in the literature
on nonhomogeneous Markov chains [17, 9] to help in the derivation of worst case
agent heading convergence rates for the leaderless version of the Vicsek problem.
Section 4 addresses a modified version of the consensus problem in which integer-
valued delays occur in the values of the headings which agents measure. In keeping
with the overall theme of this paper, the effect of measurement delays is analyzed
from a mainly graph-theoretic point of view. This enables us to significantly relax
previously derived conditions [18, 19, 3] under which consensus can be achieved in
the face of measurement delays. A comparison is made between the results of [3] and
∗
Received by the editors April 11, 2006; accepted for publication (in revised form) August 16,
2007; published electronically February 6, 2008. A preliminary version of this work can be found
in A. S. Morse, Logically switched dynamical systems, in Nonlinear and Optimal Control Theory,
Springer-Verlag, Berlin, 2008, pp. 1–84.
http://www.siam.org/journals/sicon/47-2/65702.html
†
Electrical Engineering, Yale University, P.O. Box 208267, New Haven, CT 06520 (m.cao@
yale.edu, morse@sysc.eng.yale.edu). The research of these authors was supported by the U.S. Army
Research Office, the U.S. National Science Foundation, and a gift from the Xerox Corporation.
‡
Australian National University and National ICT Australia Ltd., Locked bag 8001, Canberra
ACT 2601, Australia (brian.anderson@nicta.com.au). The research of this author was supported by
National ICT Australia, which is funded by the Australian Government’s Department of Commu-
nications, Information Technology, and the Arts, and the Australian Research Council through the
Backing Australia’s Ability initiative and the ICT Centre of Excellence Program.
601
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602 M. CAO, A. S. MORSE, AND B. D. O. ANDERSON
the main result of this paper on measurement delays, namely Theorem 2. To model
dynamics when delays are present requires a somewhat different type of stochastic
“flocking matrix” than the one which is appropriate in the delay-free case. The graphs
of the type of matrices to which we are referring are directed, just as in the delay-free
case, but do not have self-arcs at every vertex. As a result, the set of such graphs,
denoted by D, is not closed under composition. The smallest set of directed graphs
which contains D and which is closed under composition is called the set of “extended
delay graphs.” This class is explicitly characterized. Section 4 then develops the
requisite properties of extended delay graphs needed to prove Theorem 2.
Section 5 considers a modified version of the flocking problem in which each agent
independently updates its heading at times determined by its own clock. It is not
assumed that the groups’ clocks are synchronized together or that the times any one
agent updates its heading are evenly spaced. In this case, the deriving of conditions
under which all agents eventually move with the same heading requires the analysis
of the asymptotic behavior of an overall asynchronous process which models the n-
agent system. The analysis is carried out by first embedding this process in a suitably
defined synchronous discrete-time, hybrid dynamical system S. This is accomplished
using the concept of analytic synchronization outlined previously in [12, 13]. This
enables us to bring to bear results derived earlier in [6] to characterize a rich class of
system trajectories under which consensus is achieved.
In section 6 we briefly consider a modified version of the consensus problem for the
same group of n agents as before but now with one of the group’s members (say agent
1) acting as the group’s leader. The remaining agents, called followers and labelled
2 through n, do not know who the leader is or even if there is a leader. Accordingly
they continue to function as if there was no leader using the same update rules as
are used in the leaderless case. The leader, on the other hand, acting on its own,
ignores these update rules and moves with a constant heading. Using the main result
on leaderless consensus summarized in section 2, we then develop conditions under
which all follower agents eventually move in the same direction as the leader. These
conditions correct prior findings on leader following in [11] which are in error.
2. Background. As in [6], the system of interest consists of n autonomous
agents, labelled 1 through n, all moving in the plane with the same speed but with
different headings. Each agent’s heading is updated using a simple local rule based
on the average of its own heading plus the headings of its “neighbors.” Agent i’s
neighbors at time t are those agents, including itself, which are in a closed disk of
prespecified radius r
i
centered at agent i’s current position. In what follows N
i
(t)
denotes the set of labels of those agents which are neighbors of agent i at time t.
Agent i’s heading, written θ
i
, evolves in discrete time in accordance with a model of
the form
θ
i
(t +1)=
1
n
i
(t)
⎛
⎝
j∈N
i
(t)
θ
j
(t)
⎞
⎠
,(1)
where t is a discrete-time index taking values in the nonnegative integers {0, 1, 2,...},
and n
i
(t) is the number of neighbors of agent i at time t.
2.1. Neighbor graph. The explicit form of the update equations determined
by (1) depends on the relationships between neighbors which exist at time t. These
relationships can be conveniently described by a directed graph N(t) with vertex set
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REACHING A CONSENSUS 603
V = {1, 2,...,n} and arc set A(N(t)) ⊂V×Vwhich is defined so that (i, j)isan
arc or directed edge from i to j just in case agent i is a neighbor of agent j at time
t.ThusN(t) is a directed graph on n vertices with at most one arc connecting each
ordered pair of distinct vertices and with exactly one self-arc at each vertex. We write
G
sa
for the set of all such graphs and G for the set of all directed graphs with vertex
set V. It is natural to call a vertex i a neighbor of vertex j in a graph G ∈Gif (i, j)
is an arc in G.
2.2. Heading update rule. The set of agent heading update rules defined by
(1) can be written in state form. Towards this end, for each graph N ∈G
sa
define the
flocking matrix
F = D
−1
A
,(2)
where A
is the transpose of the adjacency matrix of N and D the diagonal matrix
whose jth diagonal element is the in-degree of vertex j within N. Then
θ(t +1)=F (t)θ (t),t∈{0, 1, 2,...},(3)
where θ is the heading vector θ =[θ
1
θ
2
... θ
n
]
and F (t) is the flocking matrix
of the neighbor graph N(t).
2.3. Leaderless consensus. To proceed, we need to recall a few definitions
from [6]. We call a vertex i of a directed graph G a root of G if for each other vertex
j of G, there is a path from i to j.Thusi isarootofG if it is the root of a directed
spanning tree of G. We say that G is rooted at i if i is in fact a root. Thus G is
rooted at i just in case each other vertex of G is reachable from vertex i along a path
within the graph. G is strongly rooted at i if each other vertex of G is reachable from
vertex i along a path of length 1. Thus G is strongly rooted at i if i is a neighbor of
every other vertex in the graph. A rooted graph G is a graph which possesses at least
one root. Finally, a strongly rooted graph is a graph which has at least one vertex at
which it is strongly rooted.
By the composition of two directed graphs G
p
, G
q
with the same vertex set V
we mean the graph G
q
◦ G
p
with the same vertex set V and arc set defined such that
(i, j)isanarcofG
q
◦ G
p
if for some vertex k,(i, k)isanarcofG
p
and (k, j)isan
arc of G
q
. A finite sequence of directed graphs G
1
, G
2
,...,G
q
with the same vertex
set is jointly rooted if the composition G
q
◦ G
q−1
◦ ··· ◦ G
1
is rooted. An infinite
sequence of graphs G
1
, G
2
,... with the same vertex set is repeatedly jointly rooted by
subsequences of length q if there is a positive integer q for which each finite sequence
G
qk+1
, ...,G
q(k+1)
,k≥ 0, is jointly rooted. The main result on leaderless consensus
[14, 16] is equivalent to the following result from [6].
Theorem 1. Let θ(0) be fixed. For any trajectory of the system determined by
(1) along which the sequence of neighbor graphs N(0), N(1),... is repeatedly jointly
rooted by sequences of length q, there is a constant θ
ss
, depending only on θ(0), for
which
lim
t→∞
θ(t)=θ
ss
1,(4)
where the limit is approached exponentially fast.
3. Convergence rates. The aim of this section is to derive a bound on the rate
at which θ converges.
1
There are two distinct ways to go about this, and below we
1
This section summarizes and extends some of the key findings of [7].
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
604 M. CAO, A. S. MORSE, AND B. D. O. ANDERSON
describe both. To do this we will make use of certain structural properties of the
F . As defined, each F is square and nonnegative, where by a nonnegative matrix
we mean a matrix whose entries are all nonnegative. Each F also has the property
that its row sums all equal 1 (i.e., F 1 = 1). Matrices with these two properties are
called (row) stochastic [10]. It is easy to verify that the class of all n × n stochastic
matrices is closed under multiplication. It is worth noting that because the vertices of
the graphs in G
sa
all have self-arcs, the F also have the property that their diagonal
elements are positive.
In what follows we write M ≥ N whenever M − N is a nonnegative matrix. We
also write M>Nwhenever M − N is a positive matrix, where by a positive matrix
we mean a matrix with all positive entries. For any nonnegative matrix R of any size,
we write ||R|| for the largest of the row sums of R. Note that ||R|| is the induced
infinity norm of R and consequently is submultiplicative. Moreover, ||M
1
||≤||M
2
|| if
M
1
≤ M
2
. Observe that for any n × n stochastic matrix S, ||S|| = 1 because the row
sums of a stochastic matrix all equal 1. As in [6] we write M and M for the 1 ×m
row vectors whose jth entries are the smallest and largest elements, respectively, of
the jth column of M. Note that M is the largest 1 × m nonnegative row vector c
for which M − 1c is nonnegative and that M is the smallest nonnegative row vector
c for which 1c − M is nonnegative. Note in addition that for any n × n stochastic
matrix S, one can write
S = 1S + |S| and S = 1S−|S|,(5)
where |S| and |S|are the nonnegative matrices defined by the equations
|S| = S − 1S and |S| = 1S−S,(6)
respectively. Moreover, the row sums of |S| are all equal to 1 −S1 and the row
sums of |S| are all equal to S1 − 1, and so
|||S||| =1−S1 and |||S||| = S1 − 1.(7)
In what follows we will also be interested in the matrix
|S| = |S| + |S|.(8)
This matrix satisfies
|S
| = 1(S−S)(9)
because of (5).
To prove that all θ
i
converge to a common heading, it is necessary to prove that
θ converges to a vector of the form θ
ss
1, where 1 is the n × 1 vector of 1’s. It is clear
from (3) that θ will converge to such a vector just in case, as t →∞, the matrix
product F (t) ···F (0) converges to a rank one matrix of the form 1c for some n × 1
row vector c. Thus to study how such matrix products converge it is sufficient to
study how products of stochastic matrices of the form S
j
···S
1
converge as j →∞.
As in [6], we say that a matrix product S
j
S
j−1
···S
1
converges exponentially fast at
a rate no slower than λ to a matrix of the form 1c if there are nonnegative constants
b and λ with λ<1, such that
||(S
j
···S
1
) − 1c|| ≤ bλ
j
,j≥ 1.(10)
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REACHING A CONSENSUS 605
The following fact is proved in [6].
Proposition 1. If an infinite sequence of stochastic matrices S
1
,S
2
,... satisfies
|||S
j
···S
1
||| ≤
¯
bλ
j
,j≥ 0,(11)
for some nonnegative constants
¯
b and λ<1, then the product S
j
S
j−1
···S
1
converges
exponentially fast at a rate no slower than λ to a matrix of the form 1c.
We will exploit this inequality in deriving specific convergence rates.
Any n × n stochastic matrix S determines a directed graph γ(S) with the vertex
set {1, 2,...,n} and arc set defined in such a way so that (i, j)isanarcofγ(S) from i
to j just in case the jith entry of S is nonzero. Note that the graph of any stochastic
matrix with positive diagonal elements must be in S
sa
. Since flocking matrices have
this property, their graphs must be in G
sa
. It is known [6] that for the set of n × n
stochastic matrices S
1
,S
2
,...,S
p
γ(S
p
···S
2
S
1
)=γ(S
p
) ◦···◦γ(S
2
) ◦ γ(S
1
).(12)
We will make use of the fact that for any two n ×n stochastic matrices S
1
and S
2
,
φ(S
2
S
1
) ≥ φ(S
2
)φ(S
1
),(13)
where for any nonnegative matrix M , φ(M) denotes the smallest nonzero element of
M. To prove that this is so, note first that any stochastic matrix S can be written as
S = φ(S)
¯
S, where
¯
S is a nonzero matrix whose nonzero entries are all bounded below
by 1; moreover, if S =
φ(S)
S, where
φ(S)isanumberand
S is also a nonzero matrix
whose nonzero entries are all bounded below by 1, then φ(S) ≥
φ(S). Accordingly,
write S
i
= φ(S
i
)
¯
S
i
,i∈{1, 2}, where each
¯
S
i
is a nonzero matrix whose nonzero entries
are all bounded below by 1. Since S
2
S
1
= φ(S
2
)φ(S
1
)
¯
S
2
¯
S
1
and S
2
S
1
is nonzero,
¯
S
2
¯
S
1
must be nonzero as well. Moreover, the nonzero entries of
¯
S
2
¯
S
1
must be bounded
below by 1 because the product of any two n × n matrices with all nonzero entries
bounded below by 1 must be a matrix with the same property. Therefore φ(S
2
S
1
) ≥
φ(S
2
)φ(S
1
) as claimed. An important consequence of (13) is that for any set of
stochastic matrices S
1
,S
2
,...,S
m
for which each φ(S
i
) is bounded below by a positive
number b,
φ(S
m
···S
1
) ≥ b
m
.(14)
Our goal is now to use these facts to derive an explicit convergence rate for the
situation considered by Theorem 1. We will do this in two different ways. The first
way is based on properties of stochastic matrices with strongly rooted graphs.
3.1. Strongly rooted graphs. Let F(q) denote the set of all products of q
flocking matrices whose corresponding sequences of q graphs are each jointly rooted.
In view of (12), each matrix in F(q) must have a rooted graph in G
sa
. In other words,
each matrix in F(q) has a rooted graph and is a product of q flocking matrices.
Since the set of all flocking matrices is finite, so is F(q). It is shown in [6] that the
composition of any set of at least (n − 1)
2
rooted graphs in G
sa
is strongly rooted.
This and (12) imply that the product of any (n − 1)
2
matrices in F(q) must have a
strongly rooted graph in G
sa
.Thusifwesetm =(n − 1)
2
and write (F(q))
m
for the
set of all products of m matrices from F(q), then each matrix in (F(q))
m
must have a
