Reaching a Consensus in a Dynamically Changing Environment -
Convergence Rates, Measurement Delays and Asynchronous Events
∗
M. Cao
Yale Univesity
A. S. Morse
Yale University
B. D. O. Anderson
Australia National University and
National ICT Australia
April 5, 2006
Abstract
This paper uses recently established properties of compositions of directed graphs together
with results from the theory of non-homogeneous 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 con-
structions 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.
1 Introduction
In a recent paper [2] 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 [2] to address several
issues related to the well-known Vicsek consensus problem [3] which have either not been considered
before, or have only been partially resolved.
The paper begins with a brief review in Section 2, of the basic leaderless consensus problem
treated in [2, 4, 5]. Section 3 exploits the connection between “neighbor-shared” graphs and the
elegant theory of “scrambling matrices” found in the literature on non-homogeneous Markov chains
[6, 7] 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
∗
A preliminary version of this work can be found in [1]. The research of the first two authors was supported by
the US Army Research Office, the US National Science Foundation and by a gift from the Xerox Corporation. The
research of the third author was supported by National ICT Australia, which is funded by the Australian Governments
Department of Communications, Information Technology and the Arts and the Australian Research Council through
the Backing Australias Ability initiative and the ICT Centre of Excellence Program.
1
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 [8] under
which consensus can be achieved in the face of measurement delays. A comparison is made between
the results of [8] and 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 that 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 [9, 10]. This enables us to bring to bear results
derived earlier in [2] 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
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 [2], 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 either in or on a closed
disk of pre-specified radius r
i
centered at agent i’s current position. In the sequel 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)
X
j∈N
i
(t)
θ
j
(t)
(1)
2
where t is a discrete-time index taking values in the non-negative 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 V = {1, 2, . . . n} and arc set A(N(t)) ⊂ V × V which is defined in such a
way so that (i, j) is an arc or directed edge from i to j just in case agent i is a neighbor of agent j
at time t. Thus N(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 ∈ G if (i, j) is and arc in G.
2.2 Heading Update Rule
The set of agent heading update rules defined by (1) can be written in state form. Toward this
end, for each graph N ∈ G
sa
define the flocking matrix
F = D
−1
A
0
(2)
where A
0
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
]
0
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 [2]. 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. Thus i is a root of G, 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. By a
rooted graph G is meant 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 that
graph G
q
◦G
p
with the same vertex set V and arc set defined such that (i, j) is an arc of G
q
◦G
p
if for
some vertex k, (i, k) is an arc of G
p
and (k, j) is an 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
3
G
q(k+1)
, . . . , G
qk+1
, k ≥ 0, is jointly rooted. The main result on leaderless consensus in [2] is as
follows.
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 describe both. To do this we will make use of certain
structural properties of the F . As defined, each F is square and non-negative, where by a non-
negative matrix is meant a matrix whose entries are all non-negative. 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 [13]. 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 the sequel we write M ≥ N whenever M − N is a non-negative matrix. We also write
M > N whenever M − N is a positive matrix where by a positive matrix is meant a matrix with
all positive entries. For any non-negative 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 sub-
multiplicative. 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 [2] we write bMc and
dMe for the 1 × m row vectors whose jth entries are the smallest and largest elements respectively,
of the jth column of M . Note that bM c is the largest 1 × m non-negative row vector c for which
M − 1c is non-negative and that dM e is the smallest non-negative row vector c for which 1c − M
is non-negative. Note in addition that for any n × n stochastic matrix S, one can write
S = 1bSc + b|S|c and S = 1dSe − d|S|e (5)
where b|S|c and d|S|eare the non-negative matrices
b|S|c = S − 1bSc and d|S|e = 1dSe − S (6)
respectively. Moreover the row sums of b|S|c are all equal to 1 − bSc1 and the row sums of d|S|e are
all equal to dSe1 − 1 so
||b|S|c|| = 1 − bSc1 and ||d|S|e|| = dSe1 − 1 (7)
In the sequel we will also be interested in the matrix
bd|S|ec = b|S|c + d|S|e (8)
1
This section summarizes and extends some of the key findings of [12].
4
This matrix satisfies
bd|S|ec = 1(dSe − bSc) (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 at 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 [2], 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 non-negative constants b and
λ with λ < 1, such that
||(S
j
· · · S
1
) − 1c|| ≤ bλ
j
, j ≥ 1 (10)
The following fact is proved in [2].
Proposition 1 If an infinite sequence of stochastic matrices S
1
, S
2
, . . . satisfies
||b|S
j
· · · S
1
|c|| ≤
¯
bλ
j
, j ≥ 0 (11)
for some non-negative 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 vertex set {1, 2, . . . , n}
and arc set defined is such a way so that (i, j) is an arc of γ(S) from i to j just in case the jith entry
of S is non-zero. 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 [2]
that for 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 non-negative matrix M , φ(M ) denotes the smallest non-zero element of M . To prove
that this is so note first that any stochastic matrix S can be written at S = φ(S)
¯
S where
¯
S is a
non-zero matrix whose non-zero entries are all bounded below by 1; moreover if S =
b
φ(S)
b
S where
b
φ(S) is a number and
b
S is also a non-zero matrix whose non-zero entries are all bounded below
by 1, then φ(S) ≥
b
φ(S). Accordingly, write S
i
= φ(S
i
)
¯
S
i
, i ∈ {1, 2} where each
¯
S
i
is a non-zero
matrix whose non-zero 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 non-zero,
¯
S
2
¯
S
1
must be non-zero 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 non-zero 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)
5