2010 American Control Conference
Marriott Waterfront, Baltimore, MD, USA
June 30-July 02, 2010
ThB02.3
A Complete Characterization of a Class of
Robust Linear Average Consensus Protocols
Randy A. Freeman, Thomas R. Nelson, and Kevin M. Lynch
Abstract— We provide a set of verifiable necessary and sufficient conditions for a member of a broad class of linear discretetime consensus protocols to guarantee the robust convergence
of the outputs of the networked dynamics to the correct average
consensus value.
I. I NTRODUCTION
We consider the average consensus problem: each member
of a network of agents has some assigned input value, and it
must calculate the average of the inputs of all agents. We seek
solutions which are decentralized (agents can only communicate with their immediate neighbors), scalable (each agent’s
memory, computation, and communication requirements are
independent of the total number of agents in the network),
asymptotically correct (if the inputs and the network are both
constant, then each agent’s estimate of the group average
converges to the right value with zero steady-state error), and
robust (temporary faults in communication or computation
have no bearing on the final steady-state result).
There has been much work on this type of problem in
the past several years, and we refer the reader to [1], [2],
[3], [4], [5] for details. In many cases, the agent inputs are
assigned as initial states in a networked dynamic system,
and as time progresses these states converge to the average
consensus value. However, such solutions are inherently nonrobust: a fault which causes a single incorrect state update
at a single time instant can cause the network to converge to
the incorrect consensus value. To avoid this problem, we
consider dynamic consensus schemes in which the agent
inputs are assigned not as initial states but as actual inputs
into the networked dynamic system [6], [7]. For robustness
in this case, the final value of the dynamic state should be the
correct average consensus value (at least for constant inputs
and constant networks), regardless of the value of the initial
state. In this paper we provide a complete characterization
of a class of linear discrete-time protocols which possess
this robustness property. The characterization of nonlinear
schemes, such as the consensus propagation scheme of [8],
is a topic of future research.
We begin in Section II with a review of relevant results
from graph theory, we present some results on a particularly
useful decentralized graph weighting scheme in Section III,
and we present our main result as Theorem 5 in Section IV.
This work was supported in part by NSF grant ECS-0601661 and by
the Office of Naval Research. The authors are with the Department of
Electrical Engineering and Computer Science (Freeman and Nelson) and the
Department of Mechanical Engineering (Lynch), Northwestern University,
Evanston, IL 60208, USA, [email protected],
[email protected],
[email protected].
978-1-4244-7425-7/10/$26.00 ©2010 AACC
II. L APLACIANS OF WEIGHTED DIGRAPHS
For each finite set S, we let S S denote the product S×S
minus its diagonal, so that (s,t) ∈ S S when s,t ∈ S and
s 6= t. For K = R or K = C, we let K S denote the free Kvector space over S, with the canonical basis given by the
indicator functions es of elements s ∈ S (where es (t) = 1
when s = t and es (t) = 0 when s 6= t). We endow K S with
the inner product hx, yi , ∑s∈S x(s) y∗(s) (where ∗ denotes
the conjugate transpose) so that the canonical basis is orthonormal. We let 0, 1 ∈ K S denote the constant functions
on S with values 0 and 1, respectively. We regard RS as
an ordered vector space by adopting the product order: for
x, y ∈ RS , we say x 6 y when x(s) 6 y(s) for all s ∈ S. Thus
the positive cone of RS is the set of mappings from S to the
nonnegative reals. We let L (K S ) denote the space of linear
maps from K S to K S , and we let Mx = M(x) ∈ K S denote the
value of a mapping M ∈ L (K S ) at a point x ∈ K S . We let I
denote both the identity map I ∈ L (K S ) and the identity
matrix, and we adopt the convention that M 0 = I for any
mapping M ∈ L (K S ) (or any square matrix M), including
the zero mapping. Finally, we let K ∅ denote the singleton
vector space of dimension zero.
A vertex set V is any nonempty finite set, and we let U(V)
and W(V) denote the sets of unweighted and positively
weighted simple digraphs on V, respectively. Specifically, we
identify U(V) with the power set of V V, and we say that
an unweighted digraph G ∈ U(V) has an arc from vertex a to
vertex b when (a, b) ∈ G. Also, we identify W(V) with the
positive cone of the vector space RVV , and the arc set of
a weighted digraph G ∈ W(V) is the underlying unweighted
digraph Arc(G) ∈ U(V) given by the support of G,
namely,
Arc(G) , (a, b) ∈ V V : G(a, b) > 0 .
(1)
Here G(a, b) represents the weight on an arc (a, b) ∈ Arc(G).
We regard the mapping Arc(·) as a surjection from W(V)
to U(V), and we refer to any right inverse of this surjection
as a weighting scheme on V.
in (a) and
Given a digraph G ∈ U(V) or G ∈ W(V), we let NG
out
NG (a) respectively denote the sets of in- and out-neighbors
of a vertex a ∈V (not including a itself). The unweighted inout
and out-degrees dGin (a) and
dG (a)outare
out in
in
dG (a) , NG (a) ,
dG (a) , NG
(a) .
(2)
We let Src(G) ⊂ V and Sink(G) ⊂ V denote the sets of
sources and sinks of G, respectively: a ∈ Src(G) when
dGin (a) = 0 and a ∈ Sink(G) when dGout (a) = 0. A weighted
digraph G ∈ W(V) has weighted in- and out-degrees win
G (a)
and wout
G (a) given by
win
wout
(3)
G (a) , ∑ G(b, a) ,
G (a) , ∑ G(a, b) .
b6=a
b6=a
Using terminology from [4], [9], we say that G ∈ W(V) is
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out
balanced when win
G (a) = wG (a) for all a ∈V and symmetric
when G(a, b) = G(b, a) for all (a, b) ∈ V V. All symmetric
weighted digraphs are balanced, and if G is balanced then
Src(G) = Sink(G). In what follows, the term “digraph” will
refer to a weighted digraph unless otherwise indicated.
We define the Laplacian of a digraph G ∈ W(V) to be the
V
linear map
∆G ∈ L (R ) given by
∆Gx (a) = (∆G)(x) (a) , ∑ G(a, b) x(a) − x(b) (4)
b6=a
for each x ∈ RV and a ∈ V. The map ∆ : W(V) → L (RV )
is injective, which means a digraph is characterized by its
Laplacian. We regard ∆G as a member of L (CV ) when
discussing its eigenvalues and eigenvectors, and we let
σ (G) ⊂ C denote the spectrum of ∆G. One can show that
V
the adjoint ∆∗ G
∈ L (R ) of the Laplacian is given by
∆∗Gx (a) = ∑ G(a, b)x(a) − G(b, a)x(b)
(5)
b6=a
for each x ∈ RV and a ∈ V, and it is clear that a digraph
is symmetric if and only if its Laplacian is self-adjoint.
We say that a digraph G is diagonalizable when ∆G is
diagonalizable, that is, when there is a basis of CV consisting
of eigenvectors of ∆G. Clearly all symmetric digraphs are
diagonalizable.
It follows from (4) and (5) that ∆G1 = 0 for every digraph
G ∈ W(V), and moreover ∆∗ G1 = 0 if and only if G is
balanced. Therefore rank(∆G) 6 |V| − 1 for every digraph
G ∈ W(V), and we say that G has maximal rank when
rank(∆G) = |V| − 1. To characterize the rank of ∆G, we need
the concept of a quotient digraph. Every partition P of the
vertex set V induces a quotient digraph GP ∈ W(P), where
GP (A, B) , ∑ G(a, b)
(6)
(a,b) ∈ A×B
for all A, B ∈ P P. It is clear from flow conservation that
any quotient of a balanced digraph is itself balanced. Recall
that a digraph is weakly connected when any distinct pair
of its vertices lie on some undirected path and strongly
connected when any distinct pair of its vertices lie on some
directed cycle. The acyclic quotient of G is the quotient Gaq
induced by the partition of V into its strongly connected
components. The acyclic quotient of a balanced digraph is
totally disconnected (every vertex is both a source and a
sink). Hence a balanced digraph is strongly connected if
and only if it is weakly connected, which means the phrase
“balanced and connected digraph” is unambiguous. It is
shown in [10] that
rank(∆G) + Sink(Gaq ) = |V| .
(7)
Equation (7) is a direct consequence of an oft-rediscovered
theorem on determinants [11]. Thus a digraph has maximal
rank if and only if its acyclic quotient has a single sink.
In particular, every strongly connected digraph has maximal
rank, which was proved independently as Theorem 1 in [4].
Also, it is clear that if a digraph is balanced, then it has
maximal rank if and only if it is connected.
For each vertex set V, the set of balanced digraphs
in W(V), the set of symmetric digraphs in W(V), and the set
of maximal-rank digraphs in W(V) are all convex. However,
if |V| > 3, then the set of balanced, connected, diagonalizable
digraphs in W(V) is not convex.
If we order the elements of a vertex set V, then we
can label the vertices as V = {1, . . . , n} for n = |V| and
regard the canonical basis for RV as an ordered basis. The
Laplacian of each digraph G ∈ W(V) has the following
matrix representation L ∈ Rn×n with respect to this ordered
canonical basis:
(
wout
if i = j
G (i)
Li j =
(8)
−G(i, j) if i 6= j .
We will use this representation (8) in some of our proofs.
III. I NVERSE OUT- DEGREE WEIGHTS
For each x ∈ C, we let Dx ⊂ C denote the closed unit disc
centered at x, and we note that rDx for r ∈ C is the closed
disc centered at rx with radius |r|.
Consider a complete digraph G ∈ W(V) with equal weights
(there is a constant c > 0 such that G(a, b) = c for all a 6= b).
Then the Laplacian of G has a single eigenvalue at zero, and
all other eigenvalues lie at c|V|. Thus without knowledge
of an upper bound on the digraph order |V|, there is no
way to choose c so that all digraphs whose weights are
all equal to c have Laplacians with eigenvalues guaranteed
to lie in some fixed disc rDx . Because the stability region
for discrete-time, linear, time-invariant systems is bounded
(being the interior of D0 ), it will be difficult to design
discrete-time algorithms involving equal-weight Laplacians
which are stable for digraphs of arbitrary order. Instead, we
seek weighting schemes which guarantee that all Laplacian
eigenvalues lie in a known bounded region, regardless of
the digraph order. Furthermore, to be useful for decentralized algorithms, the resulting weights should be computable
using information only from immediate neighbors. We now
propose such a weighting scheme which is similar to the
local-degree weighting scheme discussed in [3].
Given an unweighted digraph G ∈ U(V) on a vertex set V,
the associated inverse out-degree digraph IOD(G) ∈ W(V) is
given by the weights
(
1
when (a, b) ∈ G
out
out
(9)
IOD (G)(a, b) = dG (a) + dG (b)
0
when (a, b) 6∈ G .
This mapping IOD(·) is a right inverse of Arc(·) and is thus
a weighting scheme. Its image IOD(U(V)) is the set of all
digraphs on V having IOD weights.
Theorem 1: If a digraph G ∈ W(V) belongs to the convex
hull of IOD(U(V)), then σ (G) ⊂ D1 . If G itself belongs to
IOD ( U (V)), then σ (G) ⊂ D0 ∩ D1 .
The proof (omitted) follows the main idea in [12]. Note
that there exist digraphs, even symmetric ones, which belong
to the convex hull of IOD(U(V)) but whose Laplacians have
eigenvalues outside of D0 . Also, we conjecture that the
eigenvalue estimates in Theorem 1 can be improved:
Conjecture 2: If G ∈ IOD(U(V)) then σ (G) ⊂ 21 D1 .
If this conjecture is true, then 12 D1 is the smallest convex
region in the complex plane containing all eigenvalues of all
digraphs having IOD weights. Indeed, let G ∈ U(V) consist
of a single directed cycle with n = |V| vertices. Then it is
straightforward to show that the Laplacian of IOD(G) has
eigenvalues at 21 − 12 e2π jk/n for 1 6 k 6 n, which are equally
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spaced around the boundary of 12 D1 .
We next show that IOD weights are particularly useful
when digraphs are drawn from a “symmetric” probability
distribution. Let V be a vertex set, and let p : U(V) → [0, 1]
be a probability distribution (mass function) on the set U(V)
of unweighted digraphs. The probability digraph Π p ∈ W(V)
associated with p is given by the weights
Π p (a, b) = Prob (a, b) ∈ G = ∑ χG (a, b) p(G) , (10)
G∈ U(V)
where χG denotes the indicator function of G. We say
that
p is an
arc-independent distribution when the events
(a, b) ∈ G in (10) are mutually independent for distinct
ordered pairs of vertices (a, b); in this case the probability
digraph Π p completely characterizes the distribution p.
For a pair of unweighted digraphs G1 , G2 ∈ U(V), we write
G1 ↔ G2 whenthere exist
vertices a, b ∈ V such that
G1 \ G2 = (a, b)
and G2 \ G1 = (b, a) .
(11)
Thus G1 ↔ G2 when G1 and G2 have the same arcs except
for a single pair of vertices between which G1 and G2 have
arcs in opposite directions. We say that p is an arc-symmetric
distribution when G1 ↔ G2 implies p(G1 ) = p(G2 ). If p is
arc-symmetric, then the associated probability digraph Π p
will be a symmetric digraph, but the converse is not true
in general. However, if p is arc-independent, then the arcsymmetry of p is equivalent to the symmetry of Π p .
Let W : U(V) → W(V) be a weighting scheme on V, and
let p be a probability distribution over U(V). The expected
digraph E p [W] ∈ W(V) for W and p is given
by the weights
E p [W](a, b) = ∑ W(G)(a, b) p(G) .
(12)
G∈ U(V)
We note that for the unity equal weighting scheme in which
W(G)(·, ·) = χG (·, ·) for each G, the expected digraph E p [W]
coincides with the probability digraph Π p .
Theorem 3: If the distribution p is arc-symmetric, then
the expected digraph E p [IOD] is symmetric.
The proof of Theorem 3 (omitted) relies on the fact
that the IOD weights in (9) are a function of the sum of
the neighboring out-degrees. Indeed, if we were instead to
use some other weighting scheme, such as the local-degree
weighting scheme
( in [3] given by
1
when (a, b) ∈ G
out
out
(13)
W(G)(a, b) = max{dG (a), dG (b)}
0
when (a, b) 6∈ G ,
then the arc-symmetry of the distribution p need not guarantee that the expected digraph is symmetric or even balanced,
even when p is arc-independent.
IV. ROBUST AVERAGE CONSENSUS
A. Convergent matrices
A square complex matrix M is convergent when its power
sequence {M k }∞
k=1 converges to a finite constant matrix as
k → ∞.1 It is clear from the Jordan decomposition that M is
convergent if and only if all Jordan blocks associated with an
eigenvalue at λ = 1 are of size one, and all other eigenvalues
lie in the interior of D0 . Equivalently, M is convergent if and
only if it is similar to a block diagonal matrix diag(N, I) for
some square matrix N with ρ(N) < 1, where ρ(·) denotes
the spectral radius (neither N nor I need be present).
Lemma 4: Suppose M ∈ Cn×n and b ∈ Cn are given, and
consider the discrete-time dynamics
x(k + 1) = Mx(k) + b
(14)
with x(k) ∈ Cn . If M is convergent and there exists an equilibrium state x̄ = M x̄ + b, then for every initial state x(0), the
trajectory x(k) converges exponentially to some equilibrium
as k → ∞. If M is convergent and there does not exist an
equilibrium state, then the trajectory diverges from every
initial state, that is, |x(k)| → ∞ as k → ∞. Finally, if M is
not convergent, then there is some initial state from which
the trajectory does not converge.
B. Average consensus
Let G ∈ W(V) be a digraph on a vertex set V, and suppose
each vertex a ∈ V is an agent which receives a scalar input
signal ua (k) ∈ R at each discrete time step k and passes it
through a filter to produce a scalar output signal ya (k) ∈ R.
In computing its output ya (k), agent a also uses information
it receives from its out-neighbors at the current time step (so
that information flows against the arcs of G). The goal is
for each ya (k) to be an estimate of the average of all agent
inputs at time step k. We let u(k), y(k) ∈ RV be the vectors
of these inputs and outputs so that u(k)(a) = ua (k) and
y(k)(a) = ya (k) for each a ∈ V, and we write the estimation
error e(k) ∈ RV as
1 u(k), 1 · 1 .
(15)
e(k) = y(k) −
|V|
If the inputs ua (k) are changing rapidly with k and if the
diameter of G is large, then we cannot hope to keep the
error small by communicating only with neighbors (by the
time information about a particular input reaches a distant
agent, such information is no longer relevant). Thus we can
achieve our goal of keeping the error small only when the
inputs are varying slowly relative to the diameter of G.
We seek scalable filters for calculating the local estimates ya (k): the memory, computation, and communication
requirements for each agent should depend only on its outdegree, not on the total number of agents in the network. A
well-studied example of such a filter is given by the linear
update rule
ya (k + 1) = ya (k) − ∑ G(a, b) ya (k) − yb (k) .
(16)
b6=a
It is well known that if G is balanced, connected, and such
that I − ∆G is convergent, then y(k) converges to the vector
1 y(0), 1 · 1
(17)
ȳ =
|V|
as k → ∞ [4]. Thus if we have a constant input vector
u(k) ≡ ū and we choose the initial estimate y(0) = ū, then the
estimation error (15) will converge to zero as k → ∞. Unfortunately, this filter is not robust to errors in the calculation
of the update rule (16): if just one of the components ya (k)
is updated incorrectly at just one time instant, then the entire
estimate y(k) will converge to the wrong value. We can
characterize this lack of robustness in a different way by
examining a state-space version of the
filter (16):
xa (k + 1) = xa (k) − ∑ G(a, b) ya (k) − yb (k)
(18)
also refer to linear mappings in L (·) as convergent when their matrix
representations are convergent.
1 We
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b6=a
ya (k) = xa (k) + ua (k) ,
(19)
where xa (k) ∈ R is a scalar filter state variable for agent a.
One can show that, under a constant input u(k) ≡ ū and
with G as above, the output y(k) converges to
1 ȳ =
x(0) + ū, 1 · 1
(20)
|V|
as k → ∞. Here
the dependence
of ȳ on the initial state x(0)
is apparent: if x(0), 1 6= 0, then the estimate y(k) will converge to the wrong value. Such incorrect state initializations
might come from errors in inter-agent communication, the
addition or removal of agents, or incorrect local calculations.
In this paper, we seek alternative linear filters for which
the estimation error (15) converges to zero regardless of the
value of the initial filter state x(0).
C. Polynomial linear protocols
If we let x(k) ∈ RV denote the vector of filter states in the
update rule (18), so that x(k)(a) = xa (k) for each a ∈ V, then
we can combine the filter equations (18)–(19) for each agent
into the global rule
x(k + 1) = [I − ∆G]x(k) − ∆Gu(k)
(21)
y(k) = x(k) + u(k) .
(22)
We can generalize the structure of this filter by allowing the
internal state xa (k) of each agent to be a vector rather than a
scalar, and by allowing the coefficient matrices in the statespace description to be general polynomial functions of the
digraph Laplacian. Specifically, a polynomial linear protocol
is a collection Σ = [A(X), B(X),C(X), D(X)], where
`
`
A(X) , ∑ Ai X i
B(X) , ∑ Bi X i
(23)
C(X) , ∑ Ci X i
D(X) , ∑ Di X i
(24)
i=0
`
i=0
i=0
`
i=0
are polynomials in the formal symbol X with matrix coefficients Ai ∈ R p×p , Bi ∈ R p×q , Ci ∈ Rm×p , and Di ∈ Rm×q . Here
` > 0 denotes the degree of Σ, and p > 1, q > 1, and m > 1
denote its state, input, and output dimensions, respectively
(and we refer to p as the dimension of the protocol). We
regard such a system Σ as a protocol or a template as it does
not by itself constitute a dynamic system. Rather, given a
linear map M ∈ L (RS ) for some finite set S, the protocol Σ
generates the discrete-time linear system Σ(M) given by
x(k + 1) = A(M)x(k) + B(M)u(k)
(25)
y(k) = C(M)x(k) + D(M)u(k) ,
(26)
with state x(k) ∈ R p ⊗ RS , input u(k) ∈ Rq ⊗ RS , and output
y(k) ∈ Rm ⊗ RS , where
`
`
A(M) , ∑ Ai ⊗ M i
B(M) , ∑ Bi ⊗ M i
(27)
C(M) , ∑ Ci ⊗ M i
D(M) , ∑ Di ⊗ M i
(28)
i=0
`
i=0
i=0
`
i=0
(here ⊗ denotes the tensor product). In this manner, a
protocol Σ will generate the dynamics Σ(∆G) for any digraph
G ∈ W(V) on a vertex set V. For example, the dynamics
in (21)–(22) are generated by the SISO protocol of degree
` = 1 and dimension p = 1 given by the matrix coefficients
A0 = C0 = D0 = 1, A1 = B1 = −1, and B0 = C1 = D1 = 0.
Given a digraph G ∈ W(V), we can implement the dynamics Σ(∆G) using a local filter on each agent in which the
agent receives information from its out-neighbors. Indeed,
we can write x(k), u(k), and y(k) uniquely as
x(k) = ∑ xa (k) ⊗ ea ,
xa (k) ∈ R p ,
(29)
a∈V
u(k) =
∑ ua (k) ⊗ ea ,
ua (k) ∈ Rq ,
(30)
ya (k) ∈ Rm ,
(31)
a∈V
y(k) =
∑ ya (k) ⊗ ea ,
a∈V
where {ea }a∈V are the canonical basis vectors for RV . Then
each agent a ∈ V implements
the filter
xa (k)
a
z0 (k) =
∈ R p+q
(32)
ua (k)
zai (k) = ∑ G(a, b) zai−1 (k) − zbi−1 (k) for 1 6 i 6 ` (33)
b6=a
` xa (k + 1)
A Bi a
=∑ i
z (k) .
(34)
ya (k)
Ci Di i
i=0
It is straightforward to show that the combination of these
local filters (32)–(34) results in the global dynamics (25)–
(26) with M = ∆G.
We see from (33) that each time step k requires ` stages
of inter-agent communication: first, the za0 variables are
communicated to obtain the za1 variables, which are then
communicated to obtain the za2 variables, and so on. Thus at
each time step, each agent transmits at most `(p + q) scalar
values to its in-neighbors (the actual number could be smaller
if any of the Ai , Bi , Ci , or Di are zero). The same information
goes to each in-neighbor, which means the transmission is
possible using a simple local broadcast.
D. Robust average consensus protocols
Let Σ = [A(X), B(X),C(X), D(X)] be a SISO polynomial
linear protocol with dimension p > 1, and let G ∈ W(V) be
a digraph on a vertex set V. Suppose ū ∈ RV is a constant
vector of agent inputs, and suppose x̄ ∈ R p ⊗ RV and ȳ ∈ RV
are an equilibrium state and output (respectively) for the
dynamics Σ(∆G) under
this constant
input:
I − A(∆G) x̄ = B(∆G)ū
(35)
ȳ = C(∆G)x̄ + D(∆G)ū .
(36)
We seek protocols for which, under appropriate assumptions
on the digraph G, the state always converges to such an
equilibrium value x̄ and the only possible corresponding
value for ȳ is the average consensus value
1 ū, 1 · 1 .
(37)
ȳ =
|V|
Definition 1: Let G ⊂ W(V) be a family of digraphs on a
vertex set V. A SISO polynomial linear protocol Σ achieves
robust average consensus over G when for every G ∈ G, the
dynamics Σ(∆G) are such that for any initial state x(0) and
any constant input u(k) ≡ ū, the state x(k) converges to a
constant and the output y(k) converges to ȳ in (37) as k → ∞.
Before we state our main result on polynomial linear
protocols, we first introduce some notation. If we evaluate
the formal polynomials in (23)–(24) at a complex scalar
X = µ ∈ C, then A(µ), B(µ), C(µ), and D(µ) become
complex matrices of appropriate dimensions. Given these
complex matrices, we define the complex
scalar
+
H(µ) , C(µ) I − A(µ) B(µ) + D(µ) ,
(38)
where (·)+ denotes the Moore-Penrose pseudoinverse.
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Theorem 5: Let Σ = [A(X), B(X),C(X), D(X)] be a SISO
polynomial linear protocol, and let G ⊂ W(V) be a nonempty
collection of diagonalizable digraphs on a vertex set V with
|V| > 2. Then Σ achieves robust average consensus over G if
and only if all of the following are true:
(i) A(µ) is convergent for all µ ∈ σ (G).
(ii) B(µ) ∈ Col I − A(µ) for all µ ∈ σ (G),
(iii) C∗(µ) ∈ Col I − A∗(µ) for all µ ∈ σ (G),
(iv) H(0) = 1, where H(·) is from (38),
(v) H(µ) = 0 for all nonzero µ ∈ σ (G), and
(vi) each digraph in G is balanced and connected.
Using this theorem as a sufficient condition on a protocol Σ requires knowledge of a region in the complex plane
containing the spectrum σ (G) of each G ∈ G. For example,
suppose G is the collection of all balanced, connected, and
diagonalizable digraphs in IOD(U(V)); then we can conclude
from Theorem 1 that Σ achieves robust average consensus
over G if conditions (i)–(v) hold for all µ ∈ D0 ∩ D1 .
As we will see in the proof of Theorem 5, condition (i)
is equivalent to A(∆G) being convergent for all G ∈ G,
which from Lemma 4 is necessary and sufficient for the
dynamics Σ(∆G) to converge to an equilibrium from any
initial state and for any constant input vector (provided an
equilibrium exists). Conditions (ii) and (iii) are equivalent
to the statement that, for each µ ∈ σ (G), any eigenvalue
of A(µ) at λ = 1 is both uncontrollable through B(µ) and
unobservable through C(µ).
+ Conditions (iv) and (v) imply
that the matrix I − A(µ) is discontinuous as a function
of µ at µ = 0, which can only happen when I − A0 is
singular, namely, when A0 has an eigenvalue at λ = 1;
hence this eigenvalue must be uncontrollable through B0
and unobservable through C0 . This is clearly violated by the
protocol (21)–(22) for which A0 = C0 = 1. In fact, there is no
linear consensus protocol having degree ` = 1 and dimension
p = 1 which can achieve robust average consensus. Indeed,
if there were such a protocol, then necessarily A0 = D0 = 1
and B0 = C0 = 0. If also A1 = 0, then A(µ) = 1 for all µ,
and we conclude from (v) that D0 + µD1 = 0 for µ > 0,
which implies D0 = 0, a contradiction. Therefore A1 6= 0,
+
which means 1 − A(µ) = −(µA1 )−1 for µ > 0 and thus
H(µ) = D0 + µ(D1 −C1 B1 /A1 ) for µ > 0. But now (v) again
implies D0 = 0, a contradiction.
Fortunately, there do exist linear consensus protocols of
higher degree or higher dimension that achieve robust average consensus. Consider the protocol of degree ` = 1 and
dimension
p = 2 given by 1−γ 0
−k p ki
γ
A0 =
, A1 =
, B0 =
,
(39)
0
1
−ki 0
0
C0 = 1 0 ,
B1 = C1T = 0 ,
D0 = D1 = 0 , (40)
where γ, k p , and ki are positive gains. This is a discrete-time
version of the PI estimator studied in [7]. If we choose the
gains so that the matrix A(µ) = A0 + µA1 is convergent for
all µ ∈ σ (G), then it is straightforward to verify that this
protocol satisfies conditions (i)–(v) in Theorem 5. Another
example is the linear consensus protocol of degree ` = 2 and
dimension p = 1 given by
A0 = C1 = D0 = 1 ,
A2 = B1 = −1 ,
(41)
A1 = B0 = B2 = C0 = C2 = D1 = D2 = 0 ,
(42)
which yields A(µ) = 1 − µ 2 , B(µ) = −µ, C(µ) = µ, and
D(µ) = 1. If G is such that µ 2 ∈ D1 for all µ ∈ σ (G), then
this protocol satisfies conditions (i)–(v) in Theorem 5.
Although we state and prove Theorem 5 only for the case
of diagonalizable digraphs, we conjecture that the results
hold also for the non-diagonalizable case.
E. Proof of Theorem 5
We begin with two lemmas about Kronecker products;
their straightforward proofs are omitted.
Lemma 6: Given n matrices A1 , . . . , An ∈ C pi ×qi and n
vectors b1 , . . . , bn ∈ C pi , the equation
A1 ⊗ · · · ⊗ An x = b1 ⊗ · · · ⊗ bn
(43)
admits a solution x ∈ Cq , where q = ∏i qi , if and only if
either (i) bi = 0 for some i, or (ii) bi ∈ Col(Ai ) for each i.
Lemma 7: Let b1 , . . . , bn ∈ Cm be an independent collection of vectors, and let A1 , . . . , An ∈ C p×q each have rank q.
Then the matrix
A , A1 ⊗ b1 . . . An ⊗ bn ∈ Cmp×nq
(44)
has rank nq.
Let n = |V| > 2, suppose G ∈ G, and let L ∈ Rn×n denote
the matrix representation (8) of the Laplacian ∆G with respect to some order on V. Because L is diagonalizable (by assumption), it has n independent eigenvectors w1 , . . . , wn ∈ Cn
with corresponding eigenvalues λ1 , . . . , λn ∈ C, and we define
the nonsingular matrix W , [w1 . . . wn ]. For j ∈ {1, . . . , n},
let J j ∈ C p×p be a Jordan form of A(λ j ) with A(λ j ) =
Pj J j Pj−1 for some invertible matrix Pj ∈ C p×p . The matrix
J , diag(J1 , . . . , Jn ) is a Jordan form of A(L); indeed, the
np × np complex matrix
P , P1 ⊗ w1 . . . Pn ⊗ wn ,
(45)
which is invertible by Lemma 7, is such that A(L) = PJP−1 .
We conclude that A(L) is convergent if and only if A(λ j )
is convergent for each j, and it follows from Lemma 4
that condition (i) of Theorem 5 is necessary and sufficient
for the dynamics Σ(∆G) to converge under constant inputs
from every initial state and for every G ∈ G (provided these
dynamics admit an equilibrium state).
We define the polynomial F(X) , I − A(X), and for each
j ∈ {1, . . . , n} we let F(λ j ) = Φ j S j Ψ∗j be a singular value decomposition so that S j ∈ R p×p is diagonal with nonnegative
real entries listed in descending order and Φ∗j Φj = Ψ∗j Ψj = I.
We define the np ×np complex matrices Φ , Φ1 ⊗ w1 . . . Φn ⊗ wn
(46)
Ψ , Ψ1 ⊗ w1 . . . Ψn ⊗ wn ,
(47)
which are both invertible by Lemma 7. Next we define the
matrix Γ , [WW ∗ ]−1 so that W ∗ ΓW = I, and we observe that
∗
Φ∗ (I ⊗ Γ)Φ =
Ψ (I ⊗ Γ)Ψ = I. We next computeF(L)Ψ:
F(L)Ψ = F(λ1 )Ψ1 ⊗ w1 . . . F(λn )Ψn ⊗ wn
= Φ1 S1 ⊗ w1 . . . Φn Sn ⊗ wn
= (Φ1 ⊗ w1 )S1 . . . (Φn ⊗ wn )Sn = ΦS , (48)
3202
where S is the block diagonal matrix S , diag(S1 , . . . , Sn ).
We introduce the inner product h·, ·iΓ on Cnp by defining
hx, yiΓ , y∗ (I ⊗ Γ)x
(49)
np
for all x, y ∈ C . Thus Φ and Ψ have orthonormal columns
with respect to the inner product (49), and from (48) we have
F(L) = ΦSΨ−1 = ΦSΨ∗(I ⊗ Γ) .
(50)
For each j ∈ {1, . . . , n}, we label the columns of Φj and Ψj
as φ1 j , . . . , φ p j ∈ C p and ψ1 j , . . . , ψ p j ∈ C p (respectively), and
we write S j = diag(s1 j , . . . , s p j ). If we let r j , rank F(λ j )
for each j, then (50) implies
rj
n
F(L)x = ∑ ∑ sm j (φm j ⊗ w j ) x, (ψm j ⊗ w j ) Γ
(51)
j=1 m=1
for all x ∈Cnp . It is clear from (51) that
Col F(L) = span φm j ⊗ w j : 1 6 j 6 n, 1 6 m 6 r j (52)
Null F(L) = span ψm j ⊗ w j : 1 6 j 6 n, r j < m 6 p . (53)
In particular we note that
B(L) = B(L)WW ∗ Γ
`
= ∑ (Bi ⊗ Li )w1
...
(Bi ⊗ Li )wn
∗
W Γ
i=0
= B(λ1 ) ⊗ w1 . . . B(λn ) ⊗ wn W ∗ Γ ,
and we conclude
from (52) that
Col B(L) ⊂ Col F(L)
(54)
m
B(λ j ) ∈ Col F(λ j ) for all j ∈ {1,. . . , n} .
(55)
Next, we note that Null F(L) ⊂ Null C(L) if and only if
each basis vector ψm j ⊗ w j in the span (53) satisfies
`
0 = C(L)(ψm j ⊗ w j ) = ∑ (Ci ⊗ Li )(ψm j ⊗ w j )
i=0
= C(λ j )ψm j ⊗ w j = C(λ j )ψm j w j ,
and we conclude
from (53) that
Null F(L) ⊂ Null C(L)
(56)
m
∗
C (λ j ) ∈ Col F ∗(λ j ) for all j ∈ {1, . . . , n} . (57)
It follows from (55) and (57) that the dynamics Σ(∆G) admit
an equilibrium state x̄ with a unique corresponding equilibrium output ȳ for all constant inputs ū ∈ RV and all G ∈ G if
and only if conditions (ii) and (iii) of Theorem 5 hold. We
thus proceed assuming (ii) and (iii) hold, in which case the
vectors v j , F +(λ j )B(λ j ) are such that F(λ j )v j = B(λ j ) for
each j. It follows
from (54) that
B(L) = F(λ1 )v1 ⊗ w1 . . . F(λn )vn ⊗ wn W ∗ Γ
= F(L) v1 ⊗ w1 . . . vn ⊗ wn W ∗ Γ .
(58)
Thus we can
calculate
the
unique
output
equilibrium
ȳ
as
ȳ = C(L) v1 ⊗ w1 . . . vn ⊗ wn W ∗ Γū + D(L)ū
= C(λ1 )v1 w1 . . . C(λn )vn wn W ∗ Γū + D(L)WW ∗ Γū
= H(λ1 )w1 . . . H(λn )wn W ∗ Γū
n
= ∑ H(λ j ) w∗j Γū wj .
(59)
j=1
Because L1 = 0, we can assume without loss of generality
that w1 = 1 and λ1 = 0, and we can write (59) as
n
ȳ = H(0) 1T Γū 1 + ∑ H(λ j ) w∗j Γū wj .
(60)
j=2
We have left to show that ȳ in (60) has the value in (37) for
all inputs ū and all digraphs G ∈ G if and only if conditions
(iv)–(vi) hold. To this end, we first compute
n
n
j=1
j=2
1T Γ−1 = 1T WW ∗ = ∑ hwj , 1i w∗j = n1T + ∑ hwj , 1i w∗j , (61)
and we multiply both sides from the right by Γ/n to obtain
1 n
1 T
(62)
1 = 1T Γ + ∑ hwj , 1i w∗j Γ .
n
n j=2
In particular, we can multiply both sides of (62) from the
right by w1 = 1 to obtain
1 = 1T Γ 1 .
(63)
We also note that
G is balanced ⇔ LT1 = 0
⇔ 0 = 1TLW = λ1 hw1 , 1i . . . λn hwn , 1i
⇔ λ j hwj , 1i = 0 ∀ j ∈ {2, . . . , n} .
(64)
Now suppose conditions (iv)–(vi) hold; then G has maximal
rank and thus λ j 6= 0 for all j > 2, and it follows from (62)
and (64) that ȳ in (60) has the value in (37). Conversely,
suppose that ȳ in (60) has the value in (37) for every input ū.
It follows from (60) that H(0) 6= 0 and H(λ j ) = 0 for all
j > 2. Therefore λ j 6= 0 for all j > 2, which means G has
maximal rank and condition (v) holds. For the particular case
in which ū = 1, we see from (60) and (63) that ȳ = H(0)1,
and it follows that condition
(iv)
holds. At this point we
know that ȳ = 1T Γū 1 = 1Tū/n 1 for all ū, which implies
1T Γ = 1T/n. We conclude from (62) that hwj , 1i = 0 for all
j > 2, and it follows from (64) that G is balanced. A balanced
digraph with maximal rank is connected, which implies (vi).
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