Consensus analysis of large-scale nonlinear homogeneous multi-agent
formations with polynomial dynamics

arXiv:1612.01375v1 [] 5 Dec 2016

Paolo Massioni, Gérard Scorletti

Abstract—Drawing inspiration from the theory of linear “decomposable systems”, we provide a method, based on linear
matrix inequalities (LMIs), which makes it possible to prove the
convergence (or consensus) of a set of interacting agents with
polynomial dynamic. We also show that the use of a generalised
version of the famous Kalman-Yakubovic-Popov lemma allows
the development of an LMI test whose size does not depend on
the number of agents. The method is validated experimentally
on two academic examples.
Index Terms—Multi-agent systems, nonlinear systems, consensus, polynomial dynamic, sum of squares.

I. I NTRODUCTION
Large-scale systems are an emerging topic in the system
and control community, which is devoting significant efforts
on the development of analysis and control synthesis methods
for them. This deep interest can clearly be seen from the large
number of works published in the field in the last 40 years
[1], [2], [3], [23], [8], [12], [18], [20], [14], [17].
One of the main objectives of the studies is the development
and validation of “distributed control laws” for obtaining a
certain specified goal for a system of this kind. By “distributed
control”, opposed to “centralized control”, we mean a control
action that is computed locally according to the physical
spatial extension of the system, which is seen as an interconnection of simpler subsystems. One of the main problems of
large-scale systems is the “curse of dimensionality” that goes
with them, i.e. the analysis and synthesis problems related to
dynamical systems grow with the size, and for system of very
high order, such problems becomes computationally infeasible.
In the literature, if we restrict to linear systems, we can find a
few solutions [18], [8], [1] that can effectively overcome the
curse of dimensionality for a class of systems with a certain
regularity, namely for what we call “homogeneous systems”
which are made of the interconnection of a huge number of
identical subunits (also sometimes called “agents”).
In this paper we focus on this same problem, more specifically on the stability analysis of large-scale dynamical systems,
for the nonlinear case. Namely, we will consider a formation or
a multi-agent system made of a high number of identical units
interacting with one another through a symmetric graph-like
interconnection. Assuming that the dynamical equation of each
subunit is described by a polynomial in the state vector, we can
show that a linear matrix inequality (LMI) test can be devised
in order to verify the relative stability of such a formation. We
P. Massioni is with Laboratoire Ampère, UMR CNRS 5005,
INSA de Lyon, Université de Lyon, F-69621 Villeurbanne, France

paolo.massioni@insa-lyon.fr
G. Scorletti is with Laboratoire Ampère, UMR CNRS 5005,
Ecole Centrale de Lyon, Université de Lyon, F-69134 Ecully, France

gerard.scorletti@ec-lyon.fr

will also be able to formulate such a test in a form which is not
strictly depending on the formation size, making it possible to
check the stability of formations with virtually any number of
agents, basically extending the analysis results of [18], [9] to
the nonlinear (polynomial) case.
II. P RELIMINARIES
A. Notation
We denote by N the set of natural numbers, by R the set
of real numbers and by Rn×m the set of real n × m matrices.
A⊤ indicates the transpose of a matrix A, In is the identity
matrix of size n, 0n×m is a matrix of zeros of size n × m
and 1n ∈ Rn a column vector that contains 1 in all of its
entries. The notation A  0 (resp. A  0) indicates that all
the eigenvalues of the symmetric matrix A are positive (resp.
negative) or equal to zero, whereas A ≻ 0 (resp. A ≺ 0)
indicates that all such eigenvalues
are strictly positive (resp.

n
negative). The symbol
indicates the binomial coefficient,
k
for which we have
 
n!
n
=
.
k
k!(n − k)!
The symbol ⊗ indicates the Kronecker product, for which we
remind the basic properties (A ⊗ B)(C ⊗ D) = (AC ⊗ BD),
(A ⊗ B)⊤ = (A⊤ ⊗ B ⊤ ), (A ⊗ B)−1 = (A−1 ⊗ B −1 ) (with
matrices of compatible sizes). We employ the symbol ∗ to
complete symmetric matrix expressions avoiding repetitions.
B. Agent dynamic
We consider a set of N ∈ N identical agents or subsystems
of order n, which interact with one another. Each agent, if
taken alone, is supposed to be described by a polynomial
dynamic, of the kind
ẋi = fd (xi ) = Aa χi

(1)
⊤

n

where i = 1, ..., N , xi = [xi,1 , xi,2 , ..., xi,n ] ∈ R is the
state of the ith agent, fd is a polynomial function of degree
d ∈ N, Aa ∈ Rn×ρ and χi ∈ Rρ is the vector containing all the
monomials in xi up to degree d (for example, if n = 2, d = 2,
then χi = [1, xi,1 , xi,2 , x2i,1 , xi,1 xi,2 , x2i,2 ]⊤ ). The value of ρ
is given by


n+d
ρ=
.
(2)
n
This approach is based on the sum of squares (SOS) literature
[19], which basically allows the relaxation of polynomial
problems into linear algebra’s. In this context, it is possible to
express polynomials p up to degree 2d as quadratic forms with
⊤
respect to χi , i.e. p(xi ) = χ⊤
∈ Rρ×ρ .
i X χi , with X = X

2

This quadratic expression is not unique, due to the fact that
different products of monomials in χi can yield the same
result, for example x2i,1 is either x2i,1 times 1 or xi,1 times
xi,1 . This implies that there exist linearly independent slack
ρ×ρ
matrices Qk = Q⊤
, with k = 1, . . . , ι such that
k ∈ R
⊤
χi Qk χi = 0. The number of such matrices is [19]

2 
! 

1
d+n
d+n
n + 2d
ι=
+
−
.
(3)
2
d
d
2d
C. Formations
Moving from one single agent to a whole formation, we
employ a matrix P ∈ RN ×N to describe the interactions
among the agents. Basically P is a sparse matrix whose entries
in the ith row and j th column indicate whether the ith agent is
influenced by the state of the j th , according to the definition
that follows.
Definition 1 (Formation). We call a formation (of non-linear
agents with polynomial dynamics) a dynamical system of order
nN , with n, N ∈ N, described by the following dynamical
equation
ẋ = (IN ⊗ Aa + P ⊗ Ab )χ
(4)
⊤
⊤ ⊤
where x = [x⊤
∈
RnN , χ
=
1 , x2 , ..., xN ]
⊤
⊤
⊤ ⊤
ρN
N ×N
[χ1 , χ2 , ..., χN ] ∈ R , P ∈ R
and Aa , Ab ∈ Rn×ρ .

This definition extends and adapts the definition of “decomposable systems” given in [18] to polynomial dynamics.
In the linear case, a formation defined above boils down to
the dynamical equation
ẋ = (IN ⊗ Aa + P ⊗ Ab )x.

(5)

In [18] it has been shown that if P is diagonalisable, then
this system (of order nN ) is equivalent to a set of parameterdependent linear systems of order n. This is obtained with
the change of variables x = (S ⊗ In )x̂, where x̂ =
⊤
⊤ ⊤
nN
[x̂⊤
and S is the matrix diagonalising
1 , x̂2 , ..., x̂N ] ∈ R
−1
P, i.e. S PS = Λ, with Λ diagonal. This turns (5) into
x̂˙ = (IN ⊗ Aa + Λ ⊗ Ab )x̂, which is a block-diagonal system
equivalent to the set
x̂˙ i = (Aa + λi Ab )x̂i

(6)

for i = 1, ..., N , with λi the ith eigenvalue of P. This idea
of decomposing a distributed system into a set of parametervarying systems is very practical and it has inspired several
works in the domain of consensus and distributed control [13],
[26], [7], [4]. In this paper, to our knowledge, the idea is
adapted to nonlinear dynamics for the first time.
D. Problem formulation
The topic of this paper is to find a proof of convergence of
the state of the agents under a given dynamics expressed as in
(4). We do not require that each agent by itself converges to a
point, but that they all converge eventually to the same state,
which could be either an equilibrium point or a trajectory. In
order to do so, we formulate first an assumption on the pattern
matrix P.

Assumption 2. The pattern matrix P ∈ RN ⊗N in (4) is
symmetric and it has one and only one eigenvalue equal to
0, associated to the eigenvector 1N , i.e P1N = 0.
This assumption is very common in the literature, it basically ensures that the interconnection matrix is a (generalised)
graph Laplacian of a symmetric connected graph [5]. Such
matrices have real eigenvalues and eigenvectors. We can then
formulate the problem on which this paper focuses.
Problem 3. We consider (4) with initial conditions x(0) ∈
RnN . Prove that limt→∞ ||xi − xj || = 0, ∀i, j ∈ {1, ..., N }.
III. F ORMATION LYAPUNOV FUNCTION
In order to be able to prove the convergence of all the agents
to the same trajectory, we define what we call a “formation
Lyapunov function”, which will have the property of tending
to zero when the agents are converging. We summarise these
notions in a definition and a lemma.
Definition 4 (Formation Lyapunov function candidate). We
define as “formation Lyapunov function candidate” a function
!
l
X
(7)
P i ⊗ Li x = x⊤ Lx
V (x) = x⊤
i=1

L⊤
i

n×n

∈R
, l ∈ N, l 6 N . The reason for this
with Li =
special structure will be clear later on, in fact it allows the
block-diagonalisation of the Lyapunov matrix in the same way
as P can be diagonalised.
Lemma 5. Consider (4) and a formation Lyapunov function
N ×(N −1)
candidate V (x) = x⊤ Lx as in (7). Let 1⊥
be
N ∈ R
the orthogonal complement of 1N , i.e. [1N 1⊥
]
is
full
rank
N
⊥
and 1⊤
N 1N = 0.
⊤
⊥
If (1⊥
N ⊗ In ) L (1N ⊗ In ) > 0, then we have that xi = xj
∀i, j ∈ {1, ..., N } if and only if V (x) = 0.
Proof: Necessity is almost obvious: if xi = xj ∀i, j ∈
{1, ..., N }, then x = 1N ⊗ xi ; the fact that P1N = 0 implies
that V (x) = 0.
We prove the sufficiency by contradiction, i.e. we suppose
that there exist i and j for which xi 6= xj and V (x) = 0.
The vector x with the complete state must then have at
least one component which is orthogonal to the columns of
(1N ⊗ In ), because (1N ⊗ In ) contains columns with all the
corresponding agent states equal. So, based on the fact that
⊤
⊥
(1⊥
N ⊗ In ) L (1N ⊗ In ) > 0, then V (x) > 0 contradicting
the hypothesis.
IV. M AIN

RESULT

We are now ready to formulate our main result. A preliminary lemma is given first, which allows diagonalising the
Lyapunov function in the same way as a linear system is
decomposed in [18].
Lemma 6. If Assumption 2 holds, then 1) there exist a matrix
S ∈ RN ×N such that S ⊤ S = SS ⊤ = IN , and S ⊤ PS = Λ,
with Λ diagonal. Moreover, we have that 2) S ⊤ 1N = T =
[t1 t2 ... tN ], with ti ∈ RN = 0 if λi = Λi,i 6= 0.

3

Proof: The first part of the lemma is proven by the
fact that all symmetric matrices are diagonalisable by an
orthonormal matrix S (i.e. S −1 = S ⊤ ) [10]. For the second
part, consider that due to Assumption 2, 1N is an eigenvector
of P with eigenvalue 0; the matrix S contains the normalised
eigenvectors of P in its columns, and all of these eigenvectors
are orthogonal to one another because S ⊤ S = IN . So each
ti is the dot product between 1N and the ith eigenvector, and
it is non zero if and only if λi = 0.
Theorem 7. Consider (4) with given N , Aa , Ab and P
statisfying Assumption 2; moreover, we order the eigenvalues
of P so that the first eigenvalue is the one equal to zero, i.e.
λ1 = 0. If for a chosen l ∈ N, there exist τj ∈ R, and matrices
n×n
Lj = L⊤
such that
j ∈R
Pl
j
(8)
j=1 λi Lj ≻ 0
Pl
Pι
⊤
Π( j=1 τj Qj + j=1 (λji (Γ⊤ Lj Aa + A⊤
a Lj Γ + Γ Lj Γ)+
j+1
⊤
⊤
⊤
λi (Γ Lj Ab + Ab Lj Γ))Π  0
(9)
for i = 2, ... N , where Γ = [0n,1 In 0n,ρ−n−2 ] (i.e. Γxi =
χi ), Π = [0ρ−1,1 Iρ−1 ], then limt→∞ ||xi − xj || = 0, ∀i, j ∈
{1, ..., N }.
Proof: In order to assure the convergence of the agents,
we need to assure the conditions stated in Lemma 5, namely
that a function V (x) = x⊤ Lx exists, with and (1⊥
N ⊗
In )⊤ L (1⊥
⊗
I
)
>
0,
and
that
V̇
(x)
<
0
for
V
(x)
>
0.
n
N
⊤
⊥
For the condition (1⊥
⊗
I
)
L
(1
⊗
I
)
>
0,
consider
that
n
n
N
N
S contains a scaled version of 1N in its first column and 1⊥
N in
P
the rest of the matrix. Knowing that L = (S ⊗ In )( li=1 Λi ⊗
Li )(S ⊤ ⊗ In ) thanks to Lemma 6, this condition is equivalent
to (8).
For what concerns the condition V̇ (x) < 0 for V (x) > 0,
it is satisfied if V̇ (x) 6 −ǫx⊤ Lx, i.e.
⊤
χ⊤ (QN + Γ⊤
N L(IN ⊗ Aa ) + ΓN L(P ⊗ Ab )+
⊤
(IN ⊗ A⊤
)LΓ
+
(P
⊗
A
)LΓ
+
ǫΓ⊤
N
N
a
b
N LΓN )χ 6 0
(10)
where
Γ
=
(I
⊗
Γ)
(so
Γ
χ
=
x)
and
Q
=
I
⊗
N
N
N
N
N
Pι
⊤
τ
Q
(for
which,
by
definition
of
Q
,
χ
Q
χ
=
0
j
j
j
N
j=1
for all values of the τi ). By the fact that P = SΛS ⊤ and
IN = SS ⊤ (Assumption 2 and Lemma 6), using the properties
of Kronecker product (10) is equivalent to
⊤
χ̂⊤ (QN + Γ⊤
N L̂(IN ⊗ Aa ) + ΓN L̂(Λ ⊗ Ab )+
⊤
⊤
(IN ⊗ Aa )L̂ΓN + (Λ ⊗ Ab )L̂ΓN + ǫΓ⊤
N L̂ΓN )χ̂ 6 0
(11)
Pl
j
⊤
with L̂ =
j=1 Λ ⊗ Lj and χ̂ = (S ⊗ Iρ )χ. Notice in
this last inequality that the term between χ̂⊤ and χ̂ is blockdiagonal, as it is the sum of terms of the kind IN ⊗ X
or Λi ⊗ X (i ∈ N). If we define χ̂i ∈ Rρ such that
⊤
⊤ ⊤
χ̂ = [χ̂⊤
1 , χ̂2 , ..., χ̂N ] , then (11) is equivalent to
PN
Pι
Pl
j
⊤
⊤
⊤
i=1 χ̂i (
j=1 τj Qj +
j=1 λi (Γ Lj Aa + Aa Lj Γ)+
Pl
Pl
j
j+1 ⊤
⊤
⊤
j=1 λi (Γ Lj Γ))χ̂i 6 0.
j=1 λi (Γ Lj Ab + Ab Lj Γ)+ǫ
(12)
The term of the sum for i = 1 is always 0 (as we chose
λ1 = 0), so there is no contribution from it and it can be
discarded. Concerning the vectors χi , remember that they all

⊤
ρ−1
contain 1 in their first entry, i.e. χi = [1 χ̃⊤
.
i ] , χ̃i ∈ R
th
For each χ̂i , the first entry is by its definition the i entry of
the vector e = S ⊤ 1, which contains zeros in all of its entries
but the first (due to Lemma 6). So for i = 2, ..., N , we have
that χ̂i = Π⊤ Πχ̂i . So (12) is equivalent to
PN
Pι
⊤ ⊤
i=2 χ̂i Π Π(
j=1 τj Qj +
Pl
j
⊤
λ
(Γ
L
A
+
A⊤
j a
a Lj Γ)+
i
Pl j=1 j+1
(13)
⊤
(Γ Lj Ab + A⊤
b Lj Γ)+
j=1 λi
P
ǫ lj=1 λji (Γ⊤ Lj Γ))Π⊤ Πχ̂i 6 0

The set of LMIs in (9) imply (13), which concludes the proof.

This theorem allows proving the convergence of N agents
with two sets of N − 1 parameter-dependent LMIs, whose
matrix size is respectively n (i.e. the order of each agent taken
alone) and ρ − 1. This result is already interesting as it avoids
using LMIs scaling with N n, which is the global system order.
In the next section, we explore whether it is possible to further
reduce the computational complexity.
V. VARIATION ON THE MAIN RESULT
We explore the possibility of using a generalised version
of the famous Kalman-Yakubovich-Popov (KYP) lemma [22].
This lemma allows turning a parameter-depending LMI into a
parameter-independent one.
A. The Kalman-Yakubovic-Popov lemma
The Kalman-Yakubovic-Popov lemma or KYP [22] is
a widely celebrated result for dynamical systems that allows turning frequency-dependent inequalities into frequencyindependent ones, by exploiting a state-space formulation. It
turns out that such a result can be adapted and generalised to
inequalities depending on any scalar parameter. Namely, we
will use the following generalised version of the KYP.
Lemma 8 (Generalized KYP [6]). Consider
η
X
ξi Mi ,
M (ξ) = M0 +

(14)

i=1

with ξ ∈ Rl a vector of decision variables and Mi = Mi⊤ ∈
RnM ×nM , i = 1, ..., η. The quadratic constraint
⊤

φ(θ) M (ξ) φ(θ) ≺ 0 for θ ∈ [θ, θ]

(15)

⊤

is verified if and only if there exist D = D ≻ 0 and G = −G ⊤
such that
 ⊤ 


C̃
M (ξ) C̃ D̃ +
⊤
D̃
⊤



I 0
−2D
(θ + θ)D + G I 0
≺0
Ã B̃
Ã B̃
(θ + θ)D − G
−2θθD
(16)
with Ã ,B̃, C̃ and D̃ such that


Ã B̃
φ(θ) = D̃ + C̃θI(I − ÃθI)−1 B̃ = θI ⋆
, (17)
C̃ D̃
where the operator ⋆ implicitly defined above is known as the
Redheffer product [27].
The lemma applies as well if the sign ≺ in (15) is replaced
by : in this case replace ≺ with  in (16) as well.

4

B. Second main result
Let us define
λ = max {λi }, λ = min {λi }.
26i6N

(18)

26i6N

Then, for θ ∈ [λ, λ], the following set of LMIs
Pl
j
(19)
j=1 θ Lj ≻ 0
Pl
Pι
⊤
Π( j=1 τj Qj + j=1 (θj (Γ⊤ Lj Aa + A⊤
a Lj Γ + Γ Lj Γ)+
j+1
⊤
⊤
⊤
θ (Γ Lj Ab + Ab Lj Γ))Π  0
(20)
“embeds” the set of LMIs in (8) and (9) (notice that we have
moved from a discrete set of values to a continuous interval
which includes them all). Subsequently, Lemma 8 can be used
to turn the θ-dependent LMIs in (19) and (20) into parameterindependent ones. The dependence of the terms in (19) and
(20) from θ (which is ultimately λi ) is polynomial, so we need
to define
i⊤
h
(21)
φν (θ) = θceil((l+1)/2) Iν , θceil((l+1)/2)−1 Iν , ..., Iν

which requires

Ãν = Uν ⊗ In ,

C̃ν =



Iν
01×ν



⊗ In ,

B̃ν =



D̃ν =

0(ν−1)×1
1


0ν×1
1





⊗ In ,

⊗ In .

Theorem 9. Consider (4) with given N , Aa , Ab and P statisfying Assumption 2; excluding the first eigenvalue of P, which
is equal to 0, we have that λ 6 λi 6 λ, with i = 2, ... N .
If for a chosen l ∈ N, there exist τi ∈ R, and matrices
n×n
Li = L⊤
, and there exist Dνk , Gνk ∈ Rνk ×νk ,
i ∈ R
⊤
Dνk = Dνk ≻ 0 and Gνk = −Gν⊤k such that


[∗]⊤ Mk C̃νk D̃νk +



Iνk
0
(λ + λ)Dνk + Gνk
⊤ −2Dνk
≺0
[∗]
Ãνk B̃νk
∗
−2λλDνk
(23)
for k = 1, 2, with ν1 = n and ν2 = ρ − 1, where Γ =
[0n,1 In 0n,ρ−n−2 ] (i.e. Γxi = χi ), Π = [0ρ−1,1 Iρ−1 ], and
Ãνk , B̃νk , C̃νk , D̃νk , φνk are defined in (22) and (21), with
l
X

λji Lj

(24)

j=1

Pι
φν (λi )⊤ M2 φν (λi ) = Π( j=1 τj Qj +
Pl 2 j ⊤ 2
⊤
⊤
j=1 (λi (Γ Lj Aa + Aa Lj Γ + Γ Lj Γ)+
j+1
⊤
λi (Γ⊤ Lj Ab + A⊤
b Lj Γ))Π ,

VI. E XAMPLES
In order to provide a few challenging examples of application of the proposed method, we focus on a problem that is
widely studied in the nonlinear dynamics community, namely
the synchronisation of oscillators [21], [25]. The approach here
is of course numerical and different (or complementary) with
respect to the ones found in such a literature, where the objective is usually to find a control law and then analytically prove
stability. Our approach is just to propose a control law and test
numerically whether it will make the subsystems converge or
not. We consider two famous examples of nonlinear systems,
namely the Van der Pol oscillator [11] and the Lorenz attractor
[16].
A. Van der Pol oscillator

(22)

where Uν ∈ Riν×ν is a matrix containing 1’s in the first upper
diagonal and 0’s elsewhere, and ν = n for (19) and ν = ρ − 1
for (20). We are now ready to formulate our second main
result.

φν1 (λi )⊤ M1 φν1 (λi ) = −

With this second theorem, we replace the two sets of N − 1
LMIs of size n and ρ − 1, with only two LMIs of matrix size
n ceil((l + 3)/2) and ρ ceil((l + 3)/2). This is an interesting
result because the computational complexity is no longer
depending on N , i.e. the number of agents. On the other hand,
the choice of a bigger l will improve the chances of solving
the LMIs for high values of N .

(25)

then limt→∞ ||xi − xj || = 0, ∀i, j ∈ {1, ..., N }.
Proof: A direct application of Lemma 8 for M1 and M2
implies that the hypotheses of Theorem 7 are satisfied if the
hypotheses here are.

We consider a system of N agents of equation

ẋi = yi
ẏi = µ(1 − xi )yi − xi − cri

(26)

for which n = 2, with ri = −(xi−1 + yi−1 ) + 2(xi + yi ) −
(xi+1 + yi+1 ) (where the index is to be considered as modulo
N , i.e. 0 → N , N + 1 → 1). The interconnection between
each oscillator is given by the term c, a proportional feedback
gain. This feedback law has just been guessed, and we use
Theorem 9 to prove whether it works. We have coded the
related LMI problem in Matlab using Yalmip [15], choosing
µ = 0.5, N = 10, l = 6 and c = 15 (all arbitrary values). By
using SeDuMi [24] as solver, we managed to find a feasible
solution, which yields a valid formation Lyapunov function.
Figure 1 and Figure 2 show the evolution of the system during
a simulation, with the individual states shown as well as the
value of the Lyapunov function over time.
B. Lorenz attractor
We consider now system of N agents of equation

 ẋi = σl (yi − xi ) − crx,i
ẏi = xi (ρl − zi ) − yi − cry,i

żi = xi yi − βl zi − crz,i

(27)

with ri,• = − •i−1 +2 •i −•i+1 (again the index is taken
modulo N ). We set arbitarily ρl = 28, σl = 10, βl = 8/3,
N = 8, l = 6 and c = 50. This time we used Theorem 7, successfully obtaining a formation Lyapunov function. Figure 3,
Figure 4 and Figure 5 again show the evolution of the system
during a simulation, with individual states and the value of the
Lyapunov function. Notice that the Lorenz oscillator does not
converge to a limit cycle but to a chaotic trajectory.

5

5
xi

4

yi

3

50

2

40
30

0

z

states

1

-1

20

-2

10
-3

starting points

0

-4
-5
0

2

4

6

8

10

-10
-20

12

40

t

20

-10
0

Fig. 1.

0

10

x

Evolution of the state of the 10 coupled Van der Pol oscillators.

20

-20

y

Fig. 4. Tridimensional visualisation of the state of some of the coupled
Lorenz systems of the example (the trajectories eventually converge to the
consensus trajectory).

10 4
10 2

10 4

10 -2

10 2

10 -4

10 0

10 -6

10 -2

V

V

10 0

10 -8
10 -10

10 -4

0

2

4

6

8

10

12

10 -6

t
10 -8

Fig. 2. Value of the formation Lyapunov function V for the coupled Van
der Pol oscillators.

10 -10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t
50

Fig. 5. Value of the formation Lyapunov function V for the coupled Lorenz
systems.

xi
y

40

z

i
i

VII. C ONCLUSION

states

30

We have introduced a new method for proving convergence
or consensus of multi-agent system with polynomial dynamic.
This method is the generalisation of the analysis methods in
[18] and it has proven effective in test cases featuring dynamical oscillators. Further research will investigate if convex
controller synthesis results can be obtained with a similar
approach.

20

10

0

-10

-20
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t

Fig. 3.

Evolution of the state of the 8 coupled Lorenz systems.

0.8

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