Average consensus and stability analysis in networked dynamic systems
RHOUMA MLAYEH
Mathematics & Computer Sciences Department
Carthage University INSAT
LIM laboratory, Polytechnic School of Tunisia, BP 743, 2078 La Marsa, Tunisia
TUNISIA
Abstract: This paper provides protocols for finitetime average consensus and finitetime stability of systems
with controlled nonlinear dynamics innetwork under undirected fixed topology. Each node’s state is a high
dimensional vector as a solution of the highly nonlinear first order dynamics with and without drift terms. This
paper provides protocols for finitetime average consensus and finitetime stability of systems with controlled
nonlinear dynamics innetwork under undirected fixed topology. Each node’s state is high Under the proposed
interaction rules, agreements as a common average value or an average trajectory are reached, solving finitetime
average consensus and the multisystem equilibrium is controlled leading to the finitetime stability of each system
origin. Sufficient conditions are achieved using the Lyapunov techniques and the graph theory. In networked
dynamic systems, the theoretical results of the paper cover a large class of underactuated autonomous systems
as formation flight, multivehicle coordination, and heterogeneous multisystem behaviors. Some examples are
introduced in simulation which approves the proposed protocols.
KeyWords: Finitetime average consensus; finitetime stability; multisystem dynamics.
Received: July 15, 2021. Revised: November 20, 2021. Accepted: December 22, 2021. Published: January 7, 2022.
1 Introduction
For cooperative tasks using multiagent groups, the
presence of a large number of autonomous dynami
cal systems in industry requires interrelationships be
tween distributed control parameters which are de
signed at a first step to manage each agent separately.
Thus, in coordination of a team of autonomous agents,
the communication of sensors is fundamental in many
distributed control systems. For many applications
the main challenges in cooperative design for a group
of agents is to meet some objectives such that the ren
dezvous problem of multivehicle, control of train
ing, flocking, attitude synchronization and the fu
sion of sensors. A coherent movement in masses is
called consensus. Thus, the problem of consensus
plays a central role in study of multiagent systems.
In recent years this paradigm has introduced in multi
agent systems witnessed dramatic advances of var
ious distributed strategies that achieve agreements.
In [5] , the authors proposed a simple but interest
ing discretetime model of finite agents all moving
in the plane. Each agent’s motion is updated using
a local rule based on its own state and the states of
its neighbors. [6] provided a theoretical explanation
of the consensus property of the Vicsek model by us
ing graph theory and nonnegative matrix theory. For
this model each agent’s set of neighbors changes with
time as system evolves. Consequently, many seem
ingly different problems that involve interconnection
of dynamic systems in various areas of science and
engineering happen to be closely related to consensus
problems for multiagent systems. The existing con
nections are presented by [25] with application to lin
ear dynamics in network in studying of multisystem
behaviors.
The theoretical framework for posing and solving
consensus problems for networked dynamic systems
was introduced by [7] and [8]. Under dynamically
changing interaction topologies, [9] extended the re
sults of [6].
Various finitetime stabilizing control laws have been
proposed using continuous state feedback and output
feedback controllers [3]. Furthermore, the finitetime
control design has been extended to nth order systems
with both parametric and dynamic uncertainties [2].
Although the finitetime design is generally more dif
ficult than the asymptotically stabilizing control due
to the lack of effective analysis tools. Also, the non
smooth finitetime control synthesis can improve the
system behaviors in some aspects like highspeed,
control accuracy, and disturbance rejection. There
fore, it is not surprising that finitetime control ideas
have been applied to multiagent systems with first
order agent dynamics using gradient flow and Lya
punov function [10].
Finitetime consensus firstly was studied by [10],
where a nonsmooth consensus algorithm is proposed.
In the same filed [11], and in [17] authors proposed a
continuous nonlinear consensus algorithm to guaran
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tee the finitetime stability under an undirected fixed
interaction graph. [16] suggest an improvement to
the proposed algorithm proposed in [11]. The new
algorithm proposed in [16] is able to guarantee finite
time consensus under an undirected switching inter
action and a directed fixed interaction graph when
each strongly connected component of the topology is
detailbalanced. In [19], the authors study finitetime
consensus for second order dynamics with inherent
nonlinear dynamics under an undirected fixed inter
action graph. In networked dynamic systems, finite
time consensus problems that have been solved so far
are mostly only for simple agents like particle behav
iors as first or second order dynamics. In [13] [14],
the authors treated finitetime consensus for highly
nonlinear dynamic systems in network affine in con
trol inputs. Such a system is described by a nonlinear
firstorder ordinary differential relations.
While an interesting topic in consensus problem is the
average consensus problem such that the states of all
the agents converge asymptotically or in finite time
to the average of their initial states under a networked
interaction protocol, one cites the results in [20] [21]
[22] [23], our work consists to extend these results
and propose protocols for nonlinear dynamic systems
in network expected to reach an agreement that can be
a predefined average value or an average trajectory.
Moreover, we will make difference between consen
sus and stability protocols in treating the equilibrium
stability of the designed multisystem dynamics.
The paper is organized as follows. Some preliminar
ies results, the problem statement, and the finitetime
average consensus protocol are formulated in section
2. In section 3 one solves a finite time average
consensus of multisystem without drift terms. The
finitetime average consensus of multisystem with
drift is detailed in section 4. Finally, illustrative ex
amples are presented in section 5.
2 Preliminaries and problem
formulation
Throughout this paper, we use Rto denote the set
of real number. Rnis the ndimensional real vec
tor space and ||.|| denotes the Euclidian norm. Rn×n
is the set of n×nmatrices. diag{m1, m2, . . . , mn}
denotes a n×ndiagonal matrix. In∈Rn×nis
the identity matrix. The symbol ⊗is the Kronecker
product of matrices. We use sgn(.)to denote the
signum function. For a scalar x, note that φα(x) =
sgn(x)∥x∥α. We use xi= (xi
1, xi
2, . . . , xi
n)T∈Rn,
x= (x1, x2, . . . , xN)Tto denote the vector in Rn×N.
Let ϕα(xi) = (φα(xi
1), φα(xi
2), . . . , φα(xi
n))Twith
ϕα(x) = (ϕα(xi), . . . , ϕα(xN))T.
Let 1n= (1, . . . , 1)T. The exponent Tis the trans
pose.
2.1 Graph theory
In this subsection, we introduce some basic concepts
in algebraic graph theory for multiagent networks.
Let G={V,E} be a directed graph, where
V={1,2, . . . , n}is the set of nodes, node irepre
sents the ith agent, Eis the set of edges, and an edge
in Gis denoted by an ordered pair (i, j).(i, j)∈ E if
and only if the ith agent can send information to the
jth agent directly. A= [aij ]∈Rn×nis called the
weighted adjacency matrix of Gwith nonnegative el
ements, where aij >0if there is an edge between
the ith agent and jth agent and aij = 0 otherwise.
Moreover, if AT=A, then Gis also called an undi
rected graph. In this paper, we will refer to graphs
whose weights take values in the set {0,1}as binary
and those graphs whose adjacency matrices are sym
metric as symmetric.
Let D=diag{d1, . . . , dn} ∈ Rn×nbe a diagonal ma
trix, where di=
n
j=1
aij for i= 0,1, . . . , n. Hence,
we define the Laplacian of the weighted graph
L=D−A∈Rn×n
The undirected graph is called connected if there is a
path between any two vertices of the graph.
2.2 Some useful lemmas
Our main results are guided by the following Lem
mas. The reader may find more details in the associ
ated references.
Lemma 1 : Bhat and Bernstein(200). Consider the
system ˙x=f(x), f(0) = 0, x ∈Rn, there exist a
positive definite continuous function
V(x) : U⊂Rn→R, real numbers c > 0and
α∈]0,1[, and an open neighborhood U0⊂Uof the
origin such that
˙
V+c(V(x))α≤0, x ∈U0\{0}. Then V(x)con
verges to zero in finite time. In addition, the finite
settling time T⋆satisfies T⋆≤V((x(0))1−α
c(1 −α).
Lemma 2 : Hong & al (2002). Consider the follow
ing system, x= [x1, . . . , xn]T∈Rn
˙x=g(x) + ˜g(x)(1)
where g(0) = 0 and g(x) is a continuous homoge
neous vector field of degree d < 0with respect to di
lation [σ1, . . . σn],and ˜g(x) = [ ˜g1(x), . . . , ˜gn(x)]T∈
Rnsatisfies ˜g(0) = 0. Assume that x= 0 is an
asymptotically stable equilibrium of the system ˙x=
g(x). Then x= 0 is a globally finite time stable equi
librium of system (1) if
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lim
ε→0
˜g(σ1x1, . . . , σnxn)
εd+σi= 0, i = 0, . . . , n, ∀x= 0,
and the stable equilibrium x= 0 of system (1) is glob
ally asymptotically stable.
Lemma 3 : OlfatiSaber & al (2004). For a con
nected undirected graph G, the Laplacian matrix Lof
Ghas he following properties,
xTLx =1
2
n
i,j=1
aij (xi−xj)2,which implies that Lis
positive semidefinite. 0is a simple eigenvalue of L
and 1is the associated eigenvector. Assume that the
eigenvalues of Lare denoted by 0, λ2, . . . , λnsatis
fying 0≤λ2≤ · · · ≤ λn.Then the smallest eigen
value satisfies λ2>0. Furthermore, if 1Tx= 0,then
xTLx ≥λ2xTx.
Lemma 4 : Hardy & al (1952).
Let x1, x2, . . . , xn≥0and o < p ≤1. Then
n
i=1
xip≤
n
i=1
xp
i≤n1−pn
i=1
xip.
2.3 Problem statements
We solve the finitetime average consensus and stabil
ity of two type of models in networked dynamic sys
tems affine in control inputs. The first type is given
by equation (2) which describes a controlled dynamic
system without drift term. The second type is rep
resented by relation (3) which is clearly a controlled
dynamic system with drift term fi(xi).Let consider
a group of Nhighdimensional agents where each
agent’s behavior is described by a controlled nonlin
ear model without drift Σ1represented by the con
trolled dynamic (2) and system Σ2with drift as shown
by the controlled dynamic (3),∀i∈ I ={1, . . . , N }
Σ1:˙
xi=B(xi)ui.(2)
and
Σ2:˙
xi=fi(xi) + B(xi)ui.(3)
where xi∈Rn, xi= [xi
1, xi
2, . . . , xi
n]T, B(xi)∈
Rn×m, the continuous maps fi:Rn→Rn, ui∈Rm
is the control input and for 1≤k≤nand
1≤m, B(xi) = [bkl].
Definition 1 (stabilization) Given an interconnec
tion control ui(xi, xj), the origin the zero solution
xi(t) = 0 to (2)(3) is finitetime stable if the fol
lowing statements hold:
1. The zero solution of closed loop system to (2)(3)
is stable.
2. There exist a settlingtime T⋆such that
lim
t→T⋆
||xi(t)|| = 0
Definition 2 Given a controlinput uias protocol, we
say that systems in network meet a finite time average
consensus if for any system’s state initial conditions,
there exists some finite time T⋆such that:
lim
t→T⋆
||xi(t)−χ(t)|| = 0 (4)
for any i∈ I, and where χ(t) = 1
N
N
j=1
xj(t)is the
average trajectory.
χ(t)can be interpreted as the instantaneous consent
providing that serves the group objectives. χis time
varying, it can be also considered as the average tra
jectory of the group, and it is not necessary the aver
age from the multisystem initial conditions. We show
that the dynamic of χdepends strongly on the adopted
topology of the group.
Subsequently, for the multiΣ1and multiΣ2systems
one might analyze the following protocols are given
by (5)and (6). For i∈ I, the consensus protocol
candidate is given by,
ui=−C(xi)
N
j=1
aij ϕα(xi−xj)(5)
while the stabilizing input candidate is as
ui=−C(xi)
N
j=1
aij (ϕα(xi)−ϕα(xj)) (6)
where the aij elements are of the Gadjacency ma
trix, α∈]0,1[, and ϕα(.)is defined in section 2. The
control matrix C(xi)∈Rm×ndepends on the agent’s
model, and it will be defined in the following.
As we can see in protocols (5) and (6), the finite
time average consensus is closely related to finite
time stability. The main difference between the two
problems is that finitetime average consensus is to
make the multisystem converge to an agreement
value or trajectory as given by χ(t)in (4), while the
stability of each agent consists to reach an equilib
rium. The following assumption gives a conceptual
form of C(xi)with respect to the studied dynamics.
Assumption1: C(xi)is such that the matrix prod
uct B(xi)C(xi)is positive semidefinite matrix.
Throughout the paper, one denotes by
˜
B(xi) = B(xi)C(xi).
Assumption2: For a given control matrix C(xi), for
all x, y ∈Rn, we assume that
(ϕα(x)−ϕα(y))T˜
B(xi)ϕα(x−y)≥(ϕα(x)−
ϕα(y))T˜
B(xi)(ϕα(x)−ϕα(y))
Assumption3: Consider that
g(xi) = −
N
j=1
aij ˜
B(xi)ϕα(ξi−ξj)(7)
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is a locally homogeneous vector field of degree dwith
respect to dilation [σ1, . . . , σn].
3 Finitetime average consensus
The objective of this section is to solve finitetime av
erage consensus problems of multisystem based on
Σ1and Σ2descriptions. The average value is consid
ered as an agreement function of time but is not nec
essary function of multiagent initial conditions. Fur
ther, what motivates the analysis is that models given
by (3) and (2) cover many autonomous system be
haviors affine in the control vector. One may cite, au
tomated highway systems, multidrone, multisystem
of satellites or robots, etc. When we refer to the proto
col (5), the interaction topology uses undirected flow
information between nodes where each node’s vector
of states is as a solution of (3) or (2). The follow
ing two subsections treat the multiΣ1and multiΣ2
finitetime average consensus.
3.1 The multiΣ1finitetime average
consensus
For finitetime average consensus of multiΣ1one
considers, as interaction topology an undirected fixed
graph, an average vector obtained from each Σ1vec
tor of states, and the protocol candidate (5). As the
matrix Bstructure is taken identical for each Σ1then
one might think to networked homogeneous systems.
Recall that for a group where each agent is of the form
˙xi=ui,if the interconnection topology is based on an
undirected flow, then the average consensus is solved
with respect to the average of the agents initial states.
Proposition 1 Let Gbe an undirected and connected
graph, under the protocol (5) and Assumptions 1−2−
3the multiΣ1achieves a finitetime average consen
sus in the sense of (4).
Proof We introduce ξi(t) = xi(t)−χ(t)and
ξ(t) = [ξ1, ..., ξN]T. Due to the fact that aij =aji
for all 1≤i, j ≤N(undirected graph) and φαis an
odd function, we have,
˙χ(t) = 1
N
N
i=1
˙xi(t)
=−1
N
N
i,j=1
aij ˜
B(xi)ϕα(xi−xj)
=−1
2N
N
i,j=1
aij ˜
B(xi)−˜
B(xj)ϕα(xi−xj)
Introducing the protocol (5), we obtain
˙
ξi(t) = ˙xi(t)−˙χ(t)
=−
N
j=1
aij ˜
B(xi)ϕα(xi−xj)+
1
2N
N
i,j=1
aij (˜
B(xi)−˜
B(xj))ϕα(xi−xj)
=−
N
j=1
aij ˜
B(xi)ϕα(ξi−ξj)+
1
2N
N
i,j=1
aij (˜
B(xi)−˜
B(xj))ϕα(ξi−ξj)
Let ξ(t) = (ξ1, ..., ξN), we can write the last equation in the
form:
˙
ξi(t) = g(ξi) + ˜g(ξ)(8)
where
g(ξi) = −
N
j=1
aij ˜
B(xi)ϕα(ξi−ξj)
and
˜g(ξ) = 1
2N
N
i,j=1
aij ˜
B(xi)−˜
B(xj)ϕα(ξi−ξj)
By now, it remains to prove that the equilibrium of (8) is finite
time stable, and this is achieved in the subsequent two steps.
Step 1. First, the goal is to prove the finite time stability of system
˙
ξi(t) = g(ξi)(9)
Taking the Lyapunov function:
V(ξ(t)) = 1
α+ 1
N
i=1
(ξi)Tϕα(ξi)(10)
The derivative of Valong the solutions of system (9), yields
˙
V(ξ(t)) =
N
i=1 ϕα(ξi)T˙
ξi
=−
N
i,j=1
aij ϕα(ξi)T˜
B(xi)ϕα(ξi−ξj)
=−1
2
N
i,j=1
aij ϕα(ξi)−ϕα(ξj)T˜
B(xi)ϕα(ξi−ξj)
From Assumption 2, the following inequality holds,
˙
V≤ − 1
2
N
i,j=1
aij (ϕα(ξi)−ϕα(ξj))T˜
B(xi)ϕα(ξi)−ϕα(ξj)
=−1
2ϕT
α(ξ)L⊗˜
B(xi)ϕα(ξ)
=−1
4ϕT
α(ξ)Θϕα(ξ)
where Θ = 1
2L⊗˜
B(xi) + L⊗˜
BT(xi).
Let
D(xi) = diag{0n, γ2(xi), ..., γN(xi)}
such that 0n=diag{0, ..., 0} ∈ Rn×nand ∀j= 2, ..., N
γj(xi) = λj(L)ϱn(xi)where
ϱn(xi) = diag{0, µ2(xi), ..., µn(xi)} ∈ Rn×nand where
µ2(xi), ..., µn(xi)are the eigenvalues of the matrix ˜
B(xi), given
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in increasing order. λj(L)is the jème eigenvalue of L. Let
λ2(L), ..., λN(L)in increasing order. Since Gis connected (by
Lemma 3 ) λ2(L)>0. Therefore ∀xi, we have λ2µ2(xi)>0.
Further, since Θ∈RNn×N n is symmetric matrix, then there exist
an orthogonal matrix P∈RNn×N n such that Θ = PTD(xi)P.
Let zα=P ϕα(ξ), thus
˙
V≤ − 1
4zT
αDzα
≤ − 1
4λ2µ1(xi)∥zα∥2
=−1
4λ2µ1(ξi)∥ϕα(ξ)∥2
where λ2µ1(xi) = min
zα⊥1N n
zT
αDzα
zT
αzα
.
Let k=min
xi∈RNλ2µ1(xi)>0and ξ=1N⊗ξi=
(˜
ξ1, ..., ˜
ξNn)T, consequently,
˙
V≤ − k
4
Nn
i=1
|φα(˜
ξi)|2
≤ − k
4
Nn
i=1
|˜
ξi|2α
≤ − k
4Nn
i=1
|˜
ξi|α+12α
α+1
(11)
which permits to write
˙
V≤ − k
4(α+ 1) 2α
α+1 V2α
α+1 (12)
where 0<2α
α+1 <1and k
4(α+ 1) 2α
α+1 >0, by Lemma 1, the
above differential equation (9) shows that Vreaches zero in finite
time..
Step 2. From Assumption 3, the vector field g(ξi)is homogeneous
of degree dwhich is negative due to the fact that ξ= 0 is a finite
time stable equilibrium. Moreover, it is straightforward to prove
that ˜g(ξ), pour k= 1, ..., n, il est simple de vérifier que
lim
ε→0
˜gk(εσ1ξi
1, ..., εσnξi
n)
εd+σi= 0
Then by Lemma ??, the system (8) is finite time stable.
Thus, as a result the multiΣ1dynamic system with the protocol
(5) solve a finitetime average consensus. This ends the proof.
3.2 The multiΣ2finitetime average
consensus
The multiΣ2behavior is based on (3) while the con
sensus protocol candidate is given by ((5). Recall that
the Σ2dynamic as given by (3) is currently present in
controlled autonomous systems. However, the drift
term can be linear with respect to the system’s state
vector or taken in its nonlinear form. These two is
sues will be analyzed in the following with the ad
equate sufficient conditions for multiΣ2finitetime
average consensus. To do, let us first note that fi in (3)
can be different for each dynamic leading to heteroge
neous multisystem. At first, the subsequent analysis
is build on this form of fi(xi)∆
=˜
Axiwith ˜
Ais a con
stant matrix. A controlled dynamic system with linear
drift term is given by,
˙
xi=˜
Axi+B(xi)ui(13)
where ˜
A∈Rn×nwith ˜
A= [˜ap,q]1≤p,q≤n.
Proposition 2 Let Gbe an undirected and connected
graph, under the protocol (5) the multiΣ2, built from
(13), converges toward an average trajectory and
leads to a finitetime average consensus in the sense
of (4).
Proof One introduces ξi(t) = xi(t)−χ(t). The goal
is to rewrite equation (13) in closed loop depending
on ξiand to prove that ξconverges to zero in finite
time. Since aij =aji and ϕα(.)is an odd function,
then we have
˙χ(t) = 1
N
N
i=1
(˜
Axi+B(xi)ui)
=1
N
N
i=1
˜
Axi+1
N
N
i=1
B(xi)ui
=1
N
N
i=1
˜
Axi−1
2N
N
i,j=1
aij (˜
B(xi)−
˜
B(xj))ϕα(xi−xj)
Consequently,
˙
ξi=˜
Aξi−
N
j=1
aij ˜
B(xi)ϕα(ξi−ξj)+
1
2N
N
i,j=1
aij (˜
B(xi)−˜
B(xj))ϕα(ξi−ξj)
keeping the same steps of the previous proof, we
introduce ˙
ξi=h(ξi) + ˜
h(ξ)
where
h(ξi) = ˜
Aξi−
N
j=1
aij ˜
B(xi)ϕα(ξi−ξj)
˜
h(ξ) = 1
2N
N
i,j=1
aij (˜
B(xi)−˜
B(xj))ϕα(ξi−ξj)
where ˜
h(ξ)˜g(ξ)then it remains to prove the finitetime
stability of the system.
˙
ξi=h(ξi)(14)
Using the Lyapunov function (10), the time derivative
of V(ξ)along the networked system trajectories (14)
is given by
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˙
V(ξ(t)) =
N
i=1
ϕT
α(ξi)˙
ξi
=
N
i=1
ϕT
α(ξi)˜
Aξi−
N
i,j=1
aij ϕT
α(ξi)˜
B(xi)ϕα(ξi−ξj)
≤ ∥ ˜
A∥∞
N
i=1
ϕT
α(ξi)ξi−k
4(α+ 1) 2α
α+1 V2α
α+1
≤ ∥ ˜
A∥∞V(ξ(t)) −k
4(α+ 1) 2α
α+1 V2α
α+1
≤ −V(ξ(t)) 2α
α+1 [k
4(α+1) 2α
α+1 −∥ ˜
A∥∞(V(ξ(t)))1−α
α+1 ]
where ∥˜
A∥∞=max
1≤p≤n
n
q=1
|˜apq |>0. Since
1−α
α+1 >0and Vis continuous function which takes
(V(ξ)=0)there exists an open neighborhood Ωof
the origin such that the last inequality
˙
V(ξ(t)) ≤ −k
8(α+ 1) 2α
α+1 V(ξ(t)) 2α
α+1 (15)
By Lemma 1, Vreaches zero in finite time.Therfore
ξi= 0 is a finitetime stable equilibrium of system
(14) We may follow step 2 of the previous analysis to
end the proof.
In the following, we consider that the drift term in
(3) is nonlinear which also commonly present in con
trolled dynamic systems. Moreover, if the networked
dynamic systems is homogenous then
the fistructure is identical, otherwise the multi
system is considered as heterogenous. Our main re
sult in multiΣ2is built on the assumption that fi(xi)
is a convex function.
Proposition 3 Let Gbe a fixed undirected graph and
fi(xi)is convex. Under the protocol (5) a homoge
nous/heterogenous multiΣ2based on (3) converges
toward an average trajectory and leads to a finite
time average consensus in the sense of (4).
Proof Let ξi(t) = xi(t)−χ(t). As fiis assumed to
be convex, we have
fi(xi)−1
N
N
i=1
fi(xi)≤fi(xi)−fi1
N
N
i=1
xi
Moreover fiis locally Lipschitz function in an open
set Ω⊂Rncontaining ξ. Therefore
fi(xi)−1
N
N
i=1
fi(xi)≤ ∥fi(xi)−fi(χ)∥
≤c∥ξi∥
such that c > 0is the Lipschitz’s constant. Now, for
convenience the Lyapunov function is given by (10),
we prove:
˙
V(ξ(t)) =
N
i=1
(ϕα(ξi))T˙
ξi
≤c
N
i=1
ϕT
α(ξi)ξi−k
4(α+ 1) 2α
α+1 V2α
α+1
≤ −V(ξ(t)) 2α
α+1 k
4(α+ 1) 2α
α+1 −c(V(ξ(t)))1−α
α+1
Or 1−α
α+1 >0and Vis a continuous function which
takes 0of the origin V(0) = 0 there exists an open
neighborhood Ωsuch that ξ(t)∈Ω
˙
V(ξ(t)) ≤ −k
8(α+ 1) 2α
α+1 V(ξ(t)) 2α
α+1 (16)
At this stage, one concludes that the multiΣ2estab
lished from with the protocol (3) with the protocol (5)
lead to a finitetime average consensus. Note that if
the convexity property of fiis not satisfied, the alter
native is to linearize each Σ2system and use the same
procedure obtained for a multisystem built from (13).
4 The multisystem finitetime
stabilization
The finitetime stabilization problem in networked
dynamic systems consists to stabilize individually
each system’s equilibrium state under some connec
tion rules. Then we consider dynamic systems in net
work with continuous nonlinear decentralized feed
back that integrates the graph theory. The following
theoretical framework tackles first to the multiΣ1sta
bilization problem, the results will be extended after
that to the analysis of the multiΣ2stabilization prob
lem.
4.1 The multiΣ1finitetime stabilization
The multiΣ1describes the behavior of drift less sys
tems like kinematic of unicycles and attitude of satel
lites. Further, one considers here that each system is
nonlinear and not necessary fully actuated (dimension
of the input vector is fewer than the system degree of
freedom).
Proposition 4 For a given fixed underacted graph G,
the protocol (6) applied to multiΣ1solves the stabi
lizing problem in finite time.
Proof Let x= (x1, ..., xN)T∈RNn and
u= (u1, ..., uN)T∈RNm where xi∈Rnand
ui∈Rm. The networked systems (2) under the stabi
lizing protocol (17)
u=−(L⊗In)(IN⊗C(xi))ϕα(x)(17)
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Using the Kronecker product properties
˙
x= (IN⊗B(xi))u
=−(IN⊗B(xi))(L⊗In)(IN⊗C(xi))ϕα(x)
=−(L⊗˜
B(xi))ϕα(x)
(18)
It is obvious from (18) that the equilibrium is zero.
The goal is to prove that xreaches this equilibrium in
finite time. Taking the Lyapunov function V:RNn →
R+where ∀x∈RNn
V(x) = 1
1 + αxTϕα(x)(19)
which is positive definite with respect to x, Now, the
time derivative along the trajectories of (18) lead to
˙
V(x) = ϕT
α(x)dx
dt
=−ϕT
α(x)(L⊗˜
B)ϕα(x)
Let
D(xi) =
0n
γ2(xi)
...
γN(xi)
where 0n=diag{0, ..., 0} ∈ Rn×nand
∀j= 2, ..., N,γj(xi) = λj(L)ϱn(xi)where
ϱn(xi) = diag{0, µ2(xi), ..., µn(xi)} ∈ Rn×n. De
notes that µ2(xi), ..., µn(xi)are the eigenvalues of
the matrix ˜
B(xi), given in increasing order. λj(L)
is the jème eigenvalues of L. Let λ2(L), ..., λN(L)in
increasing order.. By Lemma 3, λ2(L)>0. We have
∀xi,λ2µ2(xi)>0.
Further, since L⊗˜
B∈RNn×N n is symmetric matrix,
then there exist an orthogonal matrix P∈RNn×Nn
such that L⊗˜
B=PTD(xi)P. Let zα=P ϕα(x).
Then
˙
V=−zT
αDzα
≤ −λ2µ1(xi)∥zα∥2
≤ −λ2µ1(xi)∥ϕα(x)∥2(20)
where
λ2µ1(xi) = min
zα⊥1N n
zT
αDzα
zT
αzα
.
Let k=min
xi∈RNλ2µ1(xi)>0and
x=1N⊗xi= (˜x1, ..., ˜xNn)T, we obtain
˙
V≤ −k
Nn
i=1
|φα(˜xi)|2
≤ −k
Nn
i=1
|˜xi|2α
≤ −kNn
i=1
|˜xi|α+1
2α
α+1
by Lemma 4,(21)
which leads to
˙
V≤ −k(α+ 1) 2α
α+1 V2α
α+1 (22)
Or 0<2α
α+1 <1et k(α+ 1) 2α
α+1 >0, by Lemma 1
the above differential equation shows that Vreaches
zero in finite time
T∗(x(0)) = (α+ 1)V(x(0))1−α
α+1
(1 −α)k(α+ 1) 2α
α+1
Therefore, based on (2), the multiΣ1under the pro
tocol (6) reaches zero in finitetime.
4.2 The multiΣ2finitetime stabilization
Recall that the multiΣ2system is based on the fol
lowing dynamic with nonlinear drift terms
Σ2:˙
xi=fi(xi) + B(xi)ui.(23)
where the fistructure can be taken different for each
system. In this case, we are in presence of heteroge
neous multisystem. We assume at first that
ϕT
α(xi)fi(xi)≤0.(24)
and we propose the following,
Proposition 5 Suppose that the inequality (24) is sat
isfied. For a given fixed underacted and connected
graph G, the protocol (6) associated to multiΣ2
solves the stabilizing problem in finite time.
Proof Let x∈RNn and
f(x) = (f1(x1), ..., fN(xN))T. Consider the stabi
lizing protocol (17), the multiΣ2dynamic becomes:
˙
x=f(x)−(L⊗˜
B)ϕα(x)(25)
Using the Lyapunov function (19), its time derivative
is as
˙
V(x) = ϕT
α(x)f(x)−ϕT
α(x)(L⊗˜
B)ϕα(x)(26)
From hypothesis (24), the first term in (26) is nega
tive. The remaining terms in (26) must verify the in
equality given by (22). So, we conclude that the origin
(25) is finitetime stable.
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Remark 1 In practice condition (24) on the drift
term isn’t often verified. For this propose this con
dition can be relaxed by the following proposition.
Proposition 6 If fiis locally Lipshitz function and
fi(0n) = 0n, given an underacted and connected
graph G, the multiΣ2origin from (23) and (6) is lo
cally finitetime stable.
Proof Recall that the time derivative of the Lyapunov
candidate function (19)
˙
V(x) = ϕT
α(x)f(x)−ϕT
α(x)(L⊗˜
B)ϕα(x)
≤c∥ϕT
α(x)x∥ − ϕT
α(x)(L⊗˜
B)ϕα(x)
(27)
where c > 0is the Lipshitz’s constant.
Let x=1N⊗xi= ( ˜x1, . . . , ˜xNn)T, consequently
from (21), the inequality (27) permits to write
˙
V(x)≤cNn
i=1
|˜xi|α+1−kNn
i=1
|˜xi|α+1
2α
α+1
≤ −V2α
α+1 [k(1 + α)2α
α+1 −cV 1−α
α+1 ]
(28)
where k=min
xi∈RNλ2µ1(xi)defined in the proof of
Proposition 4. Since 1−α
α+1 >0and Vis continuous
function which takes 0at the origin, there exists an
open neighborhood Ω⊂RNn of the origin that
permits to write
˙
V(x)≤ −k(α+ 1) 2α
α+1
2[V(x)] 2α
α+1 (29)
by Lemma 1, Vreaches zero at an estimated finite
time
T∗(x(0)) = (α+ 1)V(x(0))1−α
α+1
2(1 −α)k(α+ 1) 2α
α+1
Therefore, based on (23) and (6), the multiΣ2origin
is finitetime stable.
From the proposed stabilizing protocol, we may
conclude that the stability of each agent was asserted
from the networked behavior of the group. Further,
the drift term is not present in the protocol, however
along the proofs, this term is tackled by the control
and sufficient conditions on this term were introduced
to guarantee the multisystem stability. Note that in
individual dynamic system stability problem, the drift
term must be compensated by the controlinput. Here,
the stability of each agent is obtained from the stable
behavior of the group. This analysis is supported by
the following examples.
5 Illustrative examples
In order to validate the above theoretical framework,
some examples are presented in simulation and ana
lyzed. The multiunicycle kinematics is taken in view
of the multiΣ1system. Further as multiΣ2exam
ples, we propose to take a multisecondorder dynam
ics as system with linear drift term and multiple pen
dulums integrating nonidentical nonlinear drift terms.
The cited examples are expected to achieve finite
time average consensus. At the second stage of the
given numerical simulations, the networked dynam
ical systems stability is handled by tests on multi
unicycle. For consensus and stability objectives, the
undirected fixed networked topology (binary graph)
is shown by Fig.1
Figure 1: Gfor a system with 4 agents.
5.1 The multisystem finitetime consensus
results
Three illustrative examples are considered here where
the multiunicycle that represents the networked sys
tems modeled by (2), a multisystem based on second
order dynamic which imply a networked multimodel
as in (13), and a multipendulum example as in (3).
Each associated protocol is deduced from (5).
a) Average consensus in multiunicycle
Consider Nwheeled mobile robots (unicycles)
where the ith nonholonomic kinematic model is
as:
˙xi
˙yi
˙
θi
=cos(θi) 0
sin(θi) 0
0 1˙ui
˙wii= 1 . . . N
(30)
where (xi, yi, θi)denotes the position and the ori
entation in a an inertial frame. The inputs uiand
wiare the linear and angular velocities, respec
tively.
Let B=cos(θi) 0
sin(θi) 0
0 1and
C=cos(θi)sin(θi0
−sin(θi)cos(θi0Based on Proposi
tion 1, the finitetime average consensus problem
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can be achieved through the following protocol
ui=−
N
j=1
aij ϕα(xi−xj)cos(θi)−
N
j=1
aij ϕα(yi−yj)sin(θi)
(31)
wi=
N
j=1
aij ϕα(xi−xj)sin(θi)−
N
j=1
aij ϕα(yi−yj)cos(θi)
(32)
where φαis defined in section 2and aij are associ
ated to the graph in Fig. 1. The simulation results
are limited to N= 4 that integrate the following
initial conditions
(x1, y1, θ1)(t= 0) = (14,2, π)
(x2, y2, θ2)(t= 0) = (−4,2,−π
2)
(x3, y3, θ3)(t= 0) = (10,8,π
2)
(x4, y4, θ4)(t= 0) = (−10,−8,0)
0 5 10 15
−10
−5
0
5
10
15
xi
time[sec]
x1
x2
x3
x4
average(xi)
Figure 2: Average consensus of position xifor 4uni
cycles as multiΣ1
The numerical simulations are performed using
(30) and protocols (31)(32). The results of figures
Fig. 23 evolve according to the developed theo
retical results of multiΣ1. The common value is
also the average of the unicycles initial conditions.
The
||(xi, yi)−(ave(xi(0)), ave(yi(0)))|| converges in
finitetime to zero as show in figure Fig.4.
b) Average consensus in multisecondorder dy
namicss
A commonly used example in the literature is an
0 5 10 15
−8
−6
−4
−2
0
2
4
6
8
yi
time[sec]
y1
y2
y3
y4
average(yi)
Figure 3: Average consensus of position yifor 4uni
cycles as multiΣ1
0 5 10 15
0
2
4
6
8
10
12
14
16
k(xi, yi)−(ave(xi), ave(yi))k
time[sec]
Figure 4: Convergence of ∥(xi, yi)−(ave(xi)−
ave(yi))∥
agent with a secondorder dynamic (we can see
[18])
˙xi=−vi˙vi=−uii= 1, . . . , N (33)
where xi, vi∈Rare the states and ui∈R
is the control input. The dynamic (33) takes the
form given by (13) with xi=xi
vi,fi(xi) =
0 1
0 0xiand B=0
1
For the protocol (5) we take C= (1 1). From
Proposition 2 results, protocols that achieve finite
time average consensus are such that
ui=−
N
j=1
aij (φα(xi−xj) + φα(vi−vj)) (34)
Let us take N= 4. The control parameter is taken
α= 0.5, and each agent initial vector of states is
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as
(x1, x1, x3, x4)(t= 0) = (5,10,1,−5) (meter)
and
(v1, v1, v3, v4)(t= 0) = (2,−1,8,−4) (meter/second)
For i= 1, . . . , 4, xi(Fig.5) and vi(Fig.6) consent
an average trajectory.
0 5 10 15 20
−10
−5
0
5
10
15
20
25
30
xi
temps[sec]
moy(xi)
Figure 5: A reached average trajectory in positions by
4 secondorder dynamics.
0 5 10 15 20 25 30
−4
−2
0
2
4
6
8
velocity
time[sec]
y1
y2
y3
y4
average:yi
Figure 6: A reached average trajectory in velocities
by 4 secondorder dynamics.
Remark 2 Other processes can be studied, and
where the average is an agreement value of states
like a common temperature of sensors where fluc
tuations of data is important. The energy con
sumption is also an important factor for stability of
electric generators in networks. As example, for a
multisecondorder dynamics, the kinetic energies
consent an average, and this is shown by figure
Fig.7.
c) Average consensus in multipendulum dynam
ics
0 5 10 15 20 25 30
0
5
10
15
20
25
30
Ec
temps[sec]
moy(Ec)
Figure 7: The average of kinetic energies like consen
sus for 4 secondorder dynamics.
Consider a set of Npendulum with the following
model
¨
θi=−g
li
sin(θi)− − ψi
mili
˙
θi+ui(35)
where mi, gi, liand ψiare positive constants. For
this system the drift term issues from the first order
differential form (see (3)) is
fi(θi,˙
θi) =
˙
θi
−g
li
sin(θi)−−ψi
mili
˙
θi
we can easily check the convexity condition for
the drift term fi. Following to the subsequent the
oretical analysis (see Proposition 3), taking C=
(1 1), a protocol that solves the finitetime aver
age consensus for multipendulum is as
ui=−
N
j=1
aij (φα(θi−θj) + φα(˙
θi−˙
θj)) (36)
This set of N= 4 pendulums is analyzed. As het
erogenous multisystem, the 4pendulum parame
ters aren’t similar. Thus, m1= 1, m2= 2, m3=
3and m4= 4 (Kg). The standard gravity vector
is g= 9.8(m.s−2), the lengths li= 1 (m) and the
coefficient ψi= 0.1(Kg.m2.s−1).Initial condi
tions are such that θi= (−0.8,0.4,1,2,1.6) (rad)
and ˙
θi= (0,0,0,0)(rad.s−1).Clearly from fig
ures in Fig. 89, the synchronization toward the
average trajectory of 4 pendulums in angular po
sitions and velocities are obtained. It is important
to note that the average is time varying and the
multisystem of pendulums is heterogeneous with
respect to the proposed physical parameters. This
confirm the theoretical results of Proposition 3.
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0 2 4 6 8 10
−1.5
−1
−0.5
0
0.5
1
1.5
2
θi
temps[sec]
moy(θi)
Figure 8: The average of kinetic energies like consen
sus for 4 secondorder dynamics.
0 2 4 6 8 10
−5
−4
−3
−2
−1
0
1
2
3
4
5
˙
θi
temps[sec]
moy(˙
θi)
Figure 9: The timevarying average of angular veloc
ities consent by 4 pendulums.
5.2 The multisystem finite time stability
results
We consider a multiunicycle which represents the
networked system modeled by (2) (driftless). The
associated protocol is deduced from (6) and the
graph is in Fig.1. From Proposition 4, the finitetime
stability problem is achieved for the control matrix
C=cos(θi)sin(θi0
−sin(θi)cos(θi0that leads to the
stabilizing controlinputs
ui=−
N
j=1
aij ϕα(xi−xj)cos(θi)−
N
j=1
aij ϕα(yi−yj)sin(θi)
(37)
wi=
N
j=1
aij ϕα(xi−xj)sin(θi)−
N
j=1
aij ϕα(yi−yj)cos(θi)
(38)
where φαis defined in section 2and aij are asso
ciated to the graph in Fig. 1. Taking N= 4, the initial
conditions are as T
(x1, y1, θ1)(t= 0) = (4,2,π
4)
(x2, y2, θ2)(t= 0) = (12,−10,−π
2)
(x3, y3, θ3)(t= 0) = (10,−8,2π
3)
(x4, y4, θ4)(t= 0) = (−10,−14, π)
The results of stabilization are sketched in figures
Fig.1011 and the stabilizing protocols are given by
figures Fig.1213 which confirm the stability of each
unicycle at the origin with continuous control feed
back.
0 5 10 15
−10
−5
0
5
10
15
xi
time[sec]
x1
x2
x3
x4
Figure 10: Finitetime stability of xias positions of 4
unicycles
Figure 11: Finitetime stability of yias positions of 4
unicycles
6 Conclusion
For networked dynamic systems affine in the con
trol vector, two protocols are proposed and theoret
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0 5 10 15
−15
−10
−5
0
5
10
15
20
25
control input ui
time[sec]
u1
u2
u3
u4
Figure 12: Stabilizing inputs uiof 4 unicycles
0 5 10 15
−20
−15
−10
−5
0
5
10
15
control input wi
time[sec]
w1
w2
w3
w4
Figure 13: Stabilizing inputs wiof 4 unicycles
ically analyzed with respect to two types of nonlin
ear dynamic models. For a nonlinear driftless multi
system, necessary conditions on the control matrix are
derived that assert finitetime average consensus to
ward a predefined agreement value, obtained from the
multisystem initial conditions. However, for multi
system integrating drift terms, sufficient conditions
on the drift term are discussed, and when they as
sociated to the protocol solve a finitetime average
consensus where as a result an average trajectory is
followed by the group. Further, our stability results
in networked dynamic systems overcome the individ
ual stability analysis of each system where some ob
structions for the agent’s stability at the origin occur.
It is well known that an unicycle doesn’t verify the
Brockett’s necessary condition and the stabilization
at the origin isn’t possible with feedbacks that de
pend only on states. Here, due to the interconnection,
the multiunicycle stability result implies the stability
of each unicycle with smooth and bounded control
inputs. The results of the paper can be extended using
a directed graph while one may address the problem
of consensus and stability for heterogenous systems
based on the two fundamental dynamic models.
References:
[1] Lars, B., Cremean and Richard, M., Murray. Sta
bility Analysis of Interconnected Nonlinear Sys
tems Under Matrix Feedback, 42nd IEEE Confer
ence on Decision and Control, Hawaii, USA, pp.
30783082.
[2] Hong, Y. and Jiang, Z. P, Finitetime stabiliza
tion of nonlinear systems with parametric and dy
namic uncertainties, IEEE Transactions on Auto
matic, Vol.51 No.12, 2006, pp. 19501956.
[3] Bhat, S. P., and Bernstein, D. S. Finite Time Sta
bility of Continuous Autonomous Systems, SIAM
J. Control Optimization, Vol.38, No.3, 2000, pp.
751766.
[4] Hong, Y. , Xu. Y and Huang, J. Finitetime con
trol for robot manipulators, Systems and Control
Letters, Vol.46 No.4, 2002, pp. 243–253.
[5] Vicsek, T. , Czzirok, A., BenJacob, E., Cohen, I.
and Schochet, O. Novel, type of phasetransition
in a system of selfdriven particles, Phys. Rev.
Lett, Vol.75, No.6, 1995, pp. 1226�1229
[6] Jadbabaie, A. , Lin, J. and Morse, A.S, Coordi
nation of groups of mobile autonomous agents
using nearest neighbor rules, Proceeding of the
42st IEEE Conference on Decision and Control,
Vol.48, No.9, 2003, pp. 9881001.
[7] Saber, R. O., and Murray, R. M. B , Consen
sus protocols for networks of dynamic agents, in
Proc. 2003 Am. Control Conf, 2003, pp. 951 956.
[8] OlfatiSaber, R., and Murray, R. M, Consensus
problems in networks of agents with switching
topology and timedelays, IEEE Trans. Automat.
Contr., Vol49 No.9, 2004, pp. 1520�1533.
[9] Ren, W. and Beard, R. W, Consensus seeking
in multiagent systems under dynamically chang
ing interaction topologies, IEEE Trans. Automat.
Contr, Vol.50, No.5, 2005, pp. 655�661.
[10] Cortes, J, Finitetime convergent gradient flows
with applications to network consensus., Auto
matica, Vol.42, No.11, 2006, pp. 1993 2000.
[11] Hui, Q., Hadddad, W. M. and Bhat, S.P, Finite
time semi stability and consensus for nonlin
ear dynamical networksIEEE Transactions Auto
matic Control, Vol.53, No.8, 2008, pp. 1887�
1890.
[12] Xiao, F. and Wang, L, State consensus for
multiagent systems with switching topologies
and timevarying delays, International Journal of
Control, Vol.79, No.10, 2006, pp. 1277�1284.
WSEAS TRANSACTIONS on COMPUTERS
DOI: 10.37394/23205.2022.21.5
Rhouma Mlayeh
E-ISSN: 2224-2872
42
Volume 21, 2022
[13] Zoghlami, N., Beji, L., Mlayeh, R. and Abi
chou, A. Finitetime consensus and stability of
networked nonlinear systems, in Proc. of IEEE
Conference on Decision and Control. Florence,
Italy, 2013, pp. 66046609.
[14] Zoghlami, N., Beji, L., Mlayeh, R. and Abichou,
A. Finitetime consensus of networked nonlinear
systems under directed graph, The 13th European
Control Conference (ECC), Strasbourg, France,
Vol.X, No.X, 2014, pp. 546 551.
[15] Zoghlami, N., Beji, L., Mlayeh, R. and Abichou,
A. Finitetime average consensus in networked
dynamic systems, in Proc. of IEEE Conference
on Decision and Control. Los Angeles, Califor
nia, 2014, pp. 44864490.
[16] Wang, L. and Xiao, F. Finitetime consensus
problems for networks of dynamic agents, IEEE
Transactions Automatic Control, Vol.55, No.4,
2010, pp. 950�955.
[17] F. Xiao, L. Wang, J. Chen, and Y. Gao, Finite
time formation control for multiagent systems,
Automatica, Vol.45, No.11, 2009, pp. 2605�
2611.
[18] Xiaoli, W. and Yiguang, H. Finitetime consen
sus for multiagent networks with secondorder
agent dynamics, In IFAC World Congress. Soeul,
Korea, 2008, pp. 15185�15190.
[19] Yongcan and Wei, R, Finitetime Consensus for
Secondorder Multiagent Networks with Inher
ent Nonlinear Dynamics Under Fixed Graph, In
Proc. IEEE Conf. Decision and Control. Orlando,
FL, USA, 2011.
[20] Zhu. M and Martnez. Discretetime dynamic av
erage consensus, Automatica, Vol.46, No.2, 2010,
pp. 322�329.
[21] Fangcui, J. and Long, W. Finitetime weighted
average consensus with respect to a monotonic
function and its application, Control Letters,
Vol.60, No.9, 2011, pp. 718�725.
[22] Shahram, N., Masoud, S., Mohammad, B.M,
Dynamic average consensus via nonlinear proto
cols, Automatica, Vol.48, No.9, 2012, pp. 2262
2270.
[23] Shuai, L., Tao, L., Lihua, X., Minyue, F.,
and JiFeng, Z., Continuoustime and sampled
databased average consensus with logarithmic
quantizers., Automatica, Vol.49, No.11, 2013,
pp.3329 3336.
[24] Hardy, G.,Littlewood, J., and Polya, G, Inequal
ities, Cambridge University Press, 1952.
[25] OlfatiSaber, R., Fax, R., and Murray, v, Con
sensus and Cooperation in Networked Multi
Agent Systems, Proceedings of the IEEE, Vol.95,
No.1, 2007, pp. 215 233,.
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