53rd IEEE Conference on Decision and Control
December 15-17, 2014. Los Angeles, California, USA
Containment Control for Networked Lagrangian Systems Under a Directed
Graph and Communication Constraints
Abdelkader Abdessameud, Ilia G. Polushin, and Abdelhamid Tayebi
Abstract— In this paper, we study the containment control
problem of networked uncertain Lagrangian systems with
intermittent communication in the presence of communication
delays and possible information loss. Specifically, we present
an adaptive distributed control algorithm such that a team of
followers asymptotically converge to the convex hull spanned by
multiple non-stationary leaders. The interconnection between
the systems is represented by a directed graph. Sufficient
conditions are presented such that the control objective is
reached while communication between agents is allowed only
at some discrete instants of time in the presence of irregular
communication delays and packet dropout. Simulation results
are given to show the effectiveness of the proposed control
scheme.
I. I NTRODUCTION
The distributed coordination problem of multiple mechanical systems, modeled by Euler-Lagrange equations, has
received a growing interest during the last decade due to
the broad range of applications including multiple unmanned
vehicles missions and spacecraft formation flying [1]. The
main idea behind distributed coordination is to ensure a collective behavior using local interaction between the systems.
This interaction, in the form of information exchange, is
generally performed using communication between agents
depending on their interconnection graph topology. Despite
the rapid development in the coordination of linear multiagent systems (see, for instance, [2]-[3]), coordinating a team
of mechanical systems faces various challenges due to the
nonlinear dynamics, uncertainties and disturbances. These
challenges become prominent in practical situations imposing additional constraints on the interconnection topology
and the communication process between agents, which can
be discontinuous and subject to delays and packet loss.
Recent work in the coordination of networked lagrangian
systems focus on the leaderless synchronization problem,
where all systems or agents negotiate to reach an agreement
on a common final state. In particular, solutions to this
problem have been proposed in [4]-[6] in the presence of
constant communication delays and in [7]-[8] in the presence
of irregular time-varying communication delays. Another
problem that has been widely addressed is the leader-follower
problem where some agents named followers cooperatively
track a final state dictated by a group leader or multiple
This work was supported by the Natural Sciences and Engineering
Research Council of Canada (NSERC).
The authors are with the Department of Electrical and Computer Engineering, University of Western Ontario, London, Ontario, Canada. The
third author is also with the Department of Electrical Engineering, Lakehead University, Thunder Bay, Ontario, Canada. [email protected],
[email protected], [email protected]
978-1-4673-6090-6/14/$31.00 ©2014 IEEE
leaders. In the case of a single leader, cooperative tracking
algorithms have been developed for Lagrangian systems in
the presence of constant communication delays [5], [9]-[10]
under the assumption that the leader’s states are available to
all the followers. This assumption is relaxed in [11]-[12] by
allowing only a subset of followers to access the position
of a non-stationary leader, however, without communication
constraints. In the case of multiple leaders, the containment
control problem has been studied with stationary leaders
[13]-[14] and dynamic leaders [15]-[16] under different
interconnection topology graphs, yet by assuming ideal communication between the systems.
Two important remarks on the above results are worth
mentioning. First, the above delay-robust leaderless synchronization solutions impose a zero final velocity for the
team members, and the leader-follower problem with nonstationary leader(s) has not been addressed in the presence
of communication delays. An exception can be found in
[5], [9]-[10] where it is assumed that all followers have
full access to the leader’s states, which is restrictive from a
practical viewpoint. Second, all the above coordination laws
for Lagrangian systems assume continuous-time communication between agents. In practical situations, communication
between agents can be performed intermittently at some
discontinuous time-intervals or instants due to, for instance,
environmental constraints and temporary communication link
failures. In the presence of sufficiently small constant communication delays and bounded packet dropout, the authors
in [17] have proposed consensus algorithms for single integrator dynamics with intermittent communication. Coordinated control algorithms with intermittent communication
have also been proposed for multi-agent systems with more
general linear dynamics in [18]-[20] and globally Lipschitz
nonlinear systems in [21], yet without communication delays.
The contribution of this paper consists in designing a
containment control algorithm for networked uncertain Lagrangian systems with multiple non-stationary leaders under
the assumption that the communication between agents is
intermittent and subject to irregular communication delays
and possible packet loss. The main objective is to drive
the followers to the convex hull spanned by the moving
leaders, while the leaders’ states are transmitted to only a
subset of followers. Sufficient conditions are derived such
that our objective is attained without imposing an undirected
interconnection graph between agents. These conditions can
be easily satisfied with a suitable choice of the control gains.
Simulation results are given to illustrate the effectiveness of
the obtained theoretical results.
2938
II. BACKGROUND
AND
P ROBLEM F ORMULATION
A. System model
Consider a team of n agents with m followers, labeled 1
to m, and (n − m) ≥ 1 leaders labeled m + 1, . . . , n. The
dynamics of the followers are described by Euler-Lagrange
equations of the form
Mi (qi )q̈i + Ci (qi , q̇i )q̇i + Gi (qi ) = ui ,
(1)
for i ∈ F := {1, ..., m}, where qi ∈ RN is the vector of
generalized configuration coordinates, Mi (qi ) ∈ RN ×N is
the symmetric positive-definite inertia matrix, Ci (qi , q̇i )q̇i is
the vector of centrifugal/Coriolis forces, Gi (qi ) is the vector
of potential forces, and ui is the vector of torques associated
with the i-th follower. The leaders are assumed to evolve
with constant velocity, i.e., for all i ∈ L := {m + 1, . . . , n},
we assume that q̇i ≡ vi , for vi ∈ RN and some initial
conditions, where qi ∈ RN denotes the vector of generalized
configuration coordinates of the i-th leader.
Some common properties of Euler-Lagrange systems (1)
considered throughout the paper are as follows.
P.1 Mi (qi )x + Ci (qi , q̇i )y + Gi (qi ) = Yi (qi , q̇i , x, y)Θi ,
for all vectors x, y ∈ RN , where Yi (qi , q̇i , x, y) is
a regressor and Θi ∈ Rk is the vector of constant
parameters associated with the ith agent.
P.2 Ṁi (qi ) − 2Ci (qi , q̇i ) is skew symmetric.
P.3 There exists kci ≥ 0 such that |Ci (qi , x)y| ≤ kci |x| · |y|
holds for all qi , x, y ∈ RN . Here, | · | denotes the
Euclidean norm of a vector. In addition, Mi (qi ) and
Gi (qi ) are bounded uniformly with respect to qi .
The interconnection topology between the n agents is
modeled by the directed graph G = (N , E, A). The set
N := F ∪ L is the set of nodes or vertices, describing the
set of agents in the network, E ∈ N × N is the set of
ordered pairs of nodes, called edges, and A = [aij ] ∈ Rn×n
is the weighted adjacency matrix. An edge (j, i) ∈ E is
represented by a directed link (an arrow) from node j to
node i, and indicates that agent i can obtain information
from agent j but not vice versa; in this case, we say that
j and i are neighbors (even though the link between them
is directed). The weighted adjacency matrix is defined such
that aii := 0, aij > 0 if (j, i) ∈ E, and aij = 0 if (j, i) ∈
/ E.
The Laplacian matrix L := [lij ] ∈ Rn×n associated
to
the
P
directed graph G is defined such that: lii = nj=1 aij , and
lij = −aij for i 6= j. A directed path (of length p) from
j to i is a sequence of edges in a directed graph of the
form (j, l1 ), (l1 , l2 ), . . . , (lp−1 , lp ), with lp = i, where for
p > 1 the nodes j, l1 , . . . , lp−1 ∈ N are distinct. Consider
the following assumption on the interconnection graph.
Assumption 1: For each node f ∈ F , there exists at least
one node l ∈ L such that a directed path from l to f exists
in G.
Assumption 1 implies that for each follower, there exists at
least one leader having a directed path to that follower. On
the other hand, since the motion of each leader is independent
from all other agents, we have (i, l) ∈
/ E for each i ∈ N
and l ∈ L; the leaders do not receive information from any
other node in the team. Consequently, the Laplacian matrix
associated to G takes the form
L1
L2
L=
∈ Rn×n . (2)
0(n−m)×m 0(n−m)×(n−m)
The following Lemma gives some useful properties of the
laplacian matrix (2).
Lemma 1: [16] Under Assumption 1, all the eigenvalues
of L1 have positive real parts, each entry of −L−1
1 L2 is
nonnegative, and all row sums of −L−1
1 L2 are equal to one.
B. Communication Process
In this paper, it is assumed that the communication between agents is intermittent, i.e., may be performed only
over a certain sequence of discrete time instants, and is
subject to time-varying communication delays and possible
information losses. Specifically, there exists a sequence of
communication instants tk := kT ∈ R+ , k ∈ Z+ =
{0, 1, . . .}, where T > 0 is a fixed sampling period and
is common for all agents, such that each agent is allowed to
send its information to all or some of its neighbors at instants
tk , k = 0, 1, . . .. In addition, for each pair (j, i) ∈ E, there
(j,i)
exist a sequence of communication delays (τk )k∈Z+ that
take values in {R+ ∪ +∞} such that the information sent by
agent j at instant tk is available to agent i starting from the
(j,i)
(j,i)
instant tk + τk . Also, it is possible that τk
= +∞ for
some k ∈ Z+ , which corresponds to a situation where the
agent j has not sent information at instant tk to neighbor i
at all, or the corresponding information was never delivered
possibly due to packet loss in the communication channel.
The following assumption is imposed on the communication
process between neighboring agents.
Assumption 2: For each (j, i) ∈ E, there exist numbers
k ∗ ∈ N, h ≥ 0, and an infinite strictly increasing sequence
(j,i) (j,i)
K(j,i) := {k0 , k1 , . . .} ⊂ {0, 1, . . .} satisfying
(j,i)
(j,i)
(j,i)
i) k0 ≤ k ∗ , and kl+1 − kl
≤ k ∗ , l ∈ {0, 1, . . .},
(j,i)
(j,i)
ii) τk ≤ h for each k ∈ K
.
Assumption 2 implies that, for each pair (j, i) ∈ E, and per
any k ∗ consecutive sampling instants, there exists at least
one sampling instant at which agent j has sent information to
agent i, and this information has been successfully delivered
with delay less than or equal to h. Assumption 2 also implies
that, for each pair (j, i) ∈ E, the maximal interval between
two consecutive instants when agent i receives information
from agent j is less than or equal to
h∗ := k ∗ T + h.
(3)
C. Problem statement
Consider the n systems, described in Section II-A, interconnected according to a directed graph G and the communication process between agents satisfies Assumption 2. Also,
suppose that the vectors of the parameters Θi , i ∈ F , defined
in property P.1 are unknown. Let qF and qL be the column
stack vectors of qi , i ∈ F , and qi , i ∈ L, respectively, and
let XqL (t) := {qm+1 (t), . . . , qn (t)}. The objective of this
work is to design a distributed adaptive control scheme such
2939
that all followers converge to the convex hull spanned by the
leaders. Formally, it is required that
d(qi (t), χ[XqL (t) ]) → 0,
for i ∈ F ,
(4)
for t → +∞, where for a point x and a set M , d(x, M )
denotes the distance between P
x and M , i.e., d(x, M ) :=
p
inf y∈M
|x
−
y|,
and
χ[X]
:=
{
j=1 αj xj | xj ∈ X, αj ≥
Pp
0,
j=1 αj = 1} denotes the convex hull of the set
X := {x1 , . . . , xp } [16]. Under Assumption 1, the result
of Lemma 1 implies that −(L−1
1 L2 ⊗ IN )qL is within the
convex hull spanned by the leaders [16], where IN is the
N × N identity matrix. Therefore, objective (4) is reached
if one guarantees that (qF (t) + (L−1
1 L2 ⊗ IN )qL (t)) → 0
as t → +∞.
III. D ISTRIBUTED C ONTAINMENT C ONTROL D ESIGN
In this section, we present a distributed adaptive control
law that achieves (4) using intermittent communication between the systems in the presence of time-varying communication delays and possible packet loss. To this end,
suppose that the information that can be transmitted from
agent j to agent i at t = kT , for all (j, i) ∈ E, is given
(i)
by qj (k) = [k, qj (kT ), v̂j (kT )] where k is the time-stamp,
i.e., the sequence number at which information was sent, qj
is the generalized coordinates of the j-th agent (including the
leaders), v̂j = q̇j for j ∈ L, and v̂j , for j ∈ F , denotes the
estimate of the leaders’ generalized coordinates derivatives
obtained by the j-th follower according to an algorithm
described below. Moreover, for each pair (j, i) ∈ E and each
(j,i)
denote the largest integer number
time instant t ≥ 0, let kt
(i) (j,i)
such that qj (kt ) is the most recent information of agent
j that is already delivered to agent i at t, i.e.,
(j,i)
kt
(j,i)
:= max{k ∈ Z+ : kT + τk
≤ t}.
(5)
(j,i)
can be obtained by
It should be noted that the number kt
a simple comparison of the received time stamps.
Since only a subset of followers in the team have access
to the leaders’ states, each follower will estimate the leader’s
velocities using the following algorithm
v̂˙ i = σi ,
(6)
with
(
σ̇i
ξ˙i
=
=
(i)
−kiσ σi − λσi (v̂i − ξi )
Pn
(i)
−ξi + κ1i j=1 aij v̂j
t → +∞ for arbitrary initial conditions if the control gains
satisfy
Pm
j=1 aij
(1 + 2 · h∗ ),
for i ∈ F ,
(8)
µi >
κi
where h∗ is defined in (3) and µi
:=
− max (Re(µi,1 ), Re(µi,2 )), µi,1 , µi,2 are the roots of
p2 + kiσ p + λσi = 0.
Proof: The proof is omitted due to space limits.
Now, consider the following control law for the ith follower system
Γi
˙
Θ̂i
⊤
= −Πi Yi (qi , q̇i , q̇ri , q̈ri ) (q̇i − q̇ri ),
(9)
(10)
where kis > 0, Πi is a symmetric positive-definite matrix, Θ̂i
is an estimate of Θi , Yi (qi , q̇i , q̇ri , q̈ri ) is defined in Property
P.1. In (9)-(10), the vector q̇ri represents a reference velocity
given by
(11)
q̇ri := ηi + v̂i ,
with v̂i being obtained from (6)-(7), and ηi satisfying
(
η̇i = −kiη ηi − ληi (qi − ψi ),
Pn
(12)
(i)
(i)
ψ̇i = −ψi + κ1i j=1 aij (v̂j (t) + qj (t)),
(i)
where v̂j (t), aij , κi are defined above, kiη , ληi are strictly
(i)
positive scalar gains, and qj (t) represents an estimate of
the current position of the j-th agent defined as
(j,i)
(i)
qj (t) := qj (kt
(i)
(j,i)
T ) + v̂j (t) · (t − kt
T ),
(13)
(j,i)
given in (5).
with kt
Theorem 1: Consider the network of n systems described
by (1) and suppose Assumption 1 and Assumption 2 hold.
For each i ∈ F , consider the control algorithm (9)-(13), and
the distributed observer (6)-(7). Then, qF (t) + (L−1
1 L2 ⊗
IN )qL (t) → 0, as t → +∞, for arbitrary initial conditions
if the control gains satisfy (8) and
Pm
j=1 aij
µ̃i >
(1 + 2 · h∗ ),
for i ∈ F ,
(14)
κi
where µ̃i := − max (Re(µ̃i,1 ), Re(µ̃i,2 )), µ̃i,1 , µ̃i,2 are the
roots of p2 + kiη p + ληi = 0.
Proof: Let si := q̇i − q̇ri , for i ∈ F . Using (9)-(10) in
(1), we can write
(7)
(j,i)
for i ∈ F , where v̂j := v̂j (kt T ), for i ∈ N , the gains
entry of the
λσi , kiσ are strictly positive, aij is the (i, j)−thP
adjacency matrix A associated to G, and κi := nj=1 aij for
i ∈ F . Note that κi 6= 0 for i ∈ F in view of Assumption 1.
Let v̂F denote the column stack vectors of v̂i , i ∈ F , and
consider the following result.
Proposition 1: Consider system (6)-(7) for i ∈ F and
suppose Assumption 1 and Assumption 2 hold. Then, v̂˙ F , v̂F
are uniformly bounded and v̂F (t)+(L−1
1 L2 ⊗IN )q̇L → 0 as
= Yi (qi , q̇i , q̇ri , q̈ri )Θ̂i − kis (q̇i − q̇ri ),
Mi (qi )ṡi + Ci (qi , q̇i )si + kid si = Yi Θ̃i ,
(15)
˙ = −Π Y⊤ s
Θ̃
i
i i i
(16)
for i ∈ F , where Θ̃i = (Θ̂i − Θi ), i ∈ F , and the arguments
of the regressor Yi are omitted. The time derivative of the
−1
Lyapunov function Vi = 12 (s⊤
Θ̃i ), i ∈ F ,
i Mi (qi )si + Θ̃i Π
evaluated along (15)-(16) is obtained as: V̇i = −kis s⊤
i si .
Then, we know that si ∈ L2 ∩L∞ and Θ̂i ∈ L∞ . In addition,
properties P.2 and P.3 guarantee that ṡi ∈ L∞ if q̇ri , q̈ri ∈
L∞ . Also, the result of Proposition 1 guarantees that v̂˙ i , v̂i ,
i ∈ F , are uniformly bounded. Invoking Barbălat Lemma,
2940
we can show that si (t) → 0 as t → +∞, i ∈ F , if ηi , η̇i
are uniformly bounded.
The rest of the proof is based on Theorem 2 given in
the Appendix section. Consider the dynamicP
system (12) for
n
i ∈ F , and let q̃i := qi − ψi , ψ̃i := ψi − κ1i j=1 aij qj , for
i ∈ F . Using the relation si = (q̇i − ηi − v̂i ) with (12), one
can write
η̇i
q̃˙i
ψ̃˙
i
=
=
−ki ηi − λi q̃i
ηi + ψ̃i + ϕi
(17)
(18)
=
−ψ̃i − ϕi + φi
(19)
for i ∈ F , with
φi
= si −
−
ϕi
n
1 X
aij (ηj + sj )
κi j=1
n
1 X
(j,i)
aij (v̂j − v̂j (kt T )),
κi j=1
= si +
n
1 X
(i)
aij (qj − qj ).
κi j=1
(20)
(21)
Now, following similar steps as in the proof of [8, Theorem
2], we can show that the system (17)-(19), for i ∈ F , is inputto-state stable1 (ISS) with respect to its inputs φi and ϕi ,
i ∈ F . We can also verify that the overall system consisting
of all the systems (17)-(19), for i ∈ F , and having m outputs,
given by ȳi = ηi , i ∈ F , and 2m inputs ordered as: u2i :=
ϕi , u2i−1 := φi , for i ∈ F , is input-to-output stable (IOS),
0
with IOS gain matrix Γ0 = {γij
} ∈ Rm×2m given as


1/µ̃i if l = 2i − 1, i ∈ F ,
0
(22)
γil = 2/µ̃i if l = 2i, i ∈ F ,


0
otherwise.
with µ̃i being defined in the theorem.
Furthermore, using (20)-(21) with (13), the relations q̇i =
(i)
(si + ηi + v̂i ), for i ∈ F , q̇j ≡ v̂j ≡ v̂j for j ∈ L, (t −
(j,i)
kt T ) ≤ (k ∗ T + h) := h∗ , and the result of Proposition 1,
one can write
|u2i−1 (t)| ≤
|u2i (t)| ≤
m
X
aij
j=1
m
X
j=1
κi
|ηj (t)| + |δ2i−1 (t)| ,
aij · h∗
κi
|ηj (ς)| + |δ2i (t)|, (24)
sup
(j,i)
ς∈[kt
(23)
T,t]
for i ∈ F , where δj , j ∈ {1, . . . , 2m} are uniformly bounded
vectors that converge asymptotically to zero if the error
vector si converges to zero.
Consequently, one can conclude that the input vectors uj ,
j ∈ {1, . . . , 2m}, satisfy the conditions of Theorem 2 (given
in the Appendix section) with the interconnection matrix
M := {µlj } ∈ R2m×m obtained as
( aij
if l = 2i − 1, j ∈ F , i ∈ F ,
κi
µlj =
aij ·h∗
if l = 2i, j ∈ F , i ∈ F .
κi
(25)
Then, the elements of the closed-loop gain matrix Γ :=
Γ0 ·M = {γ̄ij }i,j∈F in Theorem 2 can be written as follows
γ̄ij :=
m
X
γil0 · µlj =
l=1
aij
(1 + 2 · h∗ ) ,
κi · µ̃i
(26)
which leads us to conclude that ρ(Γ) < 1 if condition (14)
is satisfied.
Therefore, all the conditions of Theorem 2 are satisfied
and one can conclude that ηi , φi and ϕi , for i ∈ F , are
uniformly bounded. In addition, the ISS property of (17)(19) guarantees that η̇i , q̃i and ψ̃i , for i ∈ F , are uniformly
bounded. Consequently, we have si (t) → 0 as t → +∞,
for i ∈ F , which guarantees that δj (t) → 0 at t → +∞,
j ∈ {1, . . . , 2m}.
As a result, one can conclude from Theorem 2 that ηi (t) →
0, φi (t) → 0, ϕi (t) → 0 for i ∈ F as t → +∞. Then, we
know that (q̇i (t) − v̂i (t) → 0 as t → +∞, for i ∈ F , which
implies, form Proposition 1 that q̇F (t)+(L−1
1 L2 ⊗IN )q̇L →
0 as t → +∞. In addition, the ISS property of system (17)(19) implies that q̃i (t) → P
0 and ψ̃i (t) → 0 as t → +∞ for
n
i ∈ F . Using the relation j=1 aij (qi − qj ) = κi (q̃i + ψ̃i ),
i ∈ F , and Lemma 1, we conclude that qF (t) + (L−1
1 L2 ⊗
IN )qL (t) → 0, as t → +∞.
In the case where the leaders are stationary, q̇i = 0 for
i ∈ L, the distributed observer (6)-(7) is not needed and the
following result holds.
Corollary 1: Consider the network of n systems described
by (1) and suppose that Assumption 2 and Assumption 1
hold. For each i ∈ F , consider the control algorithm (9)(13) with v̂k , k ∈ N , in (11)-(13) set to zero. Then, qF (t) +
(L−1
1 L2 ⊗ IN )qL → 0, as t → +∞, for arbitrary initial
conditions if the control gains satisfy (14).
IV. S IMULATION R ESULTS
In this section, we implement the proposed distributed
containment control scheme for a network of ten systems
with F = {1, . . . , 6}, L = {7, . . . , 10}. We consider N = 2
and the dynamic equations of each follower are given in
[8]. The agents in the network are interconnected according
to the directed graph G, and the communication process is
described in Section II-B with the parameter h∗ in (3) being
estimated to be smaller than or equal to 1.3 sec. Also, the
Laplacian matrix associated to G satisfies (2) with
1 The definitions of input-to-state stability (ISS) and input-to-output stability (IOS) of multiple inputs multiple outputs systems can be found in
[22].
2941




L1 = 


6
0
−2
0
0
0
0
6
0
0
0
−2
−2
−2
10
−2
0
0
0
0
0
6
0
−2
0
0
0
−2
12
−2
0
0
0
−2
0
6




,


0
0
0
0
−4
0
0
0
−4
0
−4
0
0
0
−4
0
−4
0
10

t = 30 sec



.


t = 15 sec
t=0
5
0
−5
−10
−10
0
10
20
30
40
Fig. 2. Trajectories of the followers with four moving leaders, qi =
(1) (2)
(qi , qi )⊤ for i ∈ N . The black and red squares denote, respectively,
the positions of the leaders and the followers at different instants of time.
5
(1)
q̃Fi (rad)
0
−5
−10
−15
−20
0
5
10
0
5
10
15
20
25
30
15
20
25
30
6
4
(2)
q̃Fi (rad)
We implement the control scheme in Theorem 1 with
the assumption that all the leaders move in a formation
of square shape with the same constant velocity q̇i (t) =
(1, 0.1)⊤ rad/sec, for i ∈ L and t ≥ 0. It should be noted
that the result of Theorem 1 is valid for distinct constant
leaders’ velocities, however, the convex hull spanned by
the leaders’ positions will increase to infinity in this case.
s
The control gains are selected
p as: Πi = 0.3I
p 5 , ki = 10,
ληi = 20, λσi = 20, kiη = 2 ληi , kiσ = 2 λσi . Note that
this choice
conditions (8) and (14) with
p
p of the gains satisfies
µi = λσi and µ̃i = ληi . Similarly to [16], we define
⊤ ⊤
the containment error vector as q̃F = (q̃⊤
:=
F1 , . . . , q̃F6 )
(2) ⊤
(1)
−1
qF + (L1 L2 ⊗ I2 )qL , where q̃Fi = (q̃Fi , q̃Fi ) ∈ R2 .
Fig. 1 shows that the containment error vectors for all
followers converge asymptotically to zero, which implies that
all followers converge to the convex hull spanned by the
leaders using intermittent communication and in the presence
of time-varying delays and possible packet loss. This is also
demonstrated in Fig. 2, which depicts the trajectories of all
agents in the plane.
We have also implemented the control law in Corollary 1,
where the leaders are assumed stationary. Using the same
above gains, the obtained results in this case are illustrated
in Fig. 3 and Fig. 4, which show that containment control
with stationary leaders is achieved under the constraints on
the communication process considered in this work.
qi
(2)



L2 = 


−4
−4
0
0
0
0
(rad)

2
0
−2
−4
−6
−8
time (sec)
Fig. 3.
5
The position errors of all followers with four stationary leaders.
(1)
q̃Fi (rad)
0
−5
−10
−15
−20
0
5
10
0
5
10
15
20
25
30
15
20
25
30
6
(2)
q̃Fi (rad)
4
2
0
−2
−4
−6
−8
time (sec)
Fig. 1.
The position errors of all followers with four moving leaders.
V. C ONCLUDING R EMARKS
In this work, we presented a solution to the containment
control problem of multiple uncertain Euler-Lagrange sys-
tems under a directed graph in the presence of communication constraints. In particular, we proposed a distributed
continuous-time adaptive control algorithm such that all the
followers converge to the convex hull spanned by the nonstationary leaders. We also presented sufficient conditions
such that our control objective is achieved while the information exchange between all agents is performed at irregular
discrete time-intervals in the presence of varying delays and
possible packet dropout. To the best of our knowledge, the
containment control problem with non-stationary leaders has
not been addressed under the assumptions on the interconnection topology and communication process considered in
this paper. In addition, the proposed approach carries an
important feature that resides on the fact that the derived
conditions can be easily satisfied independently from the
interconnection topology between the systems. In a future
work, we will consider the containment control problem in
the case where the leaders velocities are time-varying – a
problem that is still open even in the case of continuous-time
communication in the presence of communication delays.
2942
R EFERENCES
10
8
6
qi
(2)
(rad)
4
2
0
−2
−4
−6
−8
−10
−15
−10
−5
(1)
qi
0
5
10
15
(rad)
Fig. 4. Trajectories of the followers with four stationary leaders, qi =
(1) (2)
(qi , qi )⊤ for i ∈ N . The black and red squares denote, respectively,
the positions of the leaders and the followers at instants 0 and 20 sec.
A PPENDIX
In this section, we present a small gain theorem, Theorem 2 below, which is a version of [8, Theorem 1] and a
special case of a more general result given in [23]. Consider
an affine nonlinear system of the form
ẋ
y1
..
.
=
=
..
.
f (x) + g1 (x)u1 + . . . + gp (x)up ,
h1 (x),
..
.
yq
= hq (x),
(27)
where x ∈ RN , ui ∈ Rm̃i for i ∈ Np := {1, . . . , p}, yj ∈
Rm̄j for j ∈ Nq := {1, . . . , q}, and f (·), gi (·), for i ∈ Np ,
and hj (·), for j ∈ Nq , are locally Lipschitz functions of the
corresponding dimensions, f (0) = 0, h(0) = 0. We assume
that for any initial condition x(t0 ) and any inputs u1 (t), . . . ,
up (t) that are uniformly essentially bounded on [t0 , t1 ), the
corresponding solution x(t) is well defined for all t ∈ [t0 , t1 ].
Theorem 2: Consider a system of the form (27). Suppose
the system is input-to-output stable (IOS) with linear IOS
0
gains γij
≥ 0. Suppose also that each input uj (·), j ∈ Np ,
is a Lebesgue measurable function satisfying:
uj (t) ≡ 0,
|uj (t)| ≤
X
i∈Nq
µji ·
for t < 0,
sup
|yi (s)| + |δj (t)|,
(28)
(29)
s∈[t−ϑji (t),t]
for almost all t ≥ 0, where µji ≥ 0, all ϑji (t) are Lebesgue
measurable uniformly bounded nonnegative functions of
time, and δj (t) is an uniformly essentially
bounded signal.
0
, M := {µji },
Let Γ := Γ0 · M ∈ Rq×q , where Γ0 := γij
i ∈ Nq , j ∈ Np . If ρ (Γ) < 1, where ρ (Γ) is the spectral
radius of the matrix Γ, then the trajectories of the system
(27) with input-output constraints (28), (29) are well defined
for all t ≥ 0 and such that all the outputs yi (t), i ∈ Nq , and
all the inputs uj (·), j ∈ Np , are uniformly bounded. If, in
addition, |δj (t)| → 0 at t → +∞, j ∈ Np , then |yi (t)| → 0,
|uj (t)| → 0 as t → +∞ for i ∈ Nq and j ∈ Np .
[1] W. Ren and Y. Cao, Distributed coordination of multi-agent networks:
emergent problems, models, and issues. Springer, 2011.
[2] R. Olfati-Saber, J. A. Fax and R. M. Murray, “Consensus and
cooperation in networked multi-agent systems”, in Proceedings of the
IEEE, vol. 95, no. 1, pp. 215-233, 2007.
[3] W. Ren and R. Beard, Distributed consensus in multi-vehicle cooperative control: theory and applications, Springer, 2008.
[4] A. Abdessameud and A. Tayebi, “Synchronization of networked
Lagrangian systems with input constraints,” in Preprints of the 18th
IFAC, World Congress, Milano, Italy, 2011, pp. 2382–2387.
[5] E. Nuño, R. Ortega, L. Basañez and D. Hill, “Synchronization of networks of nonidentical Euler-Lagrange systems with uncertain parameters
and communication delays”, IEEE Transactions on Automatic Control,
vol. 56, no. 4, pp. 935–941, 2011.
[6] H. Wang, “Consensus of networked mechanical systems with communication delays: A unied framework”, IEEE Transactions on Automatic
Control, 2014, DOI: 10.1109/TAC.2013.2293413.
[7] A. Abdessameud, I. G. Polushin and A. Tayebi, “Synchronization of
multiple Euler-Lagrange systems with communication delays”, In Proc.
of The American Control Conference, Canada, 2012, pp. 3748-3753.
[8] A. Abdessameud, I. G. Polushin and A. Tayebi, “Synchronization
of Lagrangian systems with irregular communication delays”, IEEE
Transactions on Automatic Control, vol. 59, no. 1, pp. 187–193, 2014.
[9] M. W. Spong and N. Chopra, “Synchronization of networked Lagrangian systems”, In Lagrangian and Hamiltonian Methods for Nonlinear Control 2006, F. Bullo and K. Fujimoto, Eds. Berlin: Springer-Verlag,
2007, vol. 366, pp. 47–59.
[10] S.-J. Chung and J.-J. E. Slotine, “Cooperative robot control and
concurrent synchronization of Lagrangian systems”, IEEE Transactions
on Robotics, vol. 25, no. 3, pp. 686–700, 2009.
[11] J. Mei, W. Ren and G. Ma, “Distributed coordinated tracking
with a dynamic leader for multiple Euler-Lagrange systems”, IEEE
Transactions on Automatic Control, vol. 56, no. 6, pp. 1415–1421, 2011.
[12] G. Chen and F. L. Lewis, “Distributed adaptive tracking control for
synchronization of unknown networked Lagrangian systems”, IEEE
Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics,
vol. 41, no. 3, pp. 805–816, 2011.
[13] D. V. Dimarogonas, P. Tsiotras and K. J. Kyriakopoulos, “Leader–
follower cooperative attitude control of multiple rigid bodies”, Systems
& Control Letters, vol. 58, no. 6, pp. 429–435, 2009.
[14] J. Mei, W. Ren, J. Chen and G. Ma, “Distributed adaptive coordination
for multiple lagrangian systems under a directed graph without using
neighbors’ velocity information”, Automatica, vol. 49, pp. 1723–1731,
2013.
[15] Z. Meng, W. Ren and Z. You, “Distributed finite-time attitude
containment control for multiple rigid bodies”, Automatica, vol. 46, no.
12, pp. 2092–2099, 2010.
[16] J. Mei, W. Ren and G. Ma, “Distributed containment control for
Lagrangian networks with parametric uncertainties under a directed
graph”, Automatica, vol. 48, no. 4, pp. 653–659, 2012.
[17] Y. G. Sun and L. Wang, “Consensus of multi-agent systems in directed
networks with nonuniform time-varying delays”, IEEE Transactions on
Automatic Control, vol. 54, no. 7, pp. 1607–1613, 2009.
[18] Y. Gao and L. Wang, “Consensus of multiple double-integrator agents
with intermittent measurement”, International Journal of Robust and
Nonlinear Control, vol. 20, no. 10, pp. 1140–1155, 2010.
[19] G. Wen, Z. Duan, W. Yu and G. Chen, “Consensus in multi-agent
systems with communication constraints”, International Journal of
Robust and Nonlinear Control, vol. 22, no. 2, pp. 170–182, 2012.
[20] G. Wen, Z. Duan, W. Ren and G. Chen, “Distributed consensus of
multi-agent systems with general linear node dynamics and intermittent
communications”, International Journal of Robust and Nonlinear Control, 2013, DOI: 10.1002/rnc.3001.
[21] G. Wen, Z. Duan, Z. Li and G. Chen, “Consensus and its L2 gain performance of multi-agent systems with intermittent information
transmissions”, International Journal of Control, vol. 85, no. 4, pp.384–
396, 2012.
[22] E. D. Sontag, “Input to state stability: Basic concepts and results”, In
Nonlinear and optimal control theory, pages 163–220, 2008, Springer.
[23] I. G. Polushin, S. N. Dashkovskiy, A. Takhmar and and R. V. Patel,
“A small gain framework for networked cooperative force-reflecting
teleoperation”, Automatica, vol. 49, no. 2, pp. 338–348, 2013.
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