1
Model Predictive Control Schemes for Consensus in Multi-Agent
Systems with Single- and Double-Integrator Dynamics
Giancarlo Ferrari-Trecate, Member, IEEE, Luca Galbusera, Marco Pietro Enrico Marciandi, Riccardo Scattolini
Abstract—In this paper we address the problem of driving
a group of agents towards a consensus point when the agents
have a discrete-time single- or double-integrator dynamics and
the communication network is time-varying. We propose decen-
tralized Model Predictive Control (MPC) schemes that take into
account constraints on the agents’ input and show that they
guarantee consensus under mild assumptions. Since the global
cost does not decrease monotonically, it cannot be used as a
Lyapunov function for proving convergence to consensus. For
this reason, our proofs exploit geometric properties of the optimal
path followed by individual agents.
Index Terms—Networked autonomous agents, Consensus prob-
lems, Decentralized model predictive control.
I. INTRODUCTION
T
HIS paper deals with consensus problems for dynam-
ically decoupled agents described by a discrete-time
single- or double-integrator model and subject to possible
time-varying interconnections. The objective is to define de-
centralized control strategies guaranteeing that the states of all
agents converge to a common value, called consensus point,
which generally speaking depends on the agents’ state and
on the communication network. This problem is relevant in
many fields, e.g. in computer graphics, unmanned autonomous
vehicles, sensor networks and, generally speaking, in the
control of cooperating systems (see [1] and the references
therein; see also [2] for a more general definition of consensus
in presence of unknown but bounded disturbances).
In recent years, many control laws have been proposed for
networks of dynamic agents with different models and com-
munication topologies, see e.g. [3], [4], [5], [6], [7], [8], [9].
Most of them do not exploit optimal control ideas and, with the
exception of [3] and [7], do not account for input constraints,
which in many cases have to be included in the problem
formulation due to actuators limitations.
In this paper we propose innovative solutions based on Model
Predictive Control (MPC), which is a widely used approach in
view of its ability to handle control and state constraints. This
method can be applied in a distributed fashion to the control
of a group of agents by letting each agent solve, at each step, a
constrained finite-time optimal control problem involving the
state of neighboring agents. Moreover, following the so-called
Receding-Horizon principle, at each time step the controller
G. Ferrari-Trecate is with the Dipartimento di Informatica e Sistemistica,
Universit`a degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy. e-mail:
giancarlo.ferrari@unipv.it
L. Galbusera and R. Scattolini are with the Dipartimento di Elettronica
e Informazione, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133
Milano, Italy. e-mail: {galbusera, scattolini}@elet.polimi.it
M.P.E. Marciandi is with CESI Ricerca S.p.a., Via Rubattino 54, 20134
Milano, Italy. e-mail: mmarciandi@hotmail.com
Corresponding author: G. Ferrari-Trecate.
only applies the first input of the computed control sequence.
Stabilizing MPC algorithms for decentralized and distributed
systems have been proposed in [10], [11], [12], [13], [14], [15],
[16] for dynamically coupled and decoupled systems. However
these approaches heavily rely on the fact that all agents a
priori know a common set point, with the notable exception
of [17], where the properties of the proposed algorithm are
demonstrated by means of simulation experiments. On the
contrary, the methods described in this paper can be formally
proved to guarantee (asymptotic) consensus under control
constraints and for time-varying communication networks.
Specifically, we first propose an MPC solution for consensus
when agents have a single-integrator dynamics with bounded
inputs. Notably, the proof of consensus under this scheme does
not rely on the standard arguments used in predictive control
to guarantee closed-loop stability, see e.g. [18], [19], since
the global cost to be minimized by the MPC algorithm is not
monotonically decreasing and, as such, it cannot be used as
a Lyapunov function. Rather, we exploit geometric properties
of the optimal path followed by individual agents and rely on
the general results described in [3] for analyzing consensus.
As a further matter, on the basis of this technique we develop
other MPC solutions for consensus. We first modify the cost
function adopted in the previous scheme and add a specific
constraint on the state of the system. This constraint is called
“contractive” because it mimics, in a multi-agent system
domain, the state constraint proposed in [20] for the control of
nonlinear systems. The main advantage of this approach is that
it can be extended to agents with double-integrator dynamics,
even
though the techniques used for proving consensus in this
case are more sophisticated.
In our schemes, because of the Receding-Horizon technique,
the value of the consensus point depends not only on the initial
conditions of the system, but also on the sequence of agents’
states along time and on the communication network along
time.
The paper is organized as follows: Section II is devoted to
the definition of a model for the communication network and
summarizes some key results on convergence in multi-agent
systems presented in [3] and used in this paper. Section III de-
scribes the first MPC technique we propose for consensus in a
network of integrators with time-varying communication. The
contractive MPC solutions for single- and double-integrators
are presented, respectively, in Sections IV and V. Simulation
examples confirm the results obtained by applying the control
laws we propose. Section VI is devoted to conclusions. Finally,
the Appendix contains technical results used in the proofs of
consensus.
II. BASIC NOTIONS AND PRELIMINARY RESULTS
We consider a team of n agents indexed by the elements
of the set N
G
= {1, . . . , n}. The communication network is
represented by a weighted directed graph G = (N
G
, E
G
, w
G
),
where E
G
⊆ {(i, j) : i, j ∈ N
G
, j 6= i} is the set of edges and
w
G
: E
G
7→ R
>0
associates to each edge (i, j) ∈ E
G
a
strictly positive weight denoted by w
ij
. G is
bidirectional if
(i, j) ∈ E
G
⇔ (j, i) ∈ E
G
1
and unweighted if, ∀(i, j) ∈ E
G
,
w
ij
= 1. In the latter case, the graph can be simply represented
by the pair (N
G
, E
G
). From now on we will generally refer to
directed or
bidirectional graphs assuming implicitly that they
can be weighted. If, ∀i, j ∈ N
G
, (i, j) ∈ E
G
, the graph is
complete.
A node i ∈ N
G
is connected to a node j ∈ N
G
\{i} if there
is a path from i to j in the graph following the orientation
of the arcs. The graph G is strongly connected if, ∀(i, j) ∈
N
G
× N
G
, i is connected to j.
A directed graph is said to
have a spanning tree if and only if ∃i ∈ N
G
such that there
is a path from i to any other node j ∈ N
G
.
The creation and loss of communication links can be
modeled by means of a time-dependent collection of graphs
G = {G(k) = (N
G
, E
G
(k), w
G
(k)), k ∈ N} where all
graphs G(k) share the same set of nodes. G is a collection
of
bidirectional graphs if and only if, ∀k ∈ N, G(k) is
bidirectional. Otherwise, it is referred to as a collection of
directed graphs.
In the sequel, discrete-time intervals will be denoted with
[k
1
, k
2
] = {k : k
1
≤ k ≤ k
2
} assuming implicitly that 0 ≤
k
1
≤ k
2
≤ + ∞.
Definition 1 [21] A collection of graphs {G(1), . . . , G(m)} is
jointly connected if (N
G
, ∪
m
k=1
E
G
(
k)) is a strongly connected
graph. The agents are linked together across the interval
[l , m ] if the collection of graphs {G(k), k ∈ [l, m]} is jointly
connected. A node i is connected to all other nodes across a
time set T ⊆ N if i is connected to all other nodes in the
graph (N
G
,
S
k∈T
E
G
(k)).
If (j, i) ∈ E
G
we say that j is neighbor to i and the j-th
agent transmits instantaneously its state to the i-th agent. The
set of neighbors to the node i ∈ N
G
is N
i
(G) = {j ∈ N
G
:
(j, i) ∈ E
G
} and |N
i
| is the valency of the i-th node.
We define the communication matrix
˜
K(G) = [
˜
k
ij
]
i,j∈N
G
,
˜
K(G) ∈ R
n×n
, where:
˜
k
ij
=
w
ji
1+
P
h∈N
i
(G)
w
hi
if j ∈ N
i
(G),
1
1+
P
h∈N
i
(G)
w
hi
if i = j,
0 otherwise.
(1)
˜
K(G) is a stochastic matrix (i.e. it is square and nonnegative
and its row sums are equal to 1, see [21]), whose entry (i, j)
is non null if and only if i = j or (j, i) ∈ E
G
. We also
define K(G) =
˜
K(G) ⊗ I
d
where ⊗ denotes the Kronecker
product and I
d
is the identity matrix of order d. Moreover
we denote by K
i
(G) ∈ R
d×dn
the i-th block of the matrix
1
As remarked in [3], weights w
ij
and w
ji
can be different. If they are
equal, one recovers the notion of undirected graphs.
K(G), partitioned as K(G) = [K
T
1
(G) · · · K
T
n
(G)]
T
. Note
that if x(k) = [x
T
1
(k) · · · x
T
n
(k)]
T
, with x
i
(k) ∈ R
d
, one has
K
i
(G(k))x(k) ∈ C
i
(G(k)) =
Co({x
i
(k)} ∪ {x
j
(k), j ∈ N
i
(G(k))})
(2)
where Co(A) is the convex hull of the set A (see [21]).
For sake of completeness, next we summarize some results
provided in [3] that will enable us to prove consensus under
the MPC schemes we will propose in the sequel. Assume that
agents obey the general closed-loop dynamics
x(k + 1) = f (k, x(k)) (3)
where x(k) = [x
1
(k)
T
· · · x
n
(k)
T
]
T
and x
i
(k) ∈ R
d
, ∀i ∈
N
G
. The nodes of the network have reached consensus if and
only if x
i
= x
j
, ∀i, j ∈ N
G
, i 6= j. The corresponding state
value is called consensus point. The consensus subspace
is
Φ
c
=
x ∈ R
nd
: x
1
= x
2
= · · · = x
n
.
Definition 2 [3] Let
Φ ⊆ R
dn
be a set of equilibria for (3).
System (3) is globally attractive w.r.t. Φ if for each φ
1
∈ Φ,
∀c
1
, c
2
> 0 and ∀k
0
∈ N, ∃T ≥ 0 such that every solution ζ
to (3) has the following property:
kζ(k
0
)−φ
1
k < c
1
⇒ ∃φ
2
∈ Φ : kζ(k)−φ
2
k < c
2
, ∀k ≥ k
0
+T
where k·k denotes the Euclidean norm. The system is uniformly
globally attractive w.r.t. Φ if it is globally attractive w.r.t. Φ
and the constant T is independent of k
0
.
Note that, as described in [3], global attractivity implies that
all solutions to (3) converge to a point in Φ as k → +∞.
However, the vice-versa is
not true.
Definition 3 Assume that the set of equilibria associated to
the multiagent system (3) is Φ
c
. Then, consensus is asymptot-
ically reached if one of the following conditions holds:
1) G is a collection of directed graphs and (3) is uniformly
globally attractive w.r.t. Φ
c
;
2) G is a collection of
bidirectional graphs and (3) is
globally attractive w.r.t. Φ
c
,
The consensus results stated in [3] hinge on the following
assumption.
Assumption 1 For every graph G(k) ∈ G, agent i ∈ N
G
and
state x ∈ X
n
, X ⊆ R
d
, there is a compact set e
i
(G(k))(x) ⊆
X such that:
1) f
i
(x, k) ∈ e
i
(G(k))(x), ∀k ∈ N, ∀x ∈ X
n
;
2) e
i
(G(k))(x) = {x
i
} if x
i
= x
j
, ∀j ∈ N
i
(G(k));
3) whenever the states of agent i and agents j ∈ N
i
(G(k))
are not all equal, e
i
(G(k))(x) ∈ Ri(C
i
(G(k))), where
Ri(A) denotes the relative interior of the set A;
4) the set-valued function e
i
(G(k))(x) : X
n
7→ 2
X
is
continuous (2
X
is the power set of X).
The main theorem on consensus we will use is stated next
and exploits the following assumptions.
Assumption 2 For the collection G of directed graphs there
exists a non-negative integer T ≥ 0 such that, ∀k
0
∈ N, there
is a node connected to all other nodes across [k
0
, k
0
+ T ].
Assumption 3 For the collection G of
bidirectional graphs
and for all k
0
∈ N, all agents are linked together across the
interval [k
0
, +∞).
Theorem 1 [3] Let G be a collection of directed [resp.
bidi-
rectional
] graphs and assume that f in (3) verifies Assumption
1. Then, system (3) asymptotically reaches consensus if and
only if Assumption 2 [resp. Assumption 3] holds.
For further comments and examples clarifying the mini-
mality of the assumptions of Theorem 1 as well as the links
between uniform global attractiveness [resp. global attractive-
ness] and directed [resp.
bidirectional] graphs, we defer the
reader to [3].
III. MPC FOR CONSENSUS AMONG AGENTS WITH A
SINGLE-INTEGRATOR DYNAMICS
In this section we consider a system of n agents, each
one described by the following discrete-time single-integrator
model
x
i
(k + 1) = x
i
(k) + u
i
(k), i ∈ N
G
(4)
with initial condition x
i
(0) = x
i0
. The vector u
i
(k) ∈ R
d
is
control input of agent i at time k.
As an example, considering a group of autonomous vehicles
moving in a d-dimensional geometric space, x
i
(k) describes
the position of agent i. We will also assume that the communi-
cation topology is time-varying and captured by the collection
of graphs G.
Given a generic vector-valued signal y(·) and a positive
integer N , let us denote by the capitalized vector Y (k) =
[y
T
(k), · · · , y
T
(k + N − 1)]
T
the values of y(·) over the time
interval [k, k + N − 1].
Let N ≥ 1 denote the length of the prediction horizon. We
associate to the i-th agent the input sequence U
i
(k) and the
cost
J
i
(x(k), U
i
(k)) = J
x
i
(x(k), U
i
(k)) + J
u
i
(U
i
(k)) (5)
J
x
i
(x(k), U
i
(k)) = α
i
N
X
j=1
kx
i
(k + j) − z
i
(k)k
2
(6)
J
u
i
(U
i
(k)) = β
i
N−1
X
j=0
ku
i
(k + j)k
2
(7)
where
z
i
(k) = K
i
(G(k)) x(k ) (8)
defines the target point for the i-th agent and α
i
, β
i
> 0 are
weights. Note that from (2), one has z
i
(k) ∈ C
i
(G(k)) and, in
particular, z
i
(k) is just the barycenter of {x
i
(k)}
S
{x
j
(k), j ∈
N
i
(k)} when the graph is unweighted.
Cost (5) is decentralized, because the term z
i
(k) depends
only on the states of neighbors to the i-th agent at time k.
Consider the following Constrained Finite-Time Optimal Con-
trol (CFTOC) problem for agent i ∈ N
G
:
min
U
i
(k)
J
i
(x(k), U
i
(k)) (9)
subject to the following constraints:
(a) the agent dynamics (4);
(b) the input constraint
ku
i
(k + j)k ≤ u
i,max
, (10)
with u
i,max
> 0, j ∈ [0, N − 1].
Optimal inputs computed at time k will be denoted with
U
o
i
(k|k) and we will investigate the consensus properties
provided by the receding-horizon control law
u
RH
i
(k) = κ
RH
i
(k,
z
i
(k)), κ
RH
i
(k, z
i
(k)) = u
o
i
(k|k) (11)
Note that u
RH
i
(k) coincides with the first input
sample appearing in the vector U
o
i
(k|k) =
u
oT
i
(k|k), u
oT
i
(k + 1|k), · · · , u
oT
i
(k + N − 1|k)
T
. Note
also that the state-feedback κ
RH
i
is time-varying when the
communication network changes over time. Problem (9) is
always feasible since U
i
(k) = 0 is a feasible input sequence.
We are now in a position to state the main result of this
Section. Its proof is based on the application of Theorem
1 and the geometrical properties of discrete-time paths
introduced in
the Appendix.
Theorem 2 Let G be a sequence of directed [resp.
bidirec-
tional] graphs. The closed-loop multi-agent system given by
(4) and (11) asymptotically reaches consensus if and only if
Assumption 2 [resp. Assumption 3] holds.
Proof: Consider the closed-loop multi-agent system cor-
responding to equations (4) and (11), that can be represented
by
x(k + 1) = f (x(k)), (12)
where
f(x(k)) =
f
1
(x(k))
.
.
.
f
n
(x(k))
, f
i
(x(k)) = x
i
(k)+κ
RH
i
(k,
z
i
(k))
Apparently, from (9), (10) and (11) one has κ
RH
i
(k,
z
i
(k)) =
0, ∀i ∈ N
G
if and only if x(k) ∈ Φ
c
. Hence, Φ
c
collects the
consensus equilibria of (12). Next, we show that the map f
fulfills Assumption 1.
With the notations used in Assumption 1, we set X = R
d
and e
i
(G(k))(x) = {f
i
(x)}. Then point 1 of Assumption 1
is trivially verified. For point 2, when x
i
(k) = x
j
(k), ∀j ∈
N
i
(G(k)), then z
i
(k) = x
i
(k). From the definition of the
cost J
i
, it is immediate to verify that J
i
(x(k), 0) = 0 and
hence the optimal inputs U
o
i
(x(k)) are zero. This implies that
f
i
(x) = x
i
, and point 2 is verified.
The
core part of the proof consists in verifying point 3. To this
purpose we will use the tools and the terminology for paths
described in the Appendix.
If there exists j ∈ N
i
(G(k)) such that x
i
(k) 6= x
j
(k) one
has x
i
(k) 6= z
i
(k). Let X
i
(x(k), U
i
(k)) be the sequence of
states of the i-th agent generated by the initial state x(k) and
the input sequence U
i
(k), with u
i
(k + j), j ∈ [0, N − 1]
fulfilling the constraint (10).
As a first step we show that X
o
i
(x(k), U
o
i
(k)) =
[x
oT
i
(k) · · · x
oT
i
(k + N )]
T
is a path pointing towards z
i
(k).
Note that, because of the single-integrator dynamics (4), the
term J
u
i
can be rewritten as:
J
u
i
(k) = β
i
N−1
X
j=0
kx
i
(k + j + 1) − x
i
(k + j)k
2
(13)
Assume by contradiction that X
o
i
(x(k), U
o
i
(k)) is not pointing
towards z
i
(k). Theorem 5
(see the Appendix) shows that there
is an N-path
ˆ
X
i
(k) = [ˆx
T
i
(k) · · · ˆx
T
i
(k + N )]
T
with ˆx
i
(k) =
x
o
i
(k) pointing towards z
i
(k) and such that, ∀j = 0, . . . , (N −
1), both the following inequalities hold simultaneously:
kˆx
i
(k + j + 1) − z
i
(k)k ≤ kx
o
i
(k + j + 1) − z
i
(k)k (14)
kˆx
i
(k + j + 1) − ˆx
i
(k + j)k ≤ kx
o
i
(k + j + 1) − x
o
i
(k + j)k
(15)
Inequality (15) implies that the input vector
ˆ
U
i
(k), pro-
ducing
ˆ
X
i
(x(k)), fulfills the constraint (10). Moreover (14)
implies that J
x
i
(x(k),
ˆ
U
i
(k)) ≤ J
x
i
(x(k), U
o
i
(k)) and (15)
that J
u
i
(
ˆ
U
i
(k)) ≤ J
u
i
(U
o
i
(k)). Hence X
o
i
(x(k), U
o
i
(k)) is not
optimal. One therefore concludes that the optimal state path
X
o
i
(x(k), U
o
i
(k)) is necessarily pointing towards z
i
(k).
As a second step, we show that x
i
(k) 6= z
i
(k) implies
that u
o
i
(k) 6= 0. This will be proved by first showing, by
contradiction, that U
o
i
(x(k)) 6= 0. Assume that U
o
i
(x(k)) =
0 is the optimal input sequence. Then J
i
(x(k), U
o
i
(k)) =
Nα
i
kx
i
(k) − z
i
(k)k
2
. Consider the input sequence
¯
U
i
(k) =
[δ(z
i
(k) − x
i
(k))
T
0
T
· · · 0
T
]
T
, 0 < δ ≤ 1, where δ is
such that
kδ(z
i
(k) − x
i
(k))k ≤ u
i,max
, and let
¯
X(k) =
[¯x(k) · · · ¯x(k + N )] be the corresponding state path. One has
J
i
(x(k),
¯
U
i
(k)) =Nα
i
(1 − δ)
2
kx
i
(k) − z
i
(k)k
2
+ β
i
δ
2
kx
i
(k) − z
i
(k)k
2
(16)
For δ <
2Nα
i
Nα
i
+β
i
one has J(x(k),
¯
U
i
(k)) < J(x(k), U
o
i
(k)).
Such a choice is always feasible since
2Nα
i
Nα
i
+β
i
> 0. Therefore
U
o
i
(k) = 0 cannot be the optimal input sequence.
Now we prove that u
o
i
(k) 6= 0 by contradiction. Assume
that the optimal input sequence is U
o
i
(k) = [0
T
u
oT
i
(k +
1) · · · u
oT
i
(k + N − 1)]
T
, with u
oT
i
(k + 1) 6= 0. Now consider
the input
¯
U
i
(k) = [u
oT
i
(k + 1) · · · u
oT
i
(k + N − 1) 0
T
]
T
. We
observe that all input samples in
¯
U
i
(k) verify the constraint
(10), as those of U
o
i
(k) do. Then,
J(x(k),
¯
U
i
(k)) − J(x(k), U
o
i
(k))
= α
i
x
i
(k) +
N−1
X
j=0
u
o
i
(k + j) − z
i
(k)
2
− α
i
kx
i
(k) − z
i
(k)k
2
< 0
(17)
Inequality (17) follows from the fact that some input u
o
i
(k +
j), j = 0, . . . , (N − 1) is non null and every input makes the
state follow a path that points towards z
i
(k).
The property of stochasticity of the matrix K(G(k)), together
with the assumptions that N
i
(G) 6= ∅ and x
i
(k) 6= z
i
(k)
implies that z
i
(k) ∈ Ri(C
i
(G(k))). When also x
i
(k) ∈
Ri(C
i
(G(k))) it is trivial to conclude that the optimal trajectory
belongs to Ri(C
i
(G(k))). When x
i
(k) belongs to the boundary
of C
i
(G(k)), by applying the line segment principle [22] the
same conclusion follows because the optimal trajectory is
totally included in the relatively open segment connecting
x
i
(k) and z
i
(k).
Finally, also point 4 in Assumption 1 is verified. In fact,
since we have proved that the optimal state trajectory
X
o
i
(x
i
(k), U
o
i
(k)) is pointing towards z
i
(k), it is possible to
write: x
i
(k + 1) = x
i
(k) + η(k)L(k)(z
i
(k) − x
i
(k)), ∀k ∈ N,
where 0 ≤ L(k) ≤ 1 (specifically, L(k) = 0 only when
z
i
(k) = x
i
(k)) and
η(k) =
1 if L(k) kz
i
(k) − x
i
(k)k ≤ u
i,max
u
i,max
L(k)kz
i
(k)−x
i
(k)k
otherwise
It is then apparent that the function f
i
(x) is continuous, and
therefore e
i
(G(k))(x) is also continuous.
In conclusion, Assumption 1 holds and the rest of the proof
is a straightforward application of Theorem 1.
Remark 1 By suitably choosing the weights α
i
and β
i
in (6)
and (7), respectively, it is possible to tune the behavior of
each agent. For instance, given an unweighted or
bidirectional
graph, it is possible to mimic behaviors of the agents typically
obtained by using weighted or directed graphs. This feature
is illustrated in Example 1 where it is shown how to achieve,
approximately, leader-following.
Example 1 We consider a set of n = 5 agents moving in a
two dimensional space, with initial states x
1
(0) = [−30 30]
T
,
x
2
(0) = [−25 35]
T
, x
3
(0) = [65 −75]
T
, x
4
(0) = [70 −68]
T
,
x
5
(0) = [100 − 25]
T
. The prediction horizon is N = 3. The
weights in the cost function (5) are α
i
= 1, i = 1, . . . , 5,
β
1
= 100, β
i
= 1, i = 2, . . . , 5. The communication network
is described by the time-invariant unweighted
bidirectional
graph represented in Fig. 1, that corresponds to the following
communication matrix:
˜
K(G) =
1
2
1
2
0 0 0
1
3
1
3
1
3
0 0
0
1
3
1
3
1
3
0
0 0
1
3
1
3
1
3
0 0 0
1
2
1
2
(18)
The input constraints (10) are given by u
i,max
= 100, i =
1 52 43
Fig. 1. Communication network used in Examples 1, 2 and 3
.
1, . . . , 5. The simulation in Fig. 2 shows asymptotic con-
vergence towards a consensus point. Since β
1
≫ β
i
, i =
2, 3, 4, 5, agent 1 behaves approximately as a leader, hence
moving much less than the others from its initial po-
sition. In this case, input constraints are never active.
It is important to notice that in this example both cost
functions J
1
(x
o
(k), U
o
1
(k|k)) and J
o
(x
o
(k)
, U
o
(k|k)) =
0 5 10 15 20 25 30 35 40
−50
0
50
100
time
x
i,1
agent 1
agent 2
agent 3
agent 4
agent 5
0 5 10 15 20 25 30 35 40
−100
−50
0
50
time
x
i,2
Fig. 2. Example 1: Evolution of agents’ state.
P
5
i=1
J
i
(x
o
(k), U
o
i
(k|k)), where x
0
(k) is the state sequence
given by (4) and (11), are not monotonically decreasing over
time (see Fig. 3). This implies that the global cost function
J
o
(x(k)) cannot be used as a Lyapunov function and classical
arguments for proving the stability of MPC algorithms (see
e.g. [18], [19]) cannot be adopted. This justifies the necessity
of an alternative technique to prove convergence, as the one
we propose, based on geometrical concepts and on Theorem
1.
0 10 20 30 40
0
500
1000
1500
2000
2500
3000
3500
time
J
o
J
1
J
2
J
3
J
4
J
5
Fig. 3. Example 1: the bold line represents the global cost
J
o
(x
o
(k), U
o
(k|k)). The other lines represent the cost of individual agents.
Example 2 We use the same setting of Example 1, but we now
choose β
i
= 1, i = 1, . . . , 5 and u
i,max
= 5. We also assume
that the communication network is time-varying and fulfills As-
sumption 3. More in detail, we generated a random sequence
of unweighted
bidirectional graphs such that all agents are
linked together across the interval [k
0
, +∞), ∀k
0
∈ N.
As seen
in Fig. 4, consensus is asymptotically achieved, in accordance
with Theorem 2, even though input constraints are active at
times k ≤ 10, as shown in Fig. 5
. Notice that the value of
0 5 10 15 20 25 30 35 40
0
50
100
time
x
i,1
agent 1
agent 2
agent 3
agent 4
agent 5
0 5 10 15 20 25 30 35 40
−100
−50
0
50
time
x
i,2
Fig. 4. Example 2: Evolution of agents’ state.
0 5 10 15 20 25 30 35 40
−5
0
5
time
u
i,1
0 5 10 15 20 25 30 35 40
−5
0
5
time
u
i,2
0 5 10 15 20 25 30 35 40
0
5
10
time
||u
i
||
Fig. 5. Example 2: Control inputs of individual agents.
the asymptotic consensus point in this example differs from the
one in the previous one, though the initial conditions of the
system are the same in both cases.
IV. A CONTRACTIVE MPC SCHEME FOR AGENTS WITH
SINGLE-INTEGRATOR DYNAMICS
The CFTOC problem (9) does not include constraints on
the agents’ state. We now introduce an alternative MPC law
requiring periodic communication among agents and using
specific state constraints in order to guarantee consensus. This
solution has the advantage that it can be easily extended to
