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geometric insight on the invarianceproperties of poly-
topic regions in the state space.
The rest of the paper is organized as follows. In
Section 2 the constrained predictive control problem
is formulated. Section 3 considers the unbounded
interdicted region and analyzes the existence and
uniqueness of a fixed point on the frontier of the fea-
sible domain. Discussions based on the simulation re-
sults are presented in Section 4, while the conclusions
are drawn in Section 5.
The following notations will be used throughout
the paper. Denote B
n
p
= {x ∈ R
n
: kxk
p
≤ 1} as the
unit ball of norm p, where kxk
p
is the p-norm of vec-
tor x. The spectrum of a matrix M is the set of the
eigenvalues of M, denoted by Λ(M) = {λ
i
: i ∈ N} . A
point x
f
is a fixed point of a function f if and only if
f(x
f
) = x
f
(i.e. a point identical to its own image).
The boundary of a set S, denoted by ∂S is the set of
points which can be approached both from S and from
the outside of S.
2 PROBLEM STATEMENT
In the sequel, the principles of a receding horizoncon-
trol problem are recalled. Let us model the behavior
of an agent with a discrete time linear time-invariant
system:
x
k+1
= Ax
k
+ Bu
k
, (1)
where x
k
∈ R
n
is the state of the agent, u
k
∈ R
m
is
the input signal and A, B are matrices of appropriate
dimensions. It is assumed that the pair (A, B) is sta-
bilizable. The state constraints describe a polytopic
region S in the state-space:
S =
x ∈ R
n
:
˜
h
T
i
x ≤
˜
k
i
, i = 1 : n
h
, (2)
with (
˜
h
i
,
˜
k
i
) ∈ R
n
× R and n
h
the number of half-
spaces. This paper focuses on the case where
˜
k
i
> 0,
meaning that the origin is contained in the strict inte-
rior of the polytopic region, i.e. 0 ∈ S. The normaliza-
tion of the right hand side of the inequalities (2) leads
to
S =
x ∈ R
n
: h
T
i
x ≤ 1, i = 1 : n
h
, (3)
with h
i
=
˜
h
i
/
˜
k
i
∈ R
n
. Such limitations arise both from
for collision or obstacles avoidance problems. Note
that the feasible region in the solutions space is a non-
convex region defined as the complement R
n
\S. This
implies at the modeling stage a compact represen-
tation of the obstacles and/or a safety region for an
agent in terms of (3)
1
.
1
A safety region can be associated to each agent and im-
poses that the inter-agent dynamics do not overlap each in-
Let x
k+1|k
denote the value of x at time instant
k + 1, predicted upon the information available at
time k ∈ N. A finite receding horizon implemen-
tation of the optimal control law is typically based
on the real-time construction of a control sequence
u = {u
k|k
,u
k+1|k
,··· ,u
k+N−1|k
} that minimizes the fi-
nite horizon quadratic objective function:
u
∗
= arg
u
min(x
T
k+N|k
Px
k+N|k
+
N−1
∑
i=1
x
T
k+i|k
Qx
k+i|k
+
+
N−1
∑
i=0
u
T
k+i|k
Ru
k+i|k
) (4)
subject to:
(
x
k+i+1|k
= Ax
k+i|k
+ Bu
k+i|k
x
k+i|k
∈ R
n
\ S, i = 1 : N
Here Q = Q
T
0, R ≻ 0 are positive definite weight-
ing matrices, P = P
T
0 defines the terminal cost and
N denotes the prediction horizon.
It has to be mentioned that the solution of the un-
constrained finite horizon problem is the well-known
linear state-feedback control law:
u
k
= K
LQ
x
k
(5)
where K
LQ
is computed from the solution of the dis-
crete algebraic Riccati equation. Making the assump-
tion that the pair (A, B) is stabilizable, the stabilizing
LQ-optimal controller can be found before the res-
olution of the problem (4), closed-loop stability ex-
plicitly requiring that the state enters into a terminal
region (containing the origin) at the end of the predic-
tion horizon (Mayne et al., 2000). This considerations
do not fit the present framework as long as the equi-
librium point is not ”approachable”. Nevertheless, the
stability analysis is an important issue and it will be
one of the aims in the rest of the paper.
The objective is to find if the agent state either ap-
proaches a fixed point, or a periodic orbit (a ”limit cy-
cle”), or a finite set of fixed points. In a second stage
we will be interested in the description of the basin
of attraction and relate the analysis with classical sta-
bility results. For the sake of compactness, simplicity
of notation and representation, an analysis of second
order dynamical systems is proposed, i.e. the planar
case. It turns out that this situation is simple enough
to obtain useful results, and rich enough to gain some
understanding about the difficulties in higher dimen-
sions. Note that some of the 2-dimensional case argu-
ments from planar geometry will not longer be valid
in higher dimensions. Therefore, the generalization
for the n-dimensional case is still an open issue.
dividual restriction. It is important to assure that a control
action will not lead to a cycling behavior which implies en-
ergy consumption.
ON THE LIMIT BEHAVIOR OF MULTI-AGENT SYSTEMS
345