Adaptive Fuzzy Output Feedback Containment Control of
Second-Order Systems with Unknown Dynamics
WANG Wei
1
, YU Yang
1
1. School of Electrical Engineering, Liaoning University of Technology, Jinzhou 121001, P. R. China
E-mail: lgwangw@gmail.com
Abstract: In this paper, the adaptive fuzzy output feedback containment control problem is investigated for second-order systems
with unknown dynamics and immeasurable states under directed communication graphs. Fuzzy logic systems are employed to
approximate the unknown dynamics, and an adaptive fuzzy error observer is designed using the relative position information.
Based on the designed observer, an adaptive fuzzy output feedback containment control scheme is developed to realize that the
all followers are ultimately driven to dynamic convex hull formed by the multiple leaders with a bounded containment error. A
simulation example is provided to verify the effectiveness of the proposed scheme.
Key Words: Adaptive Fuzzy Control, Containment Control, Observer, Output Feedback
1 Introduction
There have been numerous results on cooperative control
of multi-agent systems due to its potential applications [1–
3]. The previous research is mainly focused on consensus
including leaderless consensus and consensus tracking with
a single leader [4–6]. In the presence of multiple leaders,
all followers are required to enter into a geometric space s-
panned by the leaders. This is called containment control,
which has drawn a great attention by many researchers in
recent years [7–14].
The reference [7] as the pioneering work in containment
control, proposed a stop-and-go based containment strate-
gy to drive a collection of mobile robots to a given target
destination. In [8], the result in [7] was extended to double-
integrator systems with stationary or dynamic leaders, re-
spectively. In [9], observer-type containment protocols were
proposed for linear systems using the relative output mea-
surements of the neighboring agents. In [10], some distribut-
ed observer-type containment schemes were designed based
on the relative output measurements of the neighboring a-
gents. However, the containment control approaches given
in [7]-[10] mainly focus on the multi-agent systems with lin-
ear dynamics.
Some researchers make contributions to the containment
control of second-order systems, taking into account the fact
that a broad class of practical systems, such as the robot ma-
nipulator, car-like mobile robots, industrial flow process, and
so forth, is modeled by second-order dynamics, In [11], con-
tainment control laws were designed for multiple rigid bod-
ies, which can realize that the attitudes of the followers were
driven to the convex hull formed by the leaders in finite time.
In [12], an adaptive containment control algorithm using
sliding-mode estimators was proposed for Lagrangian sys-
tems with parametric uncertainties. In [13], fully distribut-
ed adaptive containment controllers using sliding-mode esti-
mators were designed for multiple Lagrangian systems with
parametric uncertainties and external disturbances. In [14],
adaptive neural network containment control laws were de-
signed for Lagrangian systems with multiple dynamic lead-
This work is supported by the National Natural Science Foundation of
China under Grant 61603165.
ers in the presence of uncertainties and external disturbances.
It is worth noting that the system states are used for the con-
tainment control design in [11]-[14]. Therefore, these con-
trol schemes cannot be applied to second-order nonlinear
systems with immeasurable states. To the best of the authors’
knowledge, no adaptive containment control is available for
second-order nonlinear multi-agent systems with unknown
dynamics.
Motivated by the above observations, the output feedback
containment control problem is investigated for second-
order nonlinear systems with unknown dynamics and im-
measurable states. Fuzzy logic systems are utilized to ap-
proximate the unknown dynamics of the systems. A fuzzy
observer is designed only using the relative position infor-
mation. Then, an observer-based adaptive fuzzy output feed-
back containment control scheme is proposed to guaran-
tee that all followers converge to the dynamic convex hull
formed by the multiple leaders with a bounded containment
error.
2 Preliminaries
In this paper, the considered multi-agent systems consist
of N followers and M leaders. An agent is called as a fol-
lower if the agent has at least one neighbor. An agent is
called as a leader if the agent has no neighbor. The infor-
mation communication among the agents is described by a
directed graph containing a spanning tree G =(V, E), where
V = {n
1
, ···,n
N+M
} is a vertex set with n
i
representing
the ith agent. E = {(n
i
,n
j
) ∈V×V}an edge set, and
(n
i
,n
j
) ∈Emeans that there is a information flow from
the agent i to j. The neighbor set of the node i is denoted
by N
i
= {j|(n
i
,n
j
) ∈E}. Let A be an adjacency matrix
of the graph with the element a
ij
(i =1, ···,N + M; j =
1, ···,N+M). a
ij
=1, if (n
i
,n
j
) ∈E; otherwise a
ij
=0.
It is assumed that a
ii
=0. The Laplacian matrix is defined
as L = D −A, where D = diag{d
1
, ···,d
N
} is a diagonal
matrix, and the diagonal element d
i
=
j∈N
i
a
ij
.
It is assumed that n
1
, ···,n
N
represent the followers, and
n
N+1
, ···,n
N+M
represent the leaders. Then, the Lapla-
cian matrix can be partitioned as
L =
L
1
L
2
0
M×N
0
M×M
.
Proceedings of the 36th Chinese Control Conference
Jul
y
26-28, 2017, Dalian, China
8166
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