Distributed backstepping-based adaptive fuzzy control of multiple high-order nonlinear dynamics 65
the Brunovsky form with uncertainties is considered,
which include first- and second-order systems as spe-
cial cases. Second, a systematic controller design pro-
cedure is proposed to deal with the control problem
through combining distributed backstepping method
with adaptive fuzzy control techniques. The adaptive
fuzzy control is completely distributed. And the conver-
gence of the system errors is proved rigorously by virtue
of the Lyapunov stable theory and Barbalat’s lemma.
The subsequent sections are organized as follows: In
Sect. 2, the control problem is formally stated and the
background as well as necessary preliminaries concern-
ing the control problem are given. In Sect. 3, the cooper-
ative control laws are proposed relying on backstepping
method and adaptive fuzzy control approaches. The
unknown nonlinear functions are dealt by fuzzy logic
systems, and the external disturbances are addressed
by applying robust adaptive control method. In Sect. 4,
a four-order simulation example is provided to demon-
strate the performance of the proposed control laws.
The last section concludes this paper.
2 Preliminaries and problem statement
In this section, basic graph theory for networked
dynamics, control problem and fuzzy logic systems on
graph are introduced.
2.1 Brief graph theory for networked dynamics
With respect to networked dynamics, any control laws
must be distributed in the sense that it respects the
communication network topology. The communication
restrictions by topologies can severely limit the power
of local distributed control algorithm at each individ-
ual dynamics. The idea of a communication network
models the information flows in a multi-agent group.
A team of m high-order nonlinear dynamics labeled
as system 1 to m are considered. The communica-
tion topology among the m systems is assumed to be
bidirectional or directional, and the interactions among
the nodes are represented by an undirected or directed
graph G = (V, E, A). The topology G represents the
structure of networked system, where V is a set of the
indices of the systems and E ⊆ V × V is a set of edges
that describe the communications between the s ystems.
If ( p , j ) ∈ E, then p is neighboring to j, meaning sys-
tem j can obtain information from system p. A is a
weighted adjacency matrix with nonnegative adjacency
elements a
pj
. Moreover, it is assumed that a
pp
= 0. If
the state of system p is available to system j, then sys-
tem p is said to be a neighbor of system j. The neighbor
set of node v
j
is denoted by N
j
, where j /∈ N
j
[25].
2.2 Problem statement
In practice, there are several cases where the individual
dynamics can be described by a higher-order form. For
example, the jerk system is described by third-order
differential equations [1]. A single-link flexible joint
manipulator can be modeled by a fourth-order nonlin-
ear system. A group of unmanned aerial vehicle forma-
tion control problem, in essence, is a high-order multi-
agent system network coordination problem. In partic-
ular, the features of high order is more obvious during
the aircraft through tactical maneuvers. In addition, due
to the imprecision measurement and interactions with
complex environments, networked nonlinear systems
with uncertainties and external disturbances have to
be investigated simultaneously in practice. For those
complex nonlinear dynamics, the Brunovsky canoni-
cal form can been obtained through the linearization
method. In the process of model transformation, the
unmodeled dynamics and disturbances are embodied
in the smooth nonlinear function and external distur-
bances.
Take the single-link flexible joint manipulator as a
representative example, whose dynamics can be written
as [15]
˙x
1
= x
2
, ˙x
2
=−
MgL
I
sinx
1
−
k
I
(x
1
− x
3
),
˙x
3
= x
4
, ˙x
4
=
k
J
(x
1
− x
3
) +
1
J
u,
where I , J are, respectively, the link and the rotor iner-
tia moments, M is the link mass, k is the joint elastic
constant, L is the distance from the axis of the rota-
tion to the link center of mass and g is the gravitational
acceleration, respectively. This nonlinear dynamics can
be transformed to the higher-order normal form with
z
1
= x
1
as
˙z
1
= z
2
, ˙z
2
= z
3
, ˙z
3
= z
4
, ˙z
4
= a(z) + b(z)u,
In this paper, we consider a group of m (m ≥ 2)
systems with non-identical dynamics distributed on
an undirected communication network G. The dynam-
ics of the j-th system is described in the nonlinear
Brunovsky form as
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