IEEE Systems, Man and Cybernetics Magazine - July 2022 - 42

integer-order system [1]. The synchronization of chaotic
fractional differential systems, following these
results, is becoming a challenge because of the potential
applications in control and communication processing.
In addition, several practical systems, like
power system dc-dc converters, electrical circuits,
and permanent magnet synchronous motors (PMSMs),
can be elegantly represented and accurately modeled
using fractional calculus (see [2] and [3]). It is widely
known that the consensus of a multiagent system
(MAS) is one of the most important phenomena in
recent years; thus, it has been extensively studied [4]-
[6]. Nonlinear dynamic agents cannot be avoided in a
physical system, and they can be used in engineering
applications, for example, in robotic manipulators [7].
In control theory, considerable attention has been
given to research on the consensus of FOMASs (see,
for example, [8] and [9]).
The T-S fuzzy model offers a valuable tool for approximating
several complex nonlinear systems. The model
easily links a variety of linear subsystems with fuzzy
membership capabilities. Because of the mathematical
simplicity of T-S fuzzy analysis, the problem of control
of nonlinear dynamic systems has been investigated
extensively [10]. The authors of [11] investigated a T-S
fuzzy control problem based on linear matrix inequalities
for chaotic systems of fractional order with T-S
fuzzy model PMSM applications. Recently, several
authors have addressed the issue of consensus of T-S
fuzzy MASs [12], [13].
An impulsive control scheme does not need continuous
state information and, thus, it is simpler and more robust
in real systems [14]-[16]. The strategy of impulsive control
for a MAS thus needs to be developed. An impulsive control
approach has also been used in MAS consensus by
researchers [17], [18]. In addition, in many practical applications,
the consensus is important in finite time, especially
for control issues. Thus, finite-time coordinations
have attracted a significant number of researchers in the
field of MAS [19], [20]. Still, a few achievements are based
on the consensus problem of a T-S fuzzy FOMAS under an
ADT via impulsive control. Based on the previous analysis,
we aim to study the leader-following consensus problem
of a T-S fuzzy FOMAS via impulsive control using
ADT approaches.
The key contributions of this study are as follows. The
criteria guaranteeing the finite-time consensus are given,
and the ADT is estimated. Compared with existing
results, the proposed impulsive control strategy in this
article requires only local neighboring information. Thus,
the proposed impulsive method can reduce the running
cost and be applied easily to practical multiagent engineering
systems.
1) In [8] and [9], a continuous control scheme has been
studied for FOMASs. Compared with those works, our
study considers a T-S fuzzy impulsive control scheme,
42 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE July 2022
which can effectively reduce the cost of signal transmission
between agents.
2) A suitable Lyapunov functional is constructed in the
fractional domain, and, by utilizing fractional calculus
theory and algebraic graph theory, sufficient conditions
are designed to reach finite-time consensus of a T-S
fuzzy FOMAS. An exponential consensus with convergence
rate is also realized.
3) Finally, numerical simulations are proposed for a PMSM
chaotic model via the T-S fuzzy approach and on Chua's
circuit to exhibit the performance and applicability of
the suggested theories.
Model Description
For the algebraic graph theory and some definitions of
fractional calculus, please refer to [1], [4], and [15]. A nonlinear
FOMAS with one leader and N followers is
described by
t tg tt ut p 12
C
t pp p
11
where Dt
C
11 11 1f 1 !
b
=
pp pp
12 n
refers to the Caputo fractional derivative with
01 ,[ ,, ,] R
b
t
Tn denotes the information
state of agent ,and : RR R is a nonlinpg
" #+
ear function.
The distributed impulsive controller is designed as
3
ut || hhh
kka Np
p ()=-d ^tt ba tt^^ -pk qk
k=1
+)
wt
t hhk,,N
k
pp kk^11 !
^
where bk
h
h
0 ^
+
pq 11 ^
q!
(2)
satisfies 0 tt tt
+3.( )td)
-
d t 0= () , for t 0! Let w 0p
)
.
is the impulsive strength. The time sequence {}tk
111ff , and lim tkk
01 kk111 1
=
" 3
from the leader node to node p, w , p
WR .
= diag{} !wp
nn
#
is the impulsive control function satisfying
$ as the weight of the edge
is greater than 0 if
and only if there is an edge from the leader node to node p,
and
The FOMAS (1) is defined by T-S fuzzy as follows:
Rule l for the agent ;p
IF ()tp1
i
THEN
Z
[
\
]
]
]]
]
]
]
C
t 11(),,
+DAtt
tt
tt tb
b
T11 1
()
q!
p
t pl pk
pk
() =
!
r
=-= ni
-
p k() ()
p k kl
l
=1
a | 11pq(( )( ))
# at t
pk qk
+- k
wt t11 k N+
pp kk
N
(( )( )) ,.
!
The defuzzified output of the T-S fuzzy FOMAS under
an impulsive controller is represented as
(3)
| ((t))
p
is M ,,l1 f and iph ()t
is M ;lh
() (( ), )( ),
DN (1)
b =+ f= ,, ,,
IEEE Systems, Man and Cybernetics Magazine - July 2022

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