IEEE Systems, Man and Cybernetics Magazine - July 2022 - 44

Z
]
]
]
]
js
(, ())tt = tt
ml
[
\
]
]
]
]
js
(, ())( )( ,( ))
[, ],
tt
00
js mj s
b
(, ())( )( ,( ))
() (, ()),
(, ].
#dt tt! 01
d
p
00
+-pj ps pp
max
C #
t0
t
!
td
tt t
b 1
-
kk 1
+
(14)
h
It follows that (see [24]), for tt t01
tt tt Eb00 max
js #js ml tt b
(, ())( ,( )) (( )),
! [, ], we get
- 0
which leads to
(, ())( ,( )) (( )).
js #js ml tt b
tt tt Eb
11 00
For tt t12
! (, ], from (15), we have
js js m
b
tt tt
ml
C #
max
t0
00
t
-
(, ())( ,( ))(( )
() (, ()).
=+ -
+td
11
l
pj ps pp
m
d
max
b 1
b
Similar to the idea of (16), it follows that
js #js m
tt
22 00
b
10
In general, we have tt tkk 1
+
! (, ],
(, ())( ,( ))(( )
(( )))( () ).
tt tt 11
b
#EE tt b
mm
+maxmax
20
b
ll
max
-d
js
#js m
m
#ETb
max
(17)
Denoting {j sh js== {( ,( ))},
minmax tt
{( ,( ))} nd
tt a
we can get
() (, ())and (, ())( ).
{s ## (21)
" js js hs
tt tt tt0
,
2
1
3
(a)
1
2
4
6
(b)
Figure 1. The communication topology for (a) four
followers and one leader and (b) seven followers and
one leader.
44 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE July 2022
()<1t so the finite-time consensus issues of system (7)
can be reached.
t01
From Definition 1, 11 1
,
() ()
<< t
-
p tt02 <11and
()tp 00
3
Thus,
11 #{m
m
() ()
1
b
max
5
7
From (23) and its ADT satisfying (8), it follows that
() ()
11 1{{ t2
p tt0
-
-1^^ hh.
p tt - 6 () 11 T h
b
-+ ml
ET
1 ht ^^ hh^Eb
l
d
max -
# ()].
max
(23)
T
b xd
4
From (19)-(21), we have
()
{s #hs() +- m11 T
# E ^ml
__ 0
b
which implies that
() ()
tt m hhh
maxTb ,
ii^ ^^
h
d
max
Eb
max
l
00
T
(, ())( ,( ))(( )
().
tt tt 11 l
l
00 +- mmax
b
d
max ET
b())
E ((tt )))
-
b
max 10
(16)
js #js m
m
# Eb
obtain
T
b xa
(20)
(, ())( ,( ))(( )
().
tt tt 11 l
maxTb
00 +- mmax
l
d
g tt +
(,
00
max ETb
b())n
(19)
By the definition of ADT in [18], NT) , we
-
max 10
(15)
() |
d
#
#
maxmax
maxmax
max
EE
E
--b
b
((
))... (( )))
(( )),
1
l
l
1## #
m20
- 0
b
tt tt
tt
mm mm...12 k
b
b
mk
E ((tt ))
l
k
b
m10
b
(18)
!!+ (, ],kk 1
- #
tt t00
! (,
b
mm
mm
m
for tt tk N .+ kn# and tt T,0
+ T]; then
max
++ml
max
t0
k
1
| tt
m 1
=
mm
Cb # t pj ps p
t
b 1
-
js
#
#
#
#js md
|Eb
00
maxmax
l tt0
m
max tt b
l
m10
E ml
b((tt ))
-
1###
12
mm k
| E
max
b
m20
m 1
=
(, ())
(, ())( ()
(( )) ()
(( ))
tt
tt 11
k
max
mm
m
b
+-+
-
-
d
1
2
b xa
11 # 11{h ttpp00 0
^ +-d ll b
11 TEE
T
tt -1 7 _
b
max
() () () .T
mmm
-b
xa
maxmax
h # b
00t
<
() () i
@
t =
(22)
According to (21) and (22), we can obtain 10p ()t1<
-1 () . This conflicts with the following inequality of
() ()00 01
11 1
p tt .
<< t
-

IEEE Systems, Man and Cybernetics Magazine - July 2022

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