Page 36 - Read Online

Yang et al. Intell Robot 2022;2(3):223­43 I http://dx.doi.org/10.20517/ir.2022.19 Page 229

protocol (4), the dynamics of ( ) can be given by

∑

( + 1) = ( ) + { ( )[ ( )( ( ) − ( )) + ( )]

=1 (5)

+ ( )[ ( ) ( ) + ( )]}

0
0
with ( ) = [ ( ), ( ), ..., ( )] . Assume that all channel fading are identical, ie., ( ) = ( ),

1 2
0 ( ) = 0 ( ), ( ) = ( ), and 0 ( ) = 0 ( ) for ≥ 0, , = 1, 2, ..., . Then, the above equation can
be rewritten in compact form as follows:
(6)
( + 1) = [ ⊗ + ( ( )L + 0 ( )M ) ⊗ ] ( ) + [ ⊗ ( )] ( )

where ( ) = [ ( ), ( ), ..., ( )] , ( ) = ∑ ( ) ( ) + ( ) 0 ( ), = 1, 2, ..., .
1 2 =1
Before the subsequent analysis, some definition and assumptions are introduced.
Definition 3 The systems (6) is said to achieve leader-following consensus in mean square sense, if the system
(6) is -error mean square stable.
Definition 4 [39] The dynamic system (6) is said to be -error mean square stable, if the following conditions
hold
[ 2 ]
lim E ∥ ( )∥ = 0 (7)
→∞ (0), (0), +1 ≤ | =

+
for given the upper bound of sojourn-time ∈ N and any initial conditions (0), (0) ∈ O, ∈
{1, 2, ..., }, ( ) = 0.
Assumption 1 Every possible undirected graph G ( ), ( ) = ∈ O is connected.

Assumption2Themeanandvarianceofstochasticvariables { ( )}and { 0 ( )}are E{ ( )} = , E{ 0 ( )} =
0 , E{( ( ) − ) } = , and E{( 0 ( ) − 0 ) } = .
2
2
2
2

0
According to the above analysis and discussion, it can be found that the leader-following consensus of system
(1)underthesemi-Markovswitchingtopology G isequivalenttothemeansquarestabilityofsystem(6). Since
( ) and 0 ( ) are interference in the channel, we can treat the last term in equation (6) as a disturbance.
To tackle with the disturbance, the following control output are given
( ) = ( ⊗ ) ( ) (8)

with ( ) = ( ( ) − 0 ( )) and ( ) = [ ( ), ( ), ..., ( )] . ∈ R × is a known constant matrix.
1 2
Then, the consensus problem of system (1) is transformed into a H ∞ control problem of the system (6).
Consequently, the objective of this paper is to design the distributed consensus controller such that the follow-
ing two conditions are satisfied:
(I) when ( ) = 0, the condition (7) holds;
{ }
∞ +1 −1
∑ ∑ 2 2 2
(II) the inequality E [∥ ( )∥ − ˆ ∥ ( )∥ ] < 0 holds for zero-initial condition and any nonzero
=0 =
( ) ∈ 2 (0, +∞).