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Page 4 of 13 Ding et al. Complex Eng Syst 2023;3:7 I http://dx.doi.org/10.20517/ces.2023.06

with 0 ( ) ∈ R , 0 ( ) ∈ R the leader’s position and velocity, respectively, and 0 ( ) ∈ R representing the
control input.
Define the th follower’s consensus tracking errors as follows:
( Í
1 ( ) = ( ) − ( ) + ( ( ) − 0 ( )) ,
=1 (5)
Í
2 ( ) = ( ) − ( ) + ( ( ) − 0 ( )) ,
=1
with 1 ( ) and 2 ( ) the tracking error variables of position and velocity, represents the element of ,
determines whether there is information interaction between the leader and the followers, when > 0, agent
can receive information from the leader, otherwise, = 0.
The tracking errors can be rewritten in the compact form:

1 ( ) = ( + ) ⊗ · ( ( ) − 1 ⊗ 0 ( )), (6)
2 ( ) = ( + ) ⊗ · ( ( ) − 1 ⊗ 0 ( )),

with 1 ( ) ≜ [ ( ), · · · , ] , 2 ( ) ≜ ( ), · · · , ( ) , ( ) ≜ ( ), · · · , ( ) , ( ) ≜ ( ), · · · ,
11 1 21 2 1 1

( )] , ( ) ≜ ( ), · · · , ( ) , ≜diag{ 1 , 2 , · · · , }.

1
From the above definition, one can obtain the tracking error system as:

¤ 1 ( ) = 2 ( ), (7)
¤ 2 ( ) = ( + ) ⊗ · ( ( ) − 1 ⊗ 0 ( )).
Now, the consensus tracking problem of MASs (3)-(4) converts to the stabilization problem of the tracking
error system (7). The objective of this work is to achieve leader-follower consistency.
2.3. Fading channel
As stated in the Introduction, the transmission between followers may be inevitably suffered from the channel
fading phenomenon. In this work, the network channel is considered as a continuous one with time-varying
channel gain, the transmitted data will be modeled as the actually received information with random attenua-
tion. Hence, introduce the following memoryless multiplicative fading model:

( ) = ( ) ( ), (8)
( ) = ( ) ( ),
where ( ) and ( ) are the fading position and speed signal of the th agent received by the th agent, and
( ) and ( ) are the signal and speed signal sent by the th agent, respectively. The random coefficient

( ) ∈ (0, 1] are mutually independent random variables with mathematical expectation E ( ) = ¯ .
Assuming that fading occurs only in the channel between followers, the special case of channel fading from
the leader to the followers is not considered in this work. Hence, based on the fading information (8), the
tracking errors (5) are rewritten as:

 Í 1
 ( ) − Λ ( ) ( ) + ( ( ) − 0 ( )) ,
  ¯ 1 ( ) =
=1 ¯
(9)
Í 1
 ( ) − Λ ( ) ( ) + ( ( ) − 0 ( )) .
 ¯ 2 ( ) = =1 ¯

It can be seen that the tracking errors (9) involve the expectation of the random variable ( ), which is
introduced to compute the consistent tracking error variable among the agents more accurately.

Define ¯ 1 ( )≜[ ¯ ( ), · · · , ¯ 1 ( )] , ¯ 2 ( )≜ ¯ ( ) , · · · , ¯ 2 ( )] . Then, the compact form of tracking errors

21
11
(9) is of the following form:
Í 1
 =
 ¯ 1 ( ) ( ⊗ ) · Λ ( ) ( ) − ( + ) ⊗ 1 ⊗ 0 ( )
 =1 ¯


 −( + ) ⊗ · ( ) ,
Í 1 (10)
 ¯ 2 ( ) = ( ⊗ ) · Λ ( ) ( ) − ( + ) ⊗ 1 ⊗ 0 ( )
 =1 ¯


 −( + ) ⊗ · ( ) ,
