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Page 8 of 13 Ding et al. Complex Eng Syst 2023;3:7 I http://dx.doi.org/10.20517/ces.2023.06

Then, it follows from (25) that:

¤
E ( , ) ≤ − 1 k ( )k
(31)
≤0.
Hence, the reachability of the sliding surface ( ) = 0 can be ensured in the sense of mean square.
□
Theorem 2. Considering the MASs (3)-(4) subject to deception attacks (12) and channel fading model (8), the
consensus tracking for MASs (3)-(4) will be achieved under the proposed sliding surface (14) and the SMC
law (19)-(20).
Proof. Select the Lyapunov function:
1 1
( ) = 1 ( ) 1 ( ) + 2 ( ) 2 ( ), (32)
2 2
Its derivative is given as:
¤
( ) = ( ) ¤ 1 ( ) + ( ) ¤ 2 ( )
1 2 (33)

= ( ) 2 ( ) + ( ) ¤ 2 ( ).
1 2
When the sliding surface ( ) = 0, it follows from (14) and (7) that 2 ( ) = − 1 ( ) and ¤ 2 ( ) = − 2 ( ), then

we can obtain:
¤
( ) = ( ) 2 ( ) + ( ) ¤ 2 ( )
1 2

= − 1 ( ) 1 ( ) − ( ) 2 ( )
2 (34)
2 2
= − k 1 ( )k − k 2 ( )k
≤ 0.
Combining the results of Theorem 1, the consensus tracking of MASs (3)-(4) can be ensured under the pro-
posed sliding surface (14) and the SMC law (19)-(20). □
4. SIMULATION
Considerthesecond-orderMASswithoneleaderand4followers,wherethecommunicationtopologybetween
agents is shown in Figure 2. The blue arrows indicate that the followers receive the complete information from
the leader, while the red arrows indicate that the information interaction between followers is over fading
channel. Thereby, follower 1 and follower 2 can receive accurate information from the leader, follower 3 can
only receive the fading data from follower 1, and follower 4 can only receive the incomplete data from both
follower 1 and follower 2. For simplicity, in this simulation example, the adjacency weights between neighbor
agents are set as 1.

Then, according to the leader and followers’ topology, we can get the adjacency matrix , the diagonal matrix
, and the Laplace matrix of these MASs as follows:

 0 0 0 0   0 0 0 0 
   
    0 0 
0 0 0 0 0 0
=   , =   , = 1 1 0 0 .
1 0 0 0 −1 0 1 0
   
 1 1 0 0   −1 −1 0 2 
   

In this simulation, the initial state of the leader’s position, speed, and control input are set as 0 = [10, −10] ,

0 = [10, −10] , 0 = [ ( ), ( )] , the initial state of the followers’ position and speed are set as 1 =
[10, −2] , 1 = [20, −2] , 2 = [15, 15] , 2 = [20, 3] , 3 = [25, 5] , 3 = [15, 0] , 4 = [45, 15] ,

4 = [35, 0] ,andtheinjectionpackets ( )Ψ ( ( ), ) setas ( ) = [1, 1, 1, 1] and Ψ ( ( ), ) = 10 ( ) ( ).