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Page 6 of 13 Ding et al. Complex Eng Syst 2023;3:7 I http://dx.doi.org/10.20517/ces.2023.06

3.1. Adaptive SMC law
To cope with the impact of the deception attacks, the information about the attack is usually utilized to design
the controller. For example, when the upper bounds ( ) and ( ( ), ) of the attack are known, the de-
sign of the controller is relatively easy to implement, but the fixed upper bounds will inevitably lead to larger
conservativeness. To overcome this problem, an online estimation strategy will be employed to estimate the
time-varying and unknown attacks, based on which, an adaptive sliding mode controller will be designed to
attenuate the effect of the unknown attacks on MASs.

Design the sliding function as follows:

( ) = 1 ( ) + 2 ( ), (13)

with > 0 the sliding gain, denoted ( ) ≜ ( ), ( ), · · · , ( ) , the compact form of sliding function
1 2
(13) can be written as:
( ) = 1 ( ) + 2 ( ), (14)
From (7), we can obtain the derivative of the sliding function:

¤ ( ) = 2 ( ) + ¤ 2 ( )
(15)
= 2 ( ) + ( + ) ⊗ · ( ˇ ( ) − ⊗ 0 ( )) .
Under these constraints considered in this work, the th agent cannot receive accurate and complete informa-
tion from neighbor agents, the switching function (13) under fading channel is rewritten as:
( ) = 1 ( ) + 2 ( ). (16)
The compact form of expression (16) as:

( ) = 1 ( ) + 2 ( ). (17)

Then, construct the sliding mode controller as follow:

( ) = ( ) + ( ), (18)

where the robust term ( ) is designed as :
−1

( ) = −( + ) ⊗ · ( 1 · sgn(¯( )) + ¯ 2 ( )) + ⊗ 0 ( ), (19)
with 1 > 0, and the adaptive term ( ) is designed as:
−1

( ) = −( + ) ⊗ · (k + k ˆ ( ) ( ( ), ) · sgn(¯( ))). (20)
where ˆ ( ) is the estimation of ( ) under the following adaptive law:

ˆ ( ) = k + k · Proj( ˆ ( ), ¯ ( ) ( ( ), )), (21)
¤

with an adaptive parameter, and Proj the smooth projection [37] as:

Proj( ˆ ( ), ( ) ( ( ), ))



 ( ) ( ( ), ), if ( ˆ ( )) ≤ 0,


(22)

0

= ( ) ( ( ), ), if ( ˆ ( )) ≥ 0 and ( ˆ ( )) ( ) ( ( ), ) ≤ 0,



( ˆ ( )) ( ˆ ( ))k ( )k ( ( ), )
0
0


( ˆ ( )), otherwise,

( ) ( ( ), ) −
0
 k ( ˆ ( ))k 0
where the continuous function ( ˆ ( )) defined as:
2
2 ˆ ( )
( ˆ ( )) ≜ ( − 1 + ) (23)
ˆ 2