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

with ˆ the given bound of projection, and scalar 0 < < 1.

According to Imbedded Convex Sets Assumption [37] , we obtain:

Proj ˆ ( ), ( ) ( ( ), )
≤k ( )k ( ( ), ), (24)

and

( ˆ ( ) − ( ))(Proj( ˆ ( ), ( ) ( ( ), )) − ( ) ( ( ), )) ≤ 0 (25)

and these two conditions will be used in the following derivation.
Remark 3. In some existing reliable control methods, the known bounds of attacks are usually utilized, which
may inevitably yield larger conservativeness. To overcome this shortcoming, the online estimation mechanism
for unknown attacks/faults was proposed in some related works [33,38] . Inspired by these works, the online
estimation mechanism of the attack is integrated with the SMC technique in this work.

3.2. Consistence and Reachability
Theorem1. ConsidertheMASs(3)-(4)withchannelfading(8)anddeceptionattacks(12), undertheproposed
SMC law (19)-(20), the reachability of the sliding surface ( ) = 0 can be guaranteed in the sense of mean
square.
Proof. Choose the Lyapunov function as follows:
1 1 −1 2

( , ) = ( ) ( ) + e ( ), (26)
2 2
where ˜ ( ) = ˆ ( ) − ( ) is the estimated error with ˜ ( ) = ˆ ( ).
¤
¤
Then, by the expressions (15) and (21), the derivative of 1 ( , ) can be given as:

−1

˜
¤
¤
( , ) = ( ) ¤( ) + ( ) ˜ ( )

= ( )( 2 ( ) − ¯ 2 ( ) + ( + ) ⊗ · ( )Ψ ( ( ), )) (27)
−1
− k + k · ˆ ( ( ), ) · sgn(¯( )) − 1 sgn(¯( ))) + ( ) ˜ ( ).

˜
¤
Taking mathematical expectation to the above expression (27), one has:

¤
E[ ( , )] = ( )[ 2 ( ) − E( ¯ 2 ( )) − 1 E(sgn(¯( ))) + ( + ) ⊗

· ( )Ψ ( ( ), ) − k + k ˆ ( ( ), )E(sgn(¯( )))] (28)

+ k + k( ˆ ( ) − ( ))Proj( ˆ ( ), ¯ ( ) ( ( ), )).

It can be easily verified from expressions (5) and (9) that E ( 1 ( )) = E ( ¯ 1 ( )), E ( 2 ( )) = E ( ¯ 2 ( )). Mean-

while, it follows from (14) and (17) that E ( ( )) = E (¯( )). Then, one can obtain:

¤
E ( , ) = − 1 ( ) sgn( ( )) + ( )( + ) ⊗ · ( )Ψ ( ( ), )

+ k + k( ˆ ( ) − ( ))Proj( ˆ ( ), ( ) ( ( ), ))

− ( )k + k ˆ ( ) · ( ( ), ) sgn( ( )) (29)

≤ − 1 k ( )k + k + kk ( )k kΨ ( ( ), )k k ( )k

+ k + k( ˆ ( ) − ( ))Proj( ˆ ( ), k ( )k · ( ( ), ))

− k + k ˆ ( ) ( ( ), )k ( )k.
By the conditions k ( )k ≤ ( ), kΨ ( ( ), )k ≤ ( ( ), ), one has:

¤
E ( , ) ≤ − 1 k ( )k + k + k( ( ) − ˆ ( ))(k ( )k

(30)

· ( ( ), ) − Proj( ˆ ( ), ( ) · ( ( ), ))).