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Ding et al. Complex Eng Syst 2023;3:7 I http://dx.doi.org/10.20517/ces.2023.06 Page 5 of 13

Figure 1. System over fading network subject to attacks on agent .

where ≜ diag{ , , . . . , }, (·) being the Kronecker delta function, which compares values of and
1
2

and returns 0 when they are not equal; otherwise, it returns 1. Λ 1 ( ) ≜ diag{0, 12 ( ), · · · , 1 ( )}, · · ·,
Λ ( ) ≜ diag{ 1 ( ), 2 ( ), · · · , 0}. Considering the effect of channel fading, since only faded data is avail-
able, accurate neighbors’ information cannot be used for the controller design. In the following section 3.1,
the SMC law will be designed based on the consensus tracking errors ¯ 1 ( ) and ¯ 2 ( ).

Remark 1. There are two special cases considered in the channel fading model (8): when ( ) = 0, it means
that there is no information interaction between agents and the communication channel is blocked, that is, the
channel fading model is simplified to a packet loss model. In contrast, if ( ) = 1, it indicates that the data
transmission between agents is complete and without any attenuation.
2.4. Deception attacks
Among various cyber-attacks, the deception attack on controllers is a common form and usually satisfies the
following assumptions: the hackers can steal the state information or measurement output of the agents to
generate false data, which can then be injected into the controller. As shown in Figure 1, the hackers can attack
the controller of agent by injecting false data. Thus, the actual data received by the actuator of agent is as
follows:
ˇ ( ) = ( ) + ( )Ψ ( ( ), ). (11)

The compact form of expression (11) can be written as:

ˇ ( ) = ( ) + ( )Ψ ( ( ), ). (12)

where ( ) is the designed control input and ( )Ψ ( ( ), ) is the false data. The matrix ( ) is an unknown
and time-varying matrix that satisfies k ( )k ≤ ( ) with ( ) unknown and bounded, represents the in-
jection patterns of the false data, for example, ( ) may be a matrix composed of elements 0 and 1, that is,
sometimes false data is injected, sometimes not, to confuse users. Thus, the attack is difficult to be detected
by users. Ψ ( ( ), ) is a function of ( ), which means the false data generated via the state ( ), and satisfy
kΨ ( ( ), )k ≤ ( ( ), ) with ( ( ), ) a known nonnegative function.

Remark 2. The deception attacks considered in this work focus on the controller, that is, ( ) may be suffered
from the false data injection, such as the problem considered in some literature [30,36] and so on. The deception
attacks can also occur in the communication channel between agents, that is, ( ) may be affected by false data
injection during transmission [34] .

3. MAIN RESULTS