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Ding et al. Complex Eng Syst 2023;3:7 I http://dx.doi.org/10.20517/ces.2023.06 Page 9 of 13

濃

濄 濅

濆 濇
Figure 2. Communication topology diagram of MASs. MASs: multi-agent systems.

The path of agents
20
leader
0 follower 1
follower 2
follower 3
-20
follower 4
-40

-60

-80

-100

-120
0 50 100 150

Figure 3. The trajectories of the MASs under the robust control term ( )(19). MASs: multi-agent systems.

The sliding mode controller parameters are chosen as = 1, 1 = 0.1.

Simulation results are shown in Figures 3-7. Among them, Figure 3 shows the tracking trajectories of MASs
under the robust control term ( ) (19), and the horizontal and vertical axis represent the position state of
different dimensions, respectively. As we can see from the Figure 3, the trajectories of the agents don’t converge
to a point, which indicates that the MASs under the robust control term ( ) (19) can’t achieve consensus
under the deception attacks. For comparison, Figure 4 shows the tracking trajectories of the agents under
the proposed adaptive SMC law (18), and it is shown that the closed-loop MASs under channel fading and
deception attacks can achieve consensus tracking. Figures 5 and 6 show the position error 1 ( ) and velocity
error 2 ( ) between followers respectively. Figure 7 shows the sliding variable ( ) of the followers, respectively.

Remark 4. As shown in Figures 5-7, agents 1 and 2 have better consensus tracking performance with smaller
amplitude oscillating and smoother curves, that is, because they can receive accurate information from the
leader. In contrast, agents 3 and 4 perform worse because they cannot obtain information from the leader, but
only from neighbor agents over fading channel ( as shown in Figure 2). Even so, the proposed adaptive SMC
scheme can still guarantee consensus tracking of all followers, as shown in Figure 4.