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Page 12 of 18 Guo et al. Intell Robot 2023;3(4):596-613 I http://dx.doi.org/10.20517/ir.2023.32

which denotes that condition (33) can always be satisfied.

Therefore, based on Lyapunov stability principles, for an arbitrary initial state (0) and (0) of system (27),
one can achieve that
lim ( ) = 0
→∞
(52)
lim ( ) = 0
→∞

which are equivalent to that
lim − = 0
(53)
→∞
lim − = 0
→∞

where , ∈ {1, 2, . . . , }. The stability of system (27) can be achieved, which denotes that the consensus of
system (1) can be achieved.

The proof is thus completed.

Remark 5. According to the event-triggered function (50), the event-triggered condition is met when the
error k k exceeds the threshold, which means that the accumulated error in the information received at the
previous triggering time has exceeded the stable range, and the control input needs to be updated immediately.
Apparently, this event-triggered condition is easier to achieve in emergency situations due to the rapid and
drastic state changes of neighbors.

Theorem 2. Consider system (27) and event-triggered consensus protocol (28), the system will not exhibit the
Zeno behavior, which means that the time interval between any two events will not be less than

1 1 +
= ln (54)
k k − k k 1 + k k
k k
in which

= ( , 0) = min ( )
2 k ¯ k

Proof: Similar to the proof in [32] , we define

= k k
k k

And one has that

− k k ¤
¤ = ( − )
k k k k 2 k k
k k k ¤ k
≤ (1 + )
k k k k
= (1 + )(k k + k k )
and satisfies that

( ) ≤ ( , 0 )

in which ( , 0 ) is the solution of