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Tan et al. Complex Eng Syst 2023;3:6 I http://dx.doi.org/10.20517/ces.2023.10 Page 5 of 23

where Ω > 0 are the weighting matrix; ∈ [0, 1] are the weighting parameters of the corresponding packet
and satisfy ∑ = 1. ( ) are the memory event-triggered threshold and meet the following conditions
=1

1 0 ∑ ( ) T ( )
¤
( ) = ( 2 − ) (Δ ( )) Ω Δ ( ), (5)
( ) ( )
=1
where ( ) ∈ (0, 1] and 0 > 0 is used to regulate the release rate of sampling data. The framework of the
decentralized control for interconnected semi-Markovian jump systems with a dynamic METM is shown in
Figure 1.

Remark 1. From (5), we can obtain that the dynamic threshold ( ) is related to the error variable ( ) ( ).

Whentheerrorvariabletendstozero,forinstance,thesystemtendstobestableattheequilibrium,thedynamic
threshold converges to a constant. When ( ) > 0, ( ) is monotonically increasing, which means that the
¤
release rate of data at the sampling time will reduce. On the contrary, when ( ) < 0, ( ) is monotonically
¤
decreasing, the release rate of data at the sampling time will increase. In particular, when ( ) ≡ 0, the event-
¤
triggered condition becomes the traditional memory event-triggered condition [22] .

Remark 2. By using the historical trigger signals, a memory-base event-triggered condition is proposed in (4),
where the past events are assigned appropriate weighting values. This METM can not only save network re-
sourcesbutalso canimprove thefault toleranceof the event-triggering mechanismcompared tothe traditional
design.

We divide the sampling time interval [ ℎ + , +1 ℎ + +1 ) into + 1 parts as follows:

(6)
[ ℎ + , +1 ℎ + +1 ) = ∪ ,
=0
where = 0, 1, 2, · · · , , = min{ | ℎ+( +1)ℎ+ ≥ +1 ℎ+ +1 } and = [ ℎ+ ℎ+ , +1 ℎ+ ℎ+ℎ+ +1 ),
denotes the network induced delay. Define delay function ( ) = − ( ℎ + ℎ), and we can get

0 ≤ ≤ ( ) ≤ + ℎ ≤ , ∈ . (7)

Definetheerrorvariable ( ) ( ) = ( − +1 ℎ)− ( ℎ+ ℎ),andcombinethedelayfunction ( ) = −( ℎ+ ℎ),

then we can obtain
( − +1 ℎ) = ( ) ( ) + ( ℎ + ℎ) = ( ) ( ) + ( − ( )). (8)

The control input ( ) in system (1) can be designed as

∑ ∑
( ) = ( ) ( − +1 ℎ) = ( )[ ( ) ( ) + ( − ( ))], ∈ . (9)

=1 =1
Based on the above analysis, system (1) can be rewritten as

∑ ∑ ∑
( )
¤ ( ) = A ( ) + B ( − ( )) + B ( ) + G ( ), (10)

=1 =1 =1, ≠

where is the controller gain matrix. Next, a definition and some lemmas will be innovated to deduce the

subsequent results of this paper.
Definition 1( [16] ): Suppose ( ( ), , ≥ 0) is a functional candidate, then the infinitesimal operator = ( )
is represented as
{ ( ( + ), + )| ( ), } − ( ( ), )
= ( ( ), ) = lim . (11)
→0