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Sun et al. Intell Robot 2023;3(3):257-73 I http://dx.doi.org/10.20517/ir.2023.17 Page 263

{
¤ ( ) = ( ) + ( ) + ( )
(6)
)
( ) = ( ℎ), ∈ [ ℎ + , +1 ℎ + +1
where is the cloud controller gain to be designed.

In the following, the divisions of the sampling-like intervals for [ ℎ + , +1 ℎ + +1 ) are presented according
to [30] . Assume that
+1 − −1
Ω = ∪ Ω ℓ
ℓ=0
, ℎ + ℎ + +1 ) with ℎ = ℎ + ℓℎ.
where Ω ℓ = [ ℎ +
By defining
( ) = − ℎ (7)

for all ∈ Ω ℓ, then the piece-wise function satisfying ¤( ) = 1, and

} = .
0 ≤ = min{ , +1 } ≤ ℎ + max{ , +1

Therefore, the control action of (6) can be represented as

( ) = ( ( − ( )) − ( ℎ)) (8)

by combing (4) and (7).

Further, the NCS modeling under SS-ETC (5) is described as


 ¤ ( ) = ( ) + ( ( − ( )) − ( ℎ)) + ( )



subjects to: (9)
 



 ( ℎ)Φ ( ℎ) ≤ X( )
 || ( ℎ)|| +

where the detailed meaning of parameters is given by the aforementioned equation (5). Here, the initial state
of the ( ) can be defined as ( 0 ) for ∈ [ 0 − , 0 ).
Remark 4 In fact, (9) gives a more compact form of the event-triggered control for the NCS. From (9), it is clear
that the stability of the closed loop dynamics now depends on the control update rule through ( ℎ). If the SS-
ETC scheme, where the event-triggered condition given in the below equation of (9), keeps ( ℎ) in a small sense,
the stability of the studied dynamics would not be destroyed. This is the reason that one can stabilize (9) with a
variable-threshold ETC scheme with the same controller .
By using the proposed SS-ETC scheme (5), this paper attempts to pursue the ISS analysis and the controller
synthesis for the NCS (9) while the following objectives are achieved.

Definition 1 The NCS (8) resulting in (5) is said to be input-to-state stable (ISS), if there exist a KL-function
and K ∞-function such that

|| ( )|| ≤ (|| ( 0 )||, ) + (|| ( )|| ∞ ) (10)
for all ( ) ∈ L ∞.

3. MAIN RESULTS
In this section, the stability analysis and input to state stabilization issues are conducted with Lyapunov theory.
In addition, the co-design algorithm is also presented at the end of this section.