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Page 262 Sun et al. Intell Robot 2023;3(3):257-73 I http://dx.doi.org/10.20517/ir.2023.17

• The upper bound of ( ) is , while = 0.

• A larger leads to a smaller ( ) and vice versa. In addition, when → ∞, ( ) → 0.
• is used to avoid zero in the denominator.
Since the ETC is an error-oriented communication scheme, then ( ℎ) is generally defined as the state error
between the current sampling instant and the latest transmitted sampling instant, i.e.,

( ℎ) = ( ℎ) − ( ℎ) (4)
with ℎ = ℎ + ℓℎ, ℓ ∈ N.

Motivated by (3), the SS-ETC scheme described in Figure 1 is designed as

+1 ℎ = ℎ + min{ℓℎ| ( ℎ)Φ ( ℎ) ≥ X( )} (5)
ℓ || ( ℎ)|| +

where X( ) = ( ℎ)Φ ( ℎ). > 0 is a given constant, is a positive event-triggered parameter related to
, and Φ is a positive definite matrix to be designed.

Theeventdefinedby X( ) isimportanttothetransmissionundertheETCscheme. Itisworthnoting
|| ( ℎ)||+
that the existing parameter is used to avoid zero in the denominator when one wants to design a meaningful
event-triggered scheme (5). In addition, once the parameter is selected, the corresponding event-triggered
parameter can be designed.

Remark 2 In fact, the threshold value is important to make a trade-off between control performance and commu-
nication efficiency. The basics of the event design are to guarantee the stability of the controlled systems. But it is
not easy to seek the best threshold because of time-varying requirements on control performance and communica-
tion efficiency. Thus, one can design dynamic ETC schemes or adaptive ETC schemes in order to achieve a better
trade-off between control performance and communication efficiency.
Remark 3 From the SS-ETC (5), a dynamic event threshold is well characterized by , which is related
|| ( ℎ)||+
to the parameters , , and || ( ℎ)||. Based on the established SS-ETC scheme, we can arrive at
• The pair ( , ) gives the basis of the event-triggered parameter design while preserving the input-to-state sta-
bility (ISS). For a different , one can derive a corresponding to guarantee the existing of the controller
;
• The supplies a upper bound of event-triggered parameter when || ( ℎ)|| = 0. This indicates that a lower

transmission frequency is expected while the stability of the studied NCS is guaranteed;
• The event-triggered threshold is dynamically adjusted according to the latest available measurement || ( ℎ)||,
directly.
Thus, the main advantages of the proposed SS-ETC lie in that: 1) by comparing adaptive ETC proposed in [25] , the
lower bound of the event-triggered is not needed for the proposed SS-ETC (5); 2) by comparing the dynamic
ETC scheme [26] , the extra term ( ), which is used to adjust the event threshold, is not necessary for the proposed
SS-ETC (5); 3) It is clear that the thresholds of the proposed SS-ETC scheme (5) are always arrested in a stability
region while threshold adjusting.

Therefore, the proposed SS-ETC (5) will make a trade-off between communication efficiency and control per-
formance while the desired stability is guaranteed.

2.3. Path following control modeling under SS-ETC scheme
Underthe above assumptions, the actual control actions based on (2) with a fixed controller can be represented
as follows: