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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 265

Remark 5 Compared to the static ETC scheme proposed in [30] and adaptive ETC scheme [20] , the SS parameter
is introduced during the proof of the stability analysis. Obviously, the different event-triggered parameter can
be obtained by a different SS parameter . Thus, ∈ [0, 1) is removed during the design of the corresponding
event-triggered parameter while introducing an appropriate . By combing Remark 4, one can see that the

supplies the upper bound of the event-triggered threshold. Thus, any event-triggered thresholds less than would

not destroy the stability property of the physical systems.

3.2. Controller design with its algorithm

Theorem 2 For some positive scalars , , , ( ≤ ), , , if there exist real symmetric positive definite
matrices , , ( ∈ {1, 2} ) and arbitrary matrices for such that the follow inequalities hold

[ ] [ ]
Π 11 Π 12 2
Π = < 0, > 0 (14)
∗ Π 22 ∗ 2
where

Π 11 = [(1, 1) = + + 2 + − −2 1,
1
(1, 2) = −2 1 , (1, 3) = , (1, 5) = − ,
(1, 6) = ,
(2, 2) = −2 ( − 1 − ) − −2 2, (2, 3) = −2 ( 2 − ), (2, 4) = −2 ,
1
2

(3, 3) = −2 ( + − 2 2 ) + Φ,

(3, 4) = −2 ( 2 − ), (3, 5) = − Φ,

(4, 4) = −2 (− 2 − ), (5, 5) = Φ − Φ,
2

(6, 6) = − ].
Π 12 = [ 1 Γ, ( − ) 2 Γ].
Π 22 = [ − 2 1 , − 2 2 ],
Γ = [ , 0, , 0, − , ].
−1
Then, the NCS (9) under SS-ETC scheme (5) can be stabilized by the controller = while ISS is preserved.

Proof: The proof of this section is based on Theorem 1. By defining = , = , = ,
−1

= , Φ = Φ with ∈ {1, 2}, and = , then one can pre- and post-multiplying both side of

left equalities in (13) with [ , , , , , ] and right inequalities in (13) with [ , ]. By using
− ≤ − 2 to deal with the nonlinear terms in (13).
Remark 6 Different from the controller synthesis proposed in [27,30,34] , the controller design not only depended
on the parameter but also the parameter . By introducing the parameter , an upper bound of the event
threshold is well given. Then, the designed controller according to both and can be well used to cope with
the fluctuation of the event-triggered threshold. This is very different from the above controller design with a fixed
event-triggered threshold.

Moving forward, the co-design algorithm for finding the event-triggered parameter is presented.

Algorithm 1 Co-design of control and communication
1: Set the positive scalars , , ( ≤ ), , and the initial event triggered parameter . Give the
increasing step Δ > 0 and an optimization target < 0 ;
2: While < 0
3: = + Δ
4: Solve LMIs (14), if there is a feasible solution , , ( = 1, 2) and Φ satisfying LMIs (14), go to the
next step. Otherwise, return Step 1.
5: Return − Δ and calculate and Φ.