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Page 264 Sun et al. Intell Robot 2023;3(3):257-73 I http://dx.doi.org/10.20517/ir.2023.17

For facilitating the proofs in the following sections, the following useful lemmas, which play important roles
in our derivations, are presented first.

Lemma 1 [33] For any positive definite matrix ∈ R × with = , a scalar > 0 and a vector function

¤ : [− , 0] → R such that the following integration inequality is well defined, i.e.,
∫ [ − ]

− ¤ ( ) ¤( ) ≤ − ( ) ( ) (11)

− −
[ ]

where ( ) = ( ) ( − ) .
[ ]
Lemma 2 [34] For any matrix > 0, > 0, scalars − 2 ≤ ( ) ≤ − 1, and a vector function

¤ : [− 2 , − 1 ] → R such that the following integration inequality is well defined, i.e.,

∫
− 1

−( 2 − 1 ) ¤ ( ) ¤( ) ≤
− 2
 − −  (12)
 
−  ∗ 2 − − −  ( )
 
 ∗ ∗ 
 
]

where ( ) = [ ( − 1 ) ( − ( )) ( − 2 ) .
3.1. Stability analysis
Theorem 1 For some positive scalars , , , ( ≤ ), , , and matrix , 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 (13)
∗ Π 22 ∗ 2
where

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

(3, 3) = −2 ( + − 2 2 ) + Φ,

(3, 4) = −2 ( 2 − ), (3, 5) = − Φ,

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

(6, 6) = − ].
Π 12 = [ 1 Γ, ( − ) 2 Γ].
Π 22 = [− 1 , − 2 ],
Γ = [ , 0, , 0, − , ].

Then, the NCS (9) under SS-ETC scheme (5) is ISS.
Proof: See Appendix.

From the proof of Theorem 1, it is clear that the convergence speed of the proposed event-triggered control
algorithm is indicated by the parameter . Namely, for a given convergence speed , if a solution exists for
the LMIs (13) under the proposed SS-ETC scheme (5), the convergence speed is achieved by the designed
controller .