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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 273

¤ 2
2 ( ) = −2 2 ( ) + ( − ) ¤ ( ) 2 ¤( )
∫
−
¤ ( ) 2 ¤( )
− ( − ) 2 ( − ) (22)
−
In view of Lemma 1 and Lemma 2, the following relation holds
∫
¤ ( ) 1 ¤( ) ≤
− 2 ( − )
−
[ ] [ ] [ ]
( ) 1 − 1 ( )
− −2 (23)
( − ) − 1 1 ( − )
and
∫
−
¤ ( ) 2 ¤( ) ≤
− ( − ) 2 ( − )
−
−
 2 − 2 
 
− −2  ∗ 2 2 − −  ( ) (24)
( )
 − 2 
 ∗ ∗ 
 2 
]

where = [ ( − ) ( − ( )) ( − ) .

By summing up (18) ∼ (24) and applying ( ℎ)Φ ( ℎ) ≤ X( ), the following relation is achieved

( ) + 2 ( ) − ( ) ( ) <
¤
2 2

( )Π 11 ( ) + ¤ 1 ¤ + ( − ) ¤ 2 ¤ (25)

where ( ) = { ( ), ( − ), ( − ( )), ( − ), ( )} and Π 11 is given in (13).
Using Schur complement for (25), (13) hold when Π < 0 for all .
2
Let = ( ), = ( ) + 2 ( ) + 2 ( ) and || || = max(|| ( )||) for = 1, 2, it gives
2
√ √
1
|| ( )|| ≤ − || (0)|| + || || (26)

from (25).

Obviously, (26) shows that the NCS (9) is ISS under the SS-ETC scheme (5).