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Page 303 Peng et al. Intell Robot 2022;2(3):298­312 I http://dx.doi.org/10.20517/ir.2022.27

Remark 1 is introduced to cope with reset conditions, which is continuous on [ , +1 ), and does not grow in
the jumps when = +1 since

Õ
p

2

− ( − ) + ( +1 ) ≤ − −2
(− ( +1 ) + ( ))
(14)
+1
=1
Theorem 1 Under TOD scheduling scheme Equation (6), for given scalars , > 0, > 0, impulsive system
Equation (12) is exponentially stable with a prescribed ∞ performance , if there exist appropriate dimensions
and real matrices > 0, > 0, > 0, > 0, > 0, > 0, = 1, · · · , such that

Ξ 11 ∗ ∗ 
 
 Ξ ∗  < 0, (15)
 21 Ξ 22 
 
Ξ
 31 0 Ξ 33

> 0, (16)

Ω 11 Ω 12
Ω = < 0 (17)
∗ Ω 22
are feasible for , = 1, · · · , , where

 
Ψ 11 Ψ 12 Ψ 13 Ψ 14 Ψ 15
 

  [− 2 2 , · · · , − ], = 1
 ∗ Ψ 22 Ψ 23 0 0  

 
¯
Ξ 11 =  ∗ ∗ Ψ 33 0 0  , = [− 1 1 , · · · , − −1 −1 ], =
  

 ∗ ∗ ∗ Ψ 44 0   , · · · , − ], ≠ 1,
   [− 1 1 , · · · , − | ≠

 ∗ ∗ ∗ ∗ Ψ 55 
 
 
 { 2 , · · · , }, = 1 [ 2 2 , · · · , ], = 1


 
  
= { 1 , · · · , −1 }, = , = [ 1 1 , · · · , −1 −1 ], =
 
 
 , · · · , }, ≠ 1,  , · · · , ], ≠ 1, .
 { 1 , · · · , | ≠  [ 1 1 , · · · , | ≠
 

Ψ 11 = + + 2 + + , Ψ 13 = − , Ψ 14 = , Ψ 23 = − , = [ , , 0, , ],
1−2
Ψ 22 = 2 − − , Ψ 33 = − , Ψ 44 = , Ω 11 = − + , Ψ 12 = − − + ,
−1
= [− 1 1 , · · · , − ], Ω 22 = − −2 , Ξ 22 = {− , − 1 , · · · , − },
√ √
Ψ 15 = , Ω 12 = , Ξ 21 = [ , 1 1 1 , · · · , ], Ψ 55 = − ,
2

1 2 Í
= − + 2 , Θ = + , Ξ 31 = [ , 0, · · · , 0].

=1
Proof: Differentiating ( ) along Equation (12) and applying Wirtinger-based integral inequality [26] , we can
obtain

1 Õ p 2 p 2 2 2 2 1 Õ p 2
¤
( )| + | ( )| − | ( )| ≤ −
( ) + 2 ( ) − | ( )| − 2 | | ( )|

=1, ≠ =1, ≠
¹

2
2
− | ( )| − −2 2 −2 ¤ ( ) ¤( ) (18)

( − ) ( − ) + | ( )| + ¤ ( )Θ¤( ) −
−

Õ

+ 2 ( ) ( ) + 2 ( ) ¤ ( ) + ( )(2 + ) ( ),

=1, ≠
where ( ) = { ( ), ( − ( )), ( − ), ( ), ( )}.
¯