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Page 305 Peng et al. Intell Robot 2022;2(3):298­312 I http://dx.doi.org/10.20517/ir.2022.27

2 Í
k + ( 0 ) ( 0 ) for
Clearly, there exists the maximum of | ( )| for ∈ [ 0 − , 0 ]. Thus, ( 0 ) ≤ k 0
=1
some > 0.
Therefore, the exponential stability of the system Equation (12) with ( ) = 0 is guaranteed.

Next, we will show that

holds for any nonzero ∈ 2 [0, +∞).
(ii) under the zero initial condition, the inequality k k 2 ≤ k k 2
From Equation (19), we obtain that for ∈ [ , +1 ),

p 1 Õ p
2 2 2 ¤ 2 2 (27)
( )| +
| ( )| − | ( )| ≤ − ( ) + 2 | | ( )| .

=1, ≠
Integrating Equation (27) on from to − yields
+1
¹ −
+1 2 2 2
| ( )| − | ( )| + ( +1 ) ≤ ( ) + Θ ≤ ( ). (28)

Then, by summing Equation (28) on from 0 to , where → +∞, we have
¹ −
+1 2 2 2
(| ( )| − | ( )| ) ≤ ( 0 ). (29)
0
¯ ¯

Under the zero initial condition, +∞ ( ) ( ) ≤ +∞ ( ) ( ) . Therefore, one can derive that
2
0 0
for any nonzero ∈ 2 [0, +∞). This completes the proof.
under the zero initial condition, k k 2 ≤ k k 2
□
Remark 2 The feasibility of the linear martix inequalities has been sufficiently explained in [23,28] . Due to page
limitations, this part is omitted here.
3.2. Stability analysis under the RR scheduling scheme Eqution (7)
Construct the following Lyapunov-Krasovskii functional candidate:
( ) = Π( ) + + , ≥ − 1, ∈ [ , +1 ), (30)

where

 −1 ¯ √
 Í
 2 ( − ) | ¤ ( ) , ≠ − 1,

 =1
=
−1
 Í ¯ √
 2 ( − ) | ¤ ( ) , = − 1,

 0
 =1
−1
Õ +1 − q
2

= ( − 1) 2 ( −1) , = | − − ( )| .

=1
Similar to Theorem 5.2 in [29] , we establish the following result.
Theorem 2 Under RR scheduling scheme Equation (7), given > 0 and > 0, assume that there exist matrices
> 0, > 0, > 0, > 0, = 1, · · · , and with appropriate dimensions such that Equation (15) and
Equation (16) are feasible with = , where is given by Equation (30). Then, system Equation (12) is
−1
exponentially stable with a prescribed ∞ performance .
Proof: The detailed derivation process can refer to [29] and the proof of Theorem 1, which is omitted here due
to the limited pages. □