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Peng et al. Intell Robot 2022;2(3):298­312 I http://dx.doi.org/10.20517/ir.2022.27 Page 306

3.3. Controller Design under the TOD scheduling scheme Equation (6)

Theorem 3 Under TOD scheduling scheme Equation (6), for given matrices , , , , and scalars , >
0, > 0, > 0, system Equation (12) is exponentially stable with a prescribed ∞ performance , if there exist
˜
˜
˜
˜
˜
real matrices > 0, > 0, > 0, , = 1, · · · , , > 0, > 0, > 0 and appropriate dimensions ,ℶ,
such that
( )
Õ
ℎ
(31)
=1
∗
. . (17), (32) , (33), (34), (35)
 ˜ ∗ ∗ 
Ξ 11
 
 ˜ ˜ ∗  < 0, (32)
Ξ
 21 Ξ 22 
 ˜ ˜ 
Ξ
 31 0 Ξ 33

˜ ˜ (33)
˜ ˜ > 0,

ℎ ∗ > 0, (34)

− (ℶ − )
< 0, (35)
ℶ − −
is solvable, and the controller gain is given by = ℶ , where (32) is equivalent to Equation (32)
−1
∗
2 ˜
˜ −1
by replacing − −1 − with −ℎ , − 2 , and

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

˜
˜
˜
˜ ˜
˜ ˜
˜
2
˜
˜
˜
˜
Ψ 23 = − , ˜ = [− 1 1 , · · · , − ], Ψ 33 = − , Ψ 14 = ˜ , Ψ 44 = ˜ , Ψ 55 = − , Ψ 13 = − , = ,
√
√
˜
˜
˜
˜
−1
−1
Ξ 31 = [ , 0, · · · , 0], Ξ 22 = {− ˜ −1 , − , · · · , − }, Ξ = [ , 1 1 , · · · , ], = .
1 21
˜
Proof: Let = , = , = , = 1, · · · , . Pre- and post-multiplying Equation (15) and
−1 ˜
Equation (16) with
−1 −1 −1
{ , · · · , , , , , · · · , , , }
1