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Yang et al. Intell Robot 2022;2(3):223­43 I http://dx.doi.org/10.20517/ir.2022.19 Page 233
[ ]
( −1) ) ∗
−( ⊗
¯
Φ = (0) ≺ 0 (20)
¯
Ξ Υ
hold for any , ∈ O, ∈ N , where
[1, ]
˜ ¯
˜
¯
Ξ = [ ˜ L ˜ 0 M ] , Ψ 23 = [( ⊗ ) 0 0] , = ⊗ + ( L + 0 M ) ⊗ ,

˜
˜
(0)
L = L ⊗ , M = M ⊗ , Z = { ⊗ , ⊗ , ⊗ } ∈ R 3 ×3 , Υ = {Υ, Υ, Υ},
( )

(0)
Υ = Σ ( ⊗ − ⊗ − ⊗ ),
=1 Σ ∈O Ω
then the system (6) is leader-following consensus in mean square sense and possess a H ∞ performance index
−1
−1

ˆ . Moreover, the controller gains are given by = ( ) ( ) .

Proof Performing congruence transformations { , , Z , } to (9), { , Z } to (10), one

can obtain
 −( ⊗ ( −1) ) ∗ ∗ ∗ 
 
 0 2 
 − ˆ ∗ ∗  (21)
 ( )  ≺ 0
 Z Ξ Z Ψ 23 Z P Z ∗ 

 
 ⊗ 0 0 − 
 
[ ]
( −1)
−( ⊗ ) ∗
≺ 0 (22)
˜
(0)
Z Ξ Z P Z

According to the literature [40] , we can obtain that for the positive definite matrix ∈ R × and the real matrix

−1
∈ R × , there must be ( − ) ( − ) ⪰ 0. Performing cholesky decomposition on the matrix , there
−1

must be a lower triangular matrix ∈ R × such that = . Further, we can get ( − ) ( − ) =

⪰ 0, where = − . When the matrix is full rank, ( − ) ( − ) ≻ 0 holds. Based on

−1
−1
this, the inequalities
−( ) ( )
( ⊗ )[( ⊗ ) − ( ⊗ )] ⪰ 0
(0) −(0) (0)
[( ⊗ ) − ( ⊗ )] ( ⊗ )[( ⊗ ) − ( ⊗ )] ⪰ 0
can ensure
−( ) ( )
−( ⊗ ) ⪯ ( ⊗ ) − ( ⊗ ) − ( ⊗ ), (23)

−(0) (0)
−( ⊗ ) ⪯ ( ⊗ ) − ( ⊗ ) − ( ⊗ )

for any , ∈ O, ∈ N . Then, it can be shown that the inequalities (21) and (22) can ensure that the
[1, ]
conditions (19) and (20) hold. Furthermore, it implies that the inequalities (9) and (10) are true. By Theorem 1,
we can get that systems (6) and (8) is leader-following consensus in mean square sense with a H ∞ performance
index ˆ under the controller (4) and the controller gains = ( ) ( ) . This proof is completed.
−1

−1

Supposing that the channel inferences ( ) and ( ) in (4) are ignored, that is, ( ) = 0, ( ) = 0.

0
0
Then, we have ( ) = 0 in (6). Consequently, the leader-following consensus control of system (18) under
semi-Markov switching topology is realized in the following corollary.
( ) ( )
Corollary 2 Given a scalar ∈ N ≥1, if there exist a sets of symmetric matrices ∈ R × ≻ 0, ∈
R × ≻ 0, and matrices ∈ R × , ∈ R × , , ∈ O, ∈ N such that the inequalities hold:
[0, ]
[ ]
( −1)
−( ⊗ ) ∗
≺ 0
¯
Ξ −P −( ) − Z − Z

[ ]
( −1)
−( ⊗ ) ∗
≺ 0
¯
Ξ Υ (0)