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Yang et al. Intell Robot 2022;2(3):223­43 I http://dx.doi.org/10.20517/ir.2022.19 Page 235
[ ]
˜ ( −1) ) ∗
−( ⊗
˜
Φ = (0) ≺ 0 (26)
˜
˜
Ξ Υ
hold for any ∈ O , ∈ O , ∈ N , where
[1, ]
ˇ
˜
ˇ ˜
˜
ˇ
Ξ = [ ˇ L ˇ 0 M ] , Ψ 23 = [( ⊗ ) 0 0] , = ⊗ + ( L + 0 M ) ⊗ ,

ˇ
ˇ
˜
˜
˜
˜
˜
˜
L = L ⊗ , M = M ⊗ , Z = { ⊗ , ⊗ , ⊗ } ∈ R 3 ×3 ,
˜
˜
ˇ ˜
˜
ˇ ˇ
˜
˜
˜
˜
Ξ = [ ˇ L ˇ 0 M L ˇ 0 M ] , Υ = {Υ 1 , Υ 1 , Υ 1 , Υ 2 , Υ 2 , Υ 2 },
(0)

˜
˜
˜
˜
˜
˜
˜

(0)
(0)
Υ 1 = Σ Σ ∈O ( ) ( ⊗ − ⊗ − ⊗ ), Υ 2 = (1 − ¯ )( ⊗ − ⊗ − ⊗ ),
=1 ¯

¯
¯
( ), ¯ = ¯ + (1 − ) , = Σ ∈O ,
¯ = Σ Σ ∈O
=1
thensystems(6)and(8)isleader-followingmeansquareconsensusunderincompletelyaccessiblesemi-Markov
kernelofswitchingtopologiesandpossessa H ∞ performanceindex ˆ . Moreover, thecontrollergainsaregiven
by = ( ) ( ) .

−1 ˜ −1 ˜

Proof Since the stability proof of the system at non-switching time of topologies is independent of the semi-
Markov kernel, the corresponding proof is easy to obtain by Theorem 1. For incompletely accessible semi-
Markov kernel, only the stability of the system at the switching time is given in this theorem.
Similar to inequality (13), we have
{ }
E ( ( +1 ), ( +1 ), ( +1 )) − ( ( +1 − 1), ( +1 − 1), ( +1 − 1))
{ }
˜ ( ( +1 )) ˜ ( ( +1 −1))
=E ( +1 − 1)A ( ⊗ )A ( +1 − 1) − ( +1 − 1)( ⊗ ) ( +1 − 1)

{ }
( +1 )
˜

˜
=E ( +1 − 1)Σ Σ ∈O A ( ⊗ ( ( +1 )) )A ( +1 − 1)
+1 =1 (27)
Ω
( ( +1 −1))
− ( +1 − 1)( ⊗ ) ( +1 − 1)
{ }
( )
˜ (0) ˜
=E ( +1 − 1)Σ A ( ⊗ )A ( +1 − 1)
=1 Σ ∈O
Ω
( −1)
− ( +1 − 1)( ⊗ ) ( +1 − 1)
for ( +1 ) = , ( +1 ) = 0, ( +1 − 1) = , ( +1 − 1) = − 1 = +1 − 1. For Π( ), ∈ O , ∈ O ,
the equation (27) can be written as
{ }
E ( ( +1 ), ( +1 ), ( +1 )) − ( ( +1 − 1), ( +1 − 1), ( +1 − 1))
[

( ) ( ¯ ¯ ¯ 2 ¯ ¯ ¯ 2 ¯ ¯ ¯ )
= ( +1 − 1) Σ P + L PL + M PM
=1 Σ ∈O 0
Ω
]
( ) ( 2 ¯ ¯ ¯ 2 ¯ ¯ ¯ )
¯ ¯ ¯
+ Σ P + L PL + M PM ( +1 − 1)
=1 Σ ∈O 0
Ω
(28)
( −1)
− ( +1 − 1)( ⊗ ) ( +1 − 1)
[
( ) ( )
2 ¯ ¯ ¯

2 ¯ ¯ ¯
¯ ¯ ¯
= ( +1 − 1) Σ Σ ∈O P + L PL + M PM
=1 0
Ω

( ) Ω − ¯ ( ) ]
2 ¯ ¯ ¯
¯ ¯ ¯
2 ¯ ¯ ¯
+ Σ Σ ∈O × P + L PL + M PM ( +1 − 1)
=1 0
Ω − ¯ Ω
( −1)
− ( +1 − 1)( ⊗ ) ( +1 − 1)

¯ (0) ∈ R × ( ) = Ω − ¯ , ∈ O, it can be found that

withP = ⊗ . Noting the fact Σ Σ ∈O
=1