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Yang et al. Intell Robot 2022;2(3):223­43 I http://dx.doi.org/10.20517/ir.2022.19 Page 231

On the other hand, the stability of the system during mode switching needs to be considered. It can be derived
along the trajectory of systems (6) and (8) that

{ }
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
Ω (13)
( ( +1 −1))
− ( +1 − 1)( ⊗ ) ( +1 − 1)
[
( +1 ) ( )

2 ¯
¯
2 ¯
¯
¯
¯
= ( +1 − 1) Σ Σ ∈O P + L PL + M PM
+1 =1 0
Ω
]
( ( +1 −1))
− ( ⊗ ) ( +1 − 1)
( ( +1 )) ). By means of the condition (10) and Schur complement lemma, one can get that
with P = ( ⊗
{ }
E ( ( +1 ), ( +1 ), ( +1 )) − ( ( +1 − 1), ( +1 − 1), ( +1 − 1)) < 0.
Together with the condition (12), there exists a positive constant such that gives
{ }

E ( ( +1 ), ( +1 ), ( +1 )) − ( ( ), ( ), ( )) < − ( ) ( ) (14)
Then, it follows that
{ }
{ } 1
E ( ) ( ) < − E ( ( +1 ), ( +1 ), ( +1 )) − ( ( ), ( ), ( ))

Summing both sides of the above equation from 0 to yields
[ ]
{ } 1 { }
E Σ ( ) ( ) < ( ( 0 ), ( 0 ), ( 0 )) − E ( ( +1 ), ( +1 ), ( +1 )) (15)
=0
{ } 1
Let → ∞, then lim E Σ =0 ( ) ( ) < ( ( 0 ), ( 0 ), ( 0 )). Then, we have lim E{ ( ) ( )} =

→∞ →∞
{ 2 }
lim E{ )( ( )} = 0. Furthermore, it can be shown that lim E ∥ ( )∥ = 0. By Definition 3 and Defini-
→∞ →∞
tion 4, systems (6) and (8) with ( ) ≡ 0 is leader-following mean square consensus.
Next, the H ∞ consensus performance of the underlying system is discussed. Then, for ( ) ≠ 0, we have
{ }
2
E ( ( + 1), ( + 1), ( + 1)) | ( ), ( )= − ( ( ), ( ), ( )) + ( ) ( ) − ˆ ( ) ( )
{ }
[ ] [ ] (16)
˜
˜
=E ( )A + ( )[ ⊗ ( )] ( ⊗ ( ( +1)) ) A ( ) + [ ⊗ ( )] ( ) | ( ), ( )=

( ( )) 2
− ( )( ⊗ ) ( ) + ( )( ⊗ ) ( ⊗ ) ( ) − ˆ ( ) ( )
Denote ( ) = [ ( ) ( )] , then the equation (16) can be transformed into

2
E{Δ ( ( ), ( ), ( ))} + ( ) ( ) − ˆ ( ) ( )
{ }
2
=E ( ( + 1), ( + 1), ( + 1)) | ( ), ( )= − ( ( ), ( ), ( )) + ( ) ( ) − ˆ ( ) ( ) (17)

= ( )Θ ( )