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Page 230 Yang et al. Intell Robot 2022;2(3):223­43 I http://dx.doi.org/10.20517/ir.2022.19

3. MAIN RESULTS
3.1. Consensus and H ∞ performance analysis
In the subsection, the leader-following consensus and H ∞ performance analysis of systems (6) and (8) are
given. Sufficient conditions for mean square consensus are derived via stochastic Lyapunov function.

( )
Theorem 1 Given scalar ∈ N ≥1, if there exist a scalar ˆ > 0 and a set of symmetric matrices ∈
( )
R × ≻ 0, ∈ R × ≻ 0, ∈ N such that the following inequalities
[0, ]

 −( ⊗ ( −1) ) ∗ ∗ ∗ 
 
 2 
 0 − ˆ ∗ ∗  (9)
Ψ =  ( )  ≺ 0
 Ξ Ψ 23 P ∗ 
 
 ( ⊗ ) 0 0 
 

[ ]
( −1)
−( ⊗ ) ∗
Φ = (0) ≺ 0 (10)
˜
Ξ P
hold for any , ∈ O, ∈ N , then the system (6) is leader-following consensus in mean square sense
[1, ]
and possesses a H ∞ performance index ˆ , where
¯
¯
Ξ = [ ¯ L ¯ 0 M ] , Ψ 23 = [( ⊗ ) 0 0] , = [ ⊗ + ( L + 0 ) ⊗ ],

¯ ¯ ( ) −( ) −( ) −( ) )},
M = M ⊗ , L = L ⊗ , P = {−( ⊗ ), −( ⊗ ), −( ⊗
˜ (0) ˜ ˜ ˜ ˜ ∑ ∑ ( ) −(0) ∑ ∑ ⊗ .

P = {−P, −P, −P}, P = ( ⊗ ), Ω = ( ), =
=1 ∈O Ω =1 ∈O

Proof Construct a stochastic Lyapunov function candidate as ( ( ), ( ), ( )) = ( )( ⊗ ( ( )) ) ( ),
( )
where ( ) = ∈ O, ∈ [ , +1 ). ( ) = − represents the running time of the current mode for
topology G . Assume that ( ) = , ( +1 ) = , ( ) ≡ 0. Then, one can obtain along the solution of
systems (6) and (8)
{ }
E ( ( + 1), ( + 1), ( + 1)) | ( ), ( )= − ( ( ), ( ), ( ))
{ }
( ( +1)) ( ( ))
=E ( + 1)( ⊗ ) ( + 1) | ( ), ( )= − ( )( ⊗ ) ( )
{ }
˜ ( ( +1)) ˜ ( ( ))

=E ( )A ( ⊗ )A ( ) − ( )( ⊗ ) ( ) (11)
[
¯ ( ( +1)) ¯ 2 ¯ ( ( +1)) ¯ 2 ¯ ( ( +1))
= ( ) ( ⊗ ) + L ( ⊗ )L + M ( ⊗ )

0
]
¯ ( ( ))
× M − ( ⊗ ) ( )
˜
˜
where A = ⊗ + ( ( )L + 0 ( )M ) ⊗ , A ∈ R × . It can be found from the above equation
¯
2 ¯
¯
that Δ ( ( ), ( ), ( )) < 0 if and only if Σ ∈ R × < 0, Σ = ( ⊗ ( ( +1)) ) + L ( ⊗

( ( +1)) ¯ 2 ¯ ( ( +1)) ¯ ( ( )) ). Furthermore, it can be proved that the following
)L + M ( ⊗ )M − ( ⊗

0
inequality holds:
( ( + 1), ( + 1), ( + 1)) − ( ( ), ( ), ( )) < 0 (12)