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Page 232 Yang et al. Intell Robot 2022;2(3):223­43 I http://dx.doi.org/10.20517/ir.2022.19
[ ]

Σ + ( ⊗ ) ( ⊗ ) ∗
where Θ = ( ( +1)) 2
¯
( ⊗ ( )) ( ⊗ ) [ ⊗ ( )] ( ⊗ ( ( )) )[ ⊗ ( )] − ˆ
According to Schur complement lemma and the condition (9), it gives that Θ ∈ R 2 ×2 < 0. This implies
that

2
E{Δ ( ( ), ( ), ( ))} + ( ) ( ) − ˆ ( ) ( ) < 0
Further, we can obtain
{ } { }
2
2
2
E Σ ∞ Σ +1 −1 [∥ ( )∥ − ˆ ∥ ( )∥ ] < −(E lim ( ( ), ( ), ( )) − ( (0), (0), (0))) < 0
=0 =
→∞
under the zero initial conditions. Thus, the H ∞ performance condition (II) holds. This proof is completed.
In Theorem 1, the leader-following consensus and H ∞ performance of the system (6) with channel fading
is analyzed, in which data transmission between agents takes into account not only channel fading but also

channel interference. Assuming that the channel interference ( ) = 0, ( ) = 0, that is ( ) = 0. Then,

0
the system (6) can be reduced to
(18)
( + 1) = [ ⊗ + ( ( )L + 0 ( )M ) ⊗ ] ( )
In this case, the following corollary gives the leader-following mean square consensus analysis of the system
(18) under no-channel interference fading model.

( ) ( )
Corollary 1 Given scalar ∈ N ≥1, if there exist a sets of symmetric matrices ∈ R × ≻ 0, ∈
R × ≻ 0, ∈ N such that the following inequalities
[0, ]
[ ]
( −1)
−( ⊗ ) ∗
≺ 0
Ξ P ( )
[ ]
( −1)
−( ⊗ ) ∗
≺ 0
˜
Ξ P (0)
hold for , ∈ O, ∈ N , ∈ N , then the system (18) is leader-following mean square consensus,
[1, ] [1, ]
where Ξ, P , and P have been defined in Theorem 1.
˜
(0)
( )
The proof of Corollary 1 is similar to that of Theorem 1, so it is omitted.
3.2. Consensus controller gain design
Although Theorem 1 addresses a family of leader-following consensus conditions, these conditions cannot
be directly utilized to solve controller gains . Aiming at solving the leader-following consensus control
problem of systems (6) and (8) under switching topologies, sufficient conditions on the existence of the desired
controller gains are presented in the following theorem.

( )
Theorem 2 Given a scalar ∈ N ≥1, if there exist a scalar ˆ > 0 and sets of symmetric matrices ∈
R × ≻ 0, ( ) ∈ R × ≻ 0, and matrices ∈ R × , ∈ R × , , ∈ O, ∈ N such that the
[0, ]
following inequalities
 ( −1) ) ∗ ∗ ∗ 
 −( ⊗ 
 0 2 ∗ 
¯  − ˆ ∗  (19)
Ψ =  ¯ ¯ −( )  ≺ 0
 Ξ Ψ 23 −P − Z − Z ∗ 
 
 ( ⊗ ) 0 0 − 
 