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Page 236 Yang et al. Intell Robot 2022;2(3):223­43 I http://dx.doi.org/10.20517/ir.2022.19

Σ ( ) = 1, 0 ≤ ( ) ≤ 1, 0 ≤ ¯ < Ω ≤ 1. Further, it can be derived that

=1 Σ ∈O Ω − ¯ Ω − ¯
{ }
E ( ( +1 ), ( +1 ), ( +1 )) − ( ( +1 − 1), ( +1 − 1), ( +1 − 1))
[
( ) ( ) ( )
2 ¯ ¯ ¯

¯ ¯ ¯
2 ¯ ¯ ¯
≤ ( +1 − 1)Σ Σ ∈O Σ Σ ∈O P + L PL + M PM (29)

=1 =1 0

Ω − ¯ ¯
]
( 2 ¯ ¯ ¯ 2 ¯ ¯ ¯ ) ( −1)
¯ ¯ ¯

+ (1 − ¯ ) P + L PL + M PM − ( ⊗ ) ( +1 − 1)

0
{ }
Then, E ( ( +1 ), ( +1 ), ( +1 )) − ( ( +1 −1), ( +1 −1), ( +1 −1)) < 0, if thefollowing inequality
holds:
( ) ( )
2 ¯ ¯ ¯
2 ¯ ¯ ¯
¯ ¯ ¯
Σ Σ ∈O P + L PL + M PM
=1 0
¯
( ¯ ¯ ¯ 2 ¯ ¯ ¯ 2 ¯ ¯ ¯ ) ( −1)

+ (1 − ¯ ) P + L PL + M PM − ( ⊗ ) < 0.
0
By Schur complement lemma, the above inequality can be further transformed into
[ ]
˜ ( −1) ) ∗
−( ⊗
˜
Φ = (0) ≺ 0 (30)
ˆ
˜
Ξ P
˜ ¯ ¯ ¯ ¯ ¯ ¯ ˆ (0) ˆ ˆ ˆ ˆ ˆ ˆ ˆ
where Ξ = [ L 0 M L 0 M ] , P = {−P 1 , −P 1 , −P 1 , −P 2 , −P 2 , −P 2 }, P 1 =

ˆ

Σ Σ ∈O ( ) ( ⊗ −(0) ), P 2 = (1 − ¯ )( ⊗ −(0) ).
=1 ¯
Applying congruence transformation { , Z } to (30), we can get
ˆ

[ ]
˜ ( −1)
−( ⊗ ) ∗
≺ 0 (31)
ˆ
ˆ ˆ
(0)
˜ ˜
Z Ξ Z P Z

˜
with Z = {Z , Z }. According to condition (23), one can proof that inequality (31) can guarantee
ˆ
˜
that condition (26) can hold. The rest of the proof can be directly derived in a similar way to Theorem 1 and
Theorem 2. This proof is completed.
Remark 4 The controller design and consensus conditions proposed in Theorem 2 and Theorem 3 are based
on the same channel fading. However, in practice, the fading variables and interference of communication
channels between agents are more likely to be different, due to different complex external environments or
different geographic locations of the agents. This restricts the issues considered in this paper to a certain
extent. It is worth noting that although the problem of non-identical channel fading has been studied in [28,29] ,
theaboveliteratureonlyconsidersleaderlessmulti-agentsystems. They ignorethefadingeffects fromleader to
follower agents, and the edge Laplacian method introduced cannot be used to tackle the models considered in
this paper. Therefore, it is interesting and meaningful to investigate the non-identical channel fading problem
within the framework of the fading model proposed in this paper. No better method has been proposed to
solve the problem of non-identical channel fading under model (3). This also encourages us to continue to
study this issue in future work.
Remark 5 In Theorems 2 and 3, the fully known and incompletely available cases of the semi-Markov kernel
for switching topologies are handled respectively, and the corresponding consensus conditions are also de-

rived. Given parameters and , the minimum H ∞ performance index ˆ of the system can be calculated
according to the solution of the following optimization problems:
min ˆ 2 subject to (19) and (20), , ∈ O, ∈ N
[1, ]