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Page 228 Yang et al. Intell Robot 2022;2(3):223­43 I http://dx.doi.org/10.20517/ir.2022.19

Figure 1. Multi-agent systems with channel fading under semi-Markov switching topologies

Based on the above introduction to the semi-Markov chain, the switching topology among agents in this paper
can be denoted as G ( ). For convenience, let G ( ) = G , ∈ O.

Every topological graph G is an undirected graph. Then, the Laplacian matrix of the graph G , the adjacency
matrix of the leader and the follower are denoted by L ∈ R × , A ∈ R × , and M ∈ R × .

In practice, the communication process between an agent and its neighbors is often affected by channel noise
and fading. Motivated by the channel fading model in [28,29] , in this paper we assume that each agent obtains
relative state information from its neighbors through fading channels. Correspondingly, the channel fading
model among agents can be expressed as

( ) = ( )( ( ) − ( )) + ( ), ( ) = 0 ( )( ( ) − 0 ( )) + 0 ( ) (3)

0
where ( ), 0 ( ) represent the channel fading of the follower-to-follower and the leader-to-follower, re-
spectively. ( ) and 0 ( ) are disturbances in the channel. Based on the above channel fading model, we
design a distributed consensus controller as follows:

∑

( ) = { ( )[ ( )( ( ) − ( )) + ( )] + ( )[ ( )( ( ) − 0 ( )) + ( )]} (4)

0
0
=1
where ∈ R × is the controller gain to be determined, ∈ O.
Figure 1 shows the frame diagram of the system considered in this paper. Under the semi-Markov switching
communication network topology, the channel fading between agents varies randomly with the switching of
the topology. Each agent generates control inputs based on the relative information obtained from neighbors
through fading and interference, thereby further controlling the entire system to achieve consensus.

Remark 2 Compared with the channel fading model established in the literature [27–29] , the channel fading
model introduced in this paper not only considers the fading influence from leader to follower and follower
to follower, but also introduces the effect of channel interference on signal transmission and the influence of
topologyswitchingonthestatisticalcharacteristicsoffadingcoefficients. Thismeansthatthemodelspresented
in this paper are more general than those in the previous literature. When ( ) = 0 ( ) = 0, 0 ( ) ≡ 1
in equation (3), the model degenerates to the case in [27] . If the values of ( ) and 0 ( ) are 0 or 1 and
( ) = 0 ( ) = 0, the channel fading model (3) is reduced to a packet loss model. This also indicates that
the channel fading model proposed in this paper is more general.

Define the consensus error as ( ) = ( ) − 0 ( ), = 1, 2, ..., . Combining system (1) and consensus