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its stability and stabilization problems [39,40] . This also provides an idea for solving the consensus problem of
multi-agent systems under discrete semi-Markov switching topology. To the best of our knowledge, to date,
there have been few results on the leader-following consensus problem for discrete-time multi-agent systems
subject to channel fading under semi-Markov switching topologies. Therefore, how to design a suitable dis-
tributed control protocol and how to establish leader-following consensus criteria for multi-agent systems with
channel fading under discrete semi-Markov switching topologies are the key issues. This inspires us to carry
out this work.

Motivated by the above discussion, in this paper, the H ∞ leader-following consensus problem of discrete multi-
agent systems with channel fading is investigated under the premise of semi-Markov switching topology. The
major contributions of this paper can be highlighted as follows: (I) Compared with the literature [27,41] , a more
general channel fading model based on discrete semi-Markov switching topologies is established to charac-
terize the possible effects of inter-agent signal transmission. The influences of channel fading between leader
and follower, follower and follower agents are simultaneously considered to explore the influence of channel
fading on system consensus, rather than considering only channel fading among follower agents in litera-
ture [27,41] . (II) As mentioned in the previous paragraph, discrete semi-Markov processes are more powerful
in modeling ability and application range than Markov and continuous-time semi-Markov processes. For this
reason, different from the Markov switching and continuous-time semi-Markov switching topologies adopted
in [32–38] ,thispaperemploysadiscretesemi-Markovprocesstodescribethenetworktopologyswitchingamong
agents and switching for channel fading. A set of novel sufficient conditions to guarantee that leader-following
mean square consensus of multi-agent systems under semi-Markov switching topologies is derived via a semi-
Markov kernel approach. (III) The distributed consensus controller design scheme based on fading relative
states is proposed to solve the H ∞ leader-following consensus control problem when the semi-Markov kernel
of switching topologies is fully accessible and incompletely accessible, respectively. The rest of this paper is
organized as follows. The preliminaries and problem formulation are given in section II. Section III presents
the main results. Then, simulation examples are provided in Section IV. Finally, the conclusion and future
work are introduced in Section V.

Notations: Denote that R and R × be the sets of -dimensional vectors and × real matrices. N represents
+ and N [ 1 , 2 ] stand for the
the sets of nonnegative integers. The sets of positive integers is denoted by N . N ≥ 1
sets { ∈ N| ≥ 1 } and { ∈ N| 1 ≤ ≤ 2 }, respectively. A matrix ≻ 0(≺ 0) indicates it is positive
(negative) definite. A × identity matrix is defined as . Denote a symmetric term in a matrix by ∗. ⊗
refers to the Kronecker product. Moreover, E{·} and ∥ ∥ represent the mathematical expectation operator and
the Euclidean norm of the vectors. If not specifically stated, matrices and vectors have appropriate dimensions.

2. PRELIMINARIES AND PROBLEM FORMULATION
2.1. Graph theory
In this paper, we employed an undirected graph G = {V, E, A} to depict the information interaction topology
among agents. V = { 1 , 2 , ..., } stands for the node sets, in which is the th agent. E ⊂ V × V
represents a set of edges. The adjacency matrix associated with graph G is denoted by A = [ ] ∈ R × .
If node can receive information from node , there is an edge ( , ) between node and node . The
elements of matrix A is weighted coefficient of edge ( , ), and > 0, if ( , ) ∈ E, otherwise, = 0.
Self-loop is not considered. The set of neighbors of node can be represented as N = { ∈ V|( , ) ∈ E}.
The degree matrix of graph G is denoted as D = { } ∈ R × , where = ∑ . Then, one can
∈N
obtain that the Laplacian matrix is L = D − A. Denote matrix M = { } ∈ R × , where stands
for the information exchange of node and leader node. If node can access the information of the leader,
= 1, otherwise, = 0.