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Ding et al. Complex Eng Syst 2023;3:7 I http://dx.doi.org/10.20517/ces.2023.06 Page 3 of 13

position error and the velocity error are used to reflect the consistency of MASs, then the consensus tracking
problem of MASs can be transformed into the stability problem of the tracking error system (2) Coping with
the effect of the fading channel between followers, the incomplete fading information received by the agent is
introduced into the controller design (3) An online estimation strategy is employed to estimate the unknown
and time-varying attacks, based on which, an adaptive sliding mode controller is designed to attenuate the
effect of the attacks on MASs and (4) The distributed adaptive SMC strategy is designed to ensure the mean
square consistency of MASs, despite the communication constraints.

Notation: R and R × mean the dimension Euclidean space and the × real matrix set. The symbol |·|
denotes the Euclidean norm and ⊗ denotes the Kronecker product. Denote sgn( ) the sign symbolic function,

1 = [1, 1, · · · , 1] , 0 = [0, 0, · · · , 0] .
2. PROBLEM FORMULATION

2.1. Graph theory
Graph theory is an important tool to study MASs, which is a graph composed of several nodes and edges
connecting the node. Each agent can be represented as a node, and the information interaction between
agents can be denoted as an edge in graph theory. A directed weighted graph is represented by = {V, E}.
For MASs with one leader and agents, the node-set V = { 1 , 2 , · · · , } indicates the set of all points on
the graph and E = {( , ), , ∈ V, ≠ } represents the set of all edges. = [ ] ∈ R × is a non-negatively
weighted adjacency matrix. If > 0, it means that agent can receive information from agent ; conversely, if
= 0, agent cannot receive information from agent . Define the matrix = diag( 1 , 2 , · · · , ) to denote
Í
the communication between the leader and all followers, and the degree matrix = [ ] with = .
=1
So, we can obtain the Laplace matrix = [ ] as:
= − . (1)
with
Í , = ,
= =1 (2)
− , ≠ .
Lemma 1 [35] The matrix + is invertible if the directed graph has a directed spanning tree.

Definition 1 Consider a multi-agent system with agents and let ( ) represent the state of agent . If
→∞ k ( ) − ( )k = 0, for all , = 1, 2, · · · , , it is said that the multi-agent system can reach a con-
sensus. Furthermore, if there exists a leader whose state is 0 ( ), then →∞ k ( ) − 0 ( )k = 0, for all
, = 1, 2, · · · , , means the tracking consensus is achieved.

2.2.System model
Consider a second-order MASs consisting of a leader labeled as node 0 and followers indexed by ∈
{1, 2, · · · , }, and the th follower’s dynamic is given as:

¤ ( ) = ( ), (3)
¤ ( ) = ( ),

where ( ) ∈ R , ( ) ∈ R , ( ) ∈ R represent the th follower’s position, velocity and the control input,

respectively. According to equation (3), it is obvious that we are focused on double integrators.
The leader’s dynamic is of the following form:

¤ 0 ( ) = 0 ( ), (4)
¤ 0 ( ) = 0 ( ),