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Yang et al. Intell Robot 2022;2(3):223­43 I http://dx.doi.org/10.20517/ir.2022.19 Page 237

and
min ˆ 2 subject to (25) and (26), ∈ O , ∈ O , ∈ N
[1, ]
for fully known semi-Markov kernel and incompletely available semi-Markov kernel.

Similar to Corollary 2, the following corollary gives the mean square consensus controller design for system
(18) with incompletely accessible semi-Markov kernel.

( ) ( )
˜
˜
Corollary 3 Given a scalar ∈ N ≥1, if there exist a sets of symmetric matrices ∈ R × ≻ 0, ∈
˜
R × ≻ 0, and matrices ∈ R × ˜ × , , ∈ O, ∈ N such that the inequalities
, ∈ R
[0, ]
[ ]
˜ ( −1)
−( ⊗ ) ∗
≺ 0
˜
˜
ˇ
Ξ −P −( ) − Z − Z
[ ]
˜ ( −1)
−( ⊗ ) ∗
≺ 0
˜
˜
Ξ Υ (0)
for any ∈ O , ∈ O , ∈ N holds, then the system (18) with incompletely accessible semi-Markov
[1, ]

kernelisleader-followingmeansquareconsensusunderthecontrollergains = ( ) ( ) , where
−1 ˜ −1 ˜

P , Ξ, Z , Ξ, and Υ are defined in Theorem 1 and 3.
ˇ ˜
˜
˜
(0)
( )
The proof of Corollary 3 can be obtained in a similar way to Theorem 3, which is omitted here.
Remark 6 Theorem 2 and Theorem 3 respectively realize the distributed consensus control of multi-agent sys-
tems under the condition that the semi-Markov kernel of switched topology is fully available and incompletely
unavailable. In fact, event-triggered control and sampled data control are also excellent methods for dealing
with problems related to multi-agent systems [6,11,16] . The advantages and disadvantages of these methods can-
not be directly compared. Similarly, event-triggered control and sampled-data control methods can also be
applied to the problems considered in this paper. Naturally, event-triggered control and sampled-data control
can also be studied in a distributed framework.
4.SIMULATION RESULTS
In this section, a numerical example is provided to demonstrate the validity of the proposed results. Consider
a multi-agent system consisting of four followers and one leader with the following parameter matrices
 1.0607 0.1881 −0.4654 −0.15 0.1  0.2 0 0 
     
= −0.7425 1.1053 0.4284 , =  0.61 −1.5 , =  0 0.1 0.2 .




     
 0.1857 0.6231 −1.0774   −0.11 −0.2   0.1 0 0.1 
     
In this paper, the information exchange between agents is represented by an undirected switching topology
network. The topology graphs are shown in Figure 2. Correspondingly, the Laplacian matrices and the leader’s
adjacency matrices of each topology graph are given as
 2 −1 0 −1   1 0 0 0   1 0 −1 0   1 0 0 0 
       
       
−1 2 0 −1 0 1 0 0  0 1 0 −1 0 1 0 0
L 1 =   , M 1 =   , L 2 =   , M 2 =   ,
 0 0 1 −1 0 0 0 0 −1 0 1 0  0 0 0 0
       
 −1 −1 −1 3   0 0 0 0  0 −1 0 1   0 0 0 0
       
 2 0 −1 −1   1 0 0 0 
   
   
 0 1 0 −1 0 0 0 0
L 3 =   , M 3 =  
−1 0 2 −1 0 0 1 0
   
 −1 −1 −1 3   0 0 0 0
   