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Page 6 of 18 Guo et al. Intell Robot 2023;3(4):596-613 I http://dx.doi.org/10.20517/ir.2023.32

Therefore, system (5) can be converted to the form in continuous time gives

0 ( ) 0 0 ( )
¤ ( ) = ⊗ + ⊗
1 2 ( ) 1 2 ( ) (8)
= ⊗ ( ) + ⊗ ( )

Different from the previous consensus control methods [Similar to the form of System (5)] for the UAV system
(1), the individuals are supposed to guarantee the interaction of velocity through independent information
collection of position and velocity [the form of System (4)] when extreme cases, such as partial damage to
position sensors, are considered, which is also the difficulty and focus of this study.

3. METHODS AND RESULTS
3.1. Linear transformation of the system
First, System (4) can be transformed into the form of system (5) based on the lemma as follows:

Lemma 3 [26] . For Laplacian matrix related to the directed graph , there exists a non-singular matrix

1 ∗ . . . ∗
 
 1 ∗ . . . ∗ 
  ×
=  . . . .  ∈ R (9)
 . . . . . . . .
 
 
 1 ∗ . . . ∗ 
so that

0 ℎ
−1
= (10)
0 −1

−1
where ℎ ∈ R −1 and ∈ R ( −1)×( −1) . has the form as

 1 2 . . . 
 
 ∗ ∗ . . . ∗ 


−1 =  . . . .  ∈ R × (11)
 . . . . . . . . 
 
 . . . 
 ∗ ∗ ∗ 
where Í = 1.
=1

Therefore, non-singular linear transformations are built as = −1 ⊗ , and is defined as
−1