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Chen et al. Complex Eng Syst 2023;3:8 I http://dx.doi.org/10.20517/ces.2022.50 Page 9 of 15

( ) = ( ) ( ) + ( ) + ( )Δ ( ) (21)

 
 
2
 Õ 
   × 
   
   
 =1   × 
where ( ) =  . .  , =  . .  ,
 .   . 
   
 Õ   

   ×  ×1
 
 
 =1 
( )
 0 0 · · · 0 
 
2
 Õ 
 −1 
 0 · · · 0 
 
 =1 
 . . . . . 
 . . . . . . . . . . 
 
 −1 
( ) =  Õ −1 Õ −1 
 · · · · · · 
 
 =1 =1 
 . . . . . 
 . . . . . . 
 . . . . 
 
−1 − +1
 Õ Õ Õ 
 −1 −1 −1 
 · · · · · · 
 
 =1 =1 =1  ( )
To track the reference vehicle model as well as possible, a 3-DOF vehicle model is designed as the reference
model. According to Eq. (4), the reference discrete output vector can be obtained as follows:
( ) = ( ) ( − 1) + ( ) ( − 1) (22)

where ( ) = , ( ) ( ) ( ) , ( ) = , ( ) ( ) ( ) ,
= (1/ , 1/ , 1/ ), ( ) = − , and ( ) = . The -step predictive output vector of the
reference system can be expressed as follows:

( ) = ( + 1) ( + 2) · · · ( + ) · · · ( + ) (23)

Accordingly, the output vector of each future predictive steps of the reference system is given as follows:

( ) = ( ) + ( ) (24)

" #
2
Õ −1 Õ −1
where = 2 · · · , and = · · · . In addition, the
−1 −1
following relationship ( + 1/ ) = ( )( = 1, 2 · · · ) holds.