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Zhang et al. Intell Robot 2022;2(4):371­90 I http://dx.doi.org/10.20517/ir.2022.26 Page 381

We also need to consider the control input constraints to ensure driving comfort:

| ( )| ≤ ,max , = 1, 2 (35)

According to inequality [Equation (26)], we have:

−1
( ) ( ) ≤ 1. (36)

Therefore, we have:
2 2
max k ( )k = max k ( ( ) ( )) k
−1 2
= max k( ( ) ( ) ( ))k
1 1 2
− −
= max k( ( ) ( ) 2 ( ) 2 ( ))k
(37)
1 2 1 2
− −
≤ k( ( ) ( ) 2 )k k ( ) 2 ( )|
−1 −1
= ( ( ) ( ) ( ))[ ( ) ( ) ( )]
−1
≤ ( ) ( ) ( ).
Considering the Schur complement, the input constraint can be guaranteed by the following LMI if there exists
a symmetric matrix [38]
[ ]

≥ 0, (38)
∗
where ≤ 2 .
,max

To deal with the external disturbance, the concept of RPI is introduced to ensure the closed-loop stability of
car-following system. According to the concept of RPI and quadratic boundedness as shown in Lemma 3.1,
Ω is an RPI set if

( + 1| ) ( + 1| ) ≤ ( | ) P ( | )
holds under

( ) ( ) −1
≤ ( ) ( ),
2

where ( ) ( ) ≤ .
2
The S-procedure is used to obtain a sufficient condition as follows:

[ ]

−1
−1
−1
( + 1) ( + 1) − ( ) ( ) − ( ) ( ) − ( ) ( ) ≤ 0, (39)

2

where is a positive scalar belonging to (0, 1).
By Lemma 3.3, the above mentioned inequality can be guaranteed by the following matrix: inequality:
( ) ( )
 (−1 + ) 0 + 1 0 
 
 ( ) ( ) 
 ∗ − 2 0 2 
 ( ) 
 ∗ ∗ 1 + 2 − 0 0  ≤ 0, = 1, 2. (40)
 1 2 
 
 ∗ ∗ ∗ − 0 
 
 ∗ ∗ ∗ ∗ − 
 