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Zhang et al. Intell Robot 2022;2(4):371­90 I http://dx.doi.org/10.20517/ir.2022.26 Page 379

Definition 3.3: A function : R → R ≥0 is called an ISS-Lyapunov function for the system ( + 1) =
( ( ), ( )) if there exist K ∞-functions 1, 2, 3 and a K-function satisfying

1 (|| ( )||) ≤ ( ( )) ≤ 2 (|| ( )||),
( ( + 1)) − ( ( )) ≤ − 3 (|| ( )||) + (sup || ( )||).
≥0

Lemma 3.2 [35] : A system is said to have ISS if it admits a continuous ISS-Lyapunov function.

3.3. T­S fuzzy model­based predictive control
We adopt the following model within the framework of predictive control to represent the future dynamics of
the T-S fuzzy system for vehicle-following considering lateral stability:

( + + 1| ) = ( ) ( + | ) + ℬ ( + | ) + ℬ ( ) ( + | ), (20)

where ( + | ), ( + | ) and ( + | ) represent the predicted state, control input and external disturbances
at the ( + )th time instant, respectively, and ∈ {1, 2, · · · }. The stage cost ℓ( + | ) can be represented as

(21)
ℓ( + | ) = ( + | ) ( + | ) + ( + | ) ( + | ) − ( + | ) ( + | )
where and are the weighting matrices of the state variables of the system and control inputs, respectively.
is the weight related to the external disturbance. We simplify this notation by using ( + ), ( + ) and
( + ) to denote ( + | ), ( + | ) and ( + | ), respectively.

Consider the following objective function J
−1
∑
J ( ) = ℓ( + | ) + ( ( + | )), (22)
=0

where ℓ( + | ) is the stage cost at the predicated time instant, and the positive-definite function ( ( + | ))
is called the terminal cost. This type of cost function was proposed in Ref. [36] to develop a novel synthesis
method with enhanced robustness.

As the cost function cannot be optimized in real time due to the unknown external disturbance, the upper
limit of the cost function is minimized here. We define the following fuzzy quadratic Lyapunov function:

2
∑

( ( )) = ( ) ( ) ( ) = ( ) ( ) ( ) (23)
=1
where is a positive definite matrix. Let the Lyapunov function satisfy the following inequality constraint:

( ( + + 1)) − ( ( + )) ≤ [ ( + ) ( + ) + ( + ) ( + ) − ( + ) ( + )]. (24)
Add both sides of the inequality from = 0 to ∞, then we get
J ∞ ( ) ≤ ( ( )) − ( (∞)),
which implies that we can infer the upper bound of the objective function J ∞ from the positiveness property
of the function ( (∞)). Assume a scalar exists that satisfies

( ( )) ≤ . (25)
Defining = and applying the Schur complement operation yields the following sufficient condition
−1

for Equation (24):
[ ]
1 ( )
≥ 0. (26)
( )