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Figure 3. The whole algorithm framework.

ThenumberofsubsystemsintheT-SfuzzymodelEquation(10)isthusreducedfrom 4to 2,whichconsiderably
decreases the computational burden associated with the controller parameters in each sampling period within
the MPC framework, and facilitates real-time implementation of the model, as expected.

3. T-S FUZZY MODEL PREDICATIVE CONTROL DESIGN
In this section, we provide a detailed description of the design of an adaptive cruise controller with lateral
stability based on the T-S fuzzy MPC framework. Figure 3 shows the proposed adaptive cruise control sys-
tem divided into upper and lower layers. The upper layer calculates the required acceleration and direct yaw
moment for vehicle-following control subject to various constraints, and the lower layer calculates the vehicle
throttle opening or hydraulic cylinder pressure of the four wheels to generate control signals corresponding
to the results provided by the upper layer. Before proceeding further, some definitions and lemmas are first
stated.

3.1. Robust positively invariant set
Consider a discrete-time dynamical system

( + 1) = ( ( ), ( )), (18)

where (0, 0) = 0, ( ) ∈ R is a state vector, and ( ) is a control input or external disturbance that belongs
to a compact set D ⊂ R containing the origin. A robust positively invariant (RPI) set is defined below.

Definition 3.1: If ( ) ∈ Ω (Ω ⊂ R ), it holds that ( + 1) ∈ Ω for all ( ) ∈ D; then, Ω is called an RPI set

for System [Equation (18)].

Lemma 3.1 [34] : The following two expressions are equivalent for System [Equation (18)] with ≤ , where

2
is a positive constant:

• the ellipsoidal set Ω ≜ { ≤ }, where > 0, is a robust positively invariant set;
• the inequality ( + 1) ( + 1) ≤ holds if the external disturbance satisfies 1 ≤ .

1
2
3.2.Input­to­state stability
We define input-to-state stability (ISS) for use in the following sections.
Definition 3.2: A discrete-time system ( + 1) = ( ( ), ( )) is ISS if there exist a KL-function : R ≥0 ×
Z + → R ≥0 and a K-function satisfying

|| ( , 0 , ( ))|| ≤ (|| 0 ||, ) + (sup || ( )||), (19)
≥0
where 0 is the initial state, ( ) is the input sequence, and is the sampling time instant.