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Zhang et al. Intell Robot 2022;2(4):371­90 I http://dx.doi.org/10.20517/ir.2022.26 Page 383

Figure 4. Engine model.

From the inequalities Equation (42), Equation (44) and Lemma 3.2, we can conclude that ( ( )) is an ISS-
Lyapunov function, and the closed-loop system is ISS. Thus the proof is completed. □

4. LOWER LAYER
According to the vehicle longitudinal motions in the car-following scenario, the lower layer is divided into two
blocks for calculating the acceleration throttle opening and the brake pressure, which are designed to realize
the desired longitudinal acceleration or deceleration and direct yaw moment. To this end, a logic switch in the
lower layer is utilized to implement an accelerating or braking manoeuvre based on the desired longitudinal
acceleration and the direct yaw moment calculated by the T-S fuzzy controller in the upper layer.
The vehicle throttle opening or brake pressure on all four wheels can then be obtained.

4.1. Accelerating control
Figure4showstheenginespeedversusthetorqueatdifferentthrottleopenings, whichareshownatthebottom.
If the engine speed and the desired torque e,des are known, the corresponding expected throttle opening
des is obtained based on this look-up table:

( )
des = e,des , . (45)

Thevehiclelongitudinaldynamicsareproducedbythecombinedeffectofvehicletraction,windresistance,and
ground resistance. Therefore, the vehicle state depends strongly on the road smoothness and the magnitude
of the wind resistance coefficient during cruising. The equivalent air friction during longitudinal driving is

1 ( ) 2
= eff , (46)
2
where is the air mass density, is the coefficient of air friction, is the windward area of the ego vehicle,
is the ratio between the wheel speed and engine speed , and eff is the effective tire radius.

During vehicle acceleration, the engine torque e,des is related to the expected acceleration des is [39] as

[ 3 3 2 ]
e,des = + + ( eff ) , (47)

eff
eff
2
2
where = + + ( eff + ) is the effective moment of inertia for the engine side, is the moment of

inertia of the engine, and is the sum of all the rolling resistances related to the rolling damping coefficient