A Yaw Rate Tracking Control of Active Front Steering System 235
2.3 Linear Single Track Model
To design the controller for yaw rate tracking control of active front steering, a linear single track model as shown in Figure 2 is utilized. This model is linearized from the
two track nonlinear vehicle model based on few main assumptions: tire force operates in linear region, very small of front steer angle
f
δ
and vehicle side slip
β
,vehicle speed v is constant and two tires at front and rear axle are lumped into single tire at
the centre line of vehicle [18].
f
l
r
l
r
yf
F
v
β
f
δ
y
f
α
r
α
yr
F
Fig. 2. Single track model
The dynamics equation for the lateral and yaw motions are described as follows
yr yf
F F
r mv
+ =
+
β
11
yr r
yf f
z
F l
F l
r I
− =
12
As assumed above which tire forces are operates in linear region, front lateral tire force
yf
F and rear lateral tire force
yr
F are exhibit linear characteristics as described
in the following equations
f f
yf
C F
α
= 13
r r
yr
C F
α
= 14
where
f
C and
r
C are front and rear tire cornering stiffness respectively. For linear tire forces, sideslip angle of front and rear tire are given in the equation 15 and 16
respectively as follows
v r
l
f f
f
− −
=
β δ
α
15
v r
l
r r
+ −
= β
α
16 By re-arrange and simplify the equations 11 – 16, the differential equations of
sideslip and yaw rate variable can be simplified as linear state space model as shown in equation 17. Notice that the parameters are same as discussed in section 2.1
above.
236 M.K. Aripin et al.
f z
f f
f
z r
r f
f z
f f
r r
f f
r r
r f
I l
C mv
C r
v I
l C
l C
I l
C l
C mv
l C
l C
mv C
C r
Bu Ax
x δ
β β
+
−
− −
− +
− −
− =
+ =
2 2
2
1
17
2.4 Yaw Rate Reference Model
The main objective in yaw rate tracking control of active front steering is to bring the actual response of vehicle yaw rate close to desired response. The desired yaw rate
response is determined as a function of vehicle speed
v
and front wheel steer angle
f
δ in steady state condition as follows
f u
d
v k
l v
r δ
.
2
+ =
18 where
u
k
is known as cornering stability factor and define as follows
r f
r f
f f
r r
u
C C
l l
C l
C l
m k
+ −
= 19
However, due to lateral acceleration of the vehicle in
g
unit could not exceed the maximum road friction coefficient
µ
, the steady state value of yaw rate must be lim- ited as express in the following equation
v g
r
ss
µ ≤
20
3 Composite Nonlinear Feedback for Active Front Steering
In this section, the composite nonlinear feedback CNF control design procedures for yaw rate tracking control of active front steering AFS are presented. In general, The
CNF control technique is applicable for systems with or without external distur- bances. In this paper, the plant of vehicle dynamics is considered without an external
disturbance and all states variable are assumed available for measurement. The fol- lowing subsection will discuss the design procedures for CNF control which have
been established in [11].
A Yaw Rate Tracking Control of Active Front Steering System 237
3.1 CNF Control Design
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