86 6. ROLLOVER CONTROL STRATEGIES AND ALGORITHMS
8
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
:
x
D
P
T
u
D M
w
D a
y
Px
D A
x
C B
1
w
C B
2
u
:
(6.48)
Accordingly, the structured controller gain matrix (6.44) is also separated as
K
s1
D
0 k
1
k
1
K
s2
D
k
2
k
3
:
(6.49)
With these state-space equations, the objective functions and the controller structures,
K
s1
and K
s2
are designed to minimize J
s1
and J
s2
, respectively. K
s1
can be regarded as the
proportional controller gain matrix for yaw motion control, and K
s2
can be regarded as the
proportional—derivative controller gain matrix for roll motion control. To find K
s1
and K
s2
,
the CMA-ES is also used.
Figure 6.15 shows the Bode plots drawn on the basis of the designed LQ controllers. In
these plots, the input is the steering angle, and the outputs are the roll angle, the roll rate, the
lateral acceleration, and the yaw rate error. Since the frequency of the Fishhook maneuvre with
a maximum steering angle of 221
ı
is near 0.5 Hz, the frequency responses near 0.5 Hz should
be focused on.
As shown in Figure 6.15, the structured LQR controller and the separated LQR con-
trollers show almost the same performance. is is caused by the fact that the yaw motion and
the roll motion of the controlled vehicle were not related to each other. is result means that
the controllers for the yaw motion and the roll motion can be separately designed according to
their own objectives and that several approaches can be applied to design the controllers for the
yaw motion and the roll motion.
6.5 SLIDING MODE CONTROL METHOD
Sliding mode control (SMC) is a nonlinear control technique featuring remarkable properties
of accuracy, robustness, and easy tuning and implementation. So, sliding mode control has at-
tracted the attention of more and more scholars. Imine et al. proposed an estimator based on
the high-order sliding mode observer is developed to estimate the vehicle dynamics, such as
lateral acceleration limit and center height of gravity [70]. An antiroll controller is designed by
Chuwith smooth sliding mode control technique for vehicles in which an active antiroll suspen-
sion is installed [33].
A rollover avoidance controller with sliding mode technique to control the steering angle
of vehicle is designed. And two sliding mode control are developed first, and then optimized
them into one controller which has a better performance with the theory of fuzzy control. Taking
6.5. SLIDING MODE CONTROL METHOD 87
No Control
LQR
S
tructured LQ
Separated LQ
0
-5
-10
-15
-20
-25
-30
-35
-40
10
-2
10
-1
10
0
10
1
Frequency (Hz)
(a) Bode plot from δ to ϕ
Magnitude (dB)
No Control
LQR
S
tructured LQ
Separated LQ
10
5
0
-5
-10
-15
-20
-25
-30
10
-2
10
-1
10
0
10
1
10
2
Frequency (Hz)
(b) Bode plot from δ to ϕ̇
Magnitude (dB)
Figure 6.15: Bode plots from the steering input to each output. (Continues.)
88 6. ROLLOVER CONTROL STRATEGIES AND ALGORITHMS
No Control
LQR
Structured LQ
Separated LQ
35
34
33
32
31
30
29
28
27
10
-2
10
-1
10
0
10
1
10
2
Frequency (Hz)
(c) Bode plot from δ to a
y
Magnitude (dB)
No Control
LQR
Structured LQ
Separated LQ
15.4
15.2
15
14.8
14.6
14.4
14.2
14
13.8
13.6
10
-1
10
0
10
1
10
2
Frequency (Hz)
(d) Bode plot from δ to e
γ
Magnitude (dB)
Figure 6.15: (Continued.) Bode plots from the steering input to each output.
6.5. SLIDING MODE CONTROL METHOD 89
into account the true steering actuator may not be able to complete the output response control,
so set the control output of the rotation angle is ˙2.3
ı
, and the maximum change rate of control
output is ˙10
ı
/s (see Figure 6.16).
Sliding
Model
Controller-1
Destination
Calculating
Comprehensive
Control
Optimization
Vehicle
Model in
CarSim
Rollover
Dynamics
Sliding
Model
Contr
oller-2
a
y
ϕ
d
⤶
ϕ
⤶
δ x
y
d
y
-
+
+
-
Figure 6.16: e rollover prevention controller diagram.
1. Sliding mode control in consideration of roll rate.
Rewrite the vehicle rollover model in state form as follows.
Setting the state vector as x D
v r
P
u
, the state-space equation can be ob-
tained based on the vehicle dynamics model:
Px D Ax C Bı: (6.50)
And discretize the system with the method of approximate discretization.
Consider the sliding mode surface as:
S.x/ D Pe C c
0
e; e D
d
; (6.51)
where
d
is calculated from a saturation function:
(
d
D 0:8
ˇ
ˇ
a
y
ˇ
ˇ
> 0:35
d
D
ˇ
ˇ
a
y
ˇ
ˇ
0:35:
(6.52)
90 6. ROLLOVER CONTROL STRATEGIES AND ALGORITHMS
e control amount u
c
is obtained from the related sliding mode reaching rule and the
equations of deriving variable
P
S.x/.
U
c
D flag
U sgn.s.x/
; (6.53)
where
(
flag D 1
ˇ
ˇ
a
y
ˇ
ˇ
> 0:35
flag D 0
ˇ
ˇ
a
y
ˇ
ˇ
0:35:
(6.54)
So:
u
c
D T
s
.CB/
1
Œ
C
.
I C T
s
A
/ .
x x
d
/
C w
1
e w
2
T
s
.CB/
1
sgn.e/; (6.55)
where w
1
D 1, C D w
2
Œ
1 1 1 1
.
2. Sliding mode control in consideration of lateral displacement.
e lateral and yaw accelerations are computed as follows:
Ry D a
1
Py C a
2
Pr C a
3
ı
Rr D b
1
Pr C b
2
ı C b
3
ˇ:
(6.56)
e chose sliding mode surface is as follows:
S D Py
l
C y
l
; (6.57)
where
Py
l
D Py Py
d
(6.58)
y
l
D y y
d
; (6.59)
and Py
d
, y
d
are the first and double integration of desired lateral acceleration a
y lim
, respec-
tively.
Under the constant reaching rule:
P
S D "sgnS " > 0: (6.60)
Assume that x D
Py
Pr
, and regard ˇ as a disturbance, so Equation (6.50) can be recast
as:
Px D
"
a
1
a
2
0 b
1
#
x C
"
a
3
b
2
#
ı: (6.61)
en
.
a
1
C
/
Py C a
2
Pr C a
3
ı D Ry
d
C Py
d
"sgn."/: (6.62)
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