6.3. MODEL PREDICTION CONTROL METHOD 79
where is weight coefficient and "
1
is relaxation factor.
Because the steering angle of vehicle is limited, physical constraint must be considered
when designing the controller. So controlled quantity of steering angle should be constraint as
follows:
ı
u
min
.k C j / ı
u
.k C j / ı
u
max
.k C j /; j D 0; 1; : : : ; C
h
1: (6.29)
Because
ı
u
.k C j / D ı
u
.k C j 1/ Cı
u
.k C j /; (6.30)
assume
U
k
D 1
C
h
˝ ı
u
.k 1/ (6.31)
A
C
h
C
h
D
2
6
6
6
6
4
1 0 : : : 0
1 1 0
:
:
:
:
:
: 1
:
:
:
0
1 : : : 1 1
3
7
7
7
7
5
˝ E
1
: (6.32)
In Equation (6.32), 1
C
h
is column vector which has C
h
rows, E
1
is unit matrix and the dimension
of E
1
is 1, ˝ represents Kronecker product, and ı
u
.k 1/ represents the actual control output
at the last moment.
According to Equations (6.30) and (6.31), Equation (6.29) can be transformed as:
U
min
A
C
h
C
h
U
k
C U
k
U
max
; (6.33)
where U
min
and U
max
are the collection of maximal and minimum values of control in time
domain.
e complete objective function can be obtained by substituting Equation (6.27) into the
objective function (6.28). en, the objective function is reduced to a standard quadratic form:
J
.
.k/; ı
u
.k 1/; U.k/
/
D
U.k/
T
; "
T
H
k
U.k/
T
; "
C G
t
U.k/
T
; "
C P
t
(6.34)
where
H
k
D
‚
T
k
Q‚
k
C R 0
0
Q
; G
k
D
2E.k/Q‚
k
0
Q
; P
k
D E.k/
T
QE.k/;
where 0
Q
is the null matrix that has the same dimension as Q matrix, E.k/ is tracking error in
prediction horizon, and it can be showed as:
E.k/ D .k/.k/ Y
ref
.k/; Y
ref
.k/
D
y
ref
.k C 1/; y
ref
.k C 2/; : : : ; y
ref
.k C P
h
/
T
;
(6.35)
where y
ref
.k/ is reference output.
80 6. ROLLOVER CONTROL STRATEGIES AND ALGORITHMS
Binding constraint condition (6.29), a control increment sequence in control horizon can
be obtained by solving Equation (6.28):
U
k
D
.
ı
u
/
k
.
ı
u
/
kC1
: : :
.
ı
u
/
kCC
h
1
T
: (6.36)
e first element in the control increment sequence is used as the actual control increment
for the control system:
ı
u
.k/ D ı
u
.k 1/ C
.
ı
u
/
k
: (6.37)
In the following control cycle, the continuous cycle of the above solution can realize the
rollover control.
e control output quantity as Œ24; 24. After debugging, the parameters of MPC con-
troller are evaluated, prediction horizon P
h
D 35, control horizon C
h
D 5, weight coefficient
D 10, relaxation factor " D 10, output state coefficient Q D
Œ
Q
1
0
827
, control weight coef-
ficient R D r
1
Œ
1 0
14
, where
Q
1
D
2
6
6
4
Q
q
: : : 0
22
:
:
:
:
:
:
:
:
:
0
22
: : : Q
q
3
7
7
5
; Q
q
D
"
3:33 0
0 5
#
; r
1
D 0:3:
Under the Fishhook condition, there are two phases that are prone to roll over. ere-
fore, it is very necessary to analyze the control effect of multi-objective MPC control under this
condition. e initial velocity is 85 km/h and the maximum front-wheel angle input is six de-
grees. e comparison of control effect between multi-objective MPC control and PID control
is shown in Figure 6.14.
Figures 6.14a–d, respectively, represent the comparison of mRI, yaw rate, roll angle, and
steering angle under the Fishhook maneuver. As we can see in Figure 6.14a, the three-axis
bus rolls over at 4 s without control. Both the MPC control and PID control can prevent the
bus from rollover. Obviously, the MPC control can make the mRI smaller. In Figure 6.14b, it
shows the yaw rate with MPC control is more consistent with desired value. It suggests that the
MPC control can follow the steering intention of the driver while ensuring the control effect.
Figure 6.14c shows the roll angle is almost four degrees without control which is very dangerous.
e MPC control limits it to two degrees which is smaller than PID control. To sum up, it can
be concluded that MPC control can effectively prevent vehicle rollover.
6.3. MODEL PREDICTION CONTROL METHOD 81
MPC Control
PID Control
Uncontrolled
1
0.5
0
-0.5
-1
0 2 4 6 8 10
Time (s)
(a) Comparison of rollover index (mRI) under the Fishhook condition
mRI
MPC Control
PID Control
Uncontrolled
0.2
0.1
0
-0.1
-0.2
0 2 4 6 8 10
Time (s)
(b) Comparison of yaw rate under the Fishhook condition
Yaw Rate (deg/s)
Figure 6.14: Control effect comparison between MPC and PID control in the Fishhook condi-
tion. (Continues.)
82 6. ROLLOVER CONTROL STRATEGIES AND ALGORITHMS
MPC Control
PID Control
Uncontrolled
4
2
0
-2
-4
0 2 4 6 8 10
Time (s)
(c) Comparison of roll angle under the Fishhook condition
Roll Angle (deg)
MPC Control
PID Control
Uncontrolled
6
4
2
0
-2
-4
06
0 2 4 6 8 10
Time (s)
(d) Comparison of steering angle under the Fishhook condition
Stering Angle (deg)
Figure 6.14: (Continued.) Control effect comparison between MPC and PID control in the
Fishhook condition.
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