84 6. ROLLOVER CONTROL STRATEGIES AND ALGORITHMS
where A; B
1
, and B
2
are the coefficient matrix of the state space equation of the 3-DOF vehicle
rollover model. ese matrixes can be derived by vehicle rollover model.
To avoid rollover in cornering situations, the roll angle and the roll rate should be reduced
under the assumption that the lateral acceleration is controllable. If the weights on the roll angle
and the roll rate are set to higher values for rollover prevention, the yaw rate error increases owing
to the lateral load transfer caused by the ARCS, and this can make a vehicle lose maneuverability
or lateral stability because of the rear-sway phenomenon. On the other hand, if the weight on
the yaw rate error is set to a higher value for maneuverability or lateral stability, the roll angle
and the roll rate cannot be reduced effectively. ese effects are complementary to each other. In
other words, the ARCS can reduce the roll angle and roll rate which ESC cannot do, and ESC
can reduce the yaw rate error which the ARCS cannot do. For this reason, the weights on the
roll angle, the roll rate and the yaw rate error, i.e.,
1
,
3
, and
4
in Equation (6.38), should be
set to higher values in the LQ objective function. e values of
i
for the weights in the LQR
cost function are given in Table 6.1.
Table 6.1: Weights in the LQ cost function
η
1
η
2
η
3
η
4
η
5
η
6
0.08 rad/s 10 m/s
2
1° 3
deg/s 5000 N•m 2000 N•m
e LQR cost function in Equation (6.36) can be rewritten as
J D
Z
1
0
.Cx C Du/
T
.Cx C Du/dt
D
Z
1
0
x
T
Qx C u
T
N
T
x C x
T
N
T
u C u
T
Ru
dt;
(6.43)
where Q D C
T
C , N D C
T
D, R D D
T
D
C D
2
6
6
6
6
6
6
6
4
p
1
a
11
p
1
.a
12
C v
x
/
p
1
a
13
p
1
a
14
p
1
a
15
0
p
2
0 0
p
2
0 0
p
3
0 0
0 0 0
p
4
0
0 0 0 0 0
0 0 0 0 0
3
7
7
7
7
7
7
7
5
D D
2
6
6
6
6
6
6
6
4
0 0
0 0
0 0
0 0
p
5
0
0
p
6
3
7
7
7
7
7
7
7
5
:
In Equation (6.43), a
ij
is the j th element of the ith row of the matrix A. In the LQR, the
full-state feedback control u D Kx is used. e controller gain K is easily obtained by solving
the Riccati equation. en the LQR controller is designed.
e controllers for the yaw motion and the roll motion can be designed separately if the
ARCS can reduce the roll motion. To confirm this fact, a structured controller and a separated
controller are designed based on LQR controller.