6.4. LINEAR QUADRATIC REGULATOR CONTROL METHOD 83
6.4 LINEAR QUADRATIC REGULATOR CONTROL
METHOD
LQR is an algorithm for optimal control by taking the integral of quadratic function of state
variable as the performance index of a linear system. It greatly simplifies the computation of
real-time control. So, it also widely used in the development of rollover avoidance control sys-
tem [44, 66–68]. Yim et al. proposed a linear quadratic regulator for vehicle rollover prevention
as follows, which combined the Electronic Stability Control (ESC) and Active Roll Control
System (ARCS) [69].
First, 3-DOF model is established, consisting of a 2-DOF bicycle model to describe yaw
motion and lateral motion, and a 1-DOF roll model to describe the roll motion.
e LQR cost function for rollover prevention is defined as:
J D
Z
1
0
1
e
2
C
2
a
2
y
C
3
2
C
4
P
2
C
5
M
2
B
C
6
M
2
dt: (6.38)
In Equation (6.38), the weights
i
are set by the relation
i
D 1=
2
i
from Bryson’s rule,
where
i
represents the maximum allowable value of each term, a
y
is the lateral acceleration of
vehicle, is the roll angle, and
P
is roll rate. M
B
is the controlled yaw moment generated by
differential braking and M
is the roll moment induced by active anti-roll bar. e
is the yaw rate
error and is defined as the difference between the actual yaw rate r and the reference yaw rate
r
d
, according to
e
D r r
d
: (6.39)
e reference yaw rate
d
generated from the driver’s steering input ı is modeled with a
first-order system as
r
d
D
K
e
s C 1
ı; (6.40)
where is the time constant and K
e
is the steady-state yaw rate gain determined by the speed
of the vehicle.
e yaw dynamics are modeled separately without the roll dynamics. For the 3-DOF
vehicle model, the state x, the control input u, and the disturbance w are defined as
8
ˆ
<
ˆ
:
x D
v r r
d
P
T
u D
M
B
M
w D ı;
(6.41)
where v is lateral speed.
Based on the 3-DOF vehicle rollover model, the state-space equation of the vehicle model
is obtained as
Px D Ax C B
1
w C B
2
u; (6.42)
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.
6.4. LINEAR QUADRATIC REGULATOR CONTROL METHOD 85
In a structured controller, the control yaw moment is calculated only from the yaw rate
error, and the control roll moment is calculated only from the roll angle and the roll rate. e
controller structure is given by
K
s
D
0 k
1
k
1
0 0
0 0 0 k
2
k
3
: (6.44)
In a structured LQR controller, it is necessary to find K
s
which minimizes the LQR cost
function J . With the notation in Equation (6.43), the problem of finding K
s
is formulated as
the optimization problem.
min
K
J D trace.P /
.
A C B
2
K
s
/
T
P C P
.
A C B
2
K
s
/
C Q
s:t:
C K
T
s
N
T
C K
s
N C K
T
s
RK
s
:
(6.45)
For an arbitrary K
s
, the LQR cost function J can be easily computed by solving the Lya-
punov Equation (6.45). e evolutionary strategy with covariance matrix adaptation (CMA-
ES)12 is used to find the optimal K
s
. e structured controller given by Equation (6.44) is
designed for the single objective function (6.38). In this situation, the controller gains for the
yaw moment and the roll moment have an effect on each other. is means that the structured
controller does not strictly separate the yaw motion and the roll motion. Hence, it is necessary to
design separately a controller for yaw motion and a controller for roll motion with different ob-
jective functions. erefore, two LQR controllers are designed separately for yaw motion control
and roll motion control. For this purpose, the LQR objective function given by Equation (6.38)
is separated into J
s1
and J
s2
according to the yaw motion and the roll motion as given by
J
s1
D
Z
1
0
1
e
2
C
2
a
2
y
C
5
M
2
B
dt
J
s2
D
Z
1
0
3
2
C
4
P
2
C
6
M
2
dt: (6.46)
With these objective functions and the state-space Equations (6.42), (6.47), and (6.48)
represent the state-space equations of the yaw motion and roll motion, respectively, which are
used to design separated controllers:
8
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
:
x
D
v r r
d
u
D M
B
w
D ı
Px
D A
x
C B
1
w
C B
2
u
(6.47)
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