61
C H A P T E R 6
Rollover Control Strategies
and Algorithms
Effective control strategy and algorithm is the core of preventing vehicle rollover accidents. Dur-
ing the development of vehicle active rollover avoidance, many rollover control strategies and
algorithms have been proposed by researchers. In the following section, some common con-
trol methods used in the literature are introduced and analyzed, such as Proportional-integral-
derivative (PID) control, linear quadratic regulator (LQR) control, H-infinity control, and
Model predictive control (MPC).
6.1 PROPORTIONAL-INTEGRAL-DERIVATIVE
CONTROL METHOD
PID control is one of the most commonly used controllers, which has a wide range of applica-
tions. Regardless of whether the structure or parameters of the controlled plant are determined
and accurate, PID control can play its own control effect by adjusting the parameters. e basic
idea of PID control is that, based on the system variable error e.t/ (being the error between the
expected value and actual value of feedback variables), the control variable is calculated through
selecting the proper control gains and applied to controlled plant to reduce the error. e control
law is shown in Equation (6.1):
u.t/ D K
p
e.t/ C
1
T
i
Z
e.t/dt C T
d
de.t/
dt
: (6.1)
Figure 6.1 shows the control effect of PID controller. It is observed that PID control
can prevent the occurrence of the vehicle rollover. However, the parameter of PID controller
is difficult to be tuned in real time. So researchers usually use PID in combination with other
control algorithms.
Muniandy proposed a PI–PD-type (proportional-integral–proportional-derivative) fuzzy
controller to prevent vehicle rollover [45]. is controller added the fuzzy control into PI-PD-
type controller. Figure 6.2 shows the basic layout of PI–PD controller.
e controller is designed to preserve the linear structure of conventional PI–PD con-
troller and substitute the coefficient gains with nonlinear fuzzy functions. e output of PI–
62 6. ROLLOVER CONTROL STRATEGIES AND ALGORITHMS
1
0.8
0.6
0.4
0.2
0
0 2 4 6 8 10
(a) Rollover Index
Rollover Index
Time (s)
PID Control
Without Control
5
4
3
2
1
0
0 2 4 6 8 10
(b) Roll angle
Roll Angle of Sprung Mass/(deg)
Time (s)
PID Control
Without Control
Figure 6.1: Control effect of PID control. (Continues.)
6.1. PROPORTIONAL-INTEGRAL-DERIVATIVE CONTROL METHOD 63
7
6
5
4
3
2
1
0
0 2 4 6 8 10
(c) Steering angle
Steering Angle (deg)
Time (s)
PID Control
Without Control
15
10
5
0
-4 -3 -2 -1 0 1
(d) Phase track diagram
Yaw Rate (deg/s)
Side Slip Angle (deg)
Without Control
PID Control
Figure 6.1: (Continued.) Control effect of PID control.
64 6. ROLLOVER CONTROL STRATEGIES AND ALGORITHMS
⊗ ⊗
–
+
–
+
r e y
P
I Process
PD
u
PI
u
PI - PD
u
PO
Figure 6.2: Basic layout of PI–PD controller.
PD-type fuzzy controller, u
PIPD
.nT /, is represented by
u
PIPD
.nT / D u
PI
.nT / u
PD
.nT /; (6.2)
where u
PI
.nT / and u
PD
.nT / are the equivalent outputs from fuzzy PI and fuzzy PD controllers,
respectively. Prior to that, in Laplace domain, both conventional analogue PI and PD controllers
can be represented by
(
u
PI
.s/ D
K
c
p
C
K
c
i
s
E.s/
u
PD
.s/ D K
c
0
p
C K
c
d
Y.s/;
(6.3)
where u
PI
.s/ and u
PD
.s/ are outputs of analogue PI and PD controllers, respectively; K
c
p
, K
c
i
, and
K
c
d
are proportional, integral, and derivative gain, respectively. It can be seen that PI controller
is influenced by error signal E.s/ and PD controller is influenced by process output Y.s/. By
applying bilinear transformation, Equations (6.3) is transformed into a discrete version. Hence,
fuzzy PI controller output is written as
u
PI
.nT / D u
PI
.nT T / C K
uPI
u
PI
.nT / (6.4)
u
PD
.nT / D u
PD
.nT T / C K
uPD
u
PD
.nT /: (6.5)
By inserting Equations (6.4) and (6.5) into Equation (6.2), the output of PI–PD-type FLC
controller will be
u
PIPD
.nT / D u
PI
.nT T / C K
uPI
u
PI
.nT / Cu
PD
.nT T / C K
uPD
u
PD
.nT /: (6.6)
Both K
uPI
and K
uPD
will be determined by fuzzy rules. is controller’s layout applied in
active anti-roll-bar (ARB) is presented in Figure 6.3.
Similar to a standard fuzzy controller, membership functions and rules will be applied to
the fuzzy PI and fuzzy PD controllers. e inputs for both controllers will be roll angle error
signal. As the derivative controller receives roll angle signal directly from the system feedback
itself, it is expected that the derivative kick phenomenon can be avoided. Earlier, it has been
6.1. PROPORTIONAL-INTEGRAL-DERIVATIVE CONTROL METHOD 65
⊗ ⊗
–
+
–
+
Desired
Ro
ll angle
Desired Force
Roll Angle
e
ARB Force
FPI Controller
Active
Anti-Roll Bar
Vehicle
System
FPD Controller
u
PI
u
PI - PD
u
PD
Figure 6.3: PI–PD-type fuzzy logic controller layout.
stated that fuzzy PI controller has two inputs, which are roll angle error signal, e
p
.nT / and
rate of change of roll angle error signal, e
v
.nT /. e fuzzy PI controller output (also called as
incremental control output) is denoted as u
PI
.nT /. e inputs for fuzzy PD controllers are the
roll angle, d.nT / (desired roll angle) and rate of change of roll angle, y.nT /. Figure 6.4a,b show
the input membership functions for both PI and PD controllers. e input unit in Figure 6.4a
is in degree, while the input unit in Figure 6.4b is
ı
/s.
Figure 6.5 shows the output membership functions for both PI and PD controllers since
both outputs are represented by the same membership functions. e output unit would be
Newtons as it represents the required force by the actuator. e range of each membership
functions is determined by typical operating range of a passenger car in a real application. e
maximum value for required force is bound by the hardware capability.
A set of control rules base is created for fuzzy PI control as follows.
• RULE 1: IF e
p
negative AND e
v
negative, THEN PI-output = output negative.
• RULE 2: IF e
p
negative AND e
v
positive, THEN PI-output = output zero.
• RULE 3: IF e
p
positive AND e
v
negative, THEN PI-output = output zero.
• RULE 4: IF e
p
positive AND e
v
positive, THEN PI-output = output positive.
e membership functions have been kept simple in triangular form to reduce computing
memory usage. e structure of the membership functions for both input signals are the same
to avoid further memory allocations for the controller. e output signal for fuzzy PD controller
is denoted as u
PD
.nT /.
Fuzzy PD controller’s rules set is as follows.
• RULE 5: IF d positive AND y positive, THEN PD-output = output zero.
• RULE 6: IF d positive AND y negative, THEN PD-output = output positive.
• RULE 7: IF d negative AND y positive, THEN PD-output = output negative.
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