6.2. H-INFINITY CONTROL METHOD 67
Output Negative Output Zero Output Positive
Output Signal
∆u
PI
(nT), ∆u
PD
(nT)
1
0-400 400 1600
N
-1600
Figure 6.5: Membership functions for fuzzy PD and fuzzy PI output signals.
• RULE 8: IF d negative AND y negative, THEN PD-output = output zero.
Defuzzification process is done based on “center of mass” approach. To optimize the con-
troller’s performance, a set of parameters for the controller was obtained using trial and error
method. It is an initial step to determine the test parameters where the parameters were chosen
one by one while the changes occurring when the output were tracked. Individual values often
increase gradually until the desired plant output response is achieved. Excessive amounts of actu-
ator force and speed demands will also become the limiting factor for this method. After exten-
sive simulation tests, parameter values satisfying all simulation tests are as follows: Kp D 0:3140,
Ki D 0:0971, K
0
p D 0:0576, K
d
D 0:001, K
uPD
D 0:4968, and K
uPI
D 0:4496. e sampling
period is set to T D 0:01 second to cope with the vehicle suspension system response. Integral
windup problem is not expected from this simulation because the simulation was executed in
an ideal system assuming there is no saturation or physical limitation on the actuators and other
related hardware. In future, when actual hardware implemented in this system, limiting the
controller output according to the physical limit of the actual actuator is strongly recommended.
6.2 H-INFINITY CONTROL METHOD
To solve the rollover problem due to complex road conditions, the variation of the number of
passengers, and other external interference, H-infinity control performance is used to provide
robustness to model uncertainty and external disturbances [28, 31, 41, 46, 61–63]. Jin [15]
designed a H1 controller with differential braking as the actuator to prevent vehicle rollover
and it is optimized by genetic algorithm.
68 6. ROLLOVER CONTROL STRATEGIES AND ALGORITHMS
As shown in Figure 6.6, e is the rollover index error, and r
in
is the reference input of the
rollover index, z
1
, z
2
, and z
3
are the evaluation outputs which dependent on exogenous input, G
s
is the transfer functions which can be obtained using the state space equation of vehicle model
mentioned in Section 6.2, K
c
is the transfer function of H-infinity controller, w
d
is the system
input, RI is the rollover index, and M
B
is the corrected yaw torque.
H
∞
controller
op
timized by
GA(K
C
)
Vehicle
sy
stem
model
(G
S
)
W
1
W
2
W
3
z
1
z
2
z
3
w
d
RI
e
M
B
r
i
n
Figure 6.6: Block diagram of H-infinity controller for vehicle rollover prevention.
In order to formulate the standard structure of H-infinity controller, the weight func-
tions W
1
, W
2
, and W
3
are defined to characterize the performance objectives and the actuator
limitations.
W
1
weights the rollover index error signal. It is a constraint of the robustness of the rollover
prevention control system which can adjust the influence of external interference.
W
2
weights the yaw moment signal. It is a constraint of the amplitude of anti-yaw torque
due to differential braking.
W
3
weights the rollover index signal. It is a constraint of the stability of the rollover pre-
vention control system. It also restricts the yaw rate and the vehicle lateral velocity evolution.
From the robust H-infinity theory, if w
d
D 0, some transfer functions from input to out-
puts could be defined as:
G
1
def
D
e.s/
r
in
.s/
D
.
I C G
s
K
c
/
1
(6.7)
G
2
def
D
M
B
.s/
r
in
.s/
D K
c
.
I C G
s
K
c
/
1
(6.8)
G
3
def
D
RI.s/
r
in
.s/
D G
s
K
c
.
I C G
s
K
c
/
1
; (6.9)
where G
1
is the sensitivity function, and G
3
is the complementary sensitivity function.
6.2. H-INFINITY CONTROL METHOD 69
en, the weighted mixed-sensitivity can be obtained.
2
6
6
4
W
1
e
W
2
M
B
W
3
RI
e
3
7
7
5
D
2
6
6
4
W
1
W
1
G
s
0 W
2
0 W
3
G
s
I G
s
3
7
7
5
r
i n
M
B
: (6.10)
Appling the minimum gain theorem, in case of mixed sensitivity problem, the objective
is to find a rational function controller and to make the rollover prevention closed-loop system
stable satisfying the following inequality.
W
1
G
1
W
2
G
2
W
3
G
3
1
< 1: (6.11)
e weight functions W
1
, W
2
, and W
3
are the tuning parameters and it typically requires
some iterations to obtain weights which will yield a good controller. For a good robustness
margin and a small tracking error, a good starting point is to choose them as:
W
1
D
k
1
s C k
2
k
3
s C 1
(6.12)
W
2
D k
4
(6.13)
W
3
D
k
5
s C k
6
k
7
s C 1
: (6.14)
ere are various methods available in the literature for selection of weights. In most of
these design methods the weight functions are selected using trial and error method.
en, H-infinity controller is synthesized by loop shaping technique. But there is a dis-
advantage in this type of synthesis that trial and error procedure may not end up in a stabilizing
controller. So, the genetic algorithm is used to select the optimal weight functions and the
flowchart of H-infinity controller programming with genetic algorithm is given as shown in
Figure 6.7.
e parameters k
1
, k
2
, k
3
, k
4
, k
5
, k
6
, and k
7
are to be optimized with genetic algorithm, of
which the variation range can be limited by the dynamic model of vehicle rollover. e objective
function of the genetic optimization J is defined as the maximum absolute value of rollover
index. And the fitness function S
H
is defined as the reciprocal of objective function.
(
J
def
D max.jRIj/
S
H
def
D 1=J:
(6.15)
Also, to meet the stability criterion of H-infinity controller for vehicle rollover prevention,
the constraint condition in inequality (6.16) should be satisfied:
N
W
1
1
.j!/
C N
W
1
3
.j!/
1: (6.16)
70 6. ROLLOVER CONTROL STRATEGIES AND ALGORITHMS
Choosing the starting
point of weight functions
Limiting the variation
range of all parameters
Random generating initial population
Decoding all parameters and calculating
the weight functions
Setting the population size, the number of
generation, selection type, mutation rate,
type of crossover, and crossover size
Encoding all parameters
Obtaining the optimal solution and calculating
the optimized H-infinity controller
Tuning the H-infinity
controller
Setting the
fitness function
value to zero
Evaluating the fitness function
and the objective function
Selection, crossover, and mutation, then
generating the next population
Satisfy the
Constraint conditions?
Satisfy the
Constraint conditions?
No
No
Yes
Yes
Figure 6.7: Flowchart of H-infinity controller programming with genetic algorithm.
6.2. H-INFINITY CONTROL METHOD 71
Using the optimization algorithm shown in Figure 6.7, the optimized solution can be
obtained and the results show that k
1
D 0:5, k
2
D 2, k
3
D 1, k
4
D 1e 5, k
5
D 0:01, k
6
D 1e
5, and k
7
D 1.
e optimized H-infinity controller can be derived as Equation (6.17) based on the pa-
rameters a large passenger vehicle.
K
c
D
1:665e4s
11
C 1:524e6s
10
C 1:404e8s
9
C 5:811e9s
8
C 2:262e11s
7
C 3:79e12s
6
C6:125e13s
5
C 5:565e14s
4
C 4:502e15s
3
C 2:214e16s
2
C 6:327e16s C 4:511e16
s
11
C 101:7s
10
C 9172s
9
C 4:195e5s
8
C 1:569e7s
7
C 3:149e8s
6
C 3:965e9s
5
C5:021e10s
4
C 2:842e11s
3
C 2:208e12s
2
C 3:754e12s C 1:784e12
:
(6.17)
Furthermore, the optimized solutions and controller varies with different value of param-
eters. e gains can be calculated offline for different parameters and using a look up table they
are selected as the parameters are varied.
To minimize the effect of disturbance on the output, the sensitivity function and the
complementary sensitivity function should be reduced. Also, the system must be robust enough
to provide good performance and stability over the uncertainty. So, the following constraints
should be met:
N
Œ
G
1
.j!/
< N
W
1
1
.j!/
N
Œ
G
3
.j!/
< N
W
1
3
.j!/
: (6.18)
e constraints are based on the singular values which are good measures of the system robust-
ness. Figure 6.8 plots the singular value plot of the system with H-infinity control which shows
the relationship of amplitude and frequency of the sensitivity function, the complementary sen-
sitivity function, the performance weight function W
1
, and the robustness weight function W
3
.
As shown in Figure 6.8, the amplitude of the sensitivity function is small in low frequencies.
e singular value curve of the sensitivity function is lower than that of the performance weight
function, and the singular value curve of the complementary sensitivity function is lower than
that of the robustness weight function. So, the optimized H-infinity controller operates in a
stable environment and provides good control for the vehicle rollover system.
Two typical driving conditions are used to simulate the untripped rollover stability of the
vehicle with the controller, i.e., Fishhook and double-lane change maneuver.
Figure 6.9 compares the new rollover index of the vehicle in Fishhook case with different
control strategies, including without control, traditional proportional integral derivative (PID)
control method, and optimized H-infinity control method. e traditional PID controller is
tuned by the critical proportion method and the gains are set K
P
to 5000, K
I
to 20, and K
D
to
500. As shown in Figure 6.9, the absolute value of the rollover index of the vehicle is over 1 at
2.57 s without extra brake force, so the vehicle rolls over. While the rollover will be prevented by
differential braking force at each wheel with the traditional PID control method or the optimized
H-infinity control method. Furthermore, the maximum absolute value of the rollover index of
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