Optimized Genetic Algorithms Reduced Order Model Based RST Roll
Control of Antiroll Bar Dedicated to Semi-active Suspension
SAAD BABESSE
Department of Electrical Engineering
University of Setif1 19000 Setif, ALGERIA
FOUAD INEL
Department of Mechanical Engineering, Automatic Laboratory,
University of Skikda, ALGERIA
Abstract: Working with high-order transfer functions needs a lot of work and leads to major difficulties in
analysis, simulation, and control design. Model reduction studies the large-scale system properties and helps to
reduce these difficulties. In this paper, the genetic algorithms (GA) optimization method is used to calculate the
second reduced order model (ROM) of the original high order model (HOM) of the actuator. Here, the studied
hydraulic actuator is a single input, single output (SISO), and linear time invariant (LTI) system that can be
modeled by an eight-order transfer function with uncontrollable modes. The genetic algorithms are successfully
applied to reduce the original model order using MATLAB software. Thus, the proposed approach is applied to
both the original and suggested reduced order models to check the effectiveness of the reduction method.
Finally, a digital RST roll control based on the robust pole placement is applied for the two models, and
simulations are carried out to show the effectiveness of the control strategy
Key-Words: - Antiroll bar; Optimization; Genetic algorithms; Reduced model; Roll control; RST control.
Received: October 29, 2021. Revised: October 28, 2022. Accepted: November 29, 2022. Published: December 9, 2022.
1 Introduction
The mathematical modelling of most physical
systems results in infinite dimensional models, order
thus the complexity of the systems directed the
researchers towards the reduction of order of these
systems, not only to facilitate the analysis but too to
find a suitable approximation of the high order
systems while keeping the same important
characteristics as closely as possible.
In the literature, several methods are available: some
of them are based on original continued fraction
expansion technique,[1], the disadvantage of these
methods is the failure to retain the stability of the
original systems in the reduced order systems, and
for the improved suggested methods suffer from the
possibility of not having a reduction of order but an
increase in order. Modal-Padé methods, [2], the
major disadvantage of such methods is the difficulty
in deciding the dominant poles of the original
system. However, most of the optimal techniques
follow time-consuming, iterative procedures that
usually result in non-robustly stable models with
poor frequency response resemblance to the original
high order model in some frequency ranges.
Genetic Algorithms (GA) method has proved to be
excellent optimization tools in the past few years.
The use of such search-based optimization
algorithms in model reduction ensures that all the
model reduction objectives are realized with
minimal computational effort, [3].
MATLAB 7.9’s embedded GA toolbox was used to
build the GA model reduction approach based on L1
Norm.
In this paper, the high order system is a hydraulic
actuator dedicated to a heavy vehicle anti-roll bar
mechanism, which is modelling by a seven-order
transfer function, and by adding a lead-lag Pre-filter,
the overall system becomes with eight order transfer
function, stable, but not fully controllable.
The reduced order model obtained has a second
order transfer function, stable, and fully
controllable, these properties are suitable for
feedback controllers.
On the other side, we choose to control the system
by applying a digital polynomial RST controller
characterized by 2 D.O.F of control (one for the
input and the other for the output), thus his
robustness against the disturbances and noises.
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The studied system is a hydraulic actuator dedicated
to semi-active suspension of single unit heavy
vehicles. The suspension consists of two trailing
arms free to rotate about their axis independently of
each other. Each end of the anti-roll bar is attached
to one trailing arm whose position is determined by
the wheels and actuator positions. The actuators are
mounted between the anti-roll bar and the frame of
the trailer. By extending one actuator and retracting
the other, the anti-roll bar is twisted and a torque is
provided to counteract the moment generated by the
lateral acceleration and tilt the vehicle into the turn.
The different transfer functions are given in the
following paragraph, [4], [5]:
First, the transfer function between the displacement
transducer extension and the actuator extension
is given by:
(1)
The transfer function between the actuator extension
and the servo-valve spool displacement is
given by:
(2)
With:
The servo-valve is modeled as a 2nd order
Butterworth filter; it is a low-pass filter with a cut-
off frequency of 15Hz, which is given by:
(3)
Where:
To keep the entire system stable with a range of
given references, we add a lead-lag pre-filter, given
by:
(4)
Where: were chosen to enable a reasonable
choice of the regulator parameters:
The open loop transfer function of the overall
system is given by the following transfer function:
(5)
Where:
All the actuator parameters are listed in Table 2.
Figure 1. Bode plot of the overall hydraulic actuator.
From Fig.1, one can see that this system is stable.
However, it’s not fully controllable, since:
rank(A)=8,rank(ctrb(A,B))=3
For a stabilization purpose, one can use a reduction
method to control the system freely; this step is
well described in the following paragraph.
10-2 10-1 100101102103104
-540
-360
-180
0
180
360
Phase (deg)
Bode Diagram
Frequency (rad/sec)
-500
-400
-300
-200
-100
0
100
System: s1
Gain Margin (dB): 3.77
At frequency (rad/sec): 6.78
Closed Loop Stable? Yes
Magnitude (dB)
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2 Hydraulic Actuator Modeling
3 Reduction Model Based Genetic
Algorithms
Model reduction is a branch of systems and control
theory, which studies properties of dynamical
systems in order to reduce their complexity, while
preserving (to the possible extent) their input-output
behavior, [6]. And one can note that the use of low
order models lead to a simple design and analysis,
computational benefit, simplicity of simulation. On
the other hand, the accuracy
measure of the approximation should in some
concrete way take into consideration the difference
in behavior between the original system and the
reduced order model, so that, different norms are
used for the formulation of the model reduction
problem : H∞, H2, L1-Norm and hybrid norm, [7].
In this section, we adopt to use the L1 Norm
Model Reduction approach to reduce the 8th order
hydraulic actuator of eq. (5) into a 2nd order
reduced model, and GA’s approach will be used to
perform the model reduction.
3.1 Genetic algorithm theory
Genetic algorithm is a robust optimization technique
based on natural selection. The basic goal of GAs is
to optimize functions called fitness functions. GA-
based angle approaches differ from conventional
problem-solving methods in several ways, [8].
First, GAs work with a coding of the parameter set
rather than the parameters themselves. Second, GAs
search from a population of points rather than a
single point. Third, GAs use payoff (objective
function) information, not other auxiliary
knowledge. Finally, GAs use probabilistic transition
rules, not deterministic rules. These properties make
GAs robust, powerful, and data-independent. Its
basis in natural selection allows a GA to employ a
"survival of the fittest" strategy when searching for
optima. The use of a population of points helps the
GA avoid converging to false peaks (local optima)
in the search space. The following sections describe
GAs in more detail. Most of the information
presented here is based on:
• Chromosome: A simple GA requires the
parameter set of the optimization problem to be
encoded as a string (binary, real, etc.). These strings
are known as chromosomes. They are manipulated
by the GA in an attempt to obtain the string that
represents the optimal solution to the problem.
• Genes: A character or symbol in a GA
chromosome is called a gene. Genes are the basic
building blocks of the solution and represent the
properties which make one solution different from
the other.
• Allele: The value of a gene in a GA is called
an
allele, such as for eye color, the different possible
'settings' (e.g., blue, brown, hazel etc.) are called
alleles.
• Selection: A genetic operator used to select
individuals for reproduction.
• Crossover: A key operator used in the GA
to create new individuals by combining portions of
two parent strings.
• Crossover probability: Probability of
performing crossover operation, denoted by pc, i.e.,
the ratio of number of offspring produced in each
generation to the population size. This value of pc is
chosen generally in the range of 0.7 to 0.9.
• Mutation: An incremental change to a
member of the GA population.
• Mutation Probability: The probability of
mutating each gene in a GA chromosome, denoted
by pm. This value is chosen generally in the range
of 0.01 to 0.03.
3.2 L1 Norm Model Reduction Approach
Starting in 1977, El-Attar and Vidyasagar presented
new procedures for model reduction based on
interpreting the system impulse response (or transfer
function) as an input-output map, [8], [9].
The L1 norm of the system with transfer function
G(s) and impulse response g(t) on the other hand is
defined as, [6]:
(6)
on the other hand, the L1 norm is defined as:
(7)
Where: e(t) is the impulse response difference
between the original system and the reduced system:
(8)
This last equation was implemented in MATLAB
using trapezoidal numerical integration which
computes an approximate integral of the error
between the impulse response of the original system
and the impulse response of the reduced order
system with respect to time.
3.3 Reduction Model Using GAs
First, the settings of the GA used to perform
the reduction for the hydraulic actuator were as
in Table 1:
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Table 1. GA’s settings
Population size
150
Encoding Criteria
Double Vector
Crossover Fraction
0.8
Mutation Fraction
0.02
Elite Count
10
Stall Generations Limit
1500
Stall Time Limit
∞
Selection Function
Roulette Wheel
Crossover Function
Crossover Scattered
Mutation Function
Mutation Gaussian
Maximum Number of
Generations
1500
We use the polynomials of the high order
original model of eq.(5) as input data to the
optimization algorithm. The reduced order model is
obtained, after 1500 iterations, as:
(9)
The steady state error is 0.050
The L1-Norm of the reduced model is 4.7147225
The impulse response, the step response, and
Bode plot are shown in Fig.2, Fig.3 and Fig.4
respectively:
Figure 2. Impulse responses of original and reduced
order models.
Figure3. Step responses of original and reduced
order models.
0 2 4 6 8 10 12 14 16 18
-2
0
2
4
6
8
10
12
14
Impulse Response
Time (sec)
Amplitude
Orig_Sys
Red_Sys
0 2 4 6 8 10 12 14 16 18
0
10
20
30
40
50
60
Step Response
Time (sec)
Amplitude
Orig_Sys
Red_Sys
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Figure 4. Bode plot of original and reduced order
models.
Fig. 2 and Fig 3 show the good equivalence
between the high order model and the optimized low
order model, because of identical step responses,
and quasi-identical impulse responses with minor
errors. The impulse response of the original model
presents naturally oscillations that can make some
control difficulties.
In Fig.4, the frequency responses of the
reduced order models highly resemble those of the
original systems at low frequencies. The magnitude
of the reduced order model shows some error at
high frequencies due to the six missing states in the
reduced order model. However, since most real-time
physical systems operate at low frequencies, this
error at high frequencies tends to be acceptable and
can be ignored.
4 Digital RST Control of the Actuator
4.1 Models sampling
In the automatic control, the choice of the
sampling frequency is based on the following
formula, [10]:
Where:
is the closed loop pass-band of the
system.
So, we choose:
The discrete-time original model is obtained by
the discretization of the continuous-time model
(Eq.5) with Zero Order Holder and the sampling
time :
With:
.
The discrete-time reduced order model is:
The reduced model obtained has two stable poles
at 0.9593 and 0.9160 because their modules are less
than the unity, and it has a non-stable zero at:
3.7632.
4.2 RST Digital Control
The RST digital controllers have two degrees of
freedom (one for tracking, the other for regulation).
The design of such controller is done in two steps,
[11], [12]:
1) Calculation of the polynomials R and S
(regulation)
2) Calculation of T (tracking).
The general scheme of RST control is shown in
Fig.5:
-400
-300
-200
-100
0
100
Magnitude (dB)
10-2 10-1 100101102103104105
-540
-360
-180
0
180
360
Phase (deg)
Bode Diagram
Frequency (rad/sec)
Orig_Sys
Red_Sys
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Figure 5. Structure of RST controller
In this scheme:
: is the time delay of the plant (in our model
d=0).
: is the tracking reference model.
Before calculating the three polynomials, we
impose some specifications in time continuous to
satisfy both tracking dynamics and regulation
dynamics, and then in the next step, we do the
discretization.
In this paper, we adopt the following
specifications:
- Second order damped response in 15 samples
(i.e 1,5 s).
- Desired dominant poles in continuous time:
, that gives in discrete time:
.
We choose another dominant pole at 0.5, we find:
- Imposing unit static gain equal to 1. This is
done by choosing
Then, the tracking reference model is:
1) We can choose the regulation dynamics by
imposing the poles of the closed loop
polynomial , here:
Then, we add a pre-specified fixed part to the
polynomial to impose a
null static error.
The different polynomial orders, to obtain
a feasible controller, are:
2)
The resolution of the Equation of Bezzout
gives:
And, since the plant has a non-stable zero, the
polynomial is given as:
, so:
5 Simulations Results and Discussions
First, we define the main input of the plant
(hydraulic actuator) as the desired roll angle as
mentioned in, [13],[14], after that, we apply the
designed RST controller to the reduced order
model(ROM), and next, to the original high order
model (HOM) respectively.
0 2 4 6 8 10 12 14 16 18 20
-1
0
1
2
3
4
5
6
Response of the original and reduced models
time(s)
(a)
Response of HOM and ROM (°)
Desired roll angle
High order model response
Reduced order Model response
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0 2 4 6 8 10 12 14 16 18 20
-5
-4
-3
-2
-1
0
1
time(s)
(b)
Tracking error (°)
Tracking Error of original model
0 2 4 6 8 10 12 14 16 18 20
-6
-5
-4
-3
-2
-1
0
time(s)
(c)
Tracking error (°)
Tracking Error of reduced order model
0 2 4 6 8 10 12 14 16 18 20
-0.5
0
0.5
1
1.5
2
2.5
time(s)
(d)
command u (°)
RST Control input
0 2 4 6 8 10 12 14 16 18 20
-14
-12
-10
-8
-6
-4
-2
0x 104
time(s)
(e)
Roll moment (N.m)
Generated roll moment
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Figure 6. RST control results without external
disturbance.
(a): Response of the two models –ROM and
HOM-
(b): Tracking error of the original model, (c):
Tracking error of the reduced order model, (d):
command input (control input), (e): Resulting roll
moment, (f): Actuator extension
Figure 6(a) compares the responses of the original
high order model of the hydraulic actuator to its
optimized second order model. There is good
approximation between the two models and the RST
controller satisfies the underlined specifications
(settling time and steady state error). One can see
few fluctuations in the original model response due
to the missing poles in the establishment of the
reduced model.
The tracking errors are shown in figure 6(b) and
6(c): the steady state errors are being nullified after
a few seconds. In Fig 6(d), the roll angle demand in
degrees is illustrated. The steady state value is 0.1°.
The resulting roll moment of the actuators, left side
and right-side actuators, is given in Fig 6(e). The
steady state value is towards 105 N.m (this value
must be less than the maximum tolerated moment of
the actuator). The sign (-) indicates that the roll
moment and the roll angle are in opposite directions.
Finally, the actuator extension is shown in Fig 6(f),
where the steady state value is 2,2 cm.
Now, to check the robustness of the investigated
RST controller, we introduce an external output
disturbance, at t=5s, where its value is 10% of the
plant input.
0 2 4 6 8 10 12 14 16 18 20
-1
0
1
2
3
4
5
6
Response of the original and reduced models
time(s)
(a)
Response of HOM and ROM (°)
Desired roll angle
High order model response
Reduced order Model response
0 2 4 6 8 10 12 14 16 18 20
0
0.005
0.01
0.015
0.02
0.025
time(s)
(f)
Actuator extension ya (m)
Extension of the hydraulic actuator
0 2 4 6 8 10 12 14 16 18 20
-5
-4
-3
-2
-1
0
1
time(s)
(b)
Tracking error (°)
Tracking Error of original model
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Figure7. RST control results with external output
disturbance.
(a): Response of the two models –ROM and HOM-
(b): Tracking error of the original model, (c):
Tracking error of the reduced order model, (d):
command input (control input), (e): Resulting roll
moment, (f): Actuator extension
Fig 7(a) shows fast disturbance rejection of the
reduced order model (in 0.6s towards) relatively to
the high order model (in 2.5 s towards) this is due to
the non-stable poles of the HOM transfer function.
This observation is well seen in Fig 7(b) and Fig
7(c).
In Fig 7(d), we see that the controller compensates
the disturbance effect in the control input
(command).
The Roll moment generated is illustrated in Fig
7(e): the peak value becomes 14.104N.m, and the
steady state value is towards 122000 N.m .
At last, the actuator extension is shown in Fig
7(f), its maximum becomes 2,75 cm, and its steady
state value is towards 2,55 cm.
0 2 4 6 8 10 12 14 16 18 20
-6
-5
-4
-3
-2
-1
0
time(s)
(c)
Tracking error (°)
Tracking Error of reduced order model
0 2 4 6 8 10 12 14 16 18 20
-0.5
0
0.5
1
1.5
2
2.5
time(s)
(d)
command u (°)
RST Control input
0 2 4 6 8 10 12 14 16 18 20
-15
-10
-5
0x 104
time(s)
(e)
Roll moment (N.m)
Generated roll moment
0 2 4 6 8 10 12 14 16 18 20
0
0.005
0.01
0.015
0.02
0.025
0.03
time(s)
(f)
Actuator extension ya (m)
Extension of the hydraulic actuator
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Table 2. Hydraulic actuator parameters,[4].
6 Conclusion
In this paper, two objectives were underlined. The
first is that using the L1 norm, genetic algorithms
can be used to find optimal reduced models for a
complex high-order SISO model.Moreover, the
second is the investigation of the RST controller to
control the roll angle for the original model and the
reduced order model of a hydraulic actuator.
From the different results, it has been clearly seen
that the reduced model using genetic algorithms
keeps the main properties of its original model,
namely in low frequencies and steady state range.
Second, the RST controller provides good tracking
and moderate robustness in the face of external
disturbances.
Further, the given RST controller has a discrete
character and can be easily implemented on the real
process (the experimental truck).
References:
[1] Sinha N.K, Pille W, “A new method for order
reduction of dynamic systems”, International
Journal of Control 14(1), 1971 pp. 111-118.
[2] Marshall, S.A, “An approximate method for
reducing the order of a linear system”,
International Journal of Control, Vol. 10, 1966,
pp.642–643.
[3] Hsu, C.C. and Yu, C.H. Model Reduction of
Uncertain Interval Systems Using Genetic
Algorithms. SICE Annual Conference 2004, 1, 264-
267, 2004.
[4] Arnaud J.P. Miége, “Development of Active
Anti-Roll Control for Heavy Vehicles”, First
Year Report Submitted to the University of
Cambridge, 2000.
[5] S. Babesse, D. Ameddah, “Neuronal active anti-
roll control of a single unit heavy vehicle
associated with RST control of the hydraulic
actuator,” International Journal of heavy vehicle
Systems, IJHVS,Vol. 22, Issue. 3, 2015.
[6] Massachusetts Institute of Technology What is
Model Order Reduction? Retrieved : March 21,
2009,from:http://scripts.mit.edu/~mor/wiki/inde
x.php? title=What_is_Model_order_Reduction.
[7] D. E. Goldberg, Genetic Algorithms is Search,
Optimization, and Machine learning, Reading
MA: Addison-Wesley, 1989.
[8] Bettayeb, M. Approximation of Linear Systems:
New Approaches based on Singular Value
Decomposition. Ph.D. Thesis, University of
Southern California, Los Angeles, 1981.
[9] El-Attar, R.A. and Vidyasagar, M,” Order
Reduction by l1 and l∞-Norm Minimization”.,
IEEE Transaction on Automatic Control, AC-
23(4), 1978, 731-734.
[10] Doyle, J., Francis, B. and Tannenbaum, A,
Feedback Control Theory, Macmillan
Publishing Co, 1990.
Parameter
Description
Valu
e
Effective bulk modulus of hydraulic oil
6.89.
106
Area of piston of hydraulic actuator
0.012
3
Damping force coefficient
1000
0
Distance to the actuator from the
centerline of suspension
0.215
Distance to damper from centerline of
suspension
0.23
Distance to air spring from centerline of
suspension
0.535
Half-track width
0.93
Moment of inertia of anti-roll bar about
roll center of suspension
6.10
Moment of inertia of sprung mass about
roll center of suspension
9500
Air spring stiffness
2.37.
105
Roll stiffness of anti-roll bar
1.02.
106
Spring stiffness in Merritt’s valve-piston
model
1.103
3.107
Servo-valve total flow pressure
coefficient
4.2.1
0-11
Servo-valve flow gain coefficient
2.5
Distance to displacement transducer
from centerline of suspension
0.552
Mass of load in Merritt’s valve-piston
model
65.98
16
damping coefficient in Merritt’s valve-
piston model
539.6
201
Supply pressure of hydraulic system
210
Volume of ‘trapped’ oil at high pressure
in the hydraulic system
0.001
4
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[11] Carlos A.S. and Armando B.C, Principles and
practice of automatic process control. John
Wiley &Sons, 1997.
[12] Landau I. D, Commande des systèmes,
conception, identification, et mise en œuvre.
Lavoisiers Paris – France, 2002.
[13] Sampson D.J.M., McKevitt P.G., Cebon D,
“The development of an active roll control
system for heavy vehicles”. Proc. 16th IAVSD
Symposium on the Dynamics of Vehicles on
Roads and Tracks, Pretoria, South Africa, 1999.
[14] Babesse S, Ameddah D and Inel F,
“Comparison between RST and PID Controllers
Performance of a Reduced Order Model and the
Original Model of a Hydraulic Actuator
dedicated to a Semi-active Suspension”, World
Journal of Engineering, 2016.
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