Bond graphs and genetic algorithms for design and optimization of active dynamic systems

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 02 Issue: 12 | Dec-2013, Available @ http://www.ijret.org 523
BOND-GRAPHS AND GENETIC ALGORITHMS FOR DESIGN AND
OPTIMIZATION OF ACTIVE DYNAMIC SYSTEMS
Amara Elhoucine1
, Faiçal Miled2
, Kamel BenOthman3
1
PHD Student, 2
Assistant Professor, 3
Master of conferences, LARATSI, ENIM (National School of engineering of
Monastir), Tunisia
Houcine.Amara@enim.rnu.tn, f.miled@uha.fr, Kamelbenothman@yahoo.fr
Abstract
The aim of this work is to propose a methodology of sizing in a frame of the conception of mechatronic systems in particular and
active dynamic ones, in general. This methodology will support conceptual design step [1]. We propose a collaborative design
approach based on Bond Graph tool and Genetic algorithms. Mechatronic systems have a passive part and active part. In our
approach we establish a hard interaction between passive and active parts. This interaction will be materialized in taking into account
of control criteria in the evolution (synthesis) phase of passive part. This initiative treats functional aspects, both structural and
behavioral; while respecting interactions between passive and active parts.
In our approach we treat functional, structural and behavioral aspects in order to validate a post-project solution. Optimization needs
Behavioral models which are systematically deducted from bond-graph structural models. Thus, retained bond graph elements which
constitute passive part will be obtained, done by optimization Genetic Algorithms procedure. In this procedure Gramians of
controllability and of observability represent the fitness function. The proposed method will be applied to an automatic transmission
of a scooter and validated by a dynamic simulation.
Keywords: Modeling, dynamic system, Bond graph, Gramians of controllability, Genetic algorithms
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1. INTRODUCTION
The optimization of mechatronic systems [2] makes part of the
design research [3, 4]. Actually, design methodology use
optimization in routine design when we modify some values
of structural components and validate by simulations [5, 6]. In
our approach we use optimization procedure in a creative
design which starts form initial specification.
This procedure will be integrated in a mechatronic design
methodology [7] supported by a functional, structural and
behavioral steps. Functional step transform initial specification
in to functional specification. In the structural step we
synthesis passive part represented by bond graph. In the
behavioral step we use our optimization procedure to validate
our solutions.
The steps of design and dimensional of dynamic systems
assets are based on criteria of controller Such as
controllability, observability and inversibility. In fact, the
controllability we can achieve under optimal conditions of cost
and performance control system. The inversibility is essential
for the implementation of a number of control laws, including
the input-output decoupling and disturbance rejection [8] and
to clarify the necessary design equations.
In this article we propose a synthesis sizing method based on
dynamic and energetic criteria [9],by exploiting the
optimization of Gramians [9,10,11] for the formulation of an
optimization problem and its resolution by the method which
reveals out from artificial intelligence; in this case genetic
algorithms.
The first part of this article deals with an approach which leads
to a functional model and then structural in the direction of
bond-graph (casual model). This model is synthesized by
imposing to the recent structural criteria of controller.
The second part of this article presents a behavioral and
quantitative synthesis in order to size elements bond-graph of
the passive part. In this frame we base ourselves on both
dynamic and energetic criteria while respecting the
specifications. That’s why we are in front of a problem of
optimization of Gramians of controllability and observability,
which presents the objectif function, the optimization of which
is supported by the genetic algorithms.
The third part consists in making a dynamic simulation to
validate our method of sizing of which we determine the laws
of controllability which require the operation of the inverse
bond-graph which is based on the concept of bi-causality [12].

__________________________________________________________________________________________
2. FUNCTIONAL AND STRUCTURAL STUDY
Functional model is obtained after transforming technical
functions into bond-graph sub-models [7]. Interconnection of
sub-models from an a-causal bond graph model which
considered as a functional one [13]
The Bond Graph tool is a unified graphic language [13] for all
the domains of engineering sciences and confirmed as an
approach structured in the modeling and in the simulation of
the multidisciplinary systems. We chose as application an
automatic transmission of a scooter. Its global function is to
transform engine power into back wheel motion [7].
We propose two functional models. An initial model validated
in a precedent study [7] and a candidate model constituted by
an R, C and MSe elements. After structural study, we will
apply our optimization procedure to size R and C elements.
We chose as application an automatic transmission of a
scooter of a scooter. Its functional model is established by
three sub-functional models (Fig.1) (engine, transmission and
load) [7].
We can classify the functional models in two categories:
- Model without element E
- Model with element E
Fig. 1: Model candidate of automatic transmission of scooter
[7]
Jeq: represents the equivalent rotation inertia applied to the
back wheel. It takes into account the mass of the scooter, the
user and the rotational inertia of the wheel;
Jm: moment of inertia of the apparent engine relative to its
output shaft (Kg/m2
);
IC1: link connecting the engine with the transmission;
lc2: the connexion link between the transmission and back
wheel.
R = R(r) = -0.5.Cx..S.R1
3
.r ; element bond graph that
defines the action due to penetration of the whole scooter,
driver and passenger in the air;
Cpr = - mt g sin R1 ; torque generated by the road profile;
mt: total mass of the scooter;
: Slope value;
 : Air density;
R1: ray of the back wheel;
Cx: coefficient of resistance;
S: frontal surface of contact with the air;
r: angular velocity of back wheel.
The bond graph causal model is considered (D.Tanguy et al
2000) as a structural model. We synthesise the structural
model (Rahmani, 1993) by taking into account control criteria
(controllability, observability and invertibility).
In what follows we are going to work on a model with an
element E which will be retained as a model candidate for the
sizing. The identification of the nature of the element E is
made during the structural analysis. Indeed, the element E has
to contribute to the establishment of the structural properties
of control which the designers impose on the passive part.
The structural models are bond-graph causal models. The
causality allows creating explicitly the relations of cause and
effect, and the structure of calculation of the characteristic
equations associated to the models. For this study, we have to
retain a model which has to verify the criteria of controllability
of observability and structural inversibility. The model of our
conception is the one of the transmission, which has to
contribute to the check of these criteria. At a first level, to
have the compulsory structural properties, the element E has to
be a modulated source of effort. Then, we make an U2 control
[7] (Fig.2) associated to this element E.
Fig.2: Model candidate [7].
To improve the performances and possibly to limit the use of
power among the source of controller U2 we propose at a
second level the following model (Fig.3)
Fig. 3: Proposed Model
Engin
Load
Transmission

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2.1 Consideration of Controllability
To put the model in model BGD, it is necessary to make
duality sources. Thus the criterion of controllability is verified
with dualization of U1 which we consider as being a source of
controller (Fig.4).
Fig. 4: Model BGD (derived bond-graph)
2.2 Consideration of the Inversibility
According to the criteria of inversibility of A. Rahmani [12],
we should have the same number of inputs and outputs. The
proposed model is established by two outputs y1 and y2
(respectively the rotation speed of the engine and the wheel),
two inputs which are U1 and U2. By applying the procedures
of consideration of the inversibility, we deduct the shortest
causal paths connecting the inputs and the outputs (Fig 5).
Fig. 5: The causal roads connecting the input with output
2.3. Consideration of the Observability
The first step of the procedure of consideration of the
observability is the stake of the model bond-graph in BGI. At
first, it is necessary to verify the existence of a causal path
between every sensor and a dynamic element (I or C) and
making all the dynamic elements in complete causality during
BGI (Fig.6).
Fig.6: The causal roads every element C or I and the sensors.
The second step consists in putting the model in BGD with the
dualization of the sensors.
Fig.7: Model BGD (derived bond-graph)
To assure the criterion of observability, it is necessary to
dualise the sensor
*
1D f , the relatives to y1.
After verifying structurally our model, we pass to the
following step which consists in making a study behavioral of
our model with the aim of sizing the passive elements of the
model candidate.
3. BEHAVIORAL STUDY
In our days the sizing of the systems is made by a set of
simulation either by modifying the structure by programs
intelligent as the genetic programming [6, 14], or by
modifying the parameters of the structures while limiting itself
to a compromise between the performances wished and the
power of actuators [12] These practices engender a conception
under optimal as far as they do not take into account the
interactions between passive and active elements.
A current method of sizing [6, 15] of the systems is made by a
set of simulation either by modifying the structure by
programs intelligent as the genetic programming. Other used

__________________________________________________________________________________________
methods are based on opposite bond-graph [12] which consists
in imposing the input to find output.
Nevertheless these methods are limited juts to the sizing of
actuators. Furthermore, several criteria are used such as the
energy criteria [9].
We propose an approach for dimensioning passive part based
on genetic algorithms. We also use control an energetic
criteria such as Gramians of controllability and observability.
These criteria allow to minimize input power and to maximize
output power. For our case, we propose an approach of
dimensioning of the passive elements which is based on the
genetic algorithms in order to optimize Gramians of
controllability and observability via the energy of controller.
3.1. Proposed Method
Indeed, in the case of the dynamic systems, the minimal
energy to be supplied to the system to reach a given state is
inversely proportional in Gramian of controllability and that
the energy of output generated by a given initial state is
proportional in Gramian of observability [13].
There are several tools of optimization among which
determinists (method of gradient) [16] and the other heuristics
(Algorithms genetics) [15,17].
With the aim of optimizing these Gramians we are going to
use the algorithm genetic as a tool of optimization.
The proposed method of sizing amounts in the organization
chart following (Fig.8):
Fig.8: The proposed dimensioning procedure
To size elements R and C held in our model of the automatic
transmission of scooter we are going to follow the steps of the
figure:
 Synthesis of the equations of state from the structural
model.
 Calculation of Gramians of controllability and
observability from the model of state.
 Joint Optimization of the Gramians of observability
and of controllability by genetic algorithms.
3.2 Algorithms Genetic
The genetic algorithms are tools of optimization based on a
mechanism of disturbance, a criterion of evaluation and a
criterion of break [17]. These genetic algorithms are based on
techniques derived of the genetics and the natural evolution of
Darwin: crossings, transformations, selection, etc.
The functioning of the genetic algorithm can be presented on
the following organization chart (Fig.9)
Initialize population
evaluate
Gen=0
Gen=Gen+1
mutation
croisement
selection
Affect the ability
E<e
Gen > Max Gen
end
The begin
Fig. 9: Organization chart of functioning of algorithms

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The problem is defined by:
Minimize f (I) under the constraints:
 
min max
0j
i
i i i
h I
x I
x x x



  
With:
f: objectif function.
jh
: The constraints applied to the system.
I: The vector of conception of the independent variables.
ix : The variables of the individual in the terminology of the
genetic algorithms.
minix and maxix : borders lower and upper of the variables of
the individual.
The model bond-graph of the scooter admits the vector of the
variables of conception Independents I = [R, C].
After, we are going to apply our approach to the example of
automatic transmission of scooter.
3.3 Application
In this application, we suggest determining the parameters R
and C of the model bond-graph of the automatic transmission
of scooter (Fi.10):
Fig.10: Model Bond-graph of automatic transmission of
scooter
Model of State
x Ax Bu
y cx
 



(1)
The model of state is obtained by substituting the algebro-
differential equations in the equations of the elementary laws.
Matrix A and B and C are given by the following relations:
2 2
1 1
1 1
* *
1
*
1
0
m eq
m eq eq
m eq
m R m R m
J J c
R R R
A
J J m J c
m
J J
 
 
 
  
  
 
 
 
 
  ;
1
1
0
0 0
m
B
m
 
 
 
 
 
  ;
1
0 0
1
0 0
m
eq
J
C
J
 
 
 
 
 
 
This equation of state is the intermediary between the
structural analysis and the behavioral analysis.
Result of Optimization
The interval of existence of the parameters used by the genetic
algorithms is given by the following table.
Table .1. Intervals of existence of parameters.
Intervalle R C
Xmin 0.1 0
Xmax 0.5 1
The optimal solution, obtained by algorithm genetics (AGs)
during the minimization of the trace of Gramian of
observability and the maximization of the trace of Gramian of
controllability, is presented in the following table.
Table 2: the optimal solution for elements(R and C)
valeurs R C
X 0.4227 0.3132
After sizing elements bond- graphs (R and C) the following
step consists in validating of our methods.
4. SIMULATION AND VALIDATION
In order to validate our method of sizing based on the
optimization by genetic algorithm we are going to make a
comparative study enter both models:
 Model with elements R and C (proposed model: V1).
 Model without elements R and C (initial model: V0).
So we are going to determine the laws of the controls for every
model to clarify the powers put in sets everything. In this
executive we run the inverse bond-graph for the calculation of
the laws of controller.

__________________________________________________________________________________________
4.1 Simulation of the Initial Model (without elements
R and C)
To simulate this model we are going to determine the U1
controls (power to be applied of scooter) and U2 (torque to be
applied to the back wheel of scooter).
The determination of the law of control requires the operation
of the inverse bond-graph which based on the concept of bi-
causality. Thus the inverse bond-graph of the model without R
and C is given by the following figure (Fig.11).
Fig.11: The inverse Bond-graph of initial model V0 (without
R and C)
By substituting the algebro-differential equations in the
equations of the constraints, obtained from the model bond-
graph inverse, the controllers U1 and U2 are given by the
following equation ones.
1 1 2 2
2 2 2
m eq pr
eq pr
U J py mJ py mC mRy
U J py C Ry
   

   (2)
By imposing beforehand y1 (speed of scooter in Km/h) and y2
(rotation speed of the engine in rad/s) two dynamics of the
first one order Fig.12 and Fig.13:
Fig.12.Output y1(The speed
of the scooter (Km/h)
Fig.13.Brought out y2 (the
rotation speed of engine
(rad/s)
According to the equation (2) and by using SIMULINK we
obtain the following simulations:
Fig. 14. The U2 controller
(N.m)
Fig.15. The U1 controller
(W).
U2 is the control which represents the torque to be applied to
the back wheel of the scooter.
U1 represent the power to be applied in phase of starting up of
the scooter with the model without R and C.
4.2. Simulation of the Proposed Model (with elements
R and C)
The model bond-graph inverse with R and C is given by the
following figure (fig.16):
Fig.16.The inverse Bond-graph of proposed model (with R
and C).
By substituting the algebro-differential equations in the
equations of the constraints, the controllers necessary to insure
the equalization of the input and the output speeds, are given
by the equation (3):
 
1 1 2
2 1 1 2 2 2
( * * )
( ) * *
m eq
pr
U pJ y m J p R y
c
U m R my y Jeq p y R C y
p
   

  
       
 
(3)
We impose on the model V1 the same conditions on the
outputs (y1 and y2) as the model V0 and we simulate the
controls U1 and U2 via the inverse model (Fig.17 and Fig.18).

__________________________________________________________________________________________
Fig.17.the U2 controllers for
the model V1 (torque in
N.m).
Fig.18.the U1
controllers for the
model V1 (power in
W).
4.3 Validation
The control U2, which represents the torque to be applied to
the Back wheel of scooter, is lower with the control U2 found
in the model V0 for the same outputs y1 and y2.
By comparing models V1 and V0, we notice that U1 of the
model V1 decreases with regard to that of V0, thus there are
gains in term of power for the model V1 what shows and
confirm the interest of our approach which is based on the
addition of elements R and C, from which their values are
obtained by the optimization of the Gramians of controllability
and of observability by using the genetic algorithm on energy
criteria.
Indeed we deduce that input power of initial model is more
important than proposed model one. Our approach offers the
opportunity to explore more candidate solutions by adding
bond graph elements.
The dimensioning of these elements in based on Gramians of
controllability and of observability optimization. It allows the
improvement of the performances with a systematic method.
CONCLUSIONS
In this paper we proposed a dimensioning procedure of
mechatronic systems. This procedure uses collaborative tools
such as bond graph and genetic algorithms. We integrated our
contribution in a design methodology supported by functional
structural and behavioral steps. We used energetic and control
criteria which favorite interactions between passive and active
parts. These criteria decrease confrontation between
mechanical and control engineers in the preliminary design.
This procedure was validated on our application automatic
transmission of scooter. In this frame we demonstrated that the
proposed model is more performed than the initial model.
Thus we also can explore more candidate solutions and
integrate this procedure in professional software for
mechatronic design systems.
REFERENCES
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[14] Zhun Fan, Jiachuan Wang, Sofiane Achiche, Erik
Goodman, Ronald Rosenberg Structured synthesis of
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[16] F.S. Hover “ Gradient dynamic optimization with
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Publishing, Reading, MA.
BIOGRAPHIES:
Amara Elhoucine, PHD Student in National
School of engineering of Monastir, Monastir
,Tunisia, Houcine.Amara@enim.rnu.tn
Faiçal Miled, National School of engineering of Monastir
Monastir,Tunisia, f .miled@uha.fr
Kamel BenOthman, National School of engineering of
Monastir, Monastir, Tunisia, Kamelbenothman@yahoo.fr

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Bond graphs and genetic algorithms for design and optimization of active dynamic systems