the 5tthh international conference on...

I

TTHHEE 55TTHH IINNTTEERRNNAATTIIOONNAALL CCOONNFFEERREENNCCEE OONN IINNTTEEGGRRAATTEEDD MMOODDEELLIINNGG AANNDD AANNAALLYYSSIISS IINN

AAPPPPLLIIEEDD CCOONNTTRROOLL AANNDD AAUUTTOOMMAATTIIOONN SEPTEMBER 12-14 2011

ROME, ITALY

EDITED BY

AGOSTINO BRUZZONE GENEVIÈVE DAUPHIN-TANGUY

SERGIO JUNCO MIQUEL ANGEL PIERA

PRINTED IN RENDE (CS), ITALY, SEPTEMBER 2011

ISBN 978-88-903724-7-6

II

© 2011 DIPTEM UNIVERSITÀ DI GENOVA

RESPONSIBILITY FOR THE ACCURACY OF ALL STATEMENTS IN EACH PAPER RESTS SOLELY WITH THE AUTHOR(S). STATEMENTS ARE NOT NECESSARILY REPRESENTATIVE OF NOR ENDORSED BY THE DIPTEM, UNIVERSITY OF GENOA. PERMISSION IS GRANTED TO PHOTOCOPY PORTIONS OF THE PUBLICATION FOR PERSONAL USE AND FOR THE USE OF STUDENTS PROVIDING CREDIT IS GIVEN TO THE CONFERENCES AND PUBLICATION. PERMISSION DOES NOT EXTEND TO OTHER TYPES OF REPRODUCTION NOR TO COPYING FOR INCORPORATION INTO COMMERCIAL ADVERTISING NOR FOR ANY OTHER PROFIT – MAKING PURPOSE. OTHER PUBLICATIONS ARE ENCOURAGED TO INCLUDE 300 TO 500 WORD ABSTRACTS OR EXCERPTS FROM ANY PAPER CONTAINED IN THIS BOOK, PROVIDED CREDITS ARE GIVEN TO THE AUTHOR(S) AND THE WORKSHOP. FOR PERMISSION TO PUBLISH A COMPLETE PAPER WRITE TO: DIPTEM UNIVERSITY OF GENOA, DIRECTOR, VIA OPERA PIA 15, 16145 GENOVA, ITALY. ADDITIONAL COPIES OF THE PROCEEDINGS OF THE IMAACA ARE AVAILABLE FROM DIPTEM UNIVERSITY OF GENOA, DIRECTOR, VIA OPERA PIA 15, 16145 GENOVA, ITALY.

III

THE 5TH INTERNATIONAL CONFERENCE ON INTEGRATED MODELING AND ANALYSIS IN APPLIED

CONTROL AND AUTOMATION SEPTEMBER 12-14 2011

ROME, ITALY

ORGANIZED BY

DIPTEM – UNIVERSITY OF GENOA

LIOPHANT SIMULATION

SIMULATION TEAM

IMCS – INTERNATIONAL MEDITERRANEAN & LATIN AMERICAN COUNCIL OF SIMULATION

MECHANICAL DEPARTMENT, UNIVERSITY OF CALABRIA

MSC-LES, MODELING & SIMULATION CENTER, LABORATORY OF ENTERPRISE SOLUTIONS

MODELING AND SIMULATION CENTER OF EXCELLENCE (MSCOE)

MISS LATVIAN CENTER – RIGA TECHNICAL UNIVERSITY

LOGISIM

LSIS – LABORATOIRE DES SCIENCES DE L’INFORMATION ET DES SYSTEMES

MISS – UNIVERSITY OF PERUGIA

MISS – BRASILIAN CENTER, LAMCE-COPPE-UFRJ

MISS - MCLEOD INSTITUTE OF SIMULATION SCIENCES

M&SNET - MCLEOD MODELING AND SIMULATION NETWORK

IV

LATVIAN SIMULATION SOCIETY

ECOLE SUPERIEURE D'INGENIERIE EN SCIENCES APPLIQUEES

FACULTAD DE CIENCIAS EXACTAS. INGEGNERIA Y AGRIMENSURA

UNIVERSITY OF LA LAGUNA

CIFASIS: CONICET-UNR-UPCAM

INSTICC - INSTITUTE FOR SYSTEMS AND TECHNOLOGIES OF INFORMATION, CONTROL AND COMMUNICATION

I3M 2011 INDUSTRIAL SPONSORS

PRESAGIS

CAE

CAL-TEK

MAST

AEGIS TECHNOLOGIES

I3M 2011 MEDIA PARTNERS

MILITARY SIMULATION & TRAINING MAGAZINE

EURO MERCI

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EDITORS AGOSTINO BRUZZONE MISS-DIPTEM, UNIVERSITY OF GENOA, ITALY [email protected] GENEVIÈVE DAUPHIN-TANGUY ECOLE CENTRALE DE LILLE, FRANCE [email protected] SERGIO JUNCO UNIVERSIDAD NACIONAL DE ROSARIO, ARGENTINA [email protected] MIQUEL ANGEL PIERA AUTONOMOUS UNIVERSITY OF BARCELONA, SPAIN [email protected]

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THE INTERNATIONAL MEDITERRANEAN AND LATIN AMERICAN MODELING MULTICONFERENCE, I3M 2011

GENERAL CO-CHAIRS

AGOSTINO BRUZZONE, MISS DIPTEM, UNIVERSITY OF GENOA, ITALY MIQUEL ANGEL PIERA, AUTONOMOUS UNIVERSITY OF BARCELONA, SPAIN

PROGRAM CHAIR

FRANCESCO LONGO, MSC-LES, UNIVERSITY OF CALABRIA, ITALY THE 5TH INTERNATIONAL CONFERENCE ON INTEGRATED MODELING AND

ANALYSIS IN APPLIED CONTROL AND AUTOMATION, IMAACA 2011 CONFERENCE CHAIRS

SERGIO JUNCO, UNIVERSIDAD NACIONAL DE ROSARIO, ARGENTINA GENEVIÈVE DAUPHIN-TANGUY, ECOLE CENTRALE DE LILLE, FRANCE

VII

IMAACA 2011 INTERNATIONAL PROGRAM COMMITTEE BELKACEM OULD BOUAMAMA, USTL, FRANCE JEAN-YVES DIEULOT, POLYTECH’ LILLE, FRANCE ALEJANDRO DONAIRE, THE UNIVERSITY OF NEWCASTLE, AUSTRALIA TULGA ERSAL, UNIVERSITY OF MICHIGAN, USA FRANCESCO LONGO, UNICAL, ITALY PIETER MOSTERMAN, THE MATHWORKS, USA MUSTAPHA OULADSINE, LSIS, UPCAM, FRANCE RACHID OUTBIB, LSIS, UNIV. D’AIX-MARSEILLE III, FRANCE P. M. PATHAK, INDIAN INSTITUTE OF TECHNOLOGY, ROORKEE, INDIA RICARDO PÉREZ, PONT. UNIVERSIDAD CATÓLICA, CHILE XAVIER ROBOAM, INPT-LAPLACE, FRANCE BRUNO SICILIANO, UNIVERSITÁ DI NAPOLI, ITALY KAZUHIRO TANAKA, KYUSHU INSTITUTE OF TECHNOLOGY, FUKUOKA, JAPAN DANIEL VIASSOLO, VESTAS, HOUSTON, TX, USA COSTAS TZAFESTAS, NTUA, GREECE ANAND VAZ, NIT JALANDHAR, INDIA JORGE BALIÑO, UNIVERSITY OF SÃO PAULO, BRAZIL MARTA BASUALDO, CIFASIS-CONICET, ARGENTINA WOLFGANG BORUTZKY, BRS-UNIVERSITY OF APPLIED SCIENCES, SANKT AUGUSTIN, GERMANY LOUCAS S. LOUCA, UNIVERSITY OF CYPRUS, SCHOOL OF ENGINEERING, CYPRUS DAVID ZUMOFFEN, CIFASIS, CONICET, ARGENTINA

TRACKS AND WORKSHOP CHAIRS

BOND GRAPH MODELING, SIMULATION AND APPLICATIONS CHAIR: P.M. PATHAK AND ANAND VAZ, INDIAN INSTITUTE OF TECHNOLOGY (INDIA.) MODELLING AND CONTROL OF MECHANICAL SYSTEMS CHAIR: JEAN-YVES DIEULOT, LAGIS, LABORATOY OF AUTOMATIC CONTROL (FRANCE)

VIII

CONFERENCE CHAIRS’ MESSAGE WELCOME TO IMAACA 2011! This year the wonderful framework of the IMAACA conference is the Eternal City: we

are honored to welcome all the IMAACA authors as well as all the I3M 2011

participants to Rome.

Based on past experiences (Fes, Morocco; Buenos Aires, Argentina; Marseille, France;

and Genoa Italy) we are even more positive about the success of IMAACA 2011. The

very high scientific quality of selected papers and the invaluable work carried out (over

roughly one year) by the International Program Committee will bring experts together

for the purpose of presenting and discussing all type of application relevant control and

automation problems emphasizing the role of model analysis as integral part of the

complete design of the automated system. In fact an accurate review of the selected

papers reveals that most recent and innovative concepts, methods, techniques, and

tools, conceived in order to support an integrative interplay of modeling, identification,

simulation, system analysis and control theory in all the stages of system design, are

considered at IMAACA 2011.

In addition IMAACA will be co-located with the I3M 2011 multi-conference that includes

EMSS, MAS, HMS and DHSS therefore providing the authors with the opportunity to

attend multiple sessions and have a direct look to the most advanced and innovative

international research in the Modeling & Simulation area.

Finally, we would like to wish to all the attendees a fruitful and enjoyable IMAACA 2011

in Rome.

Sergio Junco, Universidad Nacional de Rosario, Argentina

Geneviève Dauphin-Tanguy, Ecole Centrale de Lille, France

IX

ACKNOWLEDGEMENTS

The IMAACA 2011 International Program Committee (IPC) has selected the papers for the Conference among many submissions; therefore, based on this effort, a very successful event is expected. The IMAACA 2011 IPC would like to thank all the authors as well as the reviewers for their invaluable work. A special thank goes to all the organizations, institutions and societies that have supported and technically sponsored the event. LOCAL ORGANIZATION COMMITTEE AGOSTINO G. BRUZZONE, MISS-DIPTEM, UNIVERSITY OF GENOA, ITALY ENRICO BOCCA, SIMULATION TEAM, ITALY FRANCESCO LONGO, MSC-LES, UNIVERSITY OF CALABRIA, ITALY FRANCESCA MADEO, UNIVERSITY OF GENOA, ITALY MARINA MASSEI, LIOPHANT SIMULATION, ITALY LETIZIA NICOLETTI, CAL-TEK, ITALY FEDERICO TARONE, SIMULATION TEAM, ITALY ALBERTO TREMORI, SIMULATION TEAM, ITALY

This International Conference is part of the I3M Multiconference: the Congress leading Simulation around the World and Along the Years

I3MI3M Simulation around the WorldSimulation around the World& along the Years& along the Years

I3M2010, I3M2010, FesFes, Morocco, Morocco

www.liophant.org/i3mwww.liophant.org/i3m



I3M2011, Rome, ItalyI3M2011, Rome, Italy


I3M2012, I3M2012, WienWien, Austria, Austria



I3M2007, I3M2007, BergeggiBergeggi, Italy, Italy



I3M2008, I3M2008, CalabriaCalabria, Italy, Italy



I3M2009, Tenerife, SpainI3M2009, Tenerife, Spain



I3M2004, I3M2004, LiguriaLiguria, Italy, Italy



I3M2005, Marseille, FranceI3M2005, Marseille, France



I3M2006, Barcelona, SpainI3M2006, Barcelona, Spain



HMS2002, HMS2002, MarseilleMarseille, France, France



HMS2003, Riga, LatviaHMS2003, Riga, Latvia



HMS2004, Rio de Janeiro, BrazilHMS2004, Rio de Janeiro, Brazil


EMSSEMSS Simulation around the WorldSimulation around the World& along the Years& along the Years

ESS1993, Delft, The NetherlandsESS1993, Delft, The Netherlands



ESS1994, Istanbul, TurkeyESS1994, Istanbul, Turkey



ESS1996, Genoa, ItalyESS1996, Genoa, Italy


X

XI

Index

Analysis of Controlled Semi-Markov Queueing Models Victor Kashtanov, Elizaveta Kondrashova

1

Modeling of Flexural Vibration of Rotating Shaft with in-span Discrete External Damping Vikas Rastogi, Amalendu Mukherjee, Anirban Dasgupta

8

A Recursive Nonlinear System Identification Method Based on Binary Measurements Laurent Bourgois, Jérôme Juillard

15

Reducing Computation Time in Optimization Procedure for Fixture Layout based on SwarmItFIX Xiong Li, Matteo Zoppi, Rezia Molfino

21

Robust Trajectory Tracking Problem: Cheap Control Approach Vladimir Turetsky, Valery Glizer

26

Integrated Damping Parameters and Servo Controller Design for Optimal H2 Performance in Hard Disk Drives Tingting Gao, Weijie Sun, Chunling Du, Lihua Xie

35

Analytical Redundancy Relations from Bond Graphs of Hybrid System Models Wolfgang Borutzky

43

A model for dynamic feed-forward control of a semi-active damper Andreas Unger, Enrico Pellegrini, Kay-Uwe Henning, Boris Lohmann

50

Autofocusing Slow-moving Objects Sensed by an Airborne Synthetic Aperture Radar SAR José Luis Pasciaroni, Juan Edmundo Cousseau, Nélida Beatriz Gálvez

56

A new Bond-Graph Model of Photovoltaic Cells Stephan Astier, Guillaume Fontes, Xavier Roboam, Christophe Turpin, Frédéric Gailly, Laurianne Menard

62

Modelling and Control of an Effort Feedback Actuator in Helicopter Flight Control Using Energetic Macroscopic Representation Mikaël Martin, Julien Gomand, François Malburet, Pierre-Jean Barre

68

Passive Fault Tolerant Control: A Bond Graph Approach Matías Nacusse, Sergio Junco

75

On Algebraic Approach for MSD Parametric Estimation Marouene Oueslati, Stéphane Thiery, Olivier Gibaru, Richard Bearee, George Moraru

83

Identification by Hybrid Algorithm Application to the Identification of Dynamics Parameters of Synchronous Machine Jean Claude A. Rakotoarisoa, Jean N. Razafinjaka,

92

Study of Coupled Dynamics Between Body and Legs of a Four Legged Walking Robot Krishnan L. V., Pathak P. M., Jain S. C.

98

A Final Marking Planning Method for Join Free Timed Continuous Petri nets Hanife Apaydin Özkan, Aydin Aybar

108

Multi-scale extension of discriminant PLS for fault detection and diagnosis Mohammad Sadegh Emami Roodbali, Mehdi Shahbazian

114

Modeling of aerodynamic flutter on a NACA 4412 airfoil wind blade Drishtysingh Ramdenee, H. Ibrahim , N.Barka , A.Ilinca

118

XII

Control Allocation Strategies for an Overactuated Electric Vehicle Jean-Yves Dieulot

125

Modeling of Wheel/Track Interaction with Wheel Defects and Diagnosis Methods: an Overview Rasul Fesharakifard, Antoine Dequidt, Florent Brunel, Yannick Desplanques, Olivier Coste, Patrick Joyez

131

Diagnosing photovoltaic systems: an iterative process Abed Alrahim Yassine, Stephane Ploix, Hussein Joumaa

141

Application of Nonlinear Model Predictive Control Based on Different Models to Batch Polymerization Reactor Sevil Cetinkaya, Duygu Anakli, Zehra Zeybek, Hale Hapoglu, Mustafa Alpbaz

147

Active Vibration Reducing of a One-link Manipulator Using the State Feedback Decoupling and the First Order Sliding Mode Control Mohammed Bakhti, Badr Bououlid Idrisssi

155

Development of a Tranfer-Case Control system for Rear Wheel Based 4WD Vehicles Woosung Park, Hyeongjin Ham, Kanghee Won, Hyeongcheol Lee

165

Process Monitoring Based on Measurement Space Decomposition José Luis Godoy, Jorge Ruben Vega, Jacinto Luis Marchetti

173

Task space control of grasped object in the case of multi-robots cooperating system A Khadraoui, C. Mahfoudi, A. Zaatri, K Djouani

179

Design of a trajectory generator for a hexapod walking robot C. Mahfoudi, M. Bouaziz, A Boulahia, K. Djouani, S. Rechak

192

Using Linear Programming for the Optimal Control of a Cart-Pendulum System Luiz Puglia, Fabrizio Leonardi, Marko Ackermann

200

Algebraic characterization of the invariant zeros structure of LTV bond graph models Dapeng Yang, Christophe Sueur, Geneviève Dauphin-Tanguy

206

Component Fault Detection and Isolation Comparison between Bond Graph and Algebraic Approach Samir Benmoussa, Belkacem Ould Bouamama, Rochdi Merzouki

213

Fault Detection and Isolation in Presence of Input and Output Uncertainties Using Bond Graph Approach Youcef Touati, Rochdi Merzouki, Belkacem Ould Bouamama

221

Monitoring Steady-State Gains in Large Process Systems Germán A. Bustos, José L. Godoy, Alejandro H. González, Jacinto L. Marchetti

228

Stability Analysis of 1 DOF Haptic Interface: Time Delay and Vibration Modes Effects Quoc Viet Dang, Antoine Dequidt, Laurent Vermeiren, Michel Dambrine

234

Bond Graph Modeling of Reverse-Flow Monolith Reactor for Air Pollutant Abatement Antoine Riaud, Sebastien Paul, Genevieve Dauphin-Tanguy

241

Modeling Multiphase Flow Dynamics in Offshore Petroleum Production Systems Rafael Horschutz Nemoto, Jorge Luis Baliño

250

Bond Graph Model Based for Robust Fault Diagnosis Rafika El Harabi, Belkacem Ould Bouamama, Mohamed Naceur Abdelkrim, Mohamed Ben Gayed

259

XIII

Geometric Properties of Kernel Partial Least Squares for Non-Linear Process Monitoring José L. Godoy, Germán Bustos, Alejandro H. Gonzalez, Jacinto L. Marchetti

267

Diodes and an alternative Procedure to derive equations from Bond Graph models by exploting a Linear Graph approach. Noe Villa-Villaseñor, Gilberto González-Avalos, Jesús Rico-Melgoza

273

Bond Graph Modeling of a Hydraulic Vibration System: Simulation and Control Farid Arvani, Geoff Rideout, Nick Krouglicof, Steve Butt

281

New Concept of Modeling to Predict Temperature in Oil-Hydraulic Cylinder Chamber Considering Internal Flow Kazuhiro Tanaka, Kouki Tomioka, Masaki Fuchiwaki, Katsuya Suzuki

287

Modeling and Simulation of Power Hacksaw Machine Using Bond Graph Anand Vaz, Aman Maini

293

Optimization of Actuating System for Flapping Robot Design T. Imura, M. Fuchiwaki, K. Tanaka

301

A New Approach for Industrial Diagnosis by Neuro-Fuzzy systems: Application to Manufacturing System Rafik Mahdaoui, Hayet Mouss, Djamel Mouss

307

Authors’ Index 312

Diodes and an alternative Procedure to deriveequations from Bond Graph models by exploting a

Linear Graph approach.Noé Villa-Villaseñor, Gilberto González-Ávalos and Jesús Rico-Melgoza

DEP-FIEUniversidad Michoacanade San Nicolás Hidalgo

Morelia, Michoacán, Méxicocontact Email: [email protected]

Keywords- Algebraic equations, State Space Equations,Bond Graph Modelling, Linear Graph Modelling, Tree Theory,Causal Paths, Shockley Model of the diode.Abstract�This paper deal with methods to get equations from

a Bond Graph (BG) Model. Mixed Techniques to get equationsfrom both Linear Graph (LG) and BG Models are employed. Acriterion that allows to identify the tree and the co-tree in a BGmodel is reviewed. Also, the advantages of causal paths inherentto BG are exploited. The equivalence between a LG model and itsrespective BG model is also used. A procedure to get equationsfrom a given BG model is proposed. To show the functionalityof the proposed methods, examples with diodes are solved.

I. INTRODUCTIONGraphs are simple geometrical �gures consisting of nodes

and lines that connect some of these nodes; they are some-times called �Linear Graphs� [1]. One of the most importantapplications of Graph Theory is its use in the formulation andsolution of the electrical network problem by Kirchhoff [1].Equations can be determined from a LG by �nding a normalform based on an appropriate tree [1].Bond graph is a highly structured modelling technique that

allows to analyze different kinds of physical systems in anunique way determined by a basis of uni�ed description [2].This technique allows the determination of the State SpaceEquation (SSE) in several different ways by reviewing theinteractions of the dynamic elements and the causal paths theydetermine [3], [4].Both BG and LG treats with spatially discrete physical sys-tems [5] even when only LG modelling preserves the spatialvisualization related to the model being analyzed. The spatialconstraints in a BG model are implicit in the location of the0 and 1 junctions [5].There exists several ways to get the SSE from a BG model [6],some of them exploiting the concept of causality and causalpaths.Mathematical manipulation of non linear semiconductor mod-els could be dif�cult. Here, diodes are reviewed and modeledwith a non linear constitutive relation.In this paper a procedure to obtain equations from a BG model

by exploiting the LG tree Theory is proposed. The proposedprocedure can be particularly useful in the analysis of somesystems involving algebraic loops.The contents are ordered as follows:Section II shows a reviewing of the Shockley diode model .Section III presents some current techniques to derive equa-tions from BG and LG models. Section IV shows the mainresult of this work, First a criterion that allow to identify theLG related tree in a BG model is discussed.Then, a procedurethat allow to derive the SSE from a BG model by employingthe LG related tree theory is presented. Finally three examplesare analyzed and discussed while they are solved with theproposed approach. In section V, the conclusions and �nalcomments are written.In the next Section a short review of the Shocley diode modelis presented.

II. THE DIODE MODELThe diode can be considered as a non controlled switching

device [7]. Switched systems can be viewed from severalapproaches and two main classi�cations can be made: Thevariant topology approach and the invariant topology approach[6].Most texts deal with the idealized model of the diode, i.e. avariant topology model whose representation changes betweena short circuit and an open circuit, depending on its on-offstate.The Shockley model is a non linear invariant topology

model that is presented in several power electronics textbooks.However, a procedure to derive equations from systems in-volving this Shockley model is rarely presented and authorsstart their analysis with the idealized on-off-switch approach.In this paper the Shockley model is used.The constitutive relation of the Shockley model as appears in[8] is

ID = IS

�e

VDnVT � 1

�where:� ID = Current through the diode.

273

� VD = Voltage in the diode in the anode referred to thecathode.

� IS = Reverse saturation current with a value of 1 �10�12A.

� n = Empirical constant known as emission coef�cient,that depends on the construction of the element.

� VT = Thermal voltage given by

VT =kT

q

Where q is the electron charge equal to 1:6022�10�19C;T is the absolute temperature in kelvins and k is theBoltzmann constant equal to 1:3806� 10� 23J=K.

For simplicity, some values will be �xed and it will beassumed that

ID = IS�egVD � 1

�(1)

considering a 27�C temperature and n = 2 that is a value thatcan be assigned to silicon diodes. With these considerationsthe g value is

g =1

�vT' 19:2308

The characteristic v � i curve for this constitutive relation isshown in Figure 1.

Fig. 1. The caracteristic v � i curve resulting of the constitutive relation(1).

The BG model that will be used in this paper is a resistivenon linear R element whose constitutive relation is thatpresented in equation (1).This model appears in �gure 2.

R:DFig. 2. A resistive non linear diode model.

In the next section a reviewing of some of the currenttechniques to get the equations from both BG and LG modelsis performed.

III. CURRENT METHODS OF EQUATIONS DERIVATION.There exist several techniques in order to derive equations

from both LG and BG models, and some of them are reviewedin the following paragraphs.

A. SSE derivation from a BG models through junction struc-ture matrixThere exists several approaches to develop the SSE from a

BG model [6], one of the most powerful is the related withthe junction structure matrix [3]. In Figure 3 The key vectorsand the junction structure relationships are shown.

Fig. 3. Key vectors and the junction structure.

In �gure 3, the key vectors are formed as follows:(MSe;MSf ) is the sources �eld; (L;C) is the storage �eld;(R) is the dissipative �eld; (De) is the �eld of detectors; �-nally, (0; 1; TF;GY ) are the elements of the junction structure.The vectors x and xd represent the states of the system atintegral and derivative causality, respectively. z denotes theco-energy vector and zd the derivative co-energy vector. Thevectors u and y are the input and the output, respectively. Dinand Dout show the relationships between efforts and �ows inthe dissipative �eld. The constitutive relationships are givenby,

z = Fx (2)

Dout = LDin (3)

zd = Fdxd (4)

Also the relationships of the junction structure are speci�edby, h

�x Din y

iT= S

hz Dout u

�xd

iT(5)

where the matrix of junction structure S is given by,

S =

24 S11 S12 S13 S14S21 S22 S23 S24S31 S32 S33 S34

35 (6)

zd = G1 � ST14z

By employing the junction structure matrix S and therelationships between the different �elds it is possible to writethe model as a space-state equation of the form

�x = Ax+Bu (7)

274

y = Cx+Du (8)

withA = E�1 (S11 + S12MS21)F (9)

B = E�1 (S13 + S12MS23) (10)

C = (S31 + S32MS21)F (11)

D = S33 + S32MS23 (12)

whereE = I + S14F

�1d ST14F (13)

M = (I � LS22)�1 L (14)

The dynamic elements being placed in vectors x and xd areselected depending on its causality assignment.One disadvantage of this method arises in non linear systemsmodelling, since this matrix representation is not alwayspossible to obtain.In the next subsection, other approach employed to get the

SSE from a BG model is cited.

B. Other SSE derivation method from Bond GraphAccording to [6] and [2], given a BG model, the SSE can

be achieved by proceeding in a systematic way, as follows:1) Write the structure laws in the junctions by consideringcausality.

2) Write the constitutive relationships of the elements byconsidering causality.

3) Combine these different laws in order to put on explicitlythe derivatives of the state variables as function of thestate variables and the inputs.

The key concept in both methods is causality, because theformulation of S matrix (in the �rst method) depends totallyon causal paths as well as the systematic (second) methodneeds the causality assignment already performed. Once theS matrix has been constructed, the procedure to derive theSSE of the model is just a step-by-step algorithm. In the nextsubsection a short review on causality and causal paths is done.

C. Causality and causal pathsOne of the most powerful characteristics of bond graph is

that a lot of information can be retrieved without writing anyequations, just by analyzing the causality. Physical systemsare full of interacting variable pairs [2]. If two elements arebonded, the effort causes one element to respond with �ow,while the �ow causes the �rst element to respond with effort.Thus, the cause-effect relationships for efforts and �ows arerepresented in opposite directions.

A detailed description of causal paths can be found in [9]from where the next has been extracted:De�nition 1: A causal path in bond graph is an alternation

of bonds and basic elements, called "nodes" such that:1) For the acausal graph (before establishing causality), thesequence forms a single chain.

2) All of the nodes in the path have complete and correctcausality.

3) Two bonds in a causal path have in the same nodeopposite causal orientations.

According to the variable being followed, there are twokinds of causal paths. The causal path is simple if it can becrossed by following always the same variable and the causalpath is mixed if it is necessary to perform a variable changewhile the graphic is crossed. In addition, two elements P1 andP2, belonging to the set fR;C; I; Se; Sf ; De; Dfg are causallyconnected if the input variable of one is in�uenced by theoutput variable of the other.In the next section some of the techniques used to develop

equations from a LG model are mentioned.

D. SEE derived from a LG modelThe tree theory developed originally by Kirchhoff [1] is very

useful in the formulation of equations in electrical networks.In purely resistive linear networks, two general approachescan lead to the formulation of a minimal set of equations:general node analysis and general loop analysis [10]. If a givennetwork has dynamic elements, there exists procedures toderive the SSE by using the basis of a normal tree. Accordingto [1] a normal tree can be de�ned as follows:

De�nition 2: A normal tree of a connected directed graphrepresenting a network is a tree that contains all the indepen-dent voltage (effort) sources edges, the maximum number (all)of capacitive edges, the minimum number (none) of inductiveedges, and none of the independent current (�ow) sourceedges.

The SSE equation of any linear time-invariant system repre-sented by a LG model can be derived by following a systematicprecedure [1]. In the case when a normal tree as describedin de�nition 2 cannot be constructed, the LG model includesdynamical elements that cannot be expressed independently inthe resulting SSE.In the next section a mixed approach that exploits the

LG tree theory is presented. This proposed approach worksdirectly on the BG model.

IV. DERIVING THE EQUATIONS FROM A BG MODEL BYUSING A LG TREE APPROACH

The equivalence between a LG model and a BG model isreviewed in [12]. Authors establishes a relation between a BGmodel an the tree of the related LG model by consideringcausality and adyacent causal strokes. A slightly differentcualitative analysis can be performed. Consider the electricalnetwork shown in Figure 4.In Figure 4 (a) an electric network is drawn. In Figure 4

(b) the corresponding tree (the only possible normal tree) ofthe system is shown. In Figure 4 (c) the unique related BondGraph model with integral preferred causality (BGI) can beseen. It is very important to notice the branches of the treeand the orientation of the causal strokes in the correspondingelements on the BG model. The following concepts allow to

275

1 0 1 0

I:L

Se:v

Sf:i

R:R

R:R

C:C

R RL

Cv ii i

1

2

vi

i

R1

1

i i

vC

vR2

ii

L

(a)

(b)

(c)

i

2

1

Fig. 4. Electrical network (a) and their LG (b) and BGI (c) representation.

formalize the criterion employed to identify a tree on a BGmodel.When a port-1 element is causally connected to a BG junctionstructure two situations are possible: (i) The causal stroke(effort) go from the junction structure toward the element oroutwards as in Figure 5 (a), or (ii) the causal stroke go fromthe element toward the junction structure or inwards, as inFigure 5 (b). In Figure 5, the X element is assumed to be aport-1 element.

0

1

X 0

1

X

(a) (b)Fig. 5. The two possible causal orientations of a port-1 element.

This direction on the casual strokes is important, because itbring the conditions to identify the spanning tree of a systemin a BG model. This is summarized in the next criterion:

Criterion 1: In a BGI model, each port-1 element whosecausal stroke is inwards (junction adjacent) determine a branchof the related spanning tree. The set of all these port-1elements spans the related tree. In the other hand, each port-1element whose causal stroke is outwards (element adjacent)determine a chord of the related cotree.

This criterion allow to �nd out a one to one relation betweena BG model and its associated LG model.

In LG theory, Kirchhoff Current Law (KCL) is applied overtree branches in order to get their currents as a sum of currentsof the cotree chords. This is completely equivalent to analyzethe causal paths related to each �tree branch� on the BG modelafter Criterion 1 has been applied. With this consideration, adirect procedure to derive the SSE from a BG model can bewritten as follows:

Procedure 1:1) Given an acausal BG model, assign causality as BGI.2) Determine the tree branches and the cotree chords ac-cording to criterion 1.

3) Set the effort sources and capacitive elements in the treeas tree-main-effort (TME) variables. If resistive elementsbelong to the tree, set them as tree-helper-effort (THE)variables.

4) Set the �ow sources and inductive elements in the cotreeas key-�ow (KF) variables. If resistive elements belogto the cotree, set them as nonkey-�ow (NF) variables

5) By using the element constitutive relation, write eachNF variable as a combination of all the TME variablesthat hold a direct causal path with it.

6) By using the element constitutive relation, write the eachTHE variable as a combination of all the �ow variablesthat hold a direct causal path with it.

7) Write the state equations by determining each capacitor�ow in the tree as a combination of the cotree �ows thathold a direct causal path with it. And by determiningeach inductive effort in the cotree as a combination ofthe tree efforts that hold a direct causal path with it.Finally, write the state equation by using (in the linearcase) the relations fC = C

�eC and eI = I

�fI

Remark 1: If in step 1, Integral causality cannot be assignedin all elements, it implies that a normal tree cannot beconstructed. In step 5 and 6 if it is found that direct causalpaths are held between THE and NF elements, then algebraicloops will arise in the case of linear systems or differentialalgebraic equations must be constructed in the case of nonlinear systems.The preceding Procedure allows to derive the SSE from

a BG model. If only algebraic equations are needed (resis-tive systems), the general node analisys can be performed.General node analysis mixed with BG analysis brings a verysystematic procedure to derive equations. Once a normal tree isestablished, it is always possible to �nd causal paths betweentree elements and cotree elements. The key of the procedureis to determine the �ows in the tree elements in terms ofcotree variables by following the (unique) causal paths that areformed, at the same time cotree elements can be determinedby their constitutive relationship and the (also unique) causalpaths to tree elements. The next example show how thisProcedure can be applied.

Example 1: Consider again the system in Figure 4.In Figure 6 the BGI model of Figure 4 has been redrawn andthe bonds are now numbered, as well as the tree branches are

276

1 0 1 0

I:L

Se:v

Sf:i

R:R

R:R

C:C

i

1

i

1

2

3

4

5

6

7

8

92

Fig. 6. The BGI model of system in Figure 4.

now remarked as black bonds. According to Procedure 1, the�rst step is already done. According to criterion 1, the secondstep is also performed and the bonds corresponding to the treebranches are darkened. In this terms, the bonds belonging tothe tree are 1, 4 and 9. the bonds of the cotree are 2, 6 and8. According with step 3, TME variables are e1and e4wherease9.is the unique THE variable. Step 4 allow to identify f6 andf8 as KF variables and f2 as NF variable.Step 5 of Procedure1 ask to write any cotree R element �ow (NF variable) asa combination of main efforts on the tree and the relatedconstitutive relationship. This is the case of R1. The �ow onR1 can be written as,

f2 =1

R1e2

f2 =1

R1(e1 � e3)

f2 =1

R1(e1 � e4) (15)

Step 6 requires to write e9 as a combination of �ows in thecotree. It can be seen that R2 holds direct causal paths withi1 throughf9; 8g and with L through f9; 7; 6g, therefore it canbe written

e9 = R2f9

e9 = R2 (f7 + f8)

e9 = R2(f6 + f8) (16)

Step 7 requires to write the �ow in the capacitor as acombination of cotree variables causally connected with it.Reviewing the causal paths, it can be noticed that

f4 = f3 � f5f4 = f2 � f6 (17)

Replacing (15) into (17) the result is,

f4 =1

R2(e1 � e4)� f6

f4 = � 1

R2e4 � f6 +

1

R2e1 (18)

Following step 7 I effort equation can also be derived as

e6 = e5 � e7 =e6 = e4 � e9 (19)

Replacing (16) into (19) the result is

e6 = e4 �R2(f6 + f8)e6 = e4 �R2f6 �R2f8 (20)

Finally, equations (18) and (20) can be modi�ed as�e4 = � 1

R2Ce4 �

1

Cf6 +

1

R2Ce1

�f6 =

1

Le4 �

R2Lf6 �

R2Lf8 (21)

Equation (21) is the SSE corresponding to model in Figure 4.

In the next example, a diode circuit is analyzed.Example 2: A purely resistive system: The central tap full

wave recti�er with resistive loadConsider the electrical network shown in Figure 7.

V1

V1

D1

2D

RL

Fig. 7. A full wave recti�er with resistive load.

In Figure 8 the model with causality assigned is shown

1

0

1

01

Se:v

Se:v

R:D

R:R

R:D

1

1

1

2

L

1

2

3

4

5

6

7

89

Fig. 8. Bond Graph model of the system in �gure 7 with causality assigned.

Notice in �gure 8 that bonds 1, 4 and 9 belong to the tree.The R elements in the cotree are D1 and D2. Their �owsdepending on the tree efforts and the constitutive relation (1)are as follows:

f2 = Is (ege2 � 1)

orf2 = Is

�eg(e1�e4) � 1

�(22)

277

andf6 = Is (e

ge6 � 1)

orf6 = Is

�eg(�e9�e4) � 1

�(23)

Writing one equation in this example can lead to the solutionof the entire model. This equation is

f4 = f2 + f6 (24)

Substituting equations (22) and (23) in equation (24) it results

f4 = Is

�eg(e1�e4) + eg(�e9�e4) � 2

�(25)

Also the effort in RL can be expressed as

e4 = RLf4 (26)

Substituting equation (26) in equation (25), the followingequation can be written

f4 = Is

�eg(e1�RLf4) + eg(�e9�RLf4) � 2

�(27)

Equation (27) is an implicit function of the �ow f4 in RLthat can be simulated in Matlab as follows

t=[0:1e-4:32e-3];ei=[36*sin(377*t)];e9=ei;n=2;vt=26e-3; %@T=27 celsiusis=1e-12;g=1/(n*vt)R=500;sixe=size(ei);sais=sixe(:,2);�4=zeros(1,sais);e2=zeros(1,sais);e6=zeros(1,sais);ye=zeros(1,sais);for j=1:saisfunk=@(f4)((is*(exp(g*(ei(1,j)-(R*f4)))+exp(g*(-e7(1,j)-(R*f4)))-2))-f4);varr=fzero(funk,[-.0010 .1]);�4(1,j)=double(varr);ye(1,j)=R*�4(1,j);endplot(t,ei,t,ye)xlabel('time, s');ylabel('in and out Voltages, V');

And the graphic of the Figure 9 is obtainedThe preceding example has shown how to apply the pro-

posed Procedure in purely resistive systems containing diodesrepresented by the Shockley model (1).

Now a dynamic system is analized. Consider the electricnetwork shown in Figure 10.The corresponding BGI is shown in Figure 11.

Fig. 9. The input and output voltages of the system in Figure 7.

RL

V1

D1

D4

D3

D2

C

Fig. 10. A full wave recti�er with RC load.

Se:v 0

1

1

0

1

R:R

1

0 1

C:C

R:D

L

1

2

3

4R:D

R:D

R:D

1

7

8

3

4

5

9

2

10

6

15

11

14

12

16

13

1

Fig. 11. BGI model of the system in Figure 10.

278

The tree is conformed by 2 TME variables an 1 THEvaraible, (e1, e7 and e12, respectively). All elements inthe cotree are causally conected with elements in thetree, D2 is causally connected with D1through the causalpath f10; 9; 11; 12g, and with V1 through the causal pathf10; 9; 11; 13; 1g, and effort e10 can be expressed as

e10 = �e1 + e12 (28)

Similarly, R is causally connected with C, D1 and V1, and

e16 = e7 (29)

In a similar way, the following cotree efforts can be expressedin terms of the tree variables as:

e3 = �e12 � e7 (30)e5 = e1 � e12 � e7 (31)

According to general node analysis, and the given proce-dure, KCL must be applied twice, and in the BG model thismeans that the �ow in the unknowns must be expressed interms of the �ows in the cotree. Starting with the �ow in D1,it can be seen that this element is causally connected with D2,RL, D3 and D4. This dependence can be written as

f12 = f3 + f5 � f10 (32)

due to the non linear nature of the system, the e12 tree variableis very dif�cult to eliminate, so a differential algebraic equa-tion must me constructed, and equation 32 shall be rewrittenas

�f12 + f3 + f5 � f10 = 0

by applying the diode constitutive relations, it can be seen that

Is [�ege12 + ege3 + ege5 � ege10 ] = 0 (33)

Substituting 28, 30 and 31 into 33, the �rst system equationis obtained as:

Is

h�ege12 + eg(�e12�e7) + eg(e1�e12�e7) � eg(�e1+e12)

i= 0

(34)Now, considering the �ow in C, this element is causallyconnected with D4, RL and D3, and its �ow can be writtenas

f7 = f5 + f3 � f16or

f7 = Is (ege5 � 1) + Is (ege3 � 1)�

1

RLe7 (35)

Substituting 29, 30 and 31 into 35, and isolating the derivativeof the state, the second system equation can be written as�e7 =

IsC

�eg(e1�e12�e7) + eg(�e12�e7) � 2

�� 1

RLCe7 (36)

equations 34 and 36 conform a differential algebraic system.This system is simulated in Matlab with the next code lines:clearclcmasa=[1 0;...0 0];%mass matrix to indicate DAE

options = odeset('Initialstep',1e-30,...'MStateDependence','none','Mass',masa,...'RelTol',1e-10,'AbsTol',[1e-12 1e-12]);[T,Y] = ode23t(@full_RC,[0:0.1e-5:32e-3],...[0 0],options);ent=100*sin(377*T);plot(T,ent,T,Y(:,1),'-',T,Y(:,2),'-.')%plotting the input, the D1 voltage and the output

With the fucntion full_RC de�ned asfunction dy = full_RC(t,y)%Shockley model parametersn=2; %silice 2, germanium 1vt=26e-3; %@T=27 degrees celsiusis=1e-12;g=1/(n*vt);%component valuesc=47e-6;%load capacitancer=100;%load resistenceei=100.*sin(377*t);%input voltagedy = zeros(2,1);dy(1)=((is/c)*(exp(g*(-y(2)-y(1)))+...exp(g*(ei-y(2)-y(1)))-2))-((1/(r*c))*y(1));dy(2)=-exp(g*y(2))+exp(g*(-y(2)-y(1)))+...exp(g*(ei-y(2)-y(1)))-exp(g*(y(2)-ei));

Once these codes are implemented in Matlab, the plottingin the following Figure is obtained:

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035100

80

60

40

20

0

20

40

60

80

100

Fig. 12. Simulation results of the system in Figure 11.

V. CONCLUSIONSAn alternative approach to derive equations from a BG

model has been presented. Two important tools are employed:The causality and causal paths involved in BG theory and thetree formulation of SSE employed in LG theory.The proposed approach allow to derive the SSE in a systematicway by following a step by step procedure.

279

Causal connections between tree elements and cotree elementscan be directly found with the suggested Procedure, allowingthe equation derivation of systems containing elements withnon linear constitutive relations, such as the well knownShockley diode model. General node analysis allows to an-alyze even purely resistive (algebraic) systems, and also dy-namic systems with the proposed modi�cation in BG models.

REFERENCES[1] Wai-Kai Chen, "Applied Graph Theory" 2nd revised edition, North-

Holland Publishing Company, 1976, Netherlands.[2] Ronald C. Rosenberg and Dean C. Karnopp, "Introduction to physical

system dynamics", McGraw-Hill 1983.[3] C. Sueur and G. Dauphin-Tanguy, "Structural controllabil-

ity/observability of linear systems represented by bond graphs",J. Franklin Inst., vol. 326 No. 6, P. 869-883, 1989.

[4] C. Sueur and G. Dauphin-Tanguy, "Bond-graph approach for structuralanalysis of MIMO linear systems", J. Franklin Inst. vol. 328 No. 1 P.55-70, 1991

[5] Stephen Birkett Jean Thoma and Peter Roe, �A pedagogical analysisof bond graph and linear graph physical system models�, Mathematicaland computer modelling of dynamical systems, vol. 12. no. 2-3, april -june 2006, P. 107-125, Taylor & Francis.

[6] Dauphin-Tanguy Geneviève, et al., �Les Bond Graphs�, Hermes SciencePublications, 2000, France.

[7] Ned Mohan, et al. �Power electronics�, John wiley and Sons, U.S.A.2003

[8] Muhammad H. Rashid �Power electronics Circuits, Devices, and Appli-cations�, Prentice Hall, 1993

[9] G. Dauphin-Tanguy and S. Scavarda, "Bond-graph modeling of physicalsystems", Chapman & Hall. A. J. Fossard, D. Normand-Cyrot. 1995. Vol.1, pp. 33-109.

[10] William Hayt et al. �Engineering Circuit Analysis�, 7th ed. 2007, Mc.Graw Hill.

[11] Amalendu Mukherjee, et al, �Bond Graph in Modeling, Simulation andFault Identi�cation�, CRC Press, 2006, India.

[12] Alan S. Perelson, George F. Oster, �Bond Graphs and Linear Graphs�J. of the Franklin Institute, 302 (2), pp.159-185. (1976)

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