MULTI-DOMAIN MODELLING - AN ADVANTAGE IN MODELLING,SIMULATION AND CONTROL EDUCATIONDejan DovžanBorut ZupančičGorazd KarerIgor ŠkrjancFaculty of Electrical EngineeringUniversity of LjubljanaTržaška 251000 Ljubljana, SloveniaE-mail: firstname.lastname@example.orgKEYWORDSMulti-domain modelling, object oriented modelling,helicopter, control systemsABSTRACTThe paper deals with an important aspect of continuoussystems modelling and simulation approaches, withpossibilities for multi-domain modelling. The traditionalapproach is based on block oriented schemes in whichcausal relations play an important role. However thiscausality is artificially generated in order to fulfilappropriate conditions for simulation on conventionalsequential computers. Fortunately new concepts whichare based on object oriented approaches, physicallyoriented connections and algebraic manipulations enablethe so called acausal modelling which can efficiently beused for multi-domain modelling. The advantages anddisadvantages of traditional and more advancedapproaches are discussed.The described multi-domain approach is illustrated withmodelling of a laboratory helicopter using a verypopular multi-domain modelling environment Dymolawith Modelica language. A multivariable controller wasdesigned for a wide operating range. The control signalswhich drive the main and the tail rotors enable thedesired pitch and rotation (azimuth) angles. The multidomain object oriented approach enables an efficientcoupling of elements from mechanical libraries and ablock library for the implementation of control system.INTRODUCTIONStandardization of languages for modelling andsimulation was always very important in the history.However the last standard that was really accepted wasCSSL standard (Strauss 1967). Nowadays perhaps themost promising activities are in conjunction with the socalled Modelica activities (www.modelica.org). After aninitiative of the Federation of European SimulationSocieties EUROSIM and The Society for ComputerSimulation SCS in the middle of nineties a new languageModelica (Modelica 2005, Fritzson 2004), which givesProceedings 23rd European Conference on Modelling andSimulation ECMS Javier Otamendi, Andrzej Bargiela,José Luis Montes, Luis Miguel Doncel Pedrera (Editors)ISBN: 978-0-9553018-8-9 / ISBN: 978-0-9553018-9-6 (CD)a hope to become a kind of international standard formodel exchange, was defined.A lack of object-oriented properties, which disables thereuse of already build models, is an importantdisadvantage of many modern modelling and simulationtools. Due to this reason some special-purpose toolswere developed (for mechanical, electrical, chemicalsystems, ). In modelling of complex industrial systemshowever combinations of systems from different areas,i.e. multi-domain systems, are frequently needed (e.g.mechanical, electrical, hydraulic as well as controlsystems in mechatronics, particularly within automotive,aerospace and robotics applications). There were manyattempts to connect different modelling tools, howeverthe more efficient approach is to use tools which supportmulti-domain modelling (Van Beck and Roda 2000).Modelica with appropriate working environment e.g.Dymola (Cellier 1991, Dymola 2008) is probably themost promising multi-domain modelling environmentbased on object oriented approach and ‘physical’ way ofconnections between model components.IMPORTANT FEATURES OF ADVANCED OOENVIRONMENTSBlock oriented approach versus object orientedmodelling and simulation approachIn order to allow the reuse of component models, theequations should be stated in a neutral form withoutconsideration of computational order. This is so calledacausal modelling (Zupančič et al. 2005, Zupančič andSodja 2008). However most of the general-purposesimulation software on the market such as ACSL,Simulink, assume that a system can be decomposedinto block diagram structures with causal interactions(Matko et al. 1992, Cellier 1991). In this procedure statederivatives must be expressed which leads to the explicitstate space form
x& f ( x, u , t )y g ( x, u , t )(1)ωiαiangular velocitiesangular accelerationswhich can be efficiently simulated with ODE solvers. uis an input, y is an output and x is a state. It is rare that anatural decomposition into subsystems leads to such amodel. Often a significant effort in terms of analysis andanalytical transformations is needed to obtain a problemin this form. It requires a lot of engineering skills andmanpower and it is error-prone.In Modelica and some other multi-domain modellingtools it is possible to write balance and other equationsin their natural form as a system of differential-algebraicequations, DAE0 f ( x& , x, y, u , t )(2)where x is the vector of unknowns that appearsdifferentiated in the equation and y is the vector ofunknowns that do not appear differentiated. Computeralgebra is utilized to achieve as efficient simulation codeas possible, similar when we convert the equations‘manually’ into ODE form.Object orientationWe shall not talk about general concepts in OOprogramming where well known terms as encapsulation,data abstraction, inheritance, dynamic binding, areused. From a modeller point of view, OO means that onecan build a model similar to real system: taking a pump,a pipe, a valve, and connecting them. For an efficientmodelling, modelled systems are decomposed intosubsystems (components), which are modelled assubmodels and then hierarchically connected into acomplete model. Modelling languages enable simplereuse of already build models. To reuse a certain modelin other models it should be defined as a class. Modelclasses can be defined by physical laws (energy andmass balance equations and not necessarily with statespace description (Equation 1). This contributes to abetter understanding and reusability of models.‘Physically’ oriented connectionsThe appropriate complexity of the implementation of theconnections between model components (classes) isprobably the most important property of multi-domainOO modelling tools. Connections between submodelsare based on variables, which define proper relationsand influences between movements, angles, currents,pressures, torques, forces, etc. It is similar as when realsystems are built. Figure 1 shows how three mechanicalrotational subsystems M1, M2 and M3 are connected.Several physical variables are presented in connections(connectors in Modelica):torquesτiθianglesFigure 1: Connection of three mechanical subsystemsIn general there are two types of variables, which aredefined in connectors of subsystems: variables thatbecome equal in connection points, in our exampleangles, velocities, accelerations (ACROSS variables,also potential, temperature, pressure in other types ofsystems)ω1 ω2 ω3α1 α2 α3(3)θ1 θ2 θ3and variables which sum equals zero (THROUGHvariables, in our example this is torque, but also current,force etc. in other systems - prefix FLOW in Modelica)τ1 τ2 τ3 0(4)CONNECTOR is a special structure in which all thevariables are collected. Each CONNECTOR has aname, which is composed of a submodel name and aname of a particular connector. Connections intraditional block diagram simulation languages can betreated as a subset of connections introduced byconnectors. Namely they possess only variables of thetype ACROSS, which become equal in junction points.Example of a connector definition in Modelicaconnector flangeReal theta,omega,alfa; // Type acrossflow Real tau;// Type throughend flange;By joining connectors the submodels are connected.During processing the modelling tool automaticallygenerates equations 3 and 4 from submodel connectordefinitions.Summary of features of Multi-domain OO modellingand simulation environmentsSome important features of modern multi-domain OOmodelling and simulation environments (e.g. Dymolawith Modelica) are: Modelling of various kinds of complex physicalsystems with object oriented approach.
Multi-domain tools, equivalently usable formodelling of mechanical, electrical, chemical,thermo dynamical and other systems. Possibilities to reuse already built models. Acausal model building. Hierarchical structure of models. Description of processes through physical laws(differential equations) irrespective to the type andpurpose of a model. Easy and efficient way for submodels connectionsthrough connectors (more general then input-outputconnections known in block oriented simulationtools). Symbolical and numerical solving of systems ofequations - algebraic formula manipulation.Modelica – OO modelling standardModelica (Modelica 2005, Fritzson 2004) is a modellinglanguage which supports both high level modellingusing pre-prepared complex model components anddetailed modelling by equations. Graphical diagramlayer and textual layer can be efficiently combined. Thebasic construct in Modelica is class. There are severaltypes of classes with several restrictions: class, model,block, function, package, type etc. There are alreadyseveral pre-prepared packages or libraries in standardModelica library, e.g. for block diagrams, for electricaland electronic systems, for mechanical systems, forthermal and some others. Many other libraries are publicdomain libraries. In the past we developed a Modelicaprocess systems library (Zupančič et al. 2005). Howeverin this paper we focus to the mechanical systems whichare even more appropriate for theoretical modelling asprocess systems (systems with level, flow, pressure,temperature, flow, PH, .) and therefore very convenientto show the advantages of multi-domain object orientedacausal modelling.The laboratory helicopter set-up comprises a helicopterbody carrying two motors, which drive the main and thetail rotors and a servomechanism, which shifts the centreof gravity by moving a weight along the helicopter’shorizontal axis. The helicopter body is mounted to thestand so that two degrees of freedom are enabled:- the rotation around the horizontal axis (pitch angle yψ) and- the rotation around the vertical axis (rotation angleor azimuth - yφ).Figure 2 depicts the helicopter as it finally appears inanimation scheme in Dymola/Modelca environment.The model can be described as a nonlinear multivariablesystem with three inputs: the voltage driving the mainrotor motor (u1), the voltage driving the tail rotor motor(u2) and the voltage which drives the servomotor forpositioning of the weight. In our case only the first twoinputs are taken into control. The weight is positioned toa fixed place, defining the appropriate mechanicalcharacteristic of the helicopter. The system has twooutputs: pitch angle (yψ) and rotation angle (yφ). Forceswhich effect the helicopter body Fmain , Freact and Ftailwill be explained later.EXAMPLE: MODELLING OF A LABORATORYHELICOPTERFigure 2: Laboratory helicopter with forces affecting thebody and control input/output signalsThe advantage of the multi-domain OO modelling willbe shown with a modelling and control of a laboratoryhelicopter device. Laboratory set-ups, which model realprocesses, and mathematical models have a significantrole in efficient control design and education. TheCE150 (Figure 2) is a laboratory helicopter made byHumusoft (Humosoft 2002). It is used for studyingsystem dynamics and control engineering principles andenables a wide range of practical experiments. The goalof modelling and identification is to prepare a basis forthe students’ laboratory assignments, such as designing amultivariable controller that ensures satisfactory controlin the wide operating range. There are two well knownmodelling approaches: theoretical and experimental. Inhelicopter modelling both approaches were combined.The mathematical model is described in details in Karerand Zupančič 2006.Multi-domain approach is useful because we combinetwo different areas: mechanical systems and controlsystems which are usually modelled with blockdiagrams. We chose Modelica (Modelica 2005, Fritzson2004) as mechanical systems and block diagrams areincluded in the Standard Modelica library. Dymolaenvironment (Dymola 2008) which supports Modelicawas used. For animation the body of the helicopter andboth rotors were drawn with program SolidEdge andwith DxfExport add-in exported into dxf file format.Dymola uses dxf files to create an animation of modelcomponents. The helicopter body and the stand weremodelled with the standard Modelica librariesMechanics\MultyBody and Mechanics\Rotational.The designed mechanical model was then used inMatlab/Simulink environment as Dymola block. Namely
Matlab/Simulink environment is very efficient forcontrol system design as Control, Optimization or someother tools (toolbox) can be used. The control schemedesigned in Matlab/Simulink was then implemented alsoin the Dymola/Modelica environment using standardModelica libraries Block\Continuous, Block\Sources andBlock\Math.The overall scheme of the top level model inDymola/Modelica is shown in Figure 3. The scheme isvery clear. It consists of the coordinate systemdefinition, the stand model, the helicopter body, the tailand main rotor model (rotor means motor and propeller)and the controller with two reference signals for pitchand rotation angle.friction was modelled using bearingFriction blocks,where characteristics of the friction are described withlookup tables. The functions for both frictioncharacteristics were defined using relationτ friction B1ω B2 sign(ω )(5)in which linear and Coulomb part are taken into account.The relation (5) was implemented with lookup tables inboth bearing friction classes.Tail rotorMain rotorHelicopter bodyWorldStandRotationFigure 4: The stand modelPitchReference pitchReference rotationControllerThe coordinate system is connected to the connectorworld and the body of the helicopter is connected to theconnector helicopter body. Translations fT2 and fT3 donot influence the dynamics. They are inserted for morerealistic animation for the rotational axis in z direction.Model of the helicopter bodyFigure 3: The top level model of the control systemCoordinate systemCoordinate system orientation is such that thegravitational force is in the direction of negative y axis.It is defined by the Modelica class world.Model of the standThe stand (see Figure 4) is modelled with translations(pure transformations of coordinates, straight linesbetween a and b connectors) and two bearings (actuatedrevolute joints Revol ext and Revol ext1 in Modelica).Rotational axis of the first bearing is y axis (rotation inthe horizontal plane) and of the second bearing is z axis(rotation in the vertical plane). The used actuatedrevolute joints were actually extended with additionaloutputs in order to get more simple and usefulinformation about the pitch and rotation angles. Namelythe connectors of revolute joint have a complexinformation about movements and velocities with threecomponents in (x,y,z) coordinate system. In this way theangle sensors of the set-up were modelled. The bearingThe central part of this model class (see Figure 5) is themodel class BodyShape from the MultyBody librarywhich is suitable for rigid body modelling. We have tospecify the vector from frame a to frame b resolved inframe a, the vector from frame a to the centre of mass,resolved in frame a and the mass of the rigid body.Beside we have to specify also the inertia tensor withJxx, Jyy, Jzz, Jyx, Jzx, Jzy. The inertia tensor has to bedefined with respect to a coordinate system that is in theorigin of the centre of mass of the body. In our examplewe had to identify and took into account Jyy and Jzz.With the series of translations (fTy, fTrx, fTry, fTrz) weadequately moved the rotors from the rigid bodyconnectors so that the generated forces produced byrotors had the right levers. In BodyShape model classthe animation shape can be selected. In our case theshape was drawn with a program SolidEdge. Thedrawing was exported to the dxf file format. This filewas then chosen in the animation options of theBodyShape class. Unfortunately Dymola cannotinterpret the dimensions of objects in dxf files.Therefore we added to the model a set of translations forthe proper animation (the encircled part in Figure 5).
rotor mainfTrzafTyrotor main ar lengthbfTzaabbfTrxfTraza mheli babaBodyShapeafTrybbModel of rotorsbrotor tailfTaybafTaxfTraxabaabbafTrayrotor tail afTazstandFigure 5: The helicopter body modelDetermination of parametersModelling in Dymola/Modelica environment is actuallya kind of theoretical modelling with a strong support ofModelica libraries. However many parameters have tobe experimentally determined. Some experiments werevery simple geometrical measurements, some otherswere more sophisticated. The approaches and theparameter values are presented in more details in Karerand Zupančič 2006. In this paper we shall brieflydescribed only some more sophisticated procedures.The centre of gravity was determined with some simplemechanical experiments for a predetermined position ofthe weight which enable the changeable dynamicproperties.For both bearings the constants B1 and B2, the momentof inertia tensor J of the regid body were derived withthe minimization of criterion function (6), which takesinto account the integral of the square differencebetween the real response yreal and the model responseymod with different experiments with appropriate inputsignals u1 and u2 and appropriate initial conditions.Each rotor generates two forces that affect themovements of the helicopter body (see Figure 2). Thefirst force is the ‘useful’ propulsation force in thedirection of rotational axis (Fmain of the main rotor andFtail of the tail rotor). The second force is a reactionforce, which is perpendicular to the rotational axis andacts as a kind of disturbance being a consequence of therotor rotations and air resistance (Freact of the main rotor,the reaction force of the tail rotor is neglected). Thereaction force of the main rotor must be compensated bythe propulsation force of the tail rotor. There is also agyroscopic effect caused by rotation of the main rotorwhich effects the additional torque around the horizontalaxis which was neglected in our study. From this shortdiscussion we conclude that we need a multivariablecontrol approach as each control input influences bothoutputs (especially u1).The propulsation forces can be modelled mathematicallyusing appropriate equations for interactions among theair, the shape and rotational speed of the rotor or as inour case they can be modelled experimentally. To modelthe forces we used experimentally obtained transferfunction (7) of second order with double pole and theoutput quadratic nonlinearity (10)Um (s)U (s) 2Fprop aum bummin (yreal (t , u1 , u 2 ) ymod (t , u1 , u 2 )) 2 dt(6)0Nelder-Mead optimization method in Matlab was used.Dynamics in one rotational direction (in one plane) wasdetermined when the movements in another plane wasdisabled. For rotation around around horizontal axis(movements in vertical plane) we obtainJzz 2.43·10-3 kg m2B1z 1.53·10-4 kg m2s-1B2z 1.03·10-4 kg m2s-1and for rotations around vertical axis (movements inhorizontal plane)Jyy 2.02·10-3 kg m2B1y 5.52·10-4 kg m2s-1B2y 5.96·10-4. kg m2s-1(7)(8)In Equation (7) U is the voltage driving the rotor motor(U1 or U2) and Fprop in Equation (10) is the useful forceof the rotor (Fmain or Ftail). Parameters a and b aregained from the static characteristic.We modelled the reaction force similar to the propulsionforce except that we introduced more complexnumerator dynamics (11)t maxJ , B1 , B21(Ts 1) 2Um (s)U (s) T1 r s 2 T2 r s 1(Ts 1) 22Freact ar um br um(9)(10)For the main rotor the parameters are: a 2.488 N,b 0.992 N, T 0.094 s, T1r 0.0017 s, T2r 0.1908 s,ar 0.296 N, br 0.2 N and for tail rotor: a 0.792 N,b 0.316 N, T 0.094s.The block diagram, which realizes the forces for themain rotor (Eqs. 7-10) is shown in Figure 6.Figure 7 describes the model of the rotor. The part thatis outside the encircled area is actually the partdescribed in Figure 6 which calculates the forces. Theencircled part is for the animation purposes. For thispurpose we model a bearing and a rotor with no mass.
These parts do not affect the model dynamics. Therotational speed of the rotor is proportional to thevoltage on the motor.main forcedescribe the controller tuning procedure and theappropriate parameters of the control system.frameForcenonlinearityb(s)uPIDresolv ea(s)PID ControllerRef 2reaction forcenonlinearity1b(s)frameForce1a(s)resolveframe bu2y2u1y1Lookup Table 1Lookup TableFigure 6: Block diagram that calculates two forces forthe main rotorDymolaBlockPIDspeedgainPID Controllerw refk 8Figure 8: The control system in Figure 9 shows the time responses of the pitch androtation signals. The reference of the pitch angle waschosen between -1.2 radians and 0.3 radians and thereference of the rotation angle was chosen between 0 an3 radians. We can observe that the control systembehaviour is satisfactory in spite of significant changes.Fanimation to helicopter body0.40.20force to helicopter bodypitch-0.2-0.4-0.6-0.8-1Figure 7: The model of rotation3The control system was designed using Matlab/Simulinkenvironment. In Simulink the overall mechanical modelwas presented with the so called Dymola block. Thesimulation scheme is depicted in Figure 8. The input u1into the helicopter model is the voltage driving the frontrotor and the input u2 is the voltage driving the tail rotor.These are control signals. Output y1 is the pitch angleand y2 is the rotation angle. These are controlled signals.2.521.510.50-0.5-1timeFigure 9: Controlled signals (pitch and rotation angles)and appropriate reference changes (dashed lines)CONCLUSIONBoth control loops are combinations of the feedforwardcontrol and the PID feedback control with windupprotection. Feedforward signals are obtained from thereference signals Ref1 and Ref2 which are processedwith two nonlinear functions which are realized with twolookup tables. The feedforward signals bring thecontrolled signals into the vicinity of the referencesignals. Fine corrections and steady state erroreliminations are achieved with appropriately tuned PIDcontrollers.As the emphasize of this paper is given to the multidomain object oriented modelling approach we shall notModelling and simulation is extremely important subjectin all control engineering courses. Namely it is mucheasier, cheaper and safer to experiment in simulationenvironment as on real processes. However traditionalblock oriented causal approaches should be extended bymore advanced object oriented acausal approaches andtools which are especially efficient in multi-domainapplications.In Karer and Zupančič 2006 the traditional modellingapproach using causal traditional block diagramapproach was used. In this paper however a moreadvances multi-domain and object oriented acausalmodelling was used. There is no important difference
when modelling control schemes. However there is a bigdifference when modelling physical systems. Thestructure of the block diagram model inMatlab/Simulink does not reflect the topology of thephysical system. It is not possible to transparentlyobserve the stand, the helicopter body, the rotors, therevolute joints There is a fundamental limitation ofblock diagram modelling. The blocks have anunidirectional data flow from inputs to outputs. This isthe reason why objects mentioned above cannot be dealtwith directly. It is also the reason why some parametersof the different components appear in mixed expressionsin the Matlab/Simulink model. And the developedstructure can be more or less used only for theconfiguration it was developed. The effort to producethe Simulink simulation model is incomparable with theModelica model. One needs much more time and muchmore modelling knowledge for Simulink basedapproach. There is no useful textual layer behind thediagram layer which is the important disadvantage of theSimulink model. So it is problematic to deal withcomplex and sophisticated models. The documentationis very difficult and inappropriate.modelling, simulation and control of conditions inbuildings. He was the president of EUROSIM(Federation of European Simulation Societies) in theperiod 2004-07. He is author of app. 200 conferencepapers and 40 papers in scientific journals, co-author ofone international book (published by Elsevier) from thearea of modelling and simulation.In Modelica models (Figs. 3, 4, 5), the connectionsbetween physical system and computer model are verytransparent. All the components are fully reusable inother configurations. The combination of textual anddiagram programming is efficient. So Modelica can beused for very complex problems and is also superior formodel documentation.Van Beek, D.A., J.E. Rooda. 2000. "Multi-domain modelling,simulation, and control". Proceedings of 4thInternational Conference on Automation of MixedProcesses: Hybrid Dynamical Systems (ADPM2000),Dortmund, pp. 139-146.Cellier, F.E. 1991. "Continuous System Modeling." SpringerVerlag, New York.Dymola, Multi-Engineering Modelling and Simulation. 2008.Users manual, Ver. 7.0, Dessault System, Dynasim AB,Sweden, Lund.Fritzson, P. 2004. "Principles of Object Oriented Modellingand Simulation with Modelica 2.1", IEEE Press, JohnWiley&Sons Inc., Publication, USA.Humusoft, CE 150 Helicopter Model. 2002. User's manual,Humusoft, Prague.Karer, G, Zupančič B. 2006. "Modelling and Identification ofa Laboratory Helicopter". Proceedings of the 5thMATHMOD conference. Vienna, vol. 2, 9 pages.Matko, D., R. Karba and B. Zupančič. 1992. "Simulation andModelling of Continuous Systems: A Case StudyApproach". Prentice Hall Int., New York.Modelica - A Unified Object-Oriented Language for PhysicalSystem Modelling. Feb. 2005. Language specification,Ver. 2.2, Modelica Association,http://www.modelica.org/documents /ModelicaSpec22.pdfStrauss, J.C. 1967. "The SCi Continuous System SimulationLanguage". Simulation, no.9, 281-303.Zupančič, B., G. Zauner, F. Breitenecker. 2005. "Modelingand Simulation in Process Technology with Modelica",Proceedings of the 2005 Summer Computer SimulationConference, Cherry Hill, NJ.Zupančič, B., A. Sodja. 2008. "Object Oriented Modelling ofVariable Envelope Properties in Buildings", WSEASTrans. on Systems and Control, Issue 12, Volume ol/2008/28-731.pdfWe conclude that Matlab/Simulink is superior for thedesign and implementation of control schemes as it hasmore facilities especially in conjunction with sometoolboxes – e.g. Control System Toolbox, OptimizationToolbox. Modelica is superior when modelling physicalsystems when the concept of algebraic manipulation andspecially defined connectors bring many advantages.This approach is also very useful in education. Wepropose to start modelling courses with OO acausalapproach especially when one can deal withimplemented libraries which do not demand a deeptheoretical background. And it can motivate studentsmuch more than the low level Simulink approach.AUTHOR BIOGRAPHYDEJAN DOVŽAN received the university degree fromelectrical engineering at the University of Ljubljana,Slovenia in 2008. Currently he is the Ph.D. student. Themain research area is: intelligent methods in modellingand control.http://msc.fe.uni-lj.si/Staff.aspBORUT ZUPANČIČ received Ph.D. in electricalengineering from the University of Ljubljana, Slovenia.He is full professor from 2000. His major researchinterests are: object oriented modelling, hybrid systems,GORAZD KARER received the university degreefrom electrical engineering at the University ofLjubljana, Slovenia in 2004. Currently he is the Ph.D.student. His research interests are in hybrid systems andmodel predictive control. He is author of 6 papers inscientific journals.IGOR ŠKRJANC received Ph.D. in electricalengineering from the University of Ljubljana, Slovenia,in 1996 and became full professor in 2008. His mainresearch interests are: adaptive, predictive, fuzzy, andfuzzy adaptive control systems. He is author of app.150 conference papers and 65 papers in scientificjournals.REFERENCES
mechanical, electrical, hydraulic as well as control systems in mechatronics, particularly within automotive, aerospace and robotics applications). There were many attempts to connect different modelling tools, however the more efficient approach is to use tools which support multi-domain modelling (Van Beck and Roda 2000).
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