Mixed Domain Modeling in Modelica -...

1FDL‘2002, Marseille, September 24 – 27

Mixed Domain Modeling in Modelica

C. Clauss1, H. Elmqvist3, S. E. Mattsson3, M. Otter2, P. Schwarz1

1 Fraunhofer Institute for Integrated Circuits, Design Automation Department, EAS, Dresden, Germany

2 German Aerospace Center (DLR), Oberpfaffenhofen, Germany

3 Dynasim AB, Lund, Sweden


1. Motivation

2. Modelica Fundamentals

3. Modelica Language Elements

4. Simulation Algorithm – Related to the Dymola Simulator

5. Modelica Libraries

6. Application Examples

7. Comparison: VHDL-AMS and Modelica

8. Conclusions

Contents


Modelica – yet another modeling language: why?

VHDL-AMS, Verilog-AMS, SystemC-AMS have their roots in digital electronics,with extensions to heterogeneous systems:- analog/digital, mixed-signal- HW/SW, digital signal processing- multi-domain capabilities

(e.g., in VHDL-AMS: THROUGH and ACROSS quantities instead of currents andvoltages, NATURE attribute)

Modelica combines some of their features, together with - object-oriented modeling and object-oriented programming- orientation on advanced simulation algorithms, e.g., to avoid index problems

( but this is not part of language definition )

Other languages:Dymola, gPROMS, NMF, ObjectMath, Omola, Smile, U.L.M, EcoSim, ...have their roots in control systems, robotics, hydraulics, and mechatronics

1. Motivation


• Goal: Technique for multi-domain modeling of complex systems.

• Development started in 1978 by Hilding Elmqvist at Lund Institute of Technology(Sweden).

Summary of some features

Standardisation with the freely available modeling language

• Developed by the (non-profit) Modelica Association since 1996,

• differential, algebraic and discrete equations,

• declarative (= mathematical equations) and procedural (optional) ,

• many Modelica libraries available (mostly public domain):1D/3D Mechanics, electronics, hydraulics, power systems, heat transfer,thermo-fluid pipe flow, flight dynamics, input/output, ... .


Modelica Association• Chairman Martin Otter DLR, Munich, Germany• Vice-Chairman Peter Fritzson Linköping University, Sweden • Secretary Hilding Elmqvist Dynasim AB, Lund, Sweden (former Chairman) • Treasurer Michael Tiller Ford Motor Company, Dearborn, U.S.A.

• Bernhard Bachmann Fachhochschule Bielefeld, Germany• Dag Brück Dynasim AB, Lund, Sweden• Peter Beater Gesamthochschule Paderborn, Germany• Vadim Engelson Linköping University, Sweden • Thilo Ernst GMD-FIRST, Berlin, Germany• Jorge Ferreira Universidade de Aveiro, Portugal• Rüdiger Franke ABB Heidelberg, Germany• Pavel Grozman BrisData AB, Stockholm, Sweden• Johan Gunnarsson MathCore, Linköping, Sweden• Mats Jirstrand MathCore, Linköping, Sweden• Kaj Juslin VTT, Espoo, Finland• Clemens Klein-Robbenhaar Germany• Sven Erik Mattsson Dynasim AB, Lund, Sweden• Henrik Nilsson Linköping University, Sweden• Hans Olsson Dynasim AB, Lund, Sweden• Tommy Persson Linköping University, Sweden• Levon Saldamli Linköping University, Sweden • Per Sahlin BrisData AB, Stockholm, Sweden• Andre Schneider Fraunhofer Institute for Integrated Circuits, Dresden, Germany• Peter Schwarz Fraunhofer Institute for Integrated Circuits, Dresden, Germany• Hubertus Tummescheit Lund University, Sweden• Hans-Juerg Wiesmann ABB Corporate Research Ltd, Baden, Switzerland


Homepage: www.Modelica.org

Tutorial: www.Modelica.org/current/ModelicaTutorial14.pdf

Formal specification: www.Modelica.org/current/ModelicaSpecification14.pdf

Model libraries: www.Modelica.org/library/library.html

Publications: www.Modelica.org/workshop2000/proceedings

www.Modelica.org/workshop2002/proceedings

Tools (simulators): www.Modelica.org/tools.shtml

Dymola (www.Dynasim.se)MathModelica (www.MathCore.se)


Modeling and simulation of multi-physics systems where the overall system consists of components from different domains.

For example, detailed vehicle model:

• Vehicle dynamics (3D mechanics)

• Power train (1D mechanics)

• Hydraulics

• Combustion

• Electrical/electronic systems

• Air conditioning (thermo fluid)

• Control systems (I/O blocks, statecharts, ...)

2. Modelica Fundamentals


Composition Diagrams of Modelica (= object diagrams)

component

• Every Icon represents a physical component.e.g. electrical resistance, mechanical gear, pump

connection

• The connection lines represent the actual physical connection.E.g.: electrical line, rigid mechanical connection, heat flow between components.

Interface

• Variables in the interface points define the couplingwith other objects

• A component consists of a connection of components(= hierarchical structure) and/or is described by equations


A Complete Example of a Composition Diagram


The general form in Modelica:expression = expressionR*i = u (Ohm’s law) not an assignment !

3. Modelica Language Elements

3.1 Equations

R1=10

AC=220

C=0.01

G

R2=100

L=0.1

The “unknown” depends on the model connection structure:i := u/R assignmentu := R*I assignmentR := u/I assignment

Vg

R1=10

R2=40

G

R3=40

or several “unknowns” (system of simultaneous equations)ε := R*i - u (residue -> 0)

No preference to a specialized form of equations (e.g., the „Modified NodalAnalysis“ (MNA) widely used in electronics)


type Angle = Real(quantity ="Angle” , unit="rad",displayUnit="deg");

type Torque = Real(quantity ="Torque” , unit="N.m");type Mass = Real(quantity ="Mass” , unit="kg", min=0);type Pressure = Real(quantity ="Pressure", unit ="Pa",

displayUnit="bar” , nominal=1.e5);

predefined data type(floating point number)

attributes(e.g. unit)

3.2 Variables

attributes:quantity, unit, displayUnit, min, max, start, fixed, nominal

new data type

predefined data types:Real, Integer, Boolean, String

library Modelica.SIunits:450 predefine ISO quantity types


connector Pin connector Flangeimport SI=Modelica.SIunits; import SI=Modelica.SIunits;SI.Voltage v; SI.Angle phi;flow SI.Current i; flow SI.Torque tau;

end Pin; end Flange;

3.3 Connectors (= interface definitions)

Flange flange_a;Gearbox gear;equationconnect(flange_a, gear.flange_b);…

• Group of variables describing interaction

• Connected flow variables are summed to zero

• Other connected variables are set equal


electrical 1D translational mechanics

321

31

21

0 iiivvvv

++===

321

31

21

0 fffssss

++===

connect(R1.p, R2.p);connect(R1.p, R3.p);

connect(m1.flange_a, m2.flange_a);connect(m1.flange_a, m3.flange_a);

i1

i3

i2

v1v3 v2

R1

R3

R2

f1 f2

f3

s1

s3

s2

m1 m3

m3

Generated statements:

Connectors


model IdealPlanetaryparameter Real ratio=100/50 ”# ring_teeth/sun_teeth";Flange sun "sun flange";Flange carrier "carrier flange ";Flange ring "ring flange";

equation// kinematic relationshipsun.phi - carrier.phi + ratio*(ring.phi - carrier.phi) = 0;

// torque balance (no inertias)ring.tau = ratio*sun.tau;carrier.tau + sun.tau + ring.tau = 0;

end IdealPlanetary

ideal planetary gear box (no inertias)

3.4 Component models


partial model Rigid "Base for rigid connection of two flanges"import SI=Modelica.SIunits;Flange flange_a "(left) driving flange";Flange flange_b "(right) driven flange";SI.Angle phi;

equationflange_a.phi = phi;flange_b.phi = phi;

end Rigid;

3.5 Partial models and inheritance

model Inertia "rotational inertia"extends Rigid;import SI=Modelica.SIunits;parameter SI.Inertia J;SI.AngularVelocity w;SI.AngularAcceleration a;

equationw = der(phi);a = der(w);

J*a = flange_a.tau + flange_b.tau;end Inertia;

Addition of inertia properties


SingleOutput

Source

SineSource

Function

Sine

f

f

OnePort

VoltageSource

SineVoltage

Inheritance(is-a-relation)

Composition (has-a-relation)

s

s

3.5 Partial models and inheritance (cont‘d)

VoltageSource

Source

time x y o

p

n

vFunction

Example: electrical voltage sources


3.6 Model composition

model SimpleCarimport Modelica.Mechanics.Translational.Sensors;Car.Driver driver (k=30, T=50);Car.FullGear gearbox(tableFile="ZF4HP22.dat");Car.FullEngine engine (tableFile="engine1.dat");Car.Resistance car(mass=1800, area=2.0, ...);Car.Axle axleSensors.SpeedSensor v;

equationconnect(car.flange_a , v.flange_a);connect(gearbox.flange_a, engine.flange_b);connect(gearbox.flange_b, axle.SteeringWheel);connect(axle.flange_b , car.flange_a);connect(driver.throttle , engine.inPort);connect(v.outPort , driver.speed);connect(driver.throttle , gearbox.inPort);connect(driver.brake , axle.inPort);

end SimpleCar;

annotation(extent=[20, -10; 40, 10]);


File: ...\Modelica\Electrical\Analog\Basic.mo

within Modelica.Electrical.Analog;package Basic

model Resistor...

end Resistor;

model Capacitor...

end Capacitor;...

end Basic;

\Modelica\Blocks\Electrical

\AnalogBasic.moSemiconductors.moSources.mo

...\Mechanics

Rotational.moTranslational.mo

Constants.moIcons.moMath.moSIunits.mo

...

Modelica libraries are hierarchically structured and are mapped to the same hierarchy of the file system (i.e. file name = model name):

new library

3.7 Component libraries

Example: Modelica.Electrical.Analog.Basic


• Prozedural sections (algorithm)

• if, for, while

• Functions, including well defined external C function interface

• Multidimensional arrays (similiar to Matlab)

• Multidimensional component arrays (e.g. for PDE discretization)

• Replaceable model components (replaceable/redeclare)

• Discrete equations (when; e.g. sampled data systems)

• Discontinuous equations (time- and state events)

• Variable structure equations (e.g. friction, clutches, ideal diode)

• Very general initialization (if initial() then ... end if)

• Re-initialization at events (impulse(..))

• Graphical annotations and embedding of icons

3.8 Additional language elements of Modelica


4. Simulation Algorithm – Related to the Dymola Simulator

Overall system equations = equations of all components+ equations of all connect statements

Continuous, discrete, and mixed-signal equations are handled in the same way!

The result is an implicit DAE (Differential Algebraic Equation) system

( ) 0,,, =tyxxf &

Direct use of numerical DAE solver not advisable:• dimension of y (algebraic variables) very high• large Jacobian leads to inefficient simulation • equations might need to be differentiated several times (high index problems)

Modelica was designed such that symbolic transformation algorithms can be applied.

In the following, the algorithms utilized in Dymola are sketched.


What Dymola Symbolic Solver Does for You:

• Transforms equations to efficient C-code• Sorts and equations• Solves equations symbolically (for reducing their number)• Solves or generates code for algebraic loops• Generates code to handle events• Generates symbolic Jacobians for efficient nonlinear iterations• Handles DAE with constraints (high index DAE’s)• Automatically assigns state variables• Generates code for consistent initialization of DAE

Dymola Symbolic SolverSimulator


Simulator Dymola

Output

Dymola simulation

C text

symbolical processing

graphical/textual input

OutputOutputOutput

Modelica text


model rlc Modelica.Electrical.Analog.Basic.Ground Ground1;Modelica.Electrical.Analog.Basic.Resistor R(R=100);Modelica.Electrical.Analog.Basic.Capacitor C;Modelica.Electrical.Analog.Basic.Inductor L;

equationconnect (C.n, L.n);connect (L.n, R.n);connect (C.p, L.p);connect (L.p, R.p);connect (C.p, Ground1.p);

end rlc;

Simulator Dymola

Output

Dymola simulation

C text



OutputOutputOutputOutput

Modelica text


Ground

ComponentsResistor R

OnePort for R

Capacitor COnePort for C

Inductor LOnePort for L

Connections:voltages currents

voltagescurrents

Ground1.p.v = 0

R.R*R.i = R.v R.v = R.p.v - R.n.v 0 = R.p.i + R.n.iR.i = R.p.i

C.i = C.C*der(C.v)C.v = C.p.v - C.n.v 0 = C.p.i + C.n.i

C.i = C.p.iL.L*der(L.i) = L.v

L.v = L.p.v - L.n.v 0 = L.p.i + L.n.iL.i = L.p.i

C.n.v = L.n.v L.n.v = R.n.v C.n.i + L.n.i + R.n.i = 0

C.p.v = L.p.v L.p.v = R.p.v C.p.v = Ground1.p.vC.p.i + L.p.i + R.p.i + Ground1.p.i = 0

Modelica Equation Set: 20 equations/variables


Protocol:

- translateModel("rlc")Translation startedDAE with 20 unknown scalars

and 20 scalar equations.4 constants found.0 parameter bound variables found.11 alias variables found.5 remaining time dependent variables.Finished

Simulator Dymola

Output

Dymola simulation

C text


Modelica text




Ground


OnePort for R




voltagescurrents

Ground1.p.v = 0

R.p.i = C.v / R.R R.v = R.p.v - R.n.v 0 = R.p.i + R.n.iR.i = R.p.i

der(C.v) = C.p.i / C.CC.v = C.p.v - C.n.v C.p.i = - (L.i + R.p.i)

C.i = C.p.ider(L.i) = C.v / L.L


C.n.v = L.n.v L.n.v = R.n.v C.n.i + L.n.i + R.n.i = 0

C.p.v = L.p.v L.p.v = R.p.v C.p.v = Ground1.p.vGround1.p.i = - (C.p.i + L.p.i + R.p.i)

Modelica Equation Set: 5 equations/variables


Ground


OnePort for R




voltagescurrents

Ground1.p.v = 0

R.R*R.i = R.v R.v = R.p.v - R.n.v 0 = R.p.i + R.n.iR.i = R.p.i

C.i = C.C*der(C.v)C.v = C.p.v - C.n.v 0 = C.p.i + C.n.i

C.i = C.p.iL.L*der(L.i) = C.v


C.n.v = L.n.v L.n.v = R.n.vC.C*der(C.v) + L.i + C.v / R.R

C.p.v = L.p.v L.p.v = R.p.v C.p.v = Ground1.p.vC.p.i + L.p.i + R.p.i + Ground1.p.i = 0

Modified Nodal Analysis Equation Set: 2 equations/variables


/* DSblock model generated by Dymola ... */#include .../* DSblock C-code: */ .../* Define variable names. */#define Sections_#define Ground1_p_v Variable(0)#define Ground1_p_i Variable(1)#define R_p_v Variable(2)#define L_L Parameter(2) ....

TranslatedEquations ....DynamicsSectionL_der_i = divmacro(C_v,"C.v",L_L,"L.L");R_p_i = divmacro(C_v,"C.v",R_R,"R.R");C_p_i = -(L_i+R_p_i);C_der_v = divmacro(C_p_i,"C.p.i",C_C,"C.C");

Ground1_p_i = -(C_p_i+L_i+R_p_i);EndTranslatedEquations ...

Simulator Dymola

Dymola simulation

C text


Modelica text


Output


- experiment StopTime=500 NumberOfIntervals=5000- C.v := 5 ;- simulate--------------------------------------------------------------------------Log-file of program .\dymosim(generated: Fri May 03 08:49:01 2002)

dymosim started ...Integration started at T = 0 using integration method DASSL(DAE multi-step solver (dassl/dasslrt of Petzold

modified by Dynasim))Integration terminated successfully at T = 500

CPU-time for integration : 0.13 secondsCPU-time for one GRID interval: 0.026 milli-secondsNumber of result points : 5001Number of GRID points : 5001Number of (successful) steps : 3355Number of F-evaluations : 6733Number of Jacobian-evaluations: 17Number of (model) time events : 0Number of (U) time events : 0

Simulator Dymola

Output

Dymola simulation

C text


Modelica text




Simulator Dymola

Dymola simulation

C text


Modelica text


Output


Two basic ideas for simulation acceleration:

1. New method “mixed mode integration” for real time simulation of systems which have “slow” and “fast” states:

- Slow states xS are discretized with the explicit Euler method.

- Fast states xF are discretized with the implicit Euler method.

2. Discretization formulae are inserted into model („inline integration“) and overall systems of equations are symbolically solved with Dymolas algorithmsto reduce the non-linear system and to generate analytical Jacobians !

Fn

Fn

Fn

Sn

Sn

Sn

nnnn

hh

t

xxx

xxx

yxxf0

&

&

&

⋅+=

⋅+=

=

−

−−

1

11

),,,(

• Local implicit small systems of equations• Automatic procedure for partitioning• Speed-up of 4 (diesel engine) ... 15 (detailed robot model).

Real-time simulation of stiff systems

slow

fast


5. Modelica librariesModelica Standard Library ( Modelica Association)

Electrical sublibraryBlock sublibraryRotational sublibraryTranslational sublibraryConstantsMathematical functionsSI unit types

HyLibLight Hydraulic Systems (Beater)ObjectStab ModelicaAdditions Library (DLR)Power Systems (Larsson) ExtendedPetriNetSystemDynamicsFuzzyControlATplus (Building simulation and control)Thermo fluid systems (Tummescheit)Electrical motors (Beuschel)Thermal systems (Tiller and DLR)PQLib (power quality analysis in networks)


Only commercially available:HyLib Hydraulic Systems – full version (Beater)PowerTrain (DLR) Fuel Cells (DLR)Thermal Building Behavior (U Kaiserslautern)Cooling and Heating Systems (U Hamburg-Harburg)TechThermo (DLR)Mass Flow in Process Plants (Zurich)Hybrid Electric Vehicles (U Gothenburg)Tyre Model Library (Royal Institute of Technology, S)Flight Dynamics Library (DLR)

Actual information:http://www.Modelica.org/libraries.html

Modelica libraries (cont’d)


Modelica.Thermal.Lumped



Modelica.Electrical.Analog



Modelica.Electrical.Analog.Basic




Modelica.Electrical.Analog.Ideal



Modelica.Electrical.Analog.LinesModelica.Electrical.Analog.Semiconductors

Modelica.Electrical.Analog.Sources


6. Application Examples


model Resistorextends OnePort;parameter Real R;

equationv = R*i;

end Resistor;

Example: Industrial robot (DLR, Dynasim, KUKA)

1000 nontrivial algebraic equations, 80 states.With “mixed-mode integration”:

faster as real time on 650 Mhz PC.


Example: hardware-in-the-loop simulation of automatic gearbox(various car manufacturers)

Electronic Control Unit(Hardware)

+ driver + motor + torque converter + 1D vehicle dynamics

desired clutch pressure

(Simulation)


Example: Large, detailed vehicle model (Ford Motors, Dynasim, DLR)

• ~ 250.000 equations before reduction

• ~ 25.000 nontrival algebraic equations

• 320 states

3D-Mechanics (60 joints, 70 bodies)

Motor (combustion)

Power train (automatic gear box)

Hydraulics


Example: Material Testing Machine (U Paderborn, FhG EAS)

Hydraulic machine, controlled by an electronic circuit

Number of Equations/Variables

before aftersymbolic reduction

total 1.031 487

without electronics 309 137


Different controller models tested, here: operational amplifier


7. Comparison: VHDL-AMS and ModelicaDifferences Modelica VHDL-AMS

Origin Control systems, mechatronics Electronics, IC’s

Simulation modes TR, DC TR, DC, AC, Steady-State, Noise

Netlists Components, connection of pins, Components, nodes nodes not explicitely used

Graphics Annotations (for Docum., icons) (via EDA-tools)

Compact data like MATLAB-Syntax, matrices Operator overloading

Model change Composition, configuration, Composition, configuration inheritance

Digital time scale Digital event at any time possible Minimal resolvable time

Delays Not yet elaborated Precisely defined ( MVL9, ... )

Structure of model Cut set law explicitely to formulate Defines internal branches

User’s training For experts of object oriented For VHDL experts: easy to learnlanguages: easy to learn

Graphical input: very simple


8. Conclusion

• Modelica/Dymola is well suited to model large parts of the physical components,such as 3-dim. vehicle dynamics, power train including automatic gearboxes, hydraulics, combustion, electric/electronics, air conditioning, etc.

• Matlab/Simulink is well suited for signal-flow oriented components (control system / state charts and for the technical infra-structure: download to signal processors / Real Time Workshop; many toolboxes).

• Combination of Modelica and Simulink is very attractive(Dymola can generate Simulink CMEX S-Function of Modelica models).

• VHDL-AMS (e.g. VeriasHDL and AdvanceMS) is very powerful for electronicsubsystems, but perhaps not so strong for other parts of a complex systems, such as 3-dim. mechanics, power train, combustion, ... (lack of libraries!).

• Modelica/Dymola is especially suited for real-time simulation, such ashardware-in-the-loop (HIL); VHDL-AMS/RT in preparation.

• Modelica is supported by many multi-domain libraries, VHDL-AMS by large digital (and, in future, analog electrical) libraries; SPICE compatibility in analog electronics!

How to model and simulate heterogeneous systems?

Mixed Domain Modeling in Modelica -...

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