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MetaModelica A Unified Equation-Based S emantical and Mathematical Modeling Language Adrian Pop and Peter Fritzson Programming Environment Laboratory Department of Computer and Information S cience Linkping University 2006-09-14 JMLC

  1. MetaModelica A Unified Equation-Based S emantical and Mathematical Modeling Language Adrian Pop and Peter Fritzson Programming Environment Laboratory Department of Computer and Information S cience Linköping University 2006-09-14 JMLC’ 2006, S eptember 13-15, Oxford, UK

  2. Outline � Modelica � Introduction � Language properties � Example � MetaModelica � Motivation � MetaModelica extensions to Modelica � Example � Future Work � Conclusions 2

  3. Modelica – General Formalism to Model Complex S ystems � Robotics � Automotive � Aircrafts � S atellites � Biomechanics � Power plants � Hardware-in-the-loop, real-time simulation � etc 3

  4. Modelica – The Next Generation Modeling Language � Declarat ive language � Equations and mathematical functions allow acausal modeling, high level specification, increased correctness � Mult i-domain modeling � Combine electrical, mechanical, thermodynamic, hydraulic, biological, control, event, real-time, etc... � Everyt hing is a class � S trongly typed obj ect-oriented language with a general class concept, Java & Matlab like syntax � Visual component programming � Hierarchical system architecture capabilities � Efficient , nonpropriet ary � Efficiency comparable to C; advanced equation compilation, e.g. 300 000 equations 4

  5. Modelica Language Properties � Declarat ive and Obj ect -Orient ed � Equat ion-based ; continuous and discrete equations � Parallel process modeling of concurrent applications, according to synchronous data flow principle � Funct ions with algorithms without global side-effects (but local data updates allowed) � Type syst em inspired by Abadi/ Cardelli (Theory of Obj ects) � Everyt hing is a class – Real, Integer, models, functions, packages, parameterized classes.... 5

  6. Modelica Background The Form - Equations Equations were used in the third millenium B.C. Equality sign was introduced by Robert Recorde in 1557 Newton (Principia, vol. 1, 1686) still wrote text: “The change of motion is proportional to the motive force impressed; ...” d ∑ ( m v ) F ⋅ = i dt CSSL (1967) introduced special form of “equation”: variable = expression v = INTEG(F) / m Programming languages usually do not allow equations 6

  7. Modelica Acausal Modeling Semantics • What is acausal modeling/design? • Why does it increase reuse ? The acausality makes Modelica classes more reusable than traditional classes containing assignment statements where the input-output causality is fixed. • Example: a resistor equation : R*i = v; can be used in three ways: i := v/R; v := R*i; R := v/i; 7

  8. Modelica - Reusable Class Libraries R= C= L= body= bodyBar= bar= Info Info G shaftS= inertial shaft3DS= shaft3D= Info S S cylBody= bodyShape= cylS= AC= DC= Vs S S Is S S S shaft= prismS= gear2= revS= screw S= univS planarS= sphereS freeS gear1= - S planetary= planet= S + V i S diff= ring= S prism= rev= D T Op sun= sphere cyl= univ planar= free : 1 bearing fixTooth S screw = E moveS move sphereC y barC= c= d= cSer= c= d= x torque C C barC2= fric= sensor force lineForce= lineSensor r s sd fricTab clutch= torque lineTorque= converter advanced drive translation w a t fixedBase Library Library Library S state 8

  9. Graphical Modeling - Drag and Drop Composition 9

  10. Hierarchical Composition Diagram for a Model of a Robot k2 i axis6 cut joint qddRef tn qdRef qRef k1 r3Control r3Drive1 r3Motor 1 1 i 1 S S qd axis5 l qdRef Kd S rel 0.03 joint=0 Jmotor=J spring=c S axis4 iRef qRef pSum Kv sum w Sum rate2 rate3 gear=i fric=Rv0 340.8 +1 b(s) 0.3 - +1 - a(s) S axis3 rate1 tacho2 tacho1 b(s) b(s) PT1 a(s) a(s) g5 axis2 q qd Rp2=50 Ra=250 C=0.004*D/w m Rp1=200 Rd1=100 axis1 La=(250/(2*D*w m)) Rd2=100 Ri=10 - - - Srel = n*n' + (identity(3) - n*n')*cos(q) - skew(n)*sin(q); + + diff + pow er wrela = n*qd; OpI zrela = n*qdd; Vs Rd4=100 Sb = Sa*Srel'; emf Rd3=100 r0b = r0a; y g3 x vb = Srel*va; inertial g1 wb = Srel*(wa + wrela); ab = Srel*aa; hall2 zb = Srel*(za + zrela + cross(wa, wrela)); hall1 w r fa = Srel'*fb; ta = Srel'*tb; q g4 qd g2 10

  11. Multi-Domain Modelica Model - DCMotor � A DC motor can be thought of as an electrical circuit which also contains an electromechanical component. model DCMotor Resistor R(R=100); Inductor L(L=100); VsourceDC DC(f=10); Ground G; ElectroMechanicalElement EM(k=10,J=10, b=2); Inertia load; equation R L connect(DC.p,R.n); EM connect(R.p,L.n); DC connect(L.p, EM.n); connect(EM.p, DC.n); load connect(DC.n,G.p); connect(EM.flange,load.flange); G end DCMotor 11

  12. Modelica compilation stages Modelica Source Code Modelica model Translator Flat model Analyzer Topologically sorted equations Optimizer Optimized sorted equations Code Generator C code C Compiler Executable Simulation 12

  13. Corresponding DCMotor Model Equations The following equations are automatically derived from the Modelica model: ( load component not included) 13

  14. Connector Classes, Components and Connections connector Pin Keyword flow indicates Voltage v; that currents of connected flow Current i; pins sums to zero. end Pin; A connect statement in Modelica connect(Pin1,Pin2) corresponds to Pin1.v = Pin2.v Pin1.i + Pin2.i = 0 Connection between Pin1 and Pin2 14

  15. Common Component Structure as SuperClass model TwoPin ”Superclass of elements with two electrical pins” Pin p,n; Voltage v; Current i; equation v = p.v – n.v; 0 = p.i + n.i; i = p.i; end TwoPin; 15

  16. Electrical Components Reuse TwoPin SuperClass model Resistor ”Ideal electrical resistor” extends TwoPin; parameter Real R ”Resistance”; equation R*i = u end Resistor; model Inductor ”Ideal electrical inductor” extends TwoPin; parameter Real L ”Inductance”; equation L*der(i) = u end Inductor; 16

  17. Corresponding DCMotor Model Equations The following equations are automatically derived from the Modelica model: ( load component not included) 17

  18. MetaModelica - Context � S yntax - there are many efficient parser generator tools � lex (flex), yacc (bison), ANTLR, Coco, etc. � S emant ics: � t here are no st andard efficient and easy t o use compiler-compiler t ools 18

  19. MetaModelica - Motivation � Can we adapt the Modelica equation-based style to define semantics of programming languages? � Answer: Yes! � MetaModelica is j ust a part of the answer � executable language specification based on � a model (abstract syntax tree) � semantic functions over the model � elaboration and typechecking � translation � meta-programming � transformation � etc. � Further improvement – more reuse of language specification parts when building specifications for a new language (Future Work) 19

  20. MetaModelica - Idea � We started from � The Relational Meta-Language (RML) � A system for building executable natural semantics specifications � Used to specify Java, Pascal-subset, C-subset, Mini-ML, etc. � The OpenModelica compiler for Modelica specified in RML � Idea: int egrat e RML met a-modeling and met a- programming facilit ies wit hin OpenModelica by ext ending t he Modelica language. The not ion of equat ion is used as t he unifying feat ure � Now we have � The MetaModelica language � The Modelica executable language specification (OpenModelica compiler) in MetaModelica (~114232 lines of code) � Meta-programming facilities for Modelica 20

  21. MetaModelica extensions to Modelica (I) � Modelica � classes, models, records, functions, packages � behaviour is defined by equations or/ and functions � equations � differential equations � algebraic equations � partial differential equations � difference equations � conditional equations � MetaModelica extensions � local equations � pattern equations � match expressions � lists, tuples, option and uniontypes 21

  22. MetaModelica extensions to Modelica (II) � pattern equations � unbound variables get their value by unification Env.BOOLVAL(x,y) = eval_something(env, e); � match expressions � pattern matching � case rules pattern := match expression opt ional-local-declarat ions case pat t ern-expression opt -local-declarat ions opt ional-local-equat ions then value-expr ession; case ... ... else opt ional-local-declarat ions opt ional-local-equat ions then value-expr ession; end match; 22

  23. MetaModelica – Example (I) package ExpressionEvaluator // abstract syntax declarations ... // semantic functions ... end ExpressionEvaluator; 23

  24. MetaModelica – Example (II) PLUS package ExpressionEvaluator RCONST MUL // abstract syntax declarations uniontype Exp 12 record RCONST Real x1; end RCONST; RCONST RCONST record PLUS Exp x1; Exp x2; end PLUS; record SUB Exp x1; Exp x2; end SUB; 5 13 record MUL Exp x1; Exp x2; end MUL; record DIV Exp x1; Exp x2; end DIV; record NEG Exp x1; end NEG; end Exp; Expression: 12+5*13 Representation: PLUS( // semantic functions RCONST(12), ... MUL( RCONST(5), RCONST(13) end ExpressionEvaluator; ) ) 24

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