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A new tool for visual modeling - Rand Model Designer 7.
February 18 - 20, 2015. Vienna University of Technology, Vienna, 8th Vienna International Conference on Modelling 8th Conference on Mathematical Mathematical Modelling Austria 8th Vienna Vienna International International Conference Mathematical Modelling 8th Vienna International Conference on on Mathematical Available onlineModelling at Vienna, www.sciencedirect.com February of February 18 18 --- 20, 20, 2015. 2015. Vienna Vienna University University of Technology, Technology, Vienna, February 18 20, 2015. Vienna University of Technology, Vienna, February 18 20, 2015. Vienna University of Technology, Vienna, Austria Austria Austria Austria
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IFAC-PapersOnLine 48-1 (2015) 661–662 A new tool for visual modeling - Rand Model Designer 7. A new tool for visual modeling -- Rand Model Designer 7. A visual Model Designer A new new tool tool for forAndrey visualA.modeling modeling - Rand Rand Model Designer 7. 7. Isakov *, Yury B. Kolesov *,
Yury B. Senichenkov* Andrey A. Isakov *, Yury B. Kolesov *, Andrey A. Isakov *, Yury B. Kolesov *, Andrey Andrey A. A. Isakov Isakov *, *, Yury Yury B. B. Kolesov Kolesov *, *, Yury B. Senichenkov* Yury B. Senichenkov* Yury B. Senichenkov* Yury B. Senichenkov* *National Research University St. Petersburg State Polytechnical University, Russia, 195251, St. Petersburg, Polytechnicheskaya, 29 (Tel: +7 (812) 297-1616; e-mail: senyb@ dcn.icc.spbstu.ru). *National Research University St. Petersburg State Polytechnical University, Russia, 195251, St. Petersburg, *National Research University St. Petersburg State Polytechnical University, Russia, 195251, St. Petersburg, *National *National Research Research University University St. St. Petersburg Petersburg State State Polytechnical Polytechnical University, University, Russia, Russia, 195251, 195251, St. St. Petersburg, Petersburg, Polytechnicheskaya, 29 (Tel: +7 (812) 297-1616; e-mail: senyb@ dcn.icc.spbstu.ru). Polytechnicheskaya, 29 (Tel: +7 (812) 297-1616; e-mail: senyb@ dcn.icc.spbstu.ru). Polytechnicheskaya, 29 (Tel: +7 (812) 297-1616; e-mail: senyb@ dcn.icc.spbstu.ru). Polytechnicheskaya, 29 (Tel: +7 (812) 297-1616; e-mail: senyb@ dcn.icc.spbstu.ru). Abstract: MvStudium research group has announced the new version of visual environment for modeling and simulation of complex dynamical systems Rand Model Designer 7. Itthe hasnew twoversion principal differences in comparison with previous versions: of а) Abstract: MvStudium research group has announced of visual environment for modeling and simulation Abstract: MvStudium research group has announced the new version of visual environment for modeling and simulation of Abstract: MvStudium research group has the new version environment modeling and of Abstract: MvStudium research group hasofannounced announced thewith new«input-output» version of of visual visual environment for forexternal modeling and simulation simulation of now it is possible using dynamic objects components or «contact-flow» variables for solving complex dynamical systems Rand Model Designer 7. It has two principal differences in comparison with previous versions: а) complex systems Rand Model 7. has two differences in comparison with versions: а) complex dynamical systems Rand Model Designer 7. It has two principal in with previous versions: а) complex dynamical dynamical systems Rand Model Designer Designer 7. It It b) hasglobal two principal principal differences in comparison with previous previous versions: а) problems of queueing theory («agent-based» approach); system ofdifferences equations forcomparison a local behavior of component model now it is possible using dynamic objects of components with «input-output» or «contact-flow» external variables for solving now it is possible using dynamic objects of components with «input-output» or «contact-flow» external variables for solving now it is possible using dynamic objects of components with «input-output» or «contact-flow» external variables for solving now it is possible using dynamic objects of components with «input-output» or «contact-flow» external variables for solving with hybrid behavior is always built, analyzed and transformed onsystem run time. problems of queueing theory («agent-based» approach); b) global of equations for a local behavior of component model problems problems of queueing theory («agent-based» approach); b) global system of equations for local behavior of component model problems of of queueing queueing theory theory («agent-based» («agent-based» approach); approach); b) b) global global system system of of equations equations for for aaa local local behavior behavior of of component component model model with hybrid behavior is always built, analyzed and transformed on run time. with hybrid behavior is always built, analyzed and transformed on run time. with behavior is always built, analyzed and transformed on run time. Keywords: modeling and simulation, visual environment, complex dynamical systems, hybrid systems, with hybrid hybrid© behavior is always built, analyzed and transformed on run time. 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. component modeling models, automatic building and transformingcomplex equations, algebraic-differential equations, Keywords: and simulation, visual Keywords: modeling and simulation, visual environment, dynamical systems, hybrid systems, Keywords: modeling andtriangular simulation, visual environment, environment, complex complex dynamical dynamical systems, systems, hybrid hybrid systems, systems, Keywords: modeling and simulation, visual environment, complex dynamical systems, hybrid systems, structural analysis, block form. component models, automatic building and transforming equations, algebraic-differential equations, component models, automatic building and transforming equations, algebraic-differential equations, component models, automatic building and transforming equations, algebraic-differential equations, component models, automatic building and transforming equations, algebraic-differential equations, structural analysis, block triangular form. structural analysis, block triangular form. structural structural analysis, analysis, block block triangular triangular form. form. MvStudiumGroup research group supports three visual tools detecting or transforming its structure on run time is for modeling and simulation complex dynamical systems: considered.or transforming its structure on run time is MvStudiumGroup research group supports three visual tools detecting MvStudiumGroup research group three detecting or or transforming transforming its its structure structure on on run run time time is is MvStudiumGroup research group supports three visual tools MvStudiumGroup research group supports supports three visual visual tools tools detecting detecting or transforming structure on time is Rand Model Designer (www.rand-service.com, The simplest RMD-7 user’sitsequation form isrun a form for for modeling and simulation complex dynamical systems: considered. for modeling and simulation complex dynamical systems: considered. for complex dynamical systems: for modeling modeling and and simulation simulation complex using, dynamical systems: considered. considered. www.mvstudium.com) for common TransasProf describing local behavior of isolated hybrid system. It is a Rand Model Designer (www.rand-service.com, Rand Model Designer (www.rand-service.com, The The simplest simplest RMD-7 RMD-7 user’s user’s equation equation form form is is aaaa form form for for Rand Model Designer (www.rand-service.com, The simplest RMD-7 user’s equation form is form for Rand Model Designer (www.rand-service.com, The simplest RMD-7 user’s equation form is form for (www.transas.com) for designing real time marine training system of differential equations with substitutions (1): www.mvstudium.com) for common using, TransasProf describing local behavior of isolated hybrid system. It is a www.mvstudium.com) for common using, TransasProf describing local behavior of isolated hybrid system. It is www.mvstudium.com) for common using, TransasProf describing local behavior of isolated hybrid system. It is www.mvstudium.com) for common using, TransasProf describing local behavior of isolated hybrid system. It is aaa simulators, and open-source tool time OpenMVLShell ds (www.transas.com) for designing real marine training system of differential equations with substitutions (1): (www.transas.com) for for designing designing real real time time marine marine training training system system of differential differential equations with substitutions (1): w Subst ( w, C ), F ( equations , s, w, t, C ) 0substitutions , Out ( y, s, w(1): , C ) 0, (www.transas.com) of with (www.transas.com) for designing real time marine of differential equations with substitutions (1): (https://dcn.ftk.spbstu.ru/) for education (Isakovtraining and system dt simulators, and open-source tool OpenMVLShell ds simulators, and open-source tool OpenMVLShell ds simulators, and open-source tool OpenMVLShell ds simulators, (2010, and open-source tool w Subst (( w ,,, C ), F ((( dsds ,,, sss,,, w ,,, ttt,,, C ))) 0 ,,, Out ((( yyy,,, sss,,, w ,,, C ))) 0 ,,, . Senichenkov 2011)). All ofeducation them useOpenMVLShell object-oriented w Subst w C ), F w C 0 Out w C 0 w Subst C ), w C 0 Out C 0 (https://dcn.ftk.spbstu.ru/) w Subst w , C ), F ( , s , w , t , C ) 0 , Out ( y , s w , C ) 0 (https://dcn.ftk.spbstu.ru/) for for education education (Isakov (Isakov and and k 1 (( w k 2F n m, w dt (https://dcn.ftk.spbstu.ru/) for (Isakov and dt w ; C ; Fdt , , s ; Out , y (1) , .. modeling language Model Vision Language (Kolesov and (https://dcn.ftk.spbstu.ru/) for education (Isakov dt Senichenkov (2010, 2011)). All of them use object-oriented .. Senichenkov (2010, 2011)). All of them use object-oriented dt Senichenkov (2010, (2010, 2011)). All of of them use use object-oriented ds kk 1 n Senichenkov 2011)). All them object-oriented (2007)). Model Vision Language (MVL), 1 ; C kkk 2 2 ; F , ds n ; Out , y m m ds kk 1 2 n m w , s (1) ds modeling language Model Vision Language (Kolesov and 1 k 2 n m w ; C ; F , , s ; Out , y (1) modeling language Model Vision Language (Kolesov and k 1 k 2 n m It contains algebraic-differential equations w ,, ss (1) modeling language Model Language w ;; C C ;; F F ,, dt ;; Out Out ,, yy (1) modeling Modelica language language Model Vision Vision Language (Kolesov and likewise (Fritzson (2011)),(Kolesov oriented and on dt dt Senichenkov dt Senichenkov (2007)). (2007)). Model Model Vision Vision Language Language (MVL), (MVL), It ds contains Senichenkov (2007)). Model Vision Language (MVL), n Senichenkov (2007)). Model Vision Language (MVL), algebraic-differential equations modeling hierarchical event-driven component systems. It is It contains algebraic-differential equations to state variables s , with w, t, c) 0 , respect algebraic-differential equations likewise ItF ( , s,contains contains algebraic-differential equations likewise Modelica Modelica language language (Fritzson (Fritzson (2011)), (2011)), oriented oriented on on It likewise Modelica language (Fritzson (2011)), oriented on likewise Modelica language (Fritzson (2011)), oriented on dt based on Unified Modeling Language’s de facto standards, ds n ds modeling hierarchical event-driven component systems. It is n ds ,, ss,, w modeling hierarchical hierarchical event-driven event-driven component component systems. systems. It It is is n respect to F ( ds modeling respect to state state variables variables with F w,,, ttt,,, ccc))) C0 0 ,,,and modeling hierarchical event-driven component is respect to state variables with accommodating them for hybrid systems. Hybridsystems. systemsItare F , s, w 0 constancies substitutions and w Substsss( w, Cnn),,,, with F ((( dt based dt , s, w, t, c) 0 , respect to state variables s , with based on on Unified Unified Modeling Modeling Language’s Language’s de de facto facto standards, standards, dt based on Unified Modeling Language’s de facto standards, considered as an extension of classical dynamical systems dt based on Unified Modeling Language’s de facto standards, output Out ( y,C , C, t )substitutions 0 respect w accommodating constancies (( w C accommodating them them for for hybrid hybrid systems. systems. Hybrid Hybrid systems systems are are equations constancies Cs, wand and substitutions and wto Subst Subst w,,,variables C ))) ,,, and accommodating them for hybrid systems. Hybrid are C (Senichenkov (2004), and Senichenkov (2006), accommodating them forKolesov hybrid systems. Hybrid systems systems are constancies constancies C and and substitutions substitutions w and w Subst Subst(( w w, C C ) , and considered as an extension of classical dynamical systems considered as an extension of classical dynamical systems ds respect to output variables equations Out ( y , s , w , C , t ) 0 considered as an extension of classical dynamical systems m respect to output variables equations Out ( y , s , w , C , t ) 0 considered asSenichenkov an extension of classical dynamical systems equations Kolesov and (2012 ,2014). respect to output variables Out ( y , s , w , C , t ) 0 F ( , s , w , t ) 0 . Systems of equations in form y respect to output variables equations Out ( y , s , w , C , t ) 0 (Senichenkov (Senichenkov (2004), (2004), Kolesov Kolesov and and Senichenkov Senichenkov (2006), (2006), (Senichenkov (2004), Kolesov and Senichenkov (2006), dt (Senichenkov (2004), Kolesov and Senichenkov (2006), ds concept of MVL is «active dynamical object» with The main m ds m ds ,, ss,, w Kolesov F ( ds , ttt ))) 0 Systems of equations form yyy Kolesov and and Senichenkov Senichenkov (2012 (2012 ,2014). ,2014). m F w 0 Systems of equations in form Kolesov and Senichenkov (2012 ,2014). will bem transformedin m ....automatically ,, ssof ,, w Systems in form Kolesov and Senichenkov (2012 ,2014). global behavior described by«active Behavior-Chart (modification F (((help w,,,Gear’s t) 0 0 Systems of of equations equations in with formtheF y dt dt concept of MVL is dynamical object» with The main dt concept of MVL is «active dynamical object» with The main concept of MVL MVLand is «active «active dynamicalinobject» object» with The main concept substitution to dt of is dynamical with The main of UML’s State Machine) local behaviors the form of will be automatically transformed with the help of Gear’s will be automatically transformed with the help of Gear’s global behavior described by Behavior-Chart (modification will be be automatically automatically transformed with with the the help help of of Gear’s Gear’s global behavior behavior described described by by Behavior-Chart Behavior-Chart (modification (modification will transformed global ds to algebraic-differential equations (Kolesov andin Senichenkov global behavior Machine) described by Behavior-Chart (modification substitution to of z; F ( z, s, w, t ) 0, s(0) s0 (2). substitution to of UML’s UML’s State State Machine) Machine) and and local local behaviors behaviors in in the the form form of of substitution substitution to of UML’s State and local behaviors the form of (2007)). of UML’s State Machine) and local behaviors inSenichenkov the form of dt algebraic-differential equations (Kolesov and ds ds algebraic-differential equations (Kolesov and Senichenkov ds algebraic-differential equations (Kolesov Senichenkov (2). z; F w , t) 0 , s(0 ) algebraic-differential equations (Kolesovforand and ds Building global systems of equations composition of Special cases (2). (((( zzzz,,,, ssss,,,, w 0 ssss000 form Senichenkov (2). F w 0 0 (linear (2007)). (2).(2) are zzz;;; andF variants) of F non-linear w,,, ttt ))) 0,,, sss(((0 0))) (2007)). dt 0 dt (2007)). 0 dt component’s automata is the main difficulty for component (2007)). dt Building global systems of equations for composition of ds Building global systems of equations for composition of Special cases (linear and non-linear variants) of form (2) are Building global systems of for of variants) models with hybrid behavior. Whendifficulty number of components Fand ( s, wnon-linear , t) Building global systems of equations equations for composition composition of Special Special cases cases(linear (linear and non-linear variants) of of form form (2) (2) are are Special cases (linear and non-linear variants) of form (2) are component’s automata is the main for component component’s automata is the main difficulty for component a 2 . dt component’s automata is the main difficulty for component ds component’s automata is the main difficulty for component and number component’s hybrid automata states increase, the ds ds models with hybrid behavior. When number of components F ((( sss,,,,tw ds( y G models with hybrid behavior. When number of components hybrid behavior. When of models F w ) ,,,,tttt ))))0, s(0) s0 models with with hybrid behavior. When number of components components , sF F, w w ( s, w 2 .a number of component’s possible states for number composition became dt dt 2 a 2 states increase, the and number hybrid automata dt a 2...a and number component’s hybrid automata states increase, the dt and hybrid states the and number number component’s component’s hybrid automata automata states increase, increase, the extraordinarily large. Alternative for building all possible sss00 ,,, ttt ))) 0 ,,, sss(((0 ))) G ds(((( yyyy,,,, ssss,,,, w G w 0 0 number of possible states for composition became G w 0 0 number of possible states for composition became G w t s s , ) 0 , ( 0 ) number of states for composition became F ( s, w, t ), s(0) s0000, 2.b systems is building systems possible directly number beforehand of possible possible states only for realized composition became extraordinarily dt extraordinarily large. large. Alternative Alternative for for building building all all possible possible ds extraordinarily large. Alternative for building all ds on run time. In large scale event-driven systems the number ds extraordinarily large. Alternative for building all possible F s(0 s , ( s, w , t ), ) 2 .b systems 2 systems beforehand beforehand is is building building only only realized realized systems systems directly directly F w b ), 0 2 systems beforehand is building only realized systems directly F, t((()sss,,, w w0,,, ttt ), b wF ), sss(((0 0))) sss00000 ,,, 2...b Gds ( s, 2.c dt of realized local behaviors isevent-driven large It systems is very important to systems beforehand isscale building onlytoo. realized systems directly dt dt on run time. In large the number dt on run run time. time. In In large large scale scale event-driven event-driven systems systems the the number number The linear form on automatically. block know structure solving system. It Systems withnumber on runthe time. In largeofscale event-driven systems the G s, recognized w (is ,,, ttt ))) 0 2.c G w 0 2.c of G 2.c of realized realized local local behaviors behaviors is is large large too. too. It It is is very very important important to to There are special G ((( sss,,, w wSolvers , t) 0 0 for each type of equations. 2.c of realized local behaviors is large too. is very important to The triangular structure are abundant problems in practice. Using of realized local behaviors is large too. It is very important to The linear form is recognized automatically. know the structure of solving system. Systems with block The linear form is recognized automatically. know the structure of solving system. Systems with block The linear form is recognized automatically. know the structure of solving system. Systems with block The linear form is recognized automatically. know the structure of solving system. Systems with block special role plays Solvers for Non-linear Algebraic Equations numerical methods taking block triangular structure in There are special Solvers for each type of equations. The triangular structure are abundant problems in practice. Using There Solvers for of The triangular structure are abundant problems in practice. Using There are special Solvers for each type of equations. The triangular structure are problems in Using There are areTospecial special Solvers for each each type type of equations. equations. The triangular structurespeed are abundant abundant problems in practice. practice. Using (NAE). solveSolvers NAE isfor necessary in implicit methods for account increases of computer modeling. The problem special role plays Non-linear Algebraic Equations numerical methods taking block triangular structure in special role plays Solvers for Non-linear Algebraic Equations numerical methods taking block triangular structure in special role roleDifferential plays Solvers Solvers for Non-linear Non-linear Algebraic Equations numerical methods taking block structure in special plays for Algebraic Equations numerical methods taking block triangular triangular structure in Ordinary Equations (ODE), in methods for of automatic building global equations for local behavior of (NAE). account increases speed of computer modeling. The problem (NAE). To To solve solve NAE NAE is is necessary necessary in in implicit implicit methods methods for for account increases speed of computer modeling. The problem (NAE). To solve NAE is necessary in implicit methods for account speed of modeling. problem account increases increases speed of computer computer modeling.andThe Theautomatic problem finding consistent initial conditions and solving Algebraic(NAE). To solve NAE is necessary in implicit methods for component model with hybrid behavior Ordinary Differential Equations (ODE), in methods for of automatic building global equations for local behavior of Ordinary Differential Equations (ODE), in methods for of automatic building global equations for local behavior of Ordinary Differential Equations (ODE), in methods for of automatic building global equations for local behavior of Differential Equations (ODE), in methods for of automatic building global equations for local behavior of Ordinary Differential Equations (DAE). NAE are solved with the help finding consistent initial conditions and solving Algebraiccomponent model with hybrid behavior and automatic finding consistent initial conditions and solving Algebraiccomponent model with hybrid behavior and automatic finding consistent initial conditions and solving Algebraiccomponent model with hybrid behavior and automatic consistent initial conditions and solving Algebraiccomponent model with hybrid behavior and automatic finding Differential Differential Equations Equations (DAE). (DAE). NAE NAE are are solved solved with with the the help help Differential Equations (DAE). NAE are solved with the help Differential Equations (DAE). NAE are solved with the help
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MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria Andrey A. Isakov et al. / IFAC-PapersOnLine 48-1 (2015) 661–662 662
of Newton method, and in case of its failure, Powell method is used. Solving of linear systems of algebraic equations (SLAE) is the main operation for Newton method in turn. Newton method may choose appropriate modification of Gauss method with taking structure of solved system matrix in account: dense, band, sparse, block triangular. Any techniques decreasing computational costs of numerical solution are very important. Computing matrix block triangular form if possible is well known and commonly used trick (Duff (1977)). In RMD-7 this technique is used for structural matrix of equations. Structural matrix of a system is a matrix with {0,1} elements pointing out occurrence of unknowns in equations. RMD computes block triangular form for structural matrix of any allowed types of systems (SLAE, NAE, ODE, DAE) with the help of Tarjan algorithm and solves only subsystems for strongly connected blocks. Tarjan algorithm in RMD is used for reordering substitutions to computable sequence of formulas and detecting equations («algebraic loops») among them too. Building and transformation global equations cause maximum difficulties for component models with «contactsflow» external variables («acausal blocks»). It is well known that if even local behaviors of components’ automata do not contain high-index DAE, they can appear in their composition. So it is necessary to analyze all states of composition, if we want detect and build all high-index DAE systems beforehand. RMD’s Analyzer detects high-index DAE on run time and builds new additional equation for numerical differentiation. The structure of RND’s Numerical Library and algorithm of interaction between Numerical library and Model Engine have been changed for implementing new approach to building global equations on run time. If any new event has occurred, the control program estimates necessity of rebuilding of current solved system. New system is built if it is necessary. If the current global system has full transversal: Tarjan algorithm is used for building block triangular form of structural matrix. Strongly connected components (diagonal blocks of reordered structural matrix and associated with them systems of equations) are analyzing for choosing appropriate Solvers. Solving a system of equations corresponding to diagonal block with the help of suitable Solver will be named «subtask». Next step is building the condensation of the Tarjan’s algorithm graph. The condensation is used for construction subtask queue. Subtasks may be executed sequentially or parallel. The information about computer hardware (number of processors, number of kernels for a processor) needed for creation a thread pool for parallel execution is determined automatically. Threads are loaded by subtasks which are ready for execution. The subtask readiness for execution determinates using condensation. Initially all nodes (subtasks) of the condensation are marked as «Unresolved». If an «unresolved» node has no input edges or input edges start from the nodes with solved systems then control program changes its status for «Ready to start», otherwise it will have status «Not ready to start». Solving subtask’s
system has name «Solving», after ending of solving it becomes «Solved». The calculation comes to an end when all nodes become «Solved». The new approach with block triangular form, different Solvers for each subtask, and threads was compared with old one. For comparison was used a set of models developed by Transas company (http://www.transas.com/products). The results of numerical experiment for most difficult problem are shown in Table 1. The computer used for calculations had four processors, so it was possible to create maximum four threads, but even if only one thread was used then total time of calculations decreased in two times. Table 1. Product Tanker, Cargo System, about 2500 equations.
REFERENCES Duff I.S. (1977). MA28—A set of Fortran subroutines for sparse unsymmetric linear equations. Technical Report AERE R8730 (SR-2040 8.0 edition) HMSO, London Isakov A.A., Senichenkov Yu. B. (2010). OpenMVL is a tool for modelling and simulation. //Computing, measuring, and control systems, pp. 84-89, SPBSPU, St. Petersburg. Isakov A.A., Senichenkov Yu. B. (2011). Program Complex OpenMVL for Modeling Complex Dynamical Systems // «Differential equations and control processes» Journal, Mathematics and Mechanics Faculty of Saint-Petersburg State University, St. Petersburg. Kolesov Yu.B., Senichenkov Yu.B. (2006) Modelling of systems. Dynamical and Hybrid systems. 224 pp, BHV, St. Petersburg. Kolesov Yu.B., Senichenkov Yu.B. (2007) Modelling of systems. Object-oriented approach. 256 pp, BHV, St. Petersburg. Kolesov Yu.B., Senichenkov Yu.B. (2012) Mathematical Modelling of component systems. 223 pp, SPbSTU, St. Petersburg. Kolesov Yu.B., Senichenkov Yu.B. (2014) Mathematical Modelling of hybrid systems. 236 pp, SPbSTU, St. Petersburg. Pissanetsky S. (1984). Sparse matrix technology, Аcademmic Press, London. Senichenkov Yu. B. (2004) Numerical modelling of hybrid systems.206 pp, SPbSTU, St. Petersburg. Fritzson P. (2011) Introduction to Modeling and Simulation of Technical and Physical Systems with Modelica, 232 pp, Wiley-IEEE Press.
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IFAC-PapersOnLine 48-1 (2015) 661–662 A new tool for visual modeling - Rand Model Designer 7. A new tool for visual modeling -- Rand Model Designer 7. A visual Model Designer A new new tool tool for forAndrey visualA.modeling modeling - Rand Rand Model Designer 7. 7. Isakov *, Yury B. Kolesov *,
Yury B. Senichenkov* Andrey A. Isakov *, Yury B. Kolesov *, Andrey A. Isakov *, Yury B. Kolesov *, Andrey Andrey A. A. Isakov Isakov *, *, Yury Yury B. B. Kolesov Kolesov *, *, Yury B. Senichenkov* Yury B. Senichenkov* Yury B. Senichenkov* Yury B. Senichenkov* *National Research University St. Petersburg State Polytechnical University, Russia, 195251, St. Petersburg, Polytechnicheskaya, 29 (Tel: +7 (812) 297-1616; e-mail: senyb@ dcn.icc.spbstu.ru). *National Research University St. Petersburg State Polytechnical University, Russia, 195251, St. Petersburg, *National Research University St. Petersburg State Polytechnical University, Russia, 195251, St. Petersburg, *National *National Research Research University University St. St. Petersburg Petersburg State State Polytechnical Polytechnical University, University, Russia, Russia, 195251, 195251, St. St. Petersburg, Petersburg, Polytechnicheskaya, 29 (Tel: +7 (812) 297-1616; e-mail: senyb@ dcn.icc.spbstu.ru). Polytechnicheskaya, 29 (Tel: +7 (812) 297-1616; e-mail: senyb@ dcn.icc.spbstu.ru). Polytechnicheskaya, 29 (Tel: +7 (812) 297-1616; e-mail: senyb@ dcn.icc.spbstu.ru). Polytechnicheskaya, 29 (Tel: +7 (812) 297-1616; e-mail: senyb@ dcn.icc.spbstu.ru). Abstract: MvStudium research group has announced the new version of visual environment for modeling and simulation of complex dynamical systems Rand Model Designer 7. Itthe hasnew twoversion principal differences in comparison with previous versions: of а) Abstract: MvStudium research group has announced of visual environment for modeling and simulation Abstract: MvStudium research group has announced the new version of visual environment for modeling and simulation of Abstract: MvStudium research group has the new version environment modeling and of Abstract: MvStudium research group hasofannounced announced thewith new«input-output» version of of visual visual environment for forexternal modeling and simulation simulation of now it is possible using dynamic objects components or «contact-flow» variables for solving complex dynamical systems Rand Model Designer 7. It has two principal differences in comparison with previous versions: а) complex systems Rand Model 7. has two differences in comparison with versions: а) complex dynamical systems Rand Model Designer 7. It has two principal in with previous versions: а) complex dynamical dynamical systems Rand Model Designer Designer 7. It It b) hasglobal two principal principal differences in comparison with previous previous versions: а) problems of queueing theory («agent-based» approach); system ofdifferences equations forcomparison a local behavior of component model now it is possible using dynamic objects of components with «input-output» or «contact-flow» external variables for solving now it is possible using dynamic objects of components with «input-output» or «contact-flow» external variables for solving now it is possible using dynamic objects of components with «input-output» or «contact-flow» external variables for solving now it is possible using dynamic objects of components with «input-output» or «contact-flow» external variables for solving with hybrid behavior is always built, analyzed and transformed onsystem run time. problems of queueing theory («agent-based» approach); b) global of equations for a local behavior of component model problems problems of queueing theory («agent-based» approach); b) global system of equations for local behavior of component model problems of of queueing queueing theory theory («agent-based» («agent-based» approach); approach); b) b) global global system system of of equations equations for for aaa local local behavior behavior of of component component model model with hybrid behavior is always built, analyzed and transformed on run time. with hybrid behavior is always built, analyzed and transformed on run time. with behavior is always built, analyzed and transformed on run time. Keywords: modeling and simulation, visual environment, complex dynamical systems, hybrid systems, with hybrid hybrid© behavior is always built, analyzed and transformed on run time. 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. component modeling models, automatic building and transformingcomplex equations, algebraic-differential equations, Keywords: and simulation, visual Keywords: modeling and simulation, visual environment, dynamical systems, hybrid systems, Keywords: modeling andtriangular simulation, visual environment, environment, complex complex dynamical dynamical systems, systems, hybrid hybrid systems, systems, Keywords: modeling and simulation, visual environment, complex dynamical systems, hybrid systems, structural analysis, block form. component models, automatic building and transforming equations, algebraic-differential equations, component models, automatic building and transforming equations, algebraic-differential equations, component models, automatic building and transforming equations, algebraic-differential equations, component models, automatic building and transforming equations, algebraic-differential equations, structural analysis, block triangular form. structural analysis, block triangular form. structural structural analysis, analysis, block block triangular triangular form. form. MvStudiumGroup research group supports three visual tools detecting or transforming its structure on run time is for modeling and simulation complex dynamical systems: considered.or transforming its structure on run time is MvStudiumGroup research group supports three visual tools detecting MvStudiumGroup research group three detecting or or transforming transforming its its structure structure on on run run time time is is MvStudiumGroup research group supports three visual tools MvStudiumGroup research group supports supports three visual visual tools tools detecting detecting or transforming structure on time is Rand Model Designer (www.rand-service.com, The simplest RMD-7 user’sitsequation form isrun a form for for modeling and simulation complex dynamical systems: considered. for modeling and simulation complex dynamical systems: considered. for complex dynamical systems: for modeling modeling and and simulation simulation complex using, dynamical systems: considered. considered. www.mvstudium.com) for common TransasProf describing local behavior of isolated hybrid system. It is a Rand Model Designer (www.rand-service.com, Rand Model Designer (www.rand-service.com, The The simplest simplest RMD-7 RMD-7 user’s user’s equation equation form form is is aaaa form form for for Rand Model Designer (www.rand-service.com, The simplest RMD-7 user’s equation form is form for Rand Model Designer (www.rand-service.com, The simplest RMD-7 user’s equation form is form for (www.transas.com) for designing real time marine training system of differential equations with substitutions (1): www.mvstudium.com) for common using, TransasProf describing local behavior of isolated hybrid system. It is a www.mvstudium.com) for common using, TransasProf describing local behavior of isolated hybrid system. It is www.mvstudium.com) for common using, TransasProf describing local behavior of isolated hybrid system. It is www.mvstudium.com) for common using, TransasProf describing local behavior of isolated hybrid system. It is aaa simulators, and open-source tool time OpenMVLShell ds (www.transas.com) for designing real marine training system of differential equations with substitutions (1): (www.transas.com) for for designing designing real real time time marine marine training training system system of differential differential equations with substitutions (1): w Subst ( w, C ), F ( equations , s, w, t, C ) 0substitutions , Out ( y, s, w(1): , C ) 0, (www.transas.com) of with (www.transas.com) for designing real time marine of differential equations with substitutions (1): (https://dcn.ftk.spbstu.ru/) for education (Isakovtraining and system dt simulators, and open-source tool OpenMVLShell ds simulators, and open-source tool OpenMVLShell ds simulators, and open-source tool OpenMVLShell ds simulators, (2010, and open-source tool w Subst (( w ,,, C ), F ((( dsds ,,, sss,,, w ,,, ttt,,, C ))) 0 ,,, Out ((( yyy,,, sss,,, w ,,, C ))) 0 ,,, . Senichenkov 2011)). All ofeducation them useOpenMVLShell object-oriented w Subst w C ), F w C 0 Out w C 0 w Subst C ), w C 0 Out C 0 (https://dcn.ftk.spbstu.ru/) w Subst w , C ), F ( , s , w , t , C ) 0 , Out ( y , s w , C ) 0 (https://dcn.ftk.spbstu.ru/) for for education education (Isakov (Isakov and and k 1 (( w k 2F n m, w dt (https://dcn.ftk.spbstu.ru/) for (Isakov and dt w ; C ; Fdt , , s ; Out , y (1) , .. modeling language Model Vision Language (Kolesov and (https://dcn.ftk.spbstu.ru/) for education (Isakov dt Senichenkov (2010, 2011)). All of them use object-oriented .. Senichenkov (2010, 2011)). All of them use object-oriented dt Senichenkov (2010, (2010, 2011)). All of of them use use object-oriented ds kk 1 n Senichenkov 2011)). All them object-oriented (2007)). Model Vision Language (MVL), 1 ; C kkk 2 2 ; F , ds n ; Out , y m m ds kk 1 2 n m w , s (1) ds modeling language Model Vision Language (Kolesov and 1 k 2 n m w ; C ; F , , s ; Out , y (1) modeling language Model Vision Language (Kolesov and k 1 k 2 n m It contains algebraic-differential equations w ,, ss (1) modeling language Model Language w ;; C C ;; F F ,, dt ;; Out Out ,, yy (1) modeling Modelica language language Model Vision Vision Language (Kolesov and likewise (Fritzson (2011)),(Kolesov oriented and on dt dt Senichenkov dt Senichenkov (2007)). (2007)). Model Model Vision Vision Language Language (MVL), (MVL), It ds contains Senichenkov (2007)). Model Vision Language (MVL), n Senichenkov (2007)). Model Vision Language (MVL), algebraic-differential equations modeling hierarchical event-driven component systems. It is It contains algebraic-differential equations to state variables s , with w, t, c) 0 , respect algebraic-differential equations likewise ItF ( , s,contains contains algebraic-differential equations likewise Modelica Modelica language language (Fritzson (Fritzson (2011)), (2011)), oriented oriented on on It likewise Modelica language (Fritzson (2011)), oriented on likewise Modelica language (Fritzson (2011)), oriented on dt based on Unified Modeling Language’s de facto standards, ds n ds modeling hierarchical event-driven component systems. It is n ds ,, ss,, w modeling hierarchical hierarchical event-driven event-driven component component systems. systems. It It is is n respect to F ( ds modeling respect to state state variables variables with F w,,, ttt,,, ccc))) C0 0 ,,,and modeling hierarchical event-driven component is respect to state variables with accommodating them for hybrid systems. Hybridsystems. systemsItare F , s, w 0 constancies substitutions and w Substsss( w, Cnn),,,, with F ((( dt based dt , s, w, t, c) 0 , respect to state variables s , with based on on Unified Unified Modeling Modeling Language’s Language’s de de facto facto standards, standards, dt based on Unified Modeling Language’s de facto standards, considered as an extension of classical dynamical systems dt based on Unified Modeling Language’s de facto standards, output Out ( y,C , C, t )substitutions 0 respect w accommodating constancies (( w C accommodating them them for for hybrid hybrid systems. systems. Hybrid Hybrid systems systems are are equations constancies Cs, wand and substitutions and wto Subst Subst w,,,variables C ))) ,,, and accommodating them for hybrid systems. Hybrid are C (Senichenkov (2004), and Senichenkov (2006), accommodating them forKolesov hybrid systems. Hybrid systems systems are constancies constancies C and and substitutions substitutions w and w Subst Subst(( w w, C C ) , and considered as an extension of classical dynamical systems considered as an extension of classical dynamical systems ds respect to output variables equations Out ( y , s , w , C , t ) 0 considered as an extension of classical dynamical systems m respect to output variables equations Out ( y , s , w , C , t ) 0 considered asSenichenkov an extension of classical dynamical systems equations Kolesov and (2012 ,2014). respect to output variables Out ( y , s , w , C , t ) 0 F ( , s , w , t ) 0 . Systems of equations in form y respect to output variables equations Out ( y , s , w , C , t ) 0 (Senichenkov (Senichenkov (2004), (2004), Kolesov Kolesov and and Senichenkov Senichenkov (2006), (2006), (Senichenkov (2004), Kolesov and Senichenkov (2006), dt (Senichenkov (2004), Kolesov and Senichenkov (2006), ds concept of MVL is «active dynamical object» with The main m ds m ds ,, ss,, w Kolesov F ( ds , ttt ))) 0 Systems of equations form yyy Kolesov and and Senichenkov Senichenkov (2012 (2012 ,2014). ,2014). m F w 0 Systems of equations in form Kolesov and Senichenkov (2012 ,2014). will bem transformedin m ....automatically ,, ssof ,, w Systems in form Kolesov and Senichenkov (2012 ,2014). global behavior described by«active Behavior-Chart (modification F (((help w,,,Gear’s t) 0 0 Systems of of equations equations in with formtheF y dt dt concept of MVL is dynamical object» with The main dt concept of MVL is «active dynamical object» with The main concept of MVL MVLand is «active «active dynamicalinobject» object» with The main concept substitution to dt of is dynamical with The main of UML’s State Machine) local behaviors the form of will be automatically transformed with the help of Gear’s will be automatically transformed with the help of Gear’s global behavior described by Behavior-Chart (modification will be be automatically automatically transformed with with the the help help of of Gear’s Gear’s global behavior behavior described described by by Behavior-Chart Behavior-Chart (modification (modification will transformed global ds to algebraic-differential equations (Kolesov andin Senichenkov global behavior Machine) described by Behavior-Chart (modification substitution to of z; F ( z, s, w, t ) 0, s(0) s0 (2). substitution to of UML’s UML’s State State Machine) Machine) and and local local behaviors behaviors in in the the form form of of substitution substitution to of UML’s State and local behaviors the form of (2007)). of UML’s State Machine) and local behaviors inSenichenkov the form of dt algebraic-differential equations (Kolesov and ds ds algebraic-differential equations (Kolesov and Senichenkov ds algebraic-differential equations (Kolesov Senichenkov (2). z; F w , t) 0 , s(0 ) algebraic-differential equations (Kolesovforand and ds Building global systems of equations composition of Special cases (2). (((( zzzz,,,, ssss,,,, w 0 ssss000 form Senichenkov (2). F w 0 0 (linear (2007)). (2).(2) are zzz;;; andF variants) of F non-linear w,,, ttt ))) 0,,, sss(((0 0))) (2007)). dt 0 dt (2007)). 0 dt component’s automata is the main difficulty for component (2007)). dt Building global systems of equations for composition of ds Building global systems of equations for composition of Special cases (linear and non-linear variants) of form (2) are Building global systems of for of variants) models with hybrid behavior. Whendifficulty number of components Fand ( s, wnon-linear , t) Building global systems of equations equations for composition composition of Special Special cases cases(linear (linear and non-linear variants) of of form form (2) (2) are are Special cases (linear and non-linear variants) of form (2) are component’s automata is the main for component component’s automata is the main difficulty for component a 2 . dt component’s automata is the main difficulty for component ds component’s automata is the main difficulty for component and number component’s hybrid automata states increase, the ds ds models with hybrid behavior. When number of components F ((( sss,,,,tw ds( y G models with hybrid behavior. When number of components hybrid behavior. When of models F w ) ,,,,tttt ))))0, s(0) s0 models with with hybrid behavior. When number of components components , sF F, w w ( s, w 2 .a number of component’s possible states for number composition became dt dt 2 a 2 states increase, the and number hybrid automata dt a 2...a and number component’s hybrid automata states increase, the dt and hybrid states the and number number component’s component’s hybrid automata automata states increase, increase, the extraordinarily large. Alternative for building all possible sss00 ,,, ttt ))) 0 ,,, sss(((0 ))) G ds(((( yyyy,,,, ssss,,,, w G w 0 0 number of possible states for composition became G w 0 0 number of possible states for composition became G w t s s , ) 0 , ( 0 ) number of states for composition became F ( s, w, t ), s(0) s0000, 2.b systems is building systems possible directly number beforehand of possible possible states only for realized composition became extraordinarily dt extraordinarily large. large. Alternative Alternative for for building building all all possible possible ds extraordinarily large. Alternative for building all ds on run time. In large scale event-driven systems the number ds extraordinarily large. Alternative for building all possible F s(0 s , ( s, w , t ), ) 2 .b systems 2 systems beforehand beforehand is is building building only only realized realized systems systems directly directly F w b ), 0 2 systems beforehand is building only realized systems directly F, t((()sss,,, w w0,,, ttt ), b wF ), sss(((0 0))) sss00000 ,,, 2...b Gds ( s, 2.c dt of realized local behaviors isevent-driven large It systems is very important to systems beforehand isscale building onlytoo. realized systems directly dt dt on run time. In large the number dt on run run time. time. In In large large scale scale event-driven event-driven systems systems the the number number The linear form on automatically. block know structure solving system. It Systems withnumber on runthe time. In largeofscale event-driven systems the G s, recognized w (is ,,, ttt ))) 0 2.c G w 0 2.c of G 2.c of realized realized local local behaviors behaviors is is large large too. too. It It is is very very important important to to There are special G ((( sss,,, w wSolvers , t) 0 0 for each type of equations. 2.c of realized local behaviors is large too. is very important to The triangular structure are abundant problems in practice. Using of realized local behaviors is large too. It is very important to The linear form is recognized automatically. know the structure of solving system. Systems with block The linear form is recognized automatically. know the structure of solving system. Systems with block The linear form is recognized automatically. know the structure of solving system. Systems with block The linear form is recognized automatically. know the structure of solving system. Systems with block special role plays Solvers for Non-linear Algebraic Equations numerical methods taking block triangular structure in There are special Solvers for each type of equations. The triangular structure are abundant problems in practice. Using There Solvers for of The triangular structure are abundant problems in practice. Using There are special Solvers for each type of equations. The triangular structure are problems in Using There are areTospecial special Solvers for each each type type of equations. equations. The triangular structurespeed are abundant abundant problems in practice. practice. Using (NAE). solveSolvers NAE isfor necessary in implicit methods for account increases of computer modeling. The problem special role plays Non-linear Algebraic Equations numerical methods taking block triangular structure in special role plays Solvers for Non-linear Algebraic Equations numerical methods taking block triangular structure in special role roleDifferential plays Solvers Solvers for Non-linear Non-linear Algebraic Equations numerical methods taking block structure in special plays for Algebraic Equations numerical methods taking block triangular triangular structure in Ordinary Equations (ODE), in methods for of automatic building global equations for local behavior of (NAE). account increases speed of computer modeling. The problem (NAE). To To solve solve NAE NAE is is necessary necessary in in implicit implicit methods methods for for account increases speed of computer modeling. The problem (NAE). To solve NAE is necessary in implicit methods for account speed of modeling. problem account increases increases speed of computer computer modeling.andThe Theautomatic problem finding consistent initial conditions and solving Algebraic(NAE). To solve NAE is necessary in implicit methods for component model with hybrid behavior Ordinary Differential Equations (ODE), in methods for of automatic building global equations for local behavior of Ordinary Differential Equations (ODE), in methods for of automatic building global equations for local behavior of Ordinary Differential Equations (ODE), in methods for of automatic building global equations for local behavior of Differential Equations (ODE), in methods for of automatic building global equations for local behavior of Ordinary Differential Equations (DAE). NAE are solved with the help finding consistent initial conditions and solving Algebraiccomponent model with hybrid behavior and automatic finding consistent initial conditions and solving Algebraiccomponent model with hybrid behavior and automatic finding consistent initial conditions and solving Algebraiccomponent model with hybrid behavior and automatic consistent initial conditions and solving Algebraiccomponent model with hybrid behavior and automatic finding Differential Differential Equations Equations (DAE). (DAE). NAE NAE are are solved solved with with the the help help Differential Equations (DAE). NAE are solved with the help Differential Equations (DAE). NAE are solved with the help
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MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria Andrey A. Isakov et al. / IFAC-PapersOnLine 48-1 (2015) 661–662 662
of Newton method, and in case of its failure, Powell method is used. Solving of linear systems of algebraic equations (SLAE) is the main operation for Newton method in turn. Newton method may choose appropriate modification of Gauss method with taking structure of solved system matrix in account: dense, band, sparse, block triangular. Any techniques decreasing computational costs of numerical solution are very important. Computing matrix block triangular form if possible is well known and commonly used trick (Duff (1977)). In RMD-7 this technique is used for structural matrix of equations. Structural matrix of a system is a matrix with {0,1} elements pointing out occurrence of unknowns in equations. RMD computes block triangular form for structural matrix of any allowed types of systems (SLAE, NAE, ODE, DAE) with the help of Tarjan algorithm and solves only subsystems for strongly connected blocks. Tarjan algorithm in RMD is used for reordering substitutions to computable sequence of formulas and detecting equations («algebraic loops») among them too. Building and transformation global equations cause maximum difficulties for component models with «contactsflow» external variables («acausal blocks»). It is well known that if even local behaviors of components’ automata do not contain high-index DAE, they can appear in their composition. So it is necessary to analyze all states of composition, if we want detect and build all high-index DAE systems beforehand. RMD’s Analyzer detects high-index DAE on run time and builds new additional equation for numerical differentiation. The structure of RND’s Numerical Library and algorithm of interaction between Numerical library and Model Engine have been changed for implementing new approach to building global equations on run time. If any new event has occurred, the control program estimates necessity of rebuilding of current solved system. New system is built if it is necessary. If the current global system has full transversal: Tarjan algorithm is used for building block triangular form of structural matrix. Strongly connected components (diagonal blocks of reordered structural matrix and associated with them systems of equations) are analyzing for choosing appropriate Solvers. Solving a system of equations corresponding to diagonal block with the help of suitable Solver will be named «subtask». Next step is building the condensation of the Tarjan’s algorithm graph. The condensation is used for construction subtask queue. Subtasks may be executed sequentially or parallel. The information about computer hardware (number of processors, number of kernels for a processor) needed for creation a thread pool for parallel execution is determined automatically. Threads are loaded by subtasks which are ready for execution. The subtask readiness for execution determinates using condensation. Initially all nodes (subtasks) of the condensation are marked as «Unresolved». If an «unresolved» node has no input edges or input edges start from the nodes with solved systems then control program changes its status for «Ready to start», otherwise it will have status «Not ready to start». Solving subtask’s
system has name «Solving», after ending of solving it becomes «Solved». The calculation comes to an end when all nodes become «Solved». The new approach with block triangular form, different Solvers for each subtask, and threads was compared with old one. For comparison was used a set of models developed by Transas company (http://www.transas.com/products). The results of numerical experiment for most difficult problem are shown in Table 1. The computer used for calculations had four processors, so it was possible to create maximum four threads, but even if only one thread was used then total time of calculations decreased in two times. Table 1. Product Tanker, Cargo System, about 2500 equations.
REFERENCES Duff I.S. (1977). MA28—A set of Fortran subroutines for sparse unsymmetric linear equations. Technical Report AERE R8730 (SR-2040 8.0 edition) HMSO, London Isakov A.A., Senichenkov Yu. B. (2010). OpenMVL is a tool for modelling and simulation. //Computing, measuring, and control systems, pp. 84-89, SPBSPU, St. Petersburg. Isakov A.A., Senichenkov Yu. B. (2011). Program Complex OpenMVL for Modeling Complex Dynamical Systems // «Differential equations and control processes» Journal, Mathematics and Mechanics Faculty of Saint-Petersburg State University, St. Petersburg. Kolesov Yu.B., Senichenkov Yu.B. (2006) Modelling of systems. Dynamical and Hybrid systems. 224 pp, BHV, St. Petersburg. Kolesov Yu.B., Senichenkov Yu.B. (2007) Modelling of systems. Object-oriented approach. 256 pp, BHV, St. Petersburg. Kolesov Yu.B., Senichenkov Yu.B. (2012) Mathematical Modelling of component systems. 223 pp, SPbSTU, St. Petersburg. Kolesov Yu.B., Senichenkov Yu.B. (2014) Mathematical Modelling of hybrid systems. 236 pp, SPbSTU, St. Petersburg. Pissanetsky S. (1984). Sparse matrix technology, Аcademmic Press, London. Senichenkov Yu. B. (2004) Numerical modelling of hybrid systems.206 pp, SPbSTU, St. Petersburg. Fritzson P. (2011) Introduction to Modeling and Simulation of Technical and Physical Systems with Modelica, 232 pp, Wiley-IEEE Press.
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