Mathematical simulation of control system in Modal Vision Studium

Transcription

1 JOINT ADVANCED STUDENT SCHOOL 2008, ST. PETERSBURG Mathematical simulatio of cotrol system i Modal Visio Studium Fial Report by Studet Melikov Kostati bor o address: pr. Dachiy, 24/2 Sait-Petersburg, Russia Tel.: (812) Sait-Petersburg State Polytechical Uiversity Istitute of iteratioal Educatio Programs Distributed itelliget Systems Supervisor: prof. Yuri Seichekov Submitted:

2 Cotets 1 Itroductio.3 2 History ad mai features of MVS.4 3 Differet types of dyamic systems. 5 4 Examples of modelig.6 5 Modelig of real complex dyamic system.7 6 Coclusio 1. Itroductio Dear colleagues, theme of my work Mathematical simulatio of cotrol system i Modal Visio stadium. I the give work I tell about oe of tools of mathematical modelig Modal Visio Studium. I the begiig it is told about history ad structure of the give software product. The hierarchy of iteractio of iterfaces eterig ito it is described. Also workigs out ad the early laguages of modelig used at its creatio are described. After that there is a speech about various kids of structural compoets blocks, ad the example of work of system is the resulted at the decisio of three various problems. Last example is cosidered especially i detail, the mechaism of use of properties of object-orieted programmig there is show. 2. History ad mai features of MVS MVS is the special laguage based o techology UML, the powerful tool for graphic modelig of difficult dyamic systems. UML uified modelig laguage is a ew directio i developmet of systems of the modelig, directed o uificatio of existig ad future tools of modelig. Now there is a big problem of iteractio of a various software, at trasitio from oe eviromet of workig out to aother it is ecessary to alter aew the created models, it is impossible will share results of the work with foreig colleagues sice frequetly i the differet coutries differet systems are popular. For struggle agaist such difficulties also has bee developed UML object-orieted laguage of prototypig of difficult program complexes ad computig systems. What is the difficult dyamic system? It is large, multicompoet system, it is frequet with difficult commuicatios. It ca possess ow hierarchy ad deped both o exteral evets, ad from the iteral. Because of it it ca cardially chage the behavior for very short time iterval. Such systems also are called as hybrid ad, as a rule, they hard give i to the mathematical descriptio ad modelig. History MVS totals some years, all has begu with system MVS 2.1 i The give product develops actively ad moderized. Sice 2005 it is possible to get versios 5. Demos with are accessible to H.Nekotorye ad o the Iteret free of charge, but their possibilities are a little reduced. For example, restrictio o the size of model is etered. We will cosider, what workigs out ad cocepts were icluded ito a basis at creatio of the give product.

3 First, it is oe of the first laguages of modelig SIMULA 67, developed Ole-Joha Dahl, Bjor Myhrhaug, Kriste Nygaard, ad SLAM II by Ala Pritsker. These authors had bee offered for the first time idea of creatio of special models - classes for the descriptio of objects with the similar properties, possessig iteral properties ad behavior. So there was a ew approach to modelig of difficult multicompoet systems the object-orieted approach. Also at creatio MVS workigs out were used: - V., V. Tools for modellig mixed cotiuous ad discreet systems (1975) - Booch G. I. The uited modellig laguage for object-orieted modellig (1977) - Fillipov A. Differetial equatios with discotiues right-had side (1985) - Maler O. Maa Z. A formal approach to hybrid systems (1992) Let's briefly review the hierarchical scheme of model i MVS. It is made by a priciple from simple to difficult, i.e. we are begiig from the fourth level of the iterface Graphical editor of hierarchical fuctioal diagrams? ext we go up o third level of iterface graphical editor of fuctioal diagrams. Further we create the ope hybrid automatic machie which ca be preseted two kids of blocks with cotacts ad a iput/exit. At the secod level we create o the basis of hybrid automatic machies hybrid system by meas of the graphic iterface. It is secod iterface level. Ad ow we pass o first level where with the help of simply ad coveiet iterface ad the mathematical editor the classical dyamic system so habitual for our eyes is created. So i geeral the hierarchy of costructio of model i MVS looks. MvStudium models Classical dyamical systems Simplest user s iterface 1 st level of iterface Mathematical editor Isolated systems Hybrid systems Simple graphical iterface Secod level of Hybrid automatio editor iterface Isolated systems Ope hybrid automata Hybrid automatio Hybrid automatio with with I/O cotacts Plae fuctioal diagrams Graphical editor of fuctioal diagrams Ope systems Third level of iterface Hierarchical fuctioal diagrams Graphical editor of hierarchical fuctioal diagrams Ope systems Forth level of iterface 3. Differet types of dyamic systems Let's cosider some kids of dyamic systems which ca be simulated i MVS.

4 The first type cotiuous systems. I their give classificatio it is possible to ame the elemetary but oly because other pole are very difficult hybrid systems with discrete time. The give system of the differetial equatios., which right part is defied i all space of variables х ad or is cotiuous, as well as its derivative, or satisfies to coditio Lipshits s o all argumets, is called as classical dyamic system. х set of decisios of the give system. t a idepedet material variable she plays a role of cotiuous time. dx dt 0 0 = f ( x), x R, x(0) = x x = x( t; x ) I discrete models we caot speak about the differetial equatios ay more sice there ca ot be a speech about derivative fuctio whe it depeds o discrete time. I such models we deal with systems differece the equatios. Directly discrete models caot be processed by meas of hybrid automatic machies, the special programmig laguages developed specially for work with discrete time are for this purpose ecessary. However it is possible to try so to trasform this model that begis to use probably hybrid automatic machies. For example, for sychroizatio of discrete processes it is possible to use cotiuous time. x y = x = x cosα siα 2 ( y x ) 2 ( y x ) siα cosα I dyamic systems ca arise various ad differet scale processes. Difficulties of reproductio of such systems are obvious. Sometimes it is possible to divide fast ad slow movemets ad to cosider them separately the each of models has a characteristic time scale. However so to make it is ot possible more ofte, ad the it is possible to use such receptio. The researcher solves, what processes i object to cosider log, ad what fast. Speed of fast processes is accepted by the ifiite. Such processes are authorized to chage value jump, ulike slow the idissoluble. We will cosider a usual cotrol system. The object of maagemet works costatly, it is system of cotiuous time, ad here a operatig sigal from a maagemet cotour comes periodically. It already will be discrete system. dx = f ( x( t), u) dt u u + 1 = G( x( t = u( t ), t + 1 ), u = t ), + h, The we will choose operatig sigals fast process, ad object reactio slow. Fast process chages i steps, ad gradual reactio of object is traced costatly.

5 So there are models which ame ow hybrid systems. The term «hybrid systems» was used may years as a syoym of aalogue-digital devices. For example, i dictioaries it is possible to meet a word-combiatio «hybrid computer» as the ame of digital computers i which aalogue devices are simultaeously used also. We will cosider a body throw uder a corer to horizo, we will ot oly watch flight, but also to model elastic отскок from the earth. For this purpose we will allocate some special coditio a cotact of a body of the earth, ad it allowed to replace vertical makig speed at the momet of its approach a sig with the opposite., the agai we reproduce the set system with ew etry coditios, ad so agai ad agai. It is possible to ame the mechaism geeratig it the automatic machie, the hybrid automatic machie. But the hybrid system caot be costructed, ot kowig, that such blocks ad what they happe. Geeralizatio of experiece of modelig of physical, techical ad atural systems allows to speak about two types of commuicatio. If exteral variables are coected by the focused commuicatios to these commuicatios there correspod formulas i which to variables i the left part (iputs) kow values of variables i the right part (exits) are appropriated. Iformatio ad physical commuicatios i techical systems whe special measures o a exceptio back iflueces are take are directed. The structural compoet, all which exteral variables are focused, is called as the focused block or the block the iput/exit. A example of the system costructed of blocks, the automatic cotrol system is. To odirectioal commuicatios there correspod the equatios i which it is impossible to specify required variables to the full aalysis of all cumulative system of the equatios arisig at coectio of blocks of model. A classical example of that approach is costructio ad modelig of electric chais. The blocks realizig such priciples, are called as odirectioal, or ope blocks with cotacts. I this example the scheme of the ideal diode which is difficult eough for realizig i other software products all busiess i difficult behavior of the diode which sharply chages the coditio is costructed. But as we already kow, for hybrid automatic machies it ot a problem. 4. Examples of modelig Let's pass to cosideratio of examples of modelig of difficult hybrid systems i MVS. The first example is «breakig pedulum» where cargo comes off from cord i ay poit of a trajectory. I the graphic editor it is possible to see, that 2 automatic machies are created. The first models rockig of the whole pedulum uder the simple law of a sie. But, as soo as cargo comes off, busiess is etered by the secod automatic machie describig movemet of cargo as a body, movig by gravity.

6 The followig example about possibility of statistical tests which ca be spet by meas of a card of behaviors. Here the body throw uder a corer to horizo is modeled, ad also the corer size is established at which the maximum rage is reached. By meas of the artful scheme we build the elemetary algorithm with the hybrid automatic machie i a basis which some times will throw a body uder differet corers to horizo. At first we set the equatios i the first widow, ecessary for creatio of the hybrid automatic machie with zero etry coditios, ad the we iclude it i algorithm.

7 5. Modelig of real complex dyamic system Now we will pass to the most iterestig example - to modelig of real dyamic system. We will cosider a drive of maagemet which is used for example i the plae. This system cosists while of four elemets: - Sesitive elemet which accepts a mismatch sigal - The amplifier which stregthes a operatig sigal that its tax o the egie. - The dysphasic asychroous egie - Reducer, reformative rotatio of a shaft of the egie i tur o small degree. It is classical cofiguratio of systems of such class with a egative feedback. All elemets of the give system are absolutely real, exist i practice. They have the correspodig trasfer fuctios show o this slide. These trasfer fuctios are a little simplified, the cosequece of fluctuatios of a shaft of the egie i a reducer, therefore its characteristic such simple for example is ot cosidered. Nevertheless all these trasfer fuctios are received o the basis of the real techical characteristics of this the elemet which has bee writte dow i their characteristics sheets. But give system for the preset ot full, it does ot suffice correctig devices. They will be added later for ow we will look as the system behaves ot corrected. Modelig ad the aalysis of the give system is spet by meas of two tools already kow to us MVS ad small special package CLASSIC 3.0. It is a package already eough old, but its charm that it is arrow специализирован o the decisio of such problems. Let's look at process ad results of modellig. Let's begi with program CLASSIC. This eviromet i the simple graphic iterface our scheme is made of blocks of type a iput/exit, commuicatios betwee them are the spet. The each block separately is edited i fractioally-poliomious kid trasfer fuctios register, the iput ad a exit of all system are appoited. After givig of values to all blocks we give a step idividual sigal o a system iput ad we receive result. The received trasitive characteristic says to us, that the system works astably, such result leads to its breakage.

8 Now let s model the same i MVS. Here already there is a special library with a set of stadard blocks. Here there is апериодическое a lik of the first order, the itegrator, differetial uit, a oscillatory lik, ad also geerators of various etrace sigals. We will make with their help our system the scheme turs out such here. It seems little bit more difficult previous, made i CLASSIC, but the sigal geerator i a explicit form here was simply added, ad the egie has bee set by two typical liks. We establish ecessary values of our blocks ad we start model. We receive this characteristic. Results of work of two programs coicide the system is ot balaced, therefore i half-secods after the begiig of work it fails. It is ecessary to create the device developig correctig ifluece ad reducig the trasitive characteristic of system to steady positio. Here the scheme of the corrected system. Not to block up this work I do ot show process of creatio of the device of stabilisatio. The story about this work doe by me ca occupy ot oe hour. For costructio of this device I had bee used a method of costructio of the desirable logarithmic peak characteristic, by the offered Russia scietist Besekersky. The give type of correctio is called as parallel correctio of a site of system. The additioal egative feedback, coverig the amplifier ad the egie is for this purpose created. To the egie it is coected тахогенератор the special device, reformative speed of rotatio of the egie i pressure. I.e. it is the gauge of agular speed. It has the followig trasfer fuctio. Devices of such type are called as a forcig lik of the secod order. Ad to it we iclude the calculated correctig device with such trasfer characteristic. We will be coviced of workig capacity of system. To begi with we will check up this system i CLASSIC. We add to our scheme two more blocks, we set a step idividual etrace sigal ad we teach result! Now the picture has strogly chaged the system does ot break, ad coverges gradually to size of a etrace sigal, ad very quickly all for half-secods. The size of reregulatio does ot exceed 25 %, oscillatio systems also is low. Fie result.

9 Let's simulate the same corrected system i MVS. But here there is oe complexity i library there are o the typical liks ecessary for us. But it is ot terrible i MVS it is possible to create the blocks of differet types with ay characteristics, to keep them ad to use i other models as elemets of the geeral library. Now I will tell, how it is possible to create easily the ow block i MVS o a lik example of tacogeerator. To begi with we choose system type to which the block cotiuous belogs, the we choose elemetary structure ad ope type of the block that there was a possibility to create iputs ad exits. We accept these istallatios ad we pass to the block descriptio. Aalytically by meas of retur trasformatio Laplas s we receive from trasfer fuctio tacogeerator the differetial equatio with which works MVS. It turs out the secod order, but for this program it ot a problem. Further we write dow parameters ad we appropriate it values 0. I model we ca already chage them to the ecessary. Well ad at last we appoit Х to a iput, ad At we submit o a exit. Well, it s doe! We ca isert our block ito model. Oly do ot forget it to keep. We make the same work with the correctig device ad we receive the workig scheme of the corrected system i MVS. We start our model. We set a stop coditio - the system will stop i half-secods. What do we have? Trasiet covergig, system works perfectly! Reregulatio of oly 15 %, system is stabilized all for 0.4 secods is very good result. If we discoect ow time restrictio we will see such schedule. Such cotiuous operatig mode correspods to real coditios, for example

10 we by turs tur to the right ad to the left. Our corrected computer-cotrolled system copes with the give problem. 6. Coclusio I summary it is possible to tell, that the sphere of applicatios MVS is very wide, ad the cosidered examples far ot a limit of its possibilities. Now there are very few systems i which it is possible quickly ad to create difficult multiple hybrid systems coveietly. Pleasat additio are iterfaces for two- ad three-d measured graphic display of the received results. Havig simulated system, we ca draw it i a three-dimesioal kid, adhere drawig coordiates to variables ad receive the movig three-dimesioal picture displayig received results. Especially pleases possibility directly to set the differetial equatios of the first ad secod order, that cosiderably facilitates ad accelerates work. O last example could seem what easier to work with trasfer fuctios, ad their trasfer i the differetial equatios oly distracts. Actually it ot so. If to cosider, that these trasfer fuctios have iitially bee deduced from the differetial equatios it becomes clear, that such approach cosiderably simplifies modelig.