Modelling and Simulation. Study Support. Zora Jančíková

Transcription

1 VYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA FAKULTA METALURGIE A MATERIÁLOVÉHO INŽENÝRSTVÍ Modelling and Simulation Study Support Zora Jančíková Ostrava 5

2 Title: Modelling and Simulation Code: 638-3/ Author: Zora Jančíková Edition: first, 5 Number of pages: 99 Aademi materials for the Eonomis and Management of Industrial Systems study programme at the Faulty of Metallurgy and Materials Engineering. Proofreading has not been performed. Exeution: VŠB - Tehnial University of Ostrava

3 CONTENT SYSTEMS MODELLING... 4 Physial modelling... 6 Natural physial modelling... 6 Artifiial physial modelling... 6 Mathematial modelling... 7 Analytial identifiation... 8 Experimental identifiation... Analogy... Cyberneti (funtional) modelling... 3 MODELS... 5 Classifiation of models... 5 Simulation... Examples of ompiling analytial midels of simple objets... 8 Heating material in furnae Reuperator for air heating Car suspension Aumulation of material in a landfill... 4 Tank ith a free outflo of fluid THEORY OF SIMILARITY AND MODELLING Geometri similarity Physial similarity Mehanial similarity Thermal similarity... 5 Differenes beteen onstant and invariant of similarity:... 5 Sentenes (theorems of similarity)... 5 st sentene of similarity... 5 nd sentene of similarity rd sentene of similarity Derivation of general riterial equation Analysis of relational equations Dimensional analysis COMPLEX SYSTEMS MODELLING Artifiial intelligene Neural netorks... 7 The spread of information is possible through membranes of soma and axon hih, hen irritated by an eletrial impulse of ertain threshold intensity, generate an eletrial impulse themselves. Synapti permeability is hanged at eah pass of the signal, hih gives the neuron the ability of memory Mathemati neuron models... 7 Artifiial neural netorks Multilayer neural netorks Aeleration of learning speed Basi appliation areas of neural netorks

4 STUDY INSTRUCTIONS You have obtained the study material for the part-time study of the subjet 638-3/ Modelling and Simulation of the inter semester of the study branh Automation and in Industrial Tehnologies. PREREQUISITES This subjet has prerequisites: System Theory, System Identifiation. SUBJECT OBJECTIVE AND LEARNING RESULTS Student ill be able to formulate the basi methods of simulation models realization on the digital omputer. Student ill get an overvie of the basi priniples of mathemati-physial modelling, similarity and modelling and of lassi and artifiial intelligene methods neessary for model realization. Student ill be able to reate mathematial models of seleted real proesses ith aim of lassi simulation programs and ith artifiial neural netorks exploitation. 4

5 SYSTEMS MODELLING Study time: hours Objetive: The student ill be able: to lassify different types of modelling and similarities, to desribe methods of system identifiation, to define and desribe the system analogy. Modelling is an experimental proess, in hih the surveyed system (original, objet, part) is, aording to speifi riteria, expliitly assigned to another system, physial or abstrat, hih is alled a model. Different types of modelling are onneted to different forms of similarity or analogy. Similarity an be understood as expliit mutual assignation among various systems in their struture, qualities and behaviour. Folloing similarities an be distinguished: Geometri similarity of geometri figures (they are similar if their respetive angles are equal and their respetive sides are in proportion to eah other). Physial similarity beteen systems and proesses of the same physial essene; in addition to geometri similarity, it inludes similarity of parameters and state values of the systems (thermal, kineti, dynami). Mathematial similarity beteen systems and proessed hih share the same mathematial desription. If the systems and proesses are physially different, it is alled analogy. Cyberneti (funtional) - mathematial similarity in external behaviour of systems. Aording to the above introdued types of similarities, the folloing types of modelling an be distinguished: Physial modelling natural, artifiial Mathematial modelling Cyberneti modelling (funtional) 5

6 Physial modelling Natural and artifiial physial models near the original the most ith their informational ontent and similarity. The signifiane of information gained from these models is higher than in ases of models using mathematial or yberneti similarity. Natural physial modelling During the experiment, the analysed objet as suh (the original, the produt) beomes the model, hih means that the experiment is exeuted on the original (produt) itself, and main instruments are measuring, respetively omputer, tehnology. Diret experiment on the original tends to be diffiult, time onsuming and ostly. Diret experiment on the original tends to be diffiult, time onsuming and ostly. It inludes objet exploration in various test onditions and it plays a ruial role in final evaluation of the system harateristis (eg. before putting mahines into operation). During the experiment, all the modelling sales are defined per unit (sales on the model = sales on the produt), riteria of similarity and other las of physial similarity are applied only hen proessing the results of the experiment, hih means hen generalizing the model information to other physially similar systems. Artifiial physial modelling This type of modelling is arried out using an artifiial objet, a physial model, hih is bound to the original by onditions of physial similarity physial phenomena are similar if the geometri similarity of the model and produt is respeted; and then it an be applied that in orresponding points of the model and produt, the ratio of quantities haraterizing the physial proess is alays onstant, e.g.: C C W W W // // // // // // 3 3 W C / / / F / / / W W W3 F F F3 t t t // // // // // // 3 3 t C / / / / / / t t t3 3 here,, 3 are orresponding points of the model and produt. Fous of the artifiial physial modelling as ell as of the natural modelling is in diret experiment, in this ase on the model; main instruments are again measuring and omputer tehnology. The riteria of similarity and other means of similarity theory are applied throughout the modelling proess, starting ith designing the model and ending ith proessing the results and their appliation to other physially similar systems. The results of F F F 6

7 measuring on the physial model are onverted into the produt by using sales of modelling (onstants of similarity) hih are assigned on the basis of equality of orresponding riteria of similarity for the model and produt. Experiment ith natural or physial model provides the most reliable information about the harateristis of the surveyed system. Creation of an artifiial physial model or of an experiment ith a natural model is usually more omplex, ostly and time onsuming than using models based on mathematial (or yberneti) similarities. Sometimes this problem is solved by a ompromise and an inomplete physial model of the investigated objet is reated (e.g. a redued nulear reator tank is used for partiular experiments as a model of a real tank). Mathematial modelling Mathematial model onsists of an abstrat system of mathematial relations that desribe essential features of the studied objet (system of algebrai and differential equations; transmission in Laplae, Z or Fourier transformation; transition, impulse, frequeny harateristis, et.). For mathematial desription of the harateristis and behaviour of the objet it is neessary to determine variables desribing ho the environment affets the system (inputs) and variables reording ho the system manifests itself toards its environment (outputs). Mathematial model expresses, then, the dependene of outputs on the inputs, hih is desribed by mathematial relations. These relations beome a mathematial model only after they are unambiguously assigned to a partiular proess or phenomenon. The proess of determining the mathematial desription of the system is alled the system identifiation. During the identifiation e try to gain the model in suh a form in hih it ould be used in the area hih e intend to use it in (e.g. for the synthesis of a regulatory iruit in blok-oriented simulation programs in the form of transfer G (p); for using equationoriented simulation programs in the form of differential equations, et.). The examined proess an be identified: a) analytially (analytial model) b) experimentally (experimental, empirial model) determinist and stohasti experimental identifiation methods 7

8 Analytial identifiation Analytial model is obtained using the methods of mathematial - physial analysis; e talk about the so-alled dedutive proedures (dedution - derivation of a partiular speial ase from general knoledge, las). Mathematial model is obtained by ompiling equations expressing relations beteen input, output and state variables, deriving from mass and energy balanes, from equations of physial, hemial and biologial proesses. Pivotal role in solving various tasks in tehnial and nontehnial fields lies in ordinary and partial differential equations, and, out of these, espeially those equations of mathematial physis represented by partial differential equations of the seond order, hih desribe physial proesses and fields in systems ith distributed parameters (flo of heat, liquids, gasses). In ase of assoiated physial proesses and fields, e refer to systems of these equations. Differential equation desribes the proess examined in an entirely general ase; it has an endless number of solutions. Complete mathematial model is also determined - in addition to the basi equation - by the equation for surfae and initial onditions, physial parameters and geometri shape of the area in hih the proess is arried out. All these onditions are olletively alled the terms of exatness (monovaleny) of the solution. To kno them is absolutely neessary for solving a speifi ase; nevertheless, their proper identifiation is often a diffiult problem. The initial ondition determines the value of the dependent variable u anyhere in the area M(x,y,z) in time =. u(m, ) f(m) e.g.: t t f f x, y, z x, y, z, onst. Surfae onditions reflet the link beteen the studied area and the external environment. They haraterize the hange of the dependent variable u on the surfae of the area in dependene on the time. Four types of boundary onditions are mostly used: st type Dirihlets: It represents a knon value of the funtion u on the surfae of the area in the plae M S and time. u f, s surfae S M S e.g.: t f x y, z, S S, S S 8

9 nd type Neumanns: It is given by the gradient of the funtion u on the surfae hih is the funtion of time. a u n M q S M S u e.g.: q f x 3rd type Fouriers: a M S M S - physial parameter of the area q - surfae energy flux on the surfae It is expressed by a linear relation beteen the funtion and its derivation in a diretion of the normal vetor. u a S S S S, n M b M, u M u p t x e.g.: t S t u p external environment M S b - physial parameter of the external environment 4th type: It expresses the ondition of equality of surfae energy flos at the interfae beteen to areas M and M in perfet ontat, having different physial properties (funtion u of the touhing points of the area is the same). a M a M u n u n The result of solving a differential equation is to find out suh a funtion hih learly satisfies both the differential equation and initial and surfae onditions. Advantages of the analytial model: Suh model an be haraterized as an internal desription of the behaviour of the studied/examined objet (the so-alled "hite box" an be hite or transparent), hih means it reprodues real patterns; its parameters have physial meaning (,,, ). It leads to a lear desription of the system (e gain one mathematial desription for the studied objet). This is applied in a ider range (it an be used for other physially similar systems and proesses). It enables speifiation of a mathematial model even in the ase that the objet has not been made yet (if it is just in the phases of projetion, for example) and input 9

10 and output signals of the system annot be measured. Results of suh analysis an be used to selet the optimal onept and detailed onstrution of the entire devie. Disadvantages of the analytial model: The model is usually very omplex and nonlinear. It an be used only for some simple ases in hih the physial nature of the proess is ell knon; its material and energy balanes an be ompiled and, then, on their basis, adequate equations desribing its behaviour an be ompiled. Therefore, suh ork requires a great deal of theoretial and pratial knoledge about the studied/examined objet or proess. It is neessary to apply a number of simplifying assumptions and mathematial manoeuvres; sometimes the resulting relations have to be approximated, sometimes linearized, and all that to the detriment of auray. The auray of the ahieved results is mostly not proportional to the effort hih is made for reating the model. Results of mathematial and physial analysis an be suessfully used to estimate the order of equation or to transfer the identified system during the experimental identifiation. Experimental identifiation The model is determined on the basis of experimentally obtained data on the behaviour of the given system. We speak about the so-alled indutive proedures (indution - determining general las on the basis of generalizing researhes onerning partiular, speial ases). Experiment is a purposeful ativity aimed at knoledge development. Its important feature is its repeatability under the same onditions (reproduibility). Experimental tehnique involves measuring, omputing and ontrol tehnique; nevertheless, omputer is urrently its essential element and plays three basi roles in the experiment: it olletively proesses experimental data (data measurement, olletion, evaluation, storage), orks as a model agent and manages the proess of experimentation. When talking about experimental methods of identifiation, it is assumed that e an measure input and output signals of the system diretly in the operation and the reords of these signals timing ould then be used to evaluate the dynami properties of systems. By analyzing the input and output values of the system e an get a mathematial model expressing its external desription (e.g. differential equation, transfer, et.). Suh a model an be haraterized as an external desription of the behaviour of the given system. Using experimental methods, e assume at least approximate knoledge of the struture of the objet hih e onsider to be a "blak box", hih means that the model gives no information on internal physi-hemial bonds of the system under examination. Disadvantage of these methods of identifiation is the

11 fat that the identified objet usually forms a part of a larger devie, and therefore, it annot be examined in isolation as one ould ish. When examining in onnetion ith another devie, e often fail to eliminate the effets of other variables of the devie or the effets of the disturbane variables. Deterministi methods of experimental identifiation do not take into aount effets of random variables on objets or measurement errors. They are simple and lear. If measurement is performed on the objet arefully, one obtains good results. They are ideal for single parameter systems, in partiular. For multiple parameter systems they are suitable if e an ignore values of the variables that are not being monitored or eliminate their influene (by maintaining a onstant value). Stohasti methods presume effets of random variables on the objet, or the fat that the measured values are subjet to noise. To identify the systems aording to random time ourses of input and output signals, methods of statistial dynamis suh as orrelation analysis, regression analysis and others are used. Statistial identifiation methods require muh larger sets of measured data and their proessing is possible only hen using a omputer. Using experimental method of identifiation e usually proeed in the folloing ay: for suitably hosen model struture (thus e understand the ay of mathematial expression of the dependene of output signal on input signal, e.g. in the form of differential equation, differene equation, transfer, impulse response), e estimate its parameters, that is orders and sizes of oeffiients of equations, respetively transmissions. It is usually performed by applying different methods to evaluate the reords of system response on a defined input signal. Nevertheless, results of the experiment an be used and proessed in other ays, too: a) they an be used to verify onlusions resulting from the mathematial and physial analysis of the system in pratie, or to refine the mathematial model obtained through mathematial and physial analysis, b) in some ases, they enable identifiation of onstants quantitatively expressing the proess - suh as heat transfer oeffiients during heating, et. Advantages of the experimental model: Mostly, this model has a simple form; parameters are easy to be determined (using a mathematial alulation method).

12 Disadvantages of the experimental model: It is appliable to a lesser extent than the analytial model, and it usually applies only to a partiular examined objet and mostly only to a measured area (it means that the extrapolation of the system behaviour outside the ertified area is inadmissible and so is the generalization of results obtained by using them to similar objets). From the above it is lear that both methods of identifiation - analytial and experimental - omplement eah other ell. One an say that an appropriate ombination of these methods an reate onditions for suessful identifiation. Finding the most suitable method of identifiation implies great theoretial knoledge and, in partiular, experiene as eah speifi system requires a different method of identifiation. Pratial methods mostly lie beteen the folloing to extreme ases: the analytial ay determines approximate mathematial relationships desribing the given system and, then, the parameters that appear in them an be speified experimentally. Analogy Analogy is based on similarity of mathematial desription of a physial proess. Relevant physial proess variables physially orrespond to other variables of the model. Analogy method is preferably used if e annot solve basi equations analytially. The fat that one equation mathematially aptures the essene of several physial phenomena enables one proess being modelled by another one hih is easier to be ahieved or measured. Mathematial similarity, as opposed to physial similarity, allos onstrution of models (analogons) hih are not geometrially or physially to resemble the produt. For example, in the field of thermal tehniques, hydrothermal analogy (beteen heat ondution and onvetion of liquid in a tank) or eletrothermal analogy (beteen heat ondution and transmission of eletri energy) is most often used. Hydrothermal analogy: for example, the similarity beteen the non-stationary heat ondution and onvetion of liquid in a tank: Fouriers equation of heat ondution: t t a x h h relation for hange in the height of a olumn of liquid floing in a tank: b n here: b = /K.s, s = setion of the tube, h = height of the liquid olumn, n = number of apillaries, k = hydrauli resistane

13 t 3 3 h h h 3 x x x x h t Fig. Hydrothermal analogy Eletrothermal analogy: for example, the similarity beteen the non-stationary equation of heat ondution and transmission of eletri energy. Fouriers equation of heat ondution: t t a x Eletri poer transmission: U U RC x R U R U R 3 U 3 C C C 3 U t Fig. Eletrothermal analogy Eletroanalogy has spread muh more signifiantly than the hydro-analogy beause onstrution of eletroanalogons is simpler than ith hydrauli models and also beause the theory of eletrial iruits is elaborated in depth; preision of eletrial measurement tehniques is high and the results an easily be automatially reorded. The tehnial mean for reating the eletroanalog model is represented by an analog omputer, here relevant variables of the physial proess orrespond to ontinuous eletrial values on the omputer. Cyberneti (funtional) modelling Cyberneti modelling uses models of the so-alled "blak box" - this term refers to the system not providing information on its internal struture, hoever, only on the external behaviour (e.g. models obtained by experimental methods of identifiation, models of neural netorks, fuzzy models, expert systems, geneti algorithms, et.). 3

14 Terms summary Modelling, similarity, physial modelling, mathematial modelling, yberneti modelling, identifiation by mathematial-physial analysis, experimental identifiation, analogy. 4

15 MODELS Study time: 4 hours Objetive: The student ill be able: to define the term of the model, to lassify the different models aording to different aspets, to define the term of simulation, to desribe the main stages of the modelling and simulation proess, to ork ith the basi simulation programs, to desribe the proedure for ompiling analytial models, to solve the examples of the reation of analytial models of simple objets. Model of a real system is alays assoiated ith simplifying and negleting irrelevant details of the real system beause the reality an be aptured only to a ertain degree. It is this degree that deides on ho exatly the physial or abstrat model ill reflet the behaviour of the real objet, but it also determines its omplexity and, thus, the pratial appliability. Therefore, it is important for the model to apture those essential harateristis of the real system that affet its behaviour most, and then to observe and desribe them and, on the other hand, to ignore irrelevant features and phenomena. The model then displays the essential harateristis of the system (e.g. mathematial desription of the mathematial model) that apture the dependenies beteen input and output variables of the system. This knoledge an then be used for example for managing the given system at the orresponding point in time, or even for predition (foreasting) of its behaviour and, thereby, it enables intervention to solve partiular events even before they, in fat, our. Classifiation of models Aording to the level of abstration of the real objet: Physial model is reated on the basis of physial similarity of the model and produt. It onsists of natural or artifiial material system. 5

16 Physial and mathematial model (physial analog) is a model reated on the basis of mathematial and physial similarity. Original physial proess is replaed by an analogous proess having the same mathematial desription (eletroanalogy, hydroanalogy). Mathemati model onsists of an abstrat system of mathematial relations. It is based on mathematial similarity of the model and produt: - internal similarity (mathematial analog) - analytial mathematial models - external similarity (blak-box models) - experimental mathematial models, yberneti models representing only the external behaviour of the system. Aording to nature of the proess on the model: Deterministi model orresponds to deterministi (uniquely determined) relationships beteen input and output variables, i.e. e are able to assoiate these relationships learly and aurately and to desribe them. Stohasti model either the examined system itself or the method of solution have a random harater; it means that relationships (orrelations) beteen input and output variables are not entirely speified; they are given statistially ith a ertain probability. Aording to their determination, gaining (aording to identifiation methods): Analyti model Model obtained by analytial methods of identifiation is based on mass and energy balane equations of physial, hemial and biologial proesses. Experimental model Model obtained by experimental methods of identifiation, by measurements on real objets. Depending on hether they express stati or dynami properties of systems: Stati model It expresses the dependene of output variables on the input ones in a steady state of the system. It enables us to predit in advane hat the outputs ill be under the given inputs in 6

17 the steady state, hoever, it does not say anything about ho long it ill take to ahieve the outputs. It annot be used for management. Its suess - hen used in pratie - depends on the quality of information ontained in the model (a priori); during the proess, it annot be orreted. Dynami model Complete model that desribes not only stati but also dynami properties of the system. It tells us hat the proess of the output ill be at the time hen the input and system status are given. It is expressed, for example, by differential equations, transfer, et. It is used in the field of management here transitions from one state to another are important phenomena. Aording to the degree of resemblane to the original: Complete model It provides a omplete similarity in basi forms of its operation in spae and time. Inomplete model (partial, loal) It is haraterized by partial similarity, for example similarity just in spae, or just in time. In pratie, there often ours inomplete modelling hile maintaining similarity in only ertain loations of spae or time oordinate systems. Approximate model Some dependenes on the model are negleted or they are expressed approximately; they do not respet the riteria of similarity. Error in modelling depends on proximity of expression of a parameter and its overall influene on the ourse of the proess. For example, in terms of analyti solutions of differential equations, approximate methods are used - finite differenes method and finite element method are the most important of them. Finite differenes method its priniple is a substitution of the original differential equation by a differene equation, i.e. the derivative of a funtion at a ertain loation is replaed by the differene of funtion at disrete points. These differene models are used in omputer simulations (numeri derivation, integration), for example. Finite element method - its priniple lies in replaing a ontinuous physial quantity by an approximate disrete model. This model is reated from a set of pieeise ontinuous funtions defined on a finite number of sub-areas alled elements. Elements are assumed to be interonneted in a finite number of nodal points on their borders. The ourse of a sought variable inside elements is - during solution - approximated by a seleted funtion. This must be learly determined by the values of variables at the nodal points. Observed values of the 7

18 physial quantity at the nodal points represent unknon basi parameters of the solution. These parameters speified by the numerial proedure determine the ourse of physial quantity of the area investigated. Appliation of models expressed by finite elements in modelling and simulation of systems has been groing, espeially in solving spatially omplex areas in time. Nevertheless, there is a signifiant limitation - there are extremely high demands on poer and memory of the omputer, partiularly hen dealing ith unstable (nonstationary) multidimensional tasks. Aording to the method of model information proessing: Analog model Analog model is based on a similarity of the mathematial desription of a physial proess. Relevant variables of the physial proess physially orrespond to other variables of the model. Analogy method is preferably used if e annot solve the basi equations analytially. The fat that a mathematial equation aptures the essene of several physial phenomena allos one proess to be modelled by another proess by the one hih is easily realizable and measurable. Mathematial similarity, as opposed to physial similarity, enables us to onstrut models (analogons) hih do not neessarily have to resemble the produt geometrially or physially. For example, in the field of thermal tehniques, hydrothermal analogy (beteen heat ondution and onvetion of liquid in the tank) or eletrothermal analogy (beteen heat ondution and transmission of eletri energy) is most frequently used. Eletroanalogy has spread muh more signifiantly than hydroanalogy beause the struture of eletroanalogons is simpler than ith hydrauli models and also beause the theory of eletri iruits is elaborated in depth, preision of eletrial measurement tehniques is high and the results an easily be reorded automatially. Tehnial mean for reating an eletroanalog model is an analog omputer, here the relevant variables of the physial proess orrespond to ontinuous eletrial values in the omputer. Digital model Information is expressed using digits (numbers); variables are in a disrete form. Tehnial mean for realizing digital models is a digital omputer. Hybrid model It represents a ombination of analog and digital models in order to take advantage of the properties of both. It is a universal mean enabling us to ombine digital models orking in 8

19 series and analog models orking parallelly and manage both of them either by a program or by a speially developed logial netork. Aording to the harater of mathematial desription of the model: Linear models All elements of mathematial desription of the model are linear operations. Nonlinear models At least one operation of mathematial desription is nonlinear. Aording to the purpose of the model: Reognition model It represents a mean for gaining knoledge. It is a passive role in hih the model does not influene the studied proess or objet diretly. Control model It is used to ontrol proesses, hih means that it is diretly involved in the proess or objet. It is partiularly applied in situations here neessary information on behaviour is impossible to be obtained by diret measurement (and feedbak is not applied). (t) Control u(t) Controlled y(t) model system Fig. 3 Control model Aording to the perspetive of modelling proess management: Unontrolled model Proess on the model is passive in terms of external ativity. Controlled model Proess on the model hanges atively aording to the ontrol onditions. Depending on hether they use objetive or subjetive information, or even heuristi information resulting from human experiene: Conventional model (lassial) They use objetive patterns resulting from natural las or experiments. 9

20 Nononventional models They use subjetive, heuristi knoledge resulting from human experiene for their reation. The area of artifiial intelligene: fuzzy models, expert models, models of neural netorks, geneti models. SIMULATION Basi priniple of simulation is to replae the original system by another system - knon as a simulation model, then experimenting on the simulation model and a retrospetive appliation of knoledge gained from the simulation model to the original system. This is knon as modelling of dynami systems in time ith diret or indiret effets on the examined objet (the onept of simulation an alays be replaed by a more general onept of modelling, but not vie versa). Simulation is used: to omplete the mathematial model of the studied system by verifying its auray and appliability based on its omparison ith the real system behaviour. On the basis of omparing the behaviour of the studied system and the simulation model, the redibility of the model an be assessed. The same signal u (t) is brought to the input into the real system and the model; the differene beteen the system output y (t) and the model output y m (t) is deteted, and adjustment of mathematial model parameters is performed so that the differene is minimal (Fig. ). hile experimenting ith the model, e use simulation to imitate a situation or to reate onditions hih may our in a real system during normal or emergeny orking onditions, hih enables to antiipated similar possible situations in advane. Knoledge aquired from the simulation model is then applied retroatively to the original system. ut system t y s ym T + simulation model - y M t min. Fig. 4 Speifying the simulation model parameters by omparing its behaviour ith the behaviour of the real system

21 Main stages of the modelling and simulation proess defining the system on the examined objet and finding a mathematial desription of the system using the identifiation methods (analytial, experimental), ompiling a omputer simulation model (its implementation on PC), verifying the ompliane of manifestations of the omputer simulation model and the real system, experiments ith the simulation model, applying the results of simulation experiments on the examined objet. Advantages of simulation models onditions that do not atually exist yet (emergeny situations that must not our on the real devie;, dynami properties of an objet an be identified before its implementation as suh, for example, in the designing stage) an be experimentally verified and tested, osts of the simulation on a model are muh loer than of tests on an existing system, time required to simulate a partiular situation is shorter than the time of the atual proess (simulation time an run muh faster than the real one), the simulation model enables testing of the influene of more variables ompared to testing on an existing system diretly. Simulation programs Simulation programs onvert a mathematial model (or a transformed mathematial model) into the form of the simulation model hih an be desribed by expressive means of the given simulation program or programming language. Simulation programs an be reated individually using the programming languages - this approah has many disadvantages - you have to kno partiular programming language; you need to reate and debug the program; you have to deal ith data input and output, expression of time on the model, statistial proessing, et. It is used in speial ases (on request of a ompany); it annot be used universally. As for the simulation, many simulation programs have been developed hih suit the user best - saving the users ork. Aording to the method of entering the model into the system, the simulation programs are divided into: ) blok-oriented (SIPRO, SIMULINK) ) equation-oriented (MATLAB, SIGSYS)

22 3) ombination of both (DYNAST) Equation-oriented simulation programs are used to solve mathematial models that are entered as a system of algebrai equations, differential equations, or integral equations, inluding boundary oeffiients and initial onditions. When solving differential equations of higher order, it is neessary to onvert these equations into the system of differential equations of the first order using the method of redution of derivative order (see belo). Blok-oriented simulation programs partial mathematial models (transfers) from a blok diagram are assigned to funtion bloks that perform required operation ith an input quantity in order to obtain the output quantity. Individual bloks are predefined and the user, hen reating the blok diagram, indiates the blok type, the method of onneting them, blok parameters, initial and boundary onditions. Advantage of these simulation programs is that they are user friendly; their illustration ability, bigger larity and easier searhing for errors hen debugging represent important advantages, too. Bloks are predefined in libraries of the simulation programs, hereas the user speifies: blok type method of onneting them blok parameters initial onditions boundary onditions parameters, initial onditions input u(t) Name and number of blok output y(t) Fig. 5 Blok of a simulation program It is suitable for a solution to be based on mathematial desription in the form of a transfer; and hen solving differential equations of a higher order, it is neessary to onvert these equations to the system of differential equations of the first order using the method of redution of derivative order. It is possible for modelling of a system given by a differential equation to use onversion in the Laplae transformation and to express the transfer. This proedure is simpler, but has its limitations:

23 differential equation has to be linear parameters in differential equation have to be onstant to express the transfer, initial onditions have to be equal to zero Matlab Matlab is an integrated environment for sientifi and tehnial omputing, design of algorithms, modelling, measurement and proessing of signals, and design of ontrol and ommuniation systems. The name of this program is an abbreviation of ords Matrix Laboratory. Having a look at the name, it is obvious that the most important part of the omputing ore are algorithms for operations ith matries of real and omplex numbers. Apart from simple matries, MATLAB even supports omplex data types suh as arrays of ells or multidimensional arrays. By omposing these data types, it is possible to reate arbitrarily omplex data strutures. The omputing ore itself makes use of LAPACK and ARPACK libraries by means of hih it is able to optimize its ativities to benefit from % of the omputer apaity. Data is automatially stored ith double preision. This setting an be hanged by the user in terms of auray and speed of alulations or in terms of the size of files. Fig. 6- MATLAB orking environment 3

24 Matlab is a omplete programming language; it means that it allos users to reate funtions aording to their needs. These funtions do not differ from embedded funtions by the proess of alling. Ne funtions an be interpreted diretly from the text format files alled M - file or from the pre-proessed form of P - file. The system also allos adding modules (MEX - file) ompiled into a mahine ode proessor. Matlab graphial system enables easy display and presentation of the results obtained from alulations. The data obtained an be dran in several types of graphs. These inlude todimensional (D), three-dimensional (3D) and pie harts, histograms and others. All graphial objets an be almost arbitrarily hanged in their appearane, both in their reation and after draing. Thus, three-dimensional graphs an be shaded, identifying soures of inoming light; graphs, inluding the three-dimensional ones, an be animated; ontours and transparent objets an be displayed; pseudoolor imaging an be used, and muh more. The appearane of graphial objets an also be hanged interatively using the toolbar. Handle Graphis system enables ontrol elements (buttons et.) to be inserted in the images, and thus, to reate an ative graphially ontrolled user interfae. Open Matlab struture led to reation of ne libraries and funtions, hih beame olletively alled toolboxes. These extend the use of program in relevant sientifi and tehnial fields []. Simulink Simulink is a blok-oriented simulation program for modelling and simulation of dynami systems using Matlab algorithms for solving nonlinear differential equations. Using Simulink, it is possible to reate models of linear and nonlinear systems that are ontinuous or disrete in time. The original funtion libraries intended to simulate a simple linear ontinuous and disrete system is orked up into their present form, hen they form a separate subsystem ith its on graphial - user interfae. The basis of Simulink is represented by individual bloks that are arranged thematially in blok libraries. These libraries inlude library ith most frequently used bloks, library for mathematial operations (Math): mathematial funtions, logial operators, et., libraries for entering mathematial desription of disrete systems (Disrete): transfer funtion, delay unit, et., ontinuous systems (Continuous ): derivation, integration, transfer funtion, et., libraries 4

25 of signal soures (Soures): generator of signals, sequene, jumps, pulses, et., libraries for displaying output signals and their storage (Sinks): osillosope, display, save in the file, in your orkspae, et., libraries ith nonlinear members (Disontinuities): learane, frition, delay members, et., libraries for orking ith signals (Signal Routing): merging of signals, signal distribution, onversion, et., and many others. You an also reate your on bloks. Models are hierarhial; therefore, you an reate models both by their deomposition and their omposition. System an be seen at high level, ithout a large amount of details, or e an look behind the mask of the system and, thus, a more detailed vie of the system an be displayed. This approah provides a piture of ho the model is organized and ho its parts interat. With the open arhiteture of Simulink, it is also possible to enrih the predefined libraries ith one s on bloks. Bloks in predefined libraries represent a fundamental part of stati and dynami systems and mathematial operations. By linking them, models of omplex systems are formed. From the library, bloks are moved to the indo for reation of the model by simply dragging the mouse. Here their inputs and outputs are onneted, and thus, reating the pattern desired. Then, it is neessary to set the properties of individual bloks, simulation time, step of the solution, numerial method and alulation auray. Fig. 7 Working environment of the Simulink toolbox 5

26 Upon ompletion of these steps, simulation an be performed. We have a hoie of several ays, either from the Simulink menu or by entering the ommand from the ommand indo of MATLAB. By using different bloks for display, results of the simulation an be seen hile the simulation is running. In addition, parameters an be hanged to see ho their hange affets the output. The simulation results an be inserted into the MATLAB orkspae for further proessing and visualization. Simulink an be used to solve differential equations, for alulation of transient and impulse harateristis, alulation of system responses to arbitrary input signals, for ontrol of stability, verifiation of quality of the regulatory proess and alulation of the response of ontrol iruits both ontinuous and disrete. To solve these examples, bloks and libraries from the folloing table are most often used. Tab. Bloks of the Simulink toolbox hih are useable in Modelling and Simulation ourse Library Blok Purpose Continuous Transfer Fn Integrator Derivate Entering the ontinuous transfer funtion Blok for integration Blok for derivation Math Operations Gain Sum Blok for Gain Blok for the sum Signal Routing Mux Blok for merging of the signal Sinks Sope Draing the output signal To Workspae Saving the output signal in the orkspae Soures Step Entering a step input signal Constant Entering a onstant input signal SIPRO SIPRO program is a blok-oriented simulation program developed espeially for teahing of modelling systems, automati ontrol, tehnial ybernetis and related disiplines. In developing the program the emphasis as put on simple environment hih is easy to use and on lo prodution model. As the blok diagrams are often used in the eduational proess for their larity, reating a model for SIPRO does not ause students major diffiulties. 6

27 Editor and the omputational part of the SIPRO program form inseparable parts of the integrated environment. In this environment the alulation parameters an be adjusted (time of the alulation, step of the solution, step of storage, et.); it is possible to start and stop alulation. The alulated results an be presented in a text (table), or graphially. The output an be performed both on a sreen and a printer, the simulation model and alulated results an be saved on a disk. Help, hih provides the user ith basi information onerning the integrated environment and individual bloks of the SIPRO language, is part of the system. Editor has a fixed format ith immediate syntax ontrol of the inserted data, therefore, you an onentrate mainly on reation of a logially orret model. SIPRO enables to solve simulation problems for ontinuous and disrete models, linear and nonlinear, aompanied by logial bloks/elements []. Fig. 8 Working environment of the SIPRO program The method of reduing derivative order: The method of reduing derivative order an be used for systems modelling even hen the onditions mentioned above are not met. Example: The system is given by a differential equation of the third order: y (t) + a y (t) + a y (t) + a o y(t) = b o u(t) + o 7

28 Model the given system in SIMULINK simulation program and express the ourse of transition and impulse harateristis. Proedure: The highest derivative is expressed expliitly: y (t) = b o u(t) + o - a y (t) - a y (t) - a o y(t) The derivative is gradually redued y (t) y (t) y (t) y(t) by onneting integrator bloks in a ro (using three integrators). The sum (or differene) of signals on the right side of the equation equals the value on the left side of the equation: y (t) (Fig. 9). u(t) Mux Step b Gain Sum Sum y (t) s Integrator y (t) s Integrator y(t) s Integrator y(t) a Gain Mux Sope Constant Sum a Gain Sum 3 a Gain 3 Fig. 9 The method of reduing derivative order in Simulink program EXAMPLES OF COMPILING ANALYTICAL MODELS OF SIMPLE OBJECTS Using the analytial method of identifiation e put together a mathematial model based on mathematial and physial analysis of the objet. We use engineering, tehnologial and operational data onerning the objet. Aording to physial, hemial and other las e mathematially desribe the phenomena taking plae in the objet, and thereby, e gain relationships beteen the variables monitored. Using equations of energy and material balane, equations of ontinuity, et. e are trying to determine relationships beteen input and output variables of the system. These relationships determine the mathematial model of the examined objet expressing the internal desription of the system (the so-alled hite box). To hat depth e have to explore phenomena and struture of the objet depends on the intended purpose of the partiular model. The deeper analysis is performed, the more aurate 8

29 should the mathematial model be. It ill, hoever, be more ompliated, more expensive; its derivation ill ost harder ork and its use ill be more demanding. Therefore, it is neessary to onsider hih objet details should be analyzed in order to make the assembled model suffiiently aurate, but not too omplex and ostly. Thus obtained mathematial model is "strutural", hih means that its individual relations orrespond to respetive parts of the examined objet. Strutures of the model and of the objet are similar; usually the same internal (state) variables are used in the model as in the objet. An advantage is the obvious orrelation beteen model parameters and onstrutional parameters of the objet and its dynami properties. Another advantage of the analytial approah is that e an determine the dynami properties of an objet even before its implementation. Thus, e an influene (optimize) its dynami properties by possible hanges already in the designing stage of the objet. Resulting models usually have ider validity than the models obtained by experimental methods of identifiation. Analytial approah requires not only a thorough knoledge of mathematis, but also a perfet knoledge of the field (tehnology) hih the studied objet belongs to. The analysis is often extremely diffiult; the resulting relationships are exessively ompliated and must be appropriately simplified. Auray and appliability of the model is limited if e onsider different random effets and unertainties hih appear in most of the real tehnial objets. Using the analytial ay, e get relations beteen all seleted variables in the objet. Based on these relations e an determine both state equations of dynami system defined on the examined objet and external system desriptions. Proedure for ompiling analytial models Proedure for ompiling a mathematial model an be divided into three phases. The first phase is based on seleting a set of variables and relationships beteen them by means of hih it is possible to desribe the real proess under onsideration aurately. For analyzing omplex objets deomposition of an objet into simpler parts (subsystems) is performed and binding (boundary) onditions are determined. By seleting too many variables for ompilation of the model e an reate too omplex model, the analysis of hih ould be extremely diffiult. Therefore, it is neessary to eliminate irrelevant variables, hih the proess is less sensitive to, hih represents the first simplisti phase. Hoever, e must not neglet the essential variables, hih ould be at the expense of the 9

30 adequay of the real objet model. Even the question of the appropriate deomposition of a omplex objet is not alays easy and requires some experiene and intuition. The seond stage is to build general dependenes (physial relationships) beteen seleted variables of the objet. This is the stage of reating the struture of mathematial model of the objet and one of the most diffiult stages of identifiation. This phase assumes a good knoledge of tehnial issues hih the objet belongs to by its physial nature, as ell as knoledge of related and theoretial disiplines. The analyst must have the ability to assess hih relationships are important, both in terms of proess behaviour and vie of management. Creating the struture of the model is usually based on knon physial las or various dependenies, derived or empirially established. Different types of equations are used for mathematial desription of the studied objets. It is neessary to take into aount distribution of the parameter monitored in the objet hih the proess takes plae in. Proesses that have the same values of monitored variables in the hole area of the devie are alled proesses ith onentrated parameters. Proesses ith a different value of the monitored variables aording to their plae in the objet are alled proesses ith distributed parameters. Algebrai equations are ommonly used for mathematial desription of the stationary modes of proesses (desription of stati behaviour, i.e. steady-state proesses ith onentrated parameters). Ordinary differential equations are used for mathematial desription of nonstationary modes (dynamis or transition states) of proesses ith onentrated parameters as ell as stationary modes (stati behaviour) of proesses ith distributed parameters, here values of the monitored variables depend only on one spatial oordinate. In the first ase, time is hosen as the independent variable in differential equations; in the seond ase, spatial oordinates are hosen. In mathematial desription of objets using ordinary differential equations it is essential to enter initial onditions. Partial differential equations are used for mathematial desription of the dynamis of proesses ith distributed parameters or stationary modes of suh proesses in hih distribution is in more than one spatial oordinate. In desribing the dynamis of a proess by suh equations, boundary onditions, hih are generally funtions of time, should be entered simultaneously ith the initial onditions. For stationary modes of proesses, haraterized by partial differential equations, only those initial onditions hih depend on oordinates are entered. 3

31 The last phase of analysis is alled proess model speifiation, i.e. determining the values of unknon parameters and oeffiients of a derived system of equations that determine the validity of the model for a partiular objet under speifi onditions. Mathematial model of an objet hih as aquired from theoretial analysis should alays be adjusted into a shape from hih the dynami properties of the examined objet ould be obvious, or into a shape hih ould be suitable for further use. Even hen identifying relatively simple objets elaborate and omplex models are reated, the solution of hih ould be uneonomi onsidering the number of alulations neessary. Therefore, in the next phase, it is neessary to simplify the model (approximate) hen the original mathematial relationships are replaed by simpler ones and the model is obtained in an easier shape to be used, hoever preserving those properties and relations - ithout distortion - on the objet hih e examine. Thorough physial analysis usually leads to more omplex types of equations (non-linear differential, partial differential, integral-differential, or nonlinear algebrai expressions) the diret analytial solution of hih is diffiult; these omplex equations ompliate further use of the results signifiantly. Transations ith linear models both in the analysis and synthesis are inomparably simpler than ith nonlinear models, so, if possible e attempt to linearize the nonlinear systems. We an assume that the relation beteen small hanges in input and output signals is linear, i.e. expressible by linear differential equations. The proess of simplifiation is not simple, it requires a great deal of experiene and there is a risk that the resulting model ill not be adequate to the objet. Suffiiently general and reliable method of approximation, hih guarantees the desired onformity of behaviour of both the original and simplified model, does not exist. Throughout the proess of reating the model e introdue a number of assumptions, simplifiations and approximations hih affet the auray of the model. This fat is one of the reasons hy the analytially derived model is experimentally tested as soon as possible. Further ork ith the mathematial model of the studied objet an be realized in time or frequeny domain. In the first ase, it is important for further progress hether the internal (state) variables are eliminated from the equations of the model or not. The most ommonly used proedure assumes the exlusion of state variables and the model adaptation into the form of the system of differential equations of the objet ontaining only input and output variables. In their solution e get timing of the response of the output signal to the input signal. Modern ontrol theory uses state spae methods to desribe dynamis of the objet. 3

32 The studied objet is then outside the vetor of input and output variables haraterized by the vetor of state variables that determine the state of the objet. The advantages of using state methods are mainly refleted in the investigation of omplex objets. When solving a mathematial model in the frequeny domain, the original system of equations is transposed by a suitable integral transformation. We also operate ith transfer funtion either in graphial or analytial expression. Currently, digital omputing means are used for mathematial modelling. Therefore, if e ant to solve differential equations on a digital omputer, e must make appropriate mathematial treatment. One option is to onvert the differential equation of the n-th order into the system of n equations of the first order, hih an be solved by an appropriate numerial method. In ase of ontinuous proesses modelling for digital ontrol, e onvert the ontinuous mathematial model of a ontrolled objet from a ontinuous form (e.g. in the form of differential equations or operators transmissions) into a disrete model (e.g. in the form of differential equations or disrete transmissions). For analytial mathematial models ompilation, the method of blok diagrams is useful (blok algebra); in this method e proeed as follos: ) We ompile a real, funtional and tehnial diagram of the objet modelled. ) We mark all the elements and variables ourring in the objet and the diretion of the ation of variables and signals. 3) For relationships beteen partiular variables e ompile relevant differential equations ith general onstants by a mathematial-physial analysis. Output variables are onsider as dependent; the input ones as independent variables. Input variables to individual subsystems an also be output variables of previous subsystems. In suh ase, it is sometimes advisable to proeed from the end, i.e. until e reah the input variables of the system. 4) Based on differential equations and interations of partiular variables e ompile a blok diagram. We desribe eah bloks and then mark out relevant input and output variables. 5) If the stati relations of partiular variables are linear, or if they are linearizable around the orking points, e an replae individual bloks ith operator transmissions. If the stati relations are not linear, e an use a blok diagram as a basis for modelling on the omputer. 3

33 6) If e have identified individual bloks ith transfer funtions, e an simplify the entire diagram aording to the rules of blok algebra to the operator transmission, expressing the relationship beteen operator images of relevant output and input variables. []. We ill illustrate ho to reate analytial models on a fe folloing examples of analyzing dynami properties of various objets. Heating material in furnae In Simulink program, your task is to build a mathematial model of heating the metal in a heating furnae. The exhange of heat takes plae in aordane ith Netons la of heat transfer by onvetion. The output variable is the temperature of the metal surfae k (t) [K]; the input variable is the temperature of the furnae p (t) [K]. The heated metal surfae is S [m ], its mass m [kg], speifi heat [J kg - K - ] and onvetion heat transfer oeffiient [W m - K - ] Define the operator transmission of material in the heating furnae and dra the blok diagram. The amount of heat transmitted by hot gases in time dt is given by the Netons la: dq = α. S. ( ( t) ( t) ). dt p k The transferred heat is spent on temperature hange of the metal by d k (t) and heat apaity of the metal inreases by: dq =. m. d k (t) Under the la of onservation of energy, the heat transmitted to the metal equals the hange of its heat apaity: α. S. ( ( t) ( t) ). dt =. m. d (t) p After the modifiation e get: k k. m. S. dk (t) k(t) p ( t) dt T. dk (t) k (t) p (t) dt here T =. m. S is a onstant of time. 33

34 34 The equation is onverted into the Laplae transformation and the transmission is determined: ) ( ) ( ) ( p p p p.. T p k k ) ( ) ( ) ( ) ( ) ( p T p p p U p Y p G p k Heating of the metal in the heating furnae is expressed by proportional system of the first order. Metal heating blok diagram is shon in Fig.. ) ( p T p G p (p) k (p) Fig. Metal heating blok diagram The folloing values of variables have been hosen for simulation: furnae temperature p (t) = K, heated metal servie is S = 4,44 m, its mass is m = 8 kg, speifi heat = 5 J kg - K - and onvetion heat transfer oeffiient = W m - K -. Then the time onstant T = S m.. = 9 s.

35 Fig. Simulation diagram of metal heating and depition of transition and impulse harateristis in the Simulink program Reuperator for air heating Build a mathematial model of dynami behaviour of reuperator for heating the air (Fig. ) in the Simulink program. Reuperator operates on the priniple of transmitting temperature of one medium (flue gases) to another medium (air). Air that is heated to a temperature v (t)[k] goes through a tube, in hih the flue gases flo around at a temperature p (t)[k]. Identify the operator transmission of the reuperator, and dra a blok diagram. We assume that heat exhange is arried out by ondution, from the flue gases through the tube all ith oeffiient of heat transfer by onvetion [W m - K - ] and then from the tube through its inner all ith the heat transfer oeffiient [W m - K - ]. In addition, e assume that the temperature differene on the alls of the tube is insignifiantly small (it is a thermally thin body; e neglet the heat transfer by ondution in the tube). V t [m 3 ] is the volume of the reuperator tube; t [kg m -3 ] is the density of the tube; t [J kg - K - ] is the speifi heat of the pipe; S [m ] is the outer surfae of the reuperator tube flon around by flue gases; V v [m 3 ] is the volume of air; v [kg m -3 ] is the air density; t [J kg - K - ] is the speifi heat of air, and S [m ] is the inner surfae of reuperator tube flon around by air []. 35

36 Fig. Reuperator for air heating [] The amount of heat transferred by flue gases to the reuperator tube alls in time dt is given by the Netons la: dq = α. S. ( ( t) ( t) ). dt p t The transferred heat is onsumed for hanging the temperature of the tube by d t (t) and heat apaity of the pipe ill inrease by the value of: dq = t. t.v t. d t (t) Under the la of onservation of energy the heat transmitted to the tube by flue gases equals to the hange in the tube heat apaity: α. S. ( p( t) t ( t) ). dt = t. t.v t. d t (t) The equation is further modified and, thus, e get the folloing: Vt t t. S dt ( t) t ( t) p ( t) dt dt ( t) T. t ( t) p ( t) dt here T = Vt t t S is a onstant of time. The equation is onverted into the Laplae transformation and the partial transfer of flue gases - tube is determined: T. p. t ( p) t ( p) p( p) 36

37 t ( p) G ( p) ( p) p T p The amount of heat transmitted by the all of the reuperator tube to the air in time dt is given by the Netons la: dq = α. S. ( ( t) ( t) ). dt t v The transferred heat is spent on air temperature hange by v (t) and heat apaity of air inreases by the value: dq = v. v.v v.. d v (t) Under the la of onservation of energy the heat transmitted by the all of the reuperator tube to the air equals to the hange in heat apaity of air: α. S. ( ( t) ( t) ). dt = v. v.v v. d v (t) t v The equation is further modified, thus: Vv v S v. dv(t ) v(t ) t (t ) dt dv(t ) T. v(t ) dt t ( t ) here T = Vv v S v is a onstant of time. The equation is onverted into the Laplae transformation and partial transfer of tube - air is determined: T. p. v( p ) v( p ) t( p ) v( p ) G( p ) ( p ) T t p The resulting transfer of the hole reuperator is the produt of partial transfers: v( p ) G( p ) G ( p ) G ( p ) ( p ) (T p p ).(T p ) Reuperator ats as a non-osillatory proportional system of the seond order. Blok diagram of the reuperator is shon in Fig

38 p (p) s (p) v (p) G (p) G (p) Fig. 3 Blok diagram of a reuperator [] For simulation purposes, it is advisable to ork ith inreases, e.g. from ambient temperature, i.e. v (t) = v (t) -, t (t) = t (t) -, p (t) = p (t) -. Sine it is a linear regulated system, forms of transmissions ill not hange. The folloing values of variables have been hosen for the simulation: t = s; t = 5 s; p (t) = 8 K; T = s; T = 5 s. Mux Sope Pulse Generator s+ Transfer Fn 5 s+ Transfer Fn Mux Fig. 4 Simulation diagram of a reuperator in the Simulink program: yello - p (t), purple - t (t), blue - v (t) 38

39 Car suspension Determine ho big ompression is reated on the axle by the fore ating on the ar body [5]. Create a mathematial model of a ar suspension in the shape of operator transmission and using the method for reduing the derivative order. The relationship desribing the displaement of the ar body position x(t)[m] depending on the applied fore F[N] is in the form of a linear differential equation of the seond order: m d x( t) b dt d x( t) dt k x( t) F( t) here m[kg] is the vehile eight, b [N s m - ] is the oeffiient of damping and k [N m - ] is strength of the spring. Using Laplae transformation of the relationship (3) e obtain the folloing relation: m p X(p)+ b p X(p)+ k X(p)= F(p) Transmission of the system, thus, has the folloing form: G( p) X ( p) F( p) m p b p k m k p k b p k This is a transmission of an osillating proportional member ith the inertia of the seond order. F(p) G( p) k m b p p k k X(p) Fig. 5 Blok diagram of the ar suspension To build a model using the method for reduing the derivative order e express expliitly the highest derivative: m d x( t) dt F( t) b d x( t) dt k x( t) We gradually derease derivative x (t) x (t) x(t) by onneting integrator bloks in a ro (e use to integrators). The sum (or differene) of signals on the right side of the equation equals to the value on the left side of the equation: x (t) 39

40 The folloing values of variables have been hosen for simulation: vehile eight m = 3 kg, damping oeffiient b = 3 N s m - and the strength of spring k = 6 N m - Step 3 s +3 s+6 Transfer Fn Sope simout To Workspae Step -K- Gain Sum s Integrator s Integrator Sope Sum Gain -K- Gain -K- Fig. 6 Simulation sheme of ar suspension and draing a body position defletion x(t) depending on the ating fore F v in the Simulink program 4

41 Aumulation of material in a landfill Fig. 7 shos the proess of aumulating material in a landfill, here m(t)[kg] is the total amount of material in a landfill, q (t), respetively q (t)[kg s-] is the mass of imports, respetively exports []. Build a mathematial model of the aumulation of material in a landfill in the Simulink program. ) ) u Fig. 7 Proess of aumulation of material in a landfill [] Elementary inrement of the amount of material in a landfill dm(t) for an elementary time inrement dt is desribed in the folloing balane equation: dm(t) = q (t)dt q (t)dt after modifying it e get a linear differential equation of the st order: dm( t) dt q ( t) q ( ). t The amount of material in a landfill annot be negative, i.e. m(t). In the landfill, under time t=, there has been some initial amount m() = m. By integrating the relation, hen taking into aount the initial ondition m, e obtain an equivalent expression of the proess of aumulating the material in a landfill. t m(t)= q ( t) q ( t) dt m. This relationship implies integration (aumulation) harater of the proess of aumulating the material (Fig. 8). 4

42 Fig. 8 Blok diagram of the proess of aumulating material in a landfill in a time domain [] Based on the relation expressed above, properties of the proess of material aumulation an be easily analyzed (Fig. 9): Fig. 9 Time ourse of the proess of aumulating material in a landfill [] Fig. 9 shos that steady-state output from the proess of aumulating the material ours ith any amount of material in a landfill m u in ase of q(t ) q(t ). Moreover, if there is qu qu konst, a steady state of the entire proess of material aumulation is established. It is, thus, obvious that the relationship beteen the amount of material in a landfill m u (output) and imported q u and exported q u quantity (inputs) does not exist in the steady state. Using Laplae transformation of the relation expressed above e get the folloing relation: m ( p) Q ( p) Q ( p) p p M For zero initial ondition m = then applies the folloing: M ( p) Q ( p) Q ( p p and operator transmission has the form of: ) M ( p) G( p) Q ( p) Q ( p) p 4

43 Fig. Blok diagram of aumulating material in a landfill [] For simulation the folloing values have been hosen: m = 3 kg; t = s; t = s; t = 4 s; t 3 = 5 s, q (t) = kg s -, q (t) = kg s -. Sope3 Step Step3 Produt Step4 s Integrator Sope Produt Step5 Fig. Simulation diagram of aumulating material in a landfill and draing of the imported q (t) and exported q (t) mass quantities in the Simulink program 43

44 Tank ith a free outflo of fluid Build a mathematial model of dynami behaviour of liquid in an open tank aording to Fig.. Flo volume of liquid flos into the tank q (t) [m 3.s - ] and the flo volume flos out of the tank freely q (t) [m 3.s - ] through a hole in the bottom of the tank due to gravitation. The fluid level has a onstant area of P[m ]; h(t)[m] is the ater level. Determine ho the ater level ill hange in time h(t) depending on the influx of the liquid q (t) []. For volume flo q (t), the folloing relation is valid: q ( t) a. h( ), t here a [m 5/ s - ] is a onstant oeffiient of the flo rate. Elementary inrement of volume of liquid in the tank Pdh (t) for the elementary time inrement dt is desribed in the folloing balane equation: P. dh( t) q( t). dt q( t). dt, resp. after substituting the relation (8): P. dh( t) q ( t). dt a. h( t). dt. Fig. Sheme of a tank ith a free fluid outflo [] After modifying it e get a nonlinear differential equation of the first order: dh( t) P. a. h ( t ) q ( t ) dt 44

45 ith the initial ondition h() = h. It is obvious that a nonzero initial ondition ould have been aused only by a nonzero influx q (t) for t <. If the inflo q (t) = for t <, then also h =. In a steady state all the hanges in time disappear (derivation that dh( t) dt t h( t) hu q ( t) qu, Stati harateristi equation ill have the form of: dh ( t) dt is zero), hih means h a. u q u. This relation expresses the dependene beteen the ater level h u and inflo q u in a steady state. Stati harateristis (Fig. 3) is defined only in the st quadrant beause physial limitations of liquid level are applied. h( t) h max, here h max is maximal height of the level given by the tank onstrution. Fig. 3 Stati harateristis of a tank ith a free fluid outflo 45

46 Blok diagram of a tank ith a free outflo hih orresponds to the differential equation mentioned above is shon in Fig. 4. Fig. 4 Blok diagram of a tank ith a free fluid outflo in a time domain [] For simulation the folloing values have been hosen: q (t) =.; P =.5 m ; h.= h u =. m; h max = m, a =.44 m 5/ s -. Sope Step Gain s Integrator Sope Gain -K- -K- Math Funtion sqrt q=., a=.44, P=.5, hu=. Fig. 5 Simulation sheme of a tank ith a free outflo of liquid and draing of infloing liquid level q (t) and the liquid level h (t) in the Simulink program 46

47 Terms summary Model, lassifiation of models aording to various riteria, simulation, simulation programs, the method of reduing the order of derivation, reation of analytial models of simple objets. 47

48 THEORY OF SIMILARITY AND MODELLING Study time: 4 hours Objetive: The student ill be able: to desribe the basi priniples of the theory of similarity, to derive a general riterion equations by arelational equations and dimensional analysis It is a physial and mathematial method, hih an be suessfully used for solving differential equations, hih are usually used to desribe a physial proess and then to proess the results of experiments. It an be used even hen the mathematial expression of the observed phenomenon is unknon, and a partial solution to the problem is aeptable. This method is onsidered to be a very good synthesis of theory and experiment. Using this method it is possible to obtain a mathematial desription of the investigated system in the form of a general riterial equation. Criterial equation is preferably used to desribe a physial phenomenon, the omplete physial equation of hih e annot solve diretly, or this equation is not knon. Any physial proess an be desribed by a basi equation, hih is usually a differential equation. This equation, hoever, desribes a ide lass of phenomena in general, so, it has infinitely many solutions. Moreover, to desribe a partiular phenomenon, onditions of uniqueness and monovaleny of solutions should be knon, too. Knoledge of physial equations and onditions mentioned above is suffiient to define and solve the task unambiguously. Nevertheless, this does not mean that e an alays find suh a solution by onventional mathematial means. In omplex ases, experimental solution is thus used. Based on the experiment empirial dependene beteen variables haraterizing the proess is sought. Suh empirial relationship is valid only for the onditions under hih the experiment took plae and annot be mehanially applied to other similar ations. Proessing of experimental results enabling their generalization to other ases should be based on learning about the similarities of phenomena. This fat has a tremendous importane for phenomena and proesses the physial equation of hih has not been ompiled yet. Even in 48

49 these ases it is neessary to use an experiment for solution but results an be generalized using the similarity las. Geometri similarity Geometri patterns are similar if their respetive angles are the same and the orresponding sides are proportional. Constant of similarity (proportionality) It aptures the ratio of orresponding values (Fig. 6). In similar systems it is onstant. C l l l l l3 onst. l l l l 3 L ) O Fig. 6 Produt (ork) and model - geometri similarity Physial similarity Physial phenomena are similar if the ratio of orresponding values that haraterize some ation in orresponding plaes of the model and ork is alays onstant. Mehanial similarity Kinemati similarity - similarity of media flo: in orresponding points veloities are similar (proportional, parallel, same orientation). C W W W W3 onst. W W W 3 49

50 W W W 3 W W W 3 PRODUCT MODEL Fig. 7 Produt (ork) and model kinematial similarity Dynami similarity - similarity of fores that ause similar movements: in orresponding points the ating fores are similar (proportional, parallel, same orientation). C F F F onst. F F L Thermal similarity Fig. 8 Produt (ork) and model dynami similarity Similarity in temperature, heat flo, thermal ondutivity, et. C t t t q q onst. C onst. C onst. q t t q q Invariant of similarity (riterion of similarity, similarity number, variable ) Variable expressed using relative units of measurement (dimensionless) having the same value at orresponding loations in similar systems. L L L l l l l l l L L l l l l l l l l L L 3 L 3 l l L 3 l l 3 ; L l l L l l, l l l l, l l L l ; L L 3 l L 3 We refrain from indies (and) marking hih proess the variables relate to beause in the transition from one proess to another, similar one, variables expressed using relative units of measurement ill remain unhanged (therefore, they are alled invariant, i.e. onstant). 5

51 Criteria (invariants) of similarity an be divided into to groups:. simplexes ratio of variables of the same name. omplexes ratio of variously named variables l L l l Re Differenes beteen onstant and invariant of similarity: Constant of similarity for expressing it to systems are alays needed. For variables of the same name it has a onstant value in all points of similar systems: C W W W // / W W // / W W // 3 / 3. For variously named variables it has a onstant value in similar systems: C C W C l It hanges if e replae a pair of similar phenomena by a pair of other similar phenomena: A C l l l l l3 B l l l l3 onst. C l onst. l l l l l l l l 3 3 Invariant of similarity for expressing it just one system is needed. In orresponding points of different similar systems it has the same value for variables of the same name: L l l L L l l and also for variously named variables: l l It does not hange in the transition from one phenomenon to another, similar one, i.e. it retains the same value in orresponding points for the hole group of similar proesses. Sentenes (theorems of similarity) In performing experiments e fous on: hih variables should be measured, 5

52 ho to treat the results of experiments best, hat other proesses an be onsidered similar, i.e. to determine the boundaries of validity of the experiment. The folloing sentenes of similarity anser these questions: st sentene of similarity For similar proesses the riteria in orresponding points are numerially equal. E.g. for a similar motion of to mehanial systems, seond Netons la an be used for any point of eah system: F m d d This equation must be of the same shape for both similar systems. For any material point of the st proess it applies: F m d d () For the orresponding point of the nd proess, the folloing is applied: F m d d The existene of similarity beteen the to proesses is expressed by these onditions: F C F F, m m m,, If e express variables of the st proess by variables of the nd proess, e get: F F m d m d F m F m d d () Equations () and () an be applied simultaneously if I F m (3) 5

53 this expression is alled an indiator of similarity. The equation shos that similarity onstants annot be seleted freely: only three onstants an be hosen, the fourth one is given by equation (3), hih is a mathematial expression of the first sentene of similarity: For similar proesses the indiators of similarity of proesses are equal to one. Equation (3) an be adjusted so that F m F m F Ne m This omplex is the same for all similar proesses, it is invariant, e all it mehanial similarity riterion, the so alled Netons riterion. The st sentene of similarity ansers the first question. In the experiment it is neessary to measure those variables that our in the derived riteria. nd sentene of similarity Most physial proesses are desribed by differential equations hih have been suessfully integrated only in some speial ases. It as proved that similarity riteria derived from differential equations and integrals (solutions) of these equations are the same. Integral (DR solution) an be expressed as a funtion of riteria hih e reeive from the differential equations desribing a partiular proess. Using this sentene of similarity riterial equation an be obtained as follos: differential equation is itemized for the st proess and nd proess, then the system of the nd proess is expressed using variables of the system of the st proess and the relevant similarity onstants (see the st sentene of similarity). By modifying it e get the indiators of similarity by means of hih e determine the relevant riteria. Funtional dependene beteen the riteria is determined experimentally. Criterial equation obtained is valid not only for the original differential equation but also for its integral. The nd sentene of similarity ansers the seond question: Results of the experiment should be proessed in order to determine the dependene not beteen partiular variables but beteen the riteria of similarity. 53

54 3 rd sentene of similarity This sentene ansers the third question beause it provides the boundaries of validity of the experiment. It speifies onditions for the similarity of proesses. st ondition: Similar are those proesses happening in the geometrially similar systems hih are subjet to the same relational equations (they are algebraially idential). nd ondition: Out of the sum of phenomena, hih are subjet to the same relational equation, a partiular phenomenon is defined by joining the onditions of uniqueness (monovaleny) of solutions, hih define the phenomenon from the hole set of phenomena. These onditions are the same for all similar phenomena (they are numerially in a onstant ratio); only the numerial values of their output variables ill be different. 3rd ondition: Criteria of similarity generated from the onditions of uniqueness must be equal at similar proesses. Summary: Conditions for the st proess (ork) to be similar to the nd proess (model): geometri shapes of the ork and the model must be respeted, in terms of the entire volume of the model, physial onstants desribing the proess aording to the same las as in the ork must be hanged, variables ourring under onditions of uniqueness of the ork and the model must be proportional in orresponding points of the model and the ork, onstants of similarity are to be seleted so that the determining riteria of orresponding points of the model and the ork are numerially equal. After fulfilling these onditions, the model ill be similar to the ork i.e. in all orresponding points of the model and the ork all riteria of similarity ill be numerially the same and all variables haraterizing the given proess ill be proportional to eah other. Derivation of general riterial equation To desribe a physial phenomenon, the omplete physial equation of hih e annot solve diretly or this equation is unknon, a riterial equation is used ith advantage. The proedure is that the relevant dimensional variables are replaed by similarity riteria and mutual funtional dependene beteen the riteria is determined experimentally. After the 54

55 55 experiment, riterial equations present a basi mathematial dependene hih is valid for the hole group of similar phenomena. Criterial equation is often used in those ases here a omplete physial equation is solvable analytially. Solution expressed by a riterial equation is simpler sine the number of riteria in this ase is alays loer than the number of relevant variables. The most frequently used methods of similarity theory for deriving a general riterial equation are analysis of relational equations and dimensional analysis. Analysis of relational equations It is more reliable but it requires knoledge of mathematial formulation of a partiular phenomenon. It is used in ases here the omplexity of a differential equation exludes the possibility of theoretial solution by their diret integrating. Solution is performed in suh a ay that riteria are determined from a differential equation and their mutual funtional dependene is determined by experiment. Example: Assignment: Determine a general riterial equation for modelling an unsteady flo of inompressible gas (liquid) ( = onst. in the entire volume). Solution: For the movement of inompressible gas flo, the Navier-Stokes equation an be applied; this equation expresses the influene of gravitational fore, pressure and frition on properties of the moving medium. The ontinuity equation an be applied, too; it expresses the la of onservation of eight in the medium flo. Itemizing the Navier-Stokes equation for the x-axis of the ork, then, the folloing is applied: z y x x x z x y x x x z y x x p g z y x () and the ontinuity equation: z y x z y x ()

56 56 Similar itemizing an be performed for other axis (y, z) and for the model. We introdue similarity onstants: p g l p p g g z z y y x x We express variables of the model through variables of the ork: z y x x p g z y x z y x l g l p x g x z x y x x l x (3) z y x z y x l (4) As equations () and (3) and equations () and (4) have to be idential, the folloing is applied: l l l p g l hih means that:. onst Ho l l l l riteria of homohronism It speifies the time periods for hih the proesses are similar. At steady states, here the time is not important, it is not neessary to be respeted.. onst Fr l g l g l g g l Frouds riteria It expresses the ratio of inertial and gravity fores. Not appliable for the eightless state.. onst Eu p p p p Eulers riteria

57 It expresses the ratio of pressure and inertial fores. Where there is a little differene in pressure, it does not have to be respeted. l l l l l Re onst. Reynolds riteria - kinemati visosity m s - -dynami visosity Pas Ho onst. e ould get the same riteria Ho l l For nonstationary flo of inompressible medium, final riterial equation an be ritten: f(re, Ho, Fr, Eu)= Funtional relationship beteen riteria is determined experimentally for speifi ases, i.e. speifi onditions for the uniqueness of solution, and further, the sope of validity of the experiment has to be given, hih means for hih ranges of values Re, Ho, Fr, Eu the experiment as performed (experiment annot be generalized outside the measured area). During the experiment it is, therefore, neessary to measure all the variables ating in the riteria of similarity; measurement results must be expressed in the form of riteria and funtional dependene beteen the riteria for speifi onditions of the uniqueness of solution, hih is required for eah speifi ase, must be determined experimentally. This proedure is muh simpler than if e ere experimentally searhing for the dependene beteen partiular variables, the number of hih is bigger than the similarity riteria. Dimensional analysis It is less reliable but it is a unique method here the mathematial equations desribing the phenomenon are not knon and e need to find a funtional relationship beteen variables appearing in the proess. This method may lead to inorret onlusions for the folloing reasons: e do not over all variables haraterizing a given proess, variables that are not applied in the proess an be inluded aidentally, dimensionless variables annot be ontrolled. 57

58 To derive dimensionless riteria e only need to kno the variables that influene the proess studied and their dimensions in the SI units system. SI units system inludes basi, supplementary and derived units. Basi units are: meter [m], kilogram [kg], seond [s], ampere [A], kelvin [K], mol [M] and andela [d], supplementary units are radian and steradian. Derived units are derived from basi units oherently based on a system of definitional equations. For eah variable φ dimensional formula an be ritten based on dimensional formula. In pratie, the dimensional formula is presented as a produt of basi units symbols ith their respetive exponents and it is alled the dimension of variable φ so that: [φ] = m x. kg x. s x3. A x4. K x5. mol x6. d x7 here x, x,... xn are dimensional exponents (rational numbers). Symbols of units ith exponent equal to zero are not mentioned at a partiular formula. The dimensionless variables are all exponents in the formula equal to zero. E.g. for the quantity F, hih is alled poer, in the lassial mehanis of equation the folloing is applied: F m d d It implies from the definition equation that apart from mass, length and time, other basi units do not our, therefore, in the dimensional formula the exponents ill be x4 = x5 = x6 = x7 =. Dimensional formula for poer (poer dimension) has the form as follos: [F] = [m.kg.s - ] Several methods have been proposed to derive the riterial equation using dimensional analysis. The easiest and most reliable one is the Rayleighs method. Proess: We suppose that e do not kno more about a ertain proess than the fat that it depends on n dimensional variables knon to us A, A,... A n. We kno that there is some hitherto unknon relationship among them: f (A, A A n ) = We suppose that the same proess an be expressed using a different relationship among a smaller number p of dimensionless arguments π, π,... π p : 58

59 F (π, π π p ) = here p<n Aording to the Bukinghams π theorem, the dimensionless riteria are the produt of the square (poer) of the original variables: π = A x. A x xn A n here x, x,... xn are dimensional exponents (usually small positive or negative numbers or zero). Sine π is a dimensionless variable, the right side of the equation is also dimensionless: dim π = [π] = = [A ] x. [A ] x [A n ] xn The dimension of eah variable A, A,... A n is expressed using basi variables of the SI system and the squares ith the same base of units are merged. If the equation equals to one, the merged squares of the units must equal to zero, too. Thus, e obtain a system of r equations ith n unknons, here n> r. We get the numerial values of exponents x, x,... xn by solving this system of equations. Sine n> r, there are infinitely many solutions that e obtain if e substitute arbitrary numbers for the redundant unknons and alulate the remaining unknons. Out of the infinite number of solutions, only p = n-r of mutually independent groups of solutions, i.e. the number of dimensionless riteria. Example: Assignment: Using the Bukingham π theorem speify the deisive riteria of similarity of heat onvetion in an isothermal turbulent steady flo of fluid in a tube. Solution: From the problem analysis one an see that the oeffiient of heat transfer by onvetion α k [kg.s-3.k-] is a funtion: of fluid flo veloity.[m. s - ] and of physial properties of the fluid suh as: kinemati visosity γ.[m. s - ], density [m -3.kg], speifi heat apaity p [m.s -.K - ], thermal ondutivity oeffiient λ [m.kg.s -3.K - ]. Heat transfer is also affeted by the inner diameter of the pipe d [m]. Number of dimensional variables = number of unknons xi: n = 7 59

60 Number of basi dimensions = number of equations (if the equations are linearly independent): r = 4 Number of independent dimensionless riteria = number of independent groups of solutions: p = 7-4=3 Physial equation ontains 7 relevant variables: f(α k,, γ,, p, λ, d) = The general form of riterial equation aording to Bukinghams theorem is as follos: π = α k x. x. γ x3. x4. p x5. λ x6. d x7 Size of π an be expressed using dimensions of other variables. Dimensional formula ill have the form of: [π] = [kg.s -3.K - ] x.[m. s - ] x.[m. s - ] x3.[m -3.kg] x4.[m.s -.K - ] x5.[m.kg.s -3.K - ] x6.[m] x7 = = m x+.x3-3.x4+.x5+x6+x7. kg x+x4+x6. s -3.x-x-x3-.x5-3.x6. K -x-x5-x6 = The sum of dimensional exponents for eah basi variable must be equal to zero beause the left side of the equation is equal to one: x + x 3 3x 4 + x 5 + x 6 + x 7 = x + x 4 + x 6 = 3x + x + x 3 + x 5 + 3x 6 = x + x 5 + x 6 = [m] [kg] [s] [K] First, e hek the mutual independene of equations (e.g. by Gaussian elimination method or using the determinant of a square matrix) in order to avoid errors arising from the fat that some of the equations is a ombination of other equations. The matrix system has the folloing form: = 3 Gauss elimination method: the diagonal elements are zero hile no line is zero; the matrix has the value of h = 4. 6

61 There are more unknons than equations; therefore, there are infinitely many solutions that e obtain if e substitute arbitrary numbers for the redundant unknons and e alulate the remaining unknons. From the infinite number of solutions only p = n-r = 3 of mutually independent groups of solution = number of dimensionless riteria (π, π, π 3 ). Criteria are determined by solving the system of equations for three times, i.e. e find three independent solutions. There are 7 unknons, only 4 equations, i.e. for eah alulation of a riterion 3 redundant unknons must be seleted so that the solution is unequivoal and definite. When hoosing the values of redundant unknons, it is appropriate to apply the folloing method: one unknon is laid equal to one and other ones are seleted as zero. Suh seletion has only the folloing restrition: the seleted unknons shall not depend on eah other. For instane, the values x, x 4 and x 6 annot be hosen arbitrarily and simultaneously in a single solution. We have seleted: x = ; x =, x 4 = x 3 + x 5 + x 6 + x 7 = + x 6 = 3 + x 3 +x 5 + 3x 6 = + x 5 + x 6 = from here x 3 =, x 5 =, x 6 = -, x 7 = Substituting the exponents into the expression e obtain a dimensionless riterion: π = α k. λ - k. d. d = Nu Nusselt s riterion We have seleted: x = ; x =, x6 = + x 3 3x 4 + x 5 + x 7 = x 4 = + x 3 + x 5 = x 5 = from here x 3 = -, x 4 =, x 5 =, x 7 = π =. γ -. d =.d Re Reynolds riterion We have seleted: x =, x3 =, x7 = 6

62 x + 3x 4 + x 5 + x 6 = x + + x 5 + 3x 6 = x 5 + x 6 = from here x =, x 4 =, x 5 =, x 6 = - π 3 = γ.. p. λ -.. = p Pr a Prandtls riterion It is possible to selet various other ombinations of unknons; e get again a group of three riteria from hih e get the knon riteria bak by mathematial modifiations. There is no general guidane that an be given for the seletion of unknons in individual solutions; in fat, every independent hoie leads to the goal. Hoever, it is appropriate that there is non-zero exponent only in a single solution for a dependent variable as it eliminates the dependent variable from other riteria. E.g. in this example is a dependent variable αk, the exponent if hih is x, and, therefore, in the seond and third solutions x = is hosen. The result is that the α k ours only in Nusselts riterion. Solved physis equation an be reritten as dimensionless expression: F (Nu, Re, Pr) = Funtional dependene beteen partiular riteria must be determined experimentally. Even the resulting dimensionless riteria must be mutually independent, therefore, e ill hek their mutual independene again, for example using Gaussian elimination method. Variations in seleting other exponents lead to riterial equations hih ontain different, sometimes less knon, riteria of similarity. If, for example, e hose in the seond solution: x =, x =, x 3 = 3x 4 + x 5 + x 6 + x 7 = x 4 + x 6 = + x 5 + 3x 6 = x 5 + x 6 = from here x 4 =, x 5 =, x 6 = -, x 7 = 6

63 π =.. p. λ -. d =... d p d. a Pe Péles riterion Criterial equation ill then be: F (Nu, Pe, Pr) = hile the F funtion is different than the F funtion and an be determined experimentally as ell. Terms summary Geometri similarity, physial similarity, onstant of similarity, invariant of similarity, the sentene of similarity, analysis of relational equations, dimensional analysis. 63

64 COMPLEX SYSTEMS MODELLING Study time: 5 hours Objetive: The student ill be able: define the term and fields of artifiial intelligene, to desribe the fundamentals of artifiial neural netorks, to define and desribe the utilization of artifiial neural netorks. It onerns the study of real systems properties by investigating the behaviour of their mathematial models - modelling and simulation tehniques these are no ell-knon disiplines of system siene. Mathematial models oupy a key position here. They represent not only a suitable form for the reation of knoledge about the studied objets and phenomena; hoever, along ith omputer tehnology, they also represent an effetive tool for their appliation and deeper understanding. The aim of onstruting a mathematial model is usually its use in information or ontrol systems. Development of suh systems has reently been supported by an extensive use of softare, personal omputers and modern ontrol devies, built on the basis of the no poerful and reliable eletronis. Hardare and softare resoures have ahieved ide expansion in engineering pratie oing to their high level of quality and implementation availability. Mathematial analytial and statistial models are still the most idely used lass of models and e onsider them to be onventional models. The methodology of formation of these onventional models is very extensive and it is very ell desribed in a number of speialist orks. Basi approahes are based on knoledge of physial priniples and essene of funtions of the modelled and studied systems. These models are built on the basis of the so alled deep, objetive knoledge. A ommon proedure is that a priori information about an examined proess allos you to reate a default model struture and, in subsequent phases, the stage of the proposed strutures identifiation ith the objet itself using information from experimentally obtained data (identifiation of the struture and parameters of the model) takes plae. 64

65 Mathematial models using exat and generally appliable las of nature are intrinsially aurate. Suh models are not alays ompletely adequate to the reality of the orld that is inherently impreise and more or less vague. This is espeially true in ases here it is neessary to model omplex systems diffiult to desribe and diffiult to measure. Quality of the model in suh ases is mainly given by the fat of ho the applied methodology omes to terms ith formalization and effetive use of unertainty, hih is typial for desribing omplex systems. A typial form of representation of unertainty is provided by the apparatus of mathematial statistis. Statistial approahes based on the empirial likelihood priniples, hoever, suffer from many limitations. It is mainly the ability to reflet only the vagueness of stohasti type. Furthermore, there is a problem of frequent unavailability of suffiient number of observations (time reasons, tehnial, eonomi reasons, et.). Frequently, there are problems ith the validity of a range of a priori assumptions neessary for the orretness of statistial methods. Statistial analyses, therefore, often lead to results the varianes of hih are - due to the fats mentioned above - unaeptably large. In many ases, in solving engineering problems there are situations hih are - in an effort to use objetive information only - insoluble. Main ause of this situation is a lak of information on the behaviour of suh systems studied; the existing information is often inaurate or even inomplete. A separate hapter of theoretial and tehnial ybernetis involves problems of behaviour desription and management of suh real systems hih are haraterized by great omplexity, great diffiulty (and often impossibility) of its formal mathematial desription, hih is hardly notieable and operationally problematially measurable. Classial formal approahes, based on numerial mathematial analysis, supported by the apparatus of mathematial statistis, hen applied they enounter notieable limits on its use in suh desriptions. Limits in the use of onventional mathematial means of investigation and management of omplex systems stem from their fundamentally preise las. These are regularly interpreted by relationships of lassial mathematis. Using suh approahes often leads to neessary simplifiations, hih auses a lak of adequay of mathematial desription. Zadehs la of inompatibility is often applied desribing the phenomenon of dereasing the pratial appliability of a mathematial model in the groth of omplexity of its strutures. If the ause of the mathematial model failure is a lak of information about the investigated system, methods, hih the ne branh of siene - artifiial intelligene - brings, an be 65

66 used. Priniples of its approahes lie in the use of knoledge based not on the validity of the general, objetive las of nature, but rather on the human experiene, hih means subjetive, heuristi knoledge. In many ases, espeially in the diagnosis, predition of behaviour and ontrolling of omplex, hardly reognizable and measurable systems, the onlusions of hih are obtained by an experiened operator expert, are suessfully used in pratie, as a result of the deision-making proesses in his/her brain. Although these findings are based, equally as in the ase of lassial mathematial analysis, on the omparison of not very preisely defined inputs and outputs of the monitored system, very different apparatus from the statistial one is used to exert them. It onerns an effetive use of oneptual, verbal unertainty (vagueness), hih is highly developed in the human subjet. It is also benefiial to effiiently use the fundamentally simple but effiient nonnumerial algorithms that allo a human expert to integrate objetive (i.e. deep) knoledge ith subjetive (i.e. shallo) knoledge ith the effet of ahieving a higher quality of onlusions hen solving problems, deision-making or ontrolling. Suh proedures, using oneptual unertainty to onstrut effetive model strutures (like a human expert), allo reating unonventional non-numerial language models. Suh models an exeed limits of onventional mathematial and analytial models by their appliability. Based on their use, ne and unonventional approahes to management are reated. In other ases, other tehniques inspired by ativities of the human brain or the las of genetis are used simultaneously to ahieve higher adequay of desriptions. Quality of human judgments is onditioned by the ability of effetive treatment of not entirely aurate information, hile their unertainty - vagueness - has a different harater than the stohasti unertainty. In reent deades, this apability, together ith other harateristis of human thinking as manifestations of the human intellet, has beome the fous of a ne sientifi disipline - artifiial intelligene [7]. Artifiial intelligene Definitions and fields of appliation of artifiial intelligene Siene of artifiial intelligene attempts to reate algorithms and systems manifesting, in a ertain sense, intelligent behaviour and abilities. It is a siene that involves many different development diretions. Just as it is not possible to define the very onept of intelligene 66

67 preisely, it is not possible to apture omprehensively this broad siene industry by a single exhausting definition. Probably the best seems to be the definition of Minsky [4], hih is often referred to as the engineering definition of artifiial intelligene: Artifiial intelligene is a field of siene that deals ith the reation and development of mahines that, in ertain situations, behave as e sa it in man as an expression of his intelligene. This definition says that the artifiial intelligene should be able to solve suh omplex tasks that a human ould solve only using his intelligene. Complexity an be determined by a number of all possible options that are under onsideration. In simple tasks, here the number of variants is small, it is possible to find a solution by piking and evaluating alternatives using the method of "trial and error". In more omplex tasks suh a proedure does not produe any results even using the most poerful superomputers. A solution that neglets those variants that do not give enough hanes for a solution already in the beginning an be onsidered an intelligent solution. The more of them are rightfully omitted, the less there is left to searh, and thus the solution ill be more intelligent. The mehanism that allos for some variations to be rejeted in the beginning is based on the use of knoledge. These an be obtained both by taking from a man ho is able to deal ith the task intelligently and by analyzing examples of the assignment and its intelligent solution. Knoledge an be either of exat nature (theorems, physial las), the so alled objetive (deep) knoledge, or it an be formed by heuristi knoledge, the so alled subjetive (shallo) knoledge, hih is usually not supported by a deeper theory but very often helps to find a solution (although generally does not guarantee the solution) effiiently. Exat knoledge leads mostly to mathematial or algorithmi formulation that gives the sought solution after evaluation. In ontrast, heuristi knoledge is often expressed using non-ategorial, indeterminate rules understood only as a non-exat guide to solution. The harateristis of artifiial intelligene using knoledge and omplexity of tasks as desribed above does not inlude some tasks that belong to the artifiial intelligene only marginally or are partially separated from it and have beome independent. It enables us to judge intelligene by enumeration of partial issues to be addressed in designing "intelligent" systems. Another approah an be seen on the Rihs quote [4]: Artifiial intelligene is onerned ith ho omputers an solve tasks, hih people an handle better so far. Aording to this definition of the ontent of artifiial intelligene, this ontent is diretly linked to the 67

68 urrent state of omputer siene and it is therefore expeted that ith the development of omputer tehnology the fous of artifiial intelligene ill shift and hange. Certain disadvantage of the definition disussed is the fat that it does not inlude tasks that need to be solved, but neither omputer nor human an deal ith them yet. On the other hand, it is a very brief and fairly preise definition of hat onstitutes the real ontent of artifiial intelligene as a sientifi disipline. Koteks definition is the definition of artifiial intelligene as suh harateristis of tehnial systems hih is being explored by the artifiial intelligene as a disipline: "Artifiial intelligene is a property of systems artifiially reated by man possessing the ability to reognize objets, phenomena and situations, to analyze the relationships beteen them and thus reate internal models of the orld in hih these systems exist, and on this basis then make purposeful deisions, ith the help of ability to predit the onsequenes of these deisions and disover ne regularities beteen different models or groups.". The introdution of internal models enables defining deision-making and management toards the given goal as follos: both the initial and the target states of the environment are given by their models. Aeptable ativities that an hange the state are given. The task is to find a sequene of ativities that an onvert the initial state to the target one hile respeting prespeified limits. Thus formulated problem in artifiial intelligene is alled solution of tasks. Formalization of models and ativities is inluded in the issue of knoledge representation. Using the Koteks definition [4] those partial theoretial tasks that fall ithin the artifiial intelligene an be expliitly identified and enumerated. These tasks inlude mainly the folloing: Pattern Reognition hih inludes both the Computer Vision and Natural Language Proessing, Knoledge Representation and related issues of speial softare systems for the use of in-depth knoledge of experts, namely the issue of Expert Systems and the area of Knoledge Engineering, Problem Solving, Qualitative Modelling, Mahine Learning, Planning, 68

69 Robotis, Geneti Algorithms, Neural Netorks. In terms of the artifiial intelligene as a sientifi disipline, it is a disipline the subjet of hih is not firmly defined yet and hih is laking a unifying theoretial basis; hoever, it rather refers to as to a set of methods, theoretial approahes and algorithms that are unified by the efforts to solve very omplex omputing tasks. Another typial thing for the artifiial intelligene is that by ahieving results in solving speifi problems these results do not form a part of the artifiial intelligene any longer and are smoothly transferred to other disiplines here they are applied to or are used to form ne, separate disiplines. For example, development of some learning algorithms is no an integral part of modern Theory of Automati Control. Tehniques of representation and use of knoledge form the bakbone of a modern, knoledge-based part of softare engineering. Similarly, the theory of knoledge representation using frames and sripts (using slightly different terminology) has beome fully naturalized in the orld of objet-oriented programming and objet-oriented systems. In the exat definition of the sope of artifiial intelligene, there is a ertain lak of uniformity, and the same inonsisteny prevails in the speifiation of its areas. This is largely given primarily by the history and development of individual areas. Use of artifiial intelligene for omplex systems modelling Artifiial intelligene, hih, in its partiular area, deals ith methods of deision-making and solving omplex problems, has provided the means hih appear to be very promising for desribing and managing omplex systems. The most important means inlude qualitative modelling, fuzzy models, neural netorks, geneti algorithms and their effiient ombinations. Qualitative modelling methodology uses the so-alled naive physis to onstrut models built on the integration of objetive and subjetive knoledge. Their onlusions are purely qualitative; do not assume any numerial information. Using the methods of interval mathematis, semi-qualitative models are therefore designed by means of hih numerially oriented onlusions are dedued. 69

70 The ore of fuzzy systems are fuzzy models, the methodology of hih is no relatively ell developed. For formalization of oneptual vagueness and its proessing, priniples of fuzzy multiset mathematis and multivalued language (fuzzy) logi are used. Fuzzy models are often oneived as systems of onditional prodution rules. The most ommon fuzzy systems, the expert systems, are aimed at solving speifi omplex problems. They are based on models using subjetive, heuristi knoledge of experts in a partiular problem area. Their onlusions have very narro loal validity. Signifiant expansion has been ahieved in reent years by methods inspired by biologial proesses in subjets. In an effort to partially imitate proesses ourring in the human brain and in neural system, neural netorks appeared. They are suitable for modelling the behaviour of omplex systems espeially beause their typial feature is the ability of selflearning. In terms of appliations the fuzzy-neural systems are very interesting. These fuzzyneural systems use fuzzy logi in their internal proedures and are very effiient at proessing vague information, partiularly hen adapting fuzzy models. Natural mehanism of natural seletion and the las of genetis ere the inspiration for the reation of geneti algorithms, hih are universal and robust means for solving optimization problems. In terms of identifiation and modelling of omplex systems, appliations of geneti algorithms for strutural and parametri identifiation of fuzzy models (fuzzy-geneti systems), or neural netorks (neuro-geneti, or fuzzy-neuro-geneti systems) are interesting. The approahes of modelling, in hih their learning ability, robustness and easy implementability are supported at the expense of preision, are inluded in the frameork of the so-alled "soft omputing" methodologies. This inludes, in partiular, the above mentioned methods, based on the use of fuzzy logi, neural netorks, geneti algorithms, probabilisti extrapolation and haos theory or its effetive ombination [7]. From a theoretial and pratial point of vie neural netorks proved to be the best possible approximation to the relationships beteen proess data of various tehnologial proesses. Currently, neural netorks signifiantly ontribute to modelling as ell as to adapting to the automation of tehnologial proesses. Neural netorks Researh on neural netorks as initially motivated by age-old effort to kno the human brain. Only later neurophysiologial knoledge enabled us to solve pratial problems in the 7

71 field of artifiial intelligene. These findings ere used to develop simplified mathematial models of neural netorks and they keep developing regardless of hether they model their living original. It is often useful to onfront the resulting models ith their original, or to be inspired by it to reate ne models. Basi exeutive and strutural element of a biologial neural netork is a biologial neuron. In a very simplified form, its funtion is to reeive signals from its surroundings, their proessing, reation of a response and transmitting this response further. Interonneted neurons form a neural netork. An example of a natural neural netork is the human nervous system. It provides links beteen the external environment and the organism as ell as beteen its various parts and ensures appropriate response to stimuli from the exterior and the interior states of the organism. The reation mehanism onsists of obtaining information (impulses) by individual sensors, alled reeptors, hih enable reeiving mehanial, thermal, light and hemial stimuli, their spread toards other neurons that proess suh information, and finally, impulses are sent to the relevant exeutive organs, the so alled effetors. Biologial neuron A biologial neuron is the basi funtional element of the nervous system. It is a speialized ell designed for transmission, proessing and storage of important information for the realization of vital funtions of the organism. Individual parts of nerve ells and the ay it is onneted ith other nerve ells is shon in Fig. 9. Cell body Synapses Axon Dendrits Fig. 9 Simplified diagram of a biologial neuron and the ay it onnets ith other neurons [9] 7

72 A neuron onsists of a body, alled a soma, hih the input transmission hannel in the form of dendrites leads to; the output is provided by axon. There are the terminals hih branh off from the axon; the terminals end ith a membrane hih meets the dendrites of other neurons. The transfer of information beteen axon terminals and dendrites is arried out via interneuron interfae, alled synapses. Synapses are of to kinds. Some may be exitatory; in this ase, they enable the spread of exitation and its amplifiation; others an be inhibitory, ausing inhibition of exitation. Memory of the nervous system orks on the priniple of enoding the synapti onnetions. The spread of information is possible through membranes of soma and axon hih, hen irritated by an eletrial impulse of ertain threshold intensity, generate an eletrial impulse themselves. Synapti permeability is hanged at eah pass of the signal, hih gives the neuron the ability of memory. Mathemati neuron models Formal neuron We have just desribed in a very simple ay a substane of the biologial neuron. No lets try to do a formal desription hat means to reate a model of a neuron defined in suh a ay. We ill start ith Fig. 3 here the basi symbols are defined: Fig. 3 Diagram of a formal neuron 7

73 The formal neuron has n of generally real inputs x,,x n orresponding to dendrites. All inputs are assessed by appropriate synapti eights,, n hih are generally also real. Weights determine the transmission rate of the input signal. The eighed sum of input values presents the inner potential of the neuron z: n z i x i i h y z Output (state) of the neuron y modelling the eletri impulse of the axon is generally given by a non-linear transfer funtion σ, the argument of hih is the inner potential of z. This output is then also the input to other neurons, as shon in Fig. 3. Neuron i xi Neuronjx j Axon Fibres Synapses Neuron n x Neuron k xk Synapti eight k Dendrits Fig. 3 Diagram of an artifiial neuron inspired by biologial model and the ay of its onnetion ith other neurons [9] The most idely used transfer funtions involve: sharp nonlinearity (jump funtion), linear features or saturated linear funtion standard (logisti) sigmoid, hyperboli tangent. Biologial neurons are omplex and annot be desribed analytially. Therefore, only simplified models are used. Transfer funtion of the artifiial neuron is based on the approximation of pulse modulation of a real neuron potential ave, hih spreads by the axon fibre neuron. Transfer funtions in the neurons models simply model the properties of biologial neurons. They an take different forms depending on hih properties of biologial 73

74 neurons e are trying to imitate by a partiular model or reation of hih neural netorks e are planning to use these models for. Researh on suitability of different types of transfer funtions for different types of neural netorks and appliations is the subjet of many sientifi papers. Used transfer funtion has a major impat on the quality of neural netork ativity. Y Z Fig. 3 Nonlinear jump transfer funtion of pereptron The nonlinear jump funtion (Fig. 3) as historially used as the first. As for the exlusive use of the transfer funtion aross the netork, it is possible to use only bistable output of neurons, hih may be disadvantageous for a number of pratial appliations. The advantage of this approah is the ability to teah the netork the Bools display in a single step. Y Z Fig. 33 Linear transfer funtion of the neuron Another possible type of transfer funtion is the linear funtion or a funtion linear in parts (Fig. 33). Several researhes have proved that using these funtions the netork quality 74

75 onerning the speed of adaptation and the ability of generalization is loer than the one of the netorks using nonlinear transfer funtion. The presene of linear transfer funtions entails one disadvantage: these funtions are not fault tolerant, i.e. that hen an unduly large value omes to the input of the neuron, then, the argument is transferred to the output in proportion to the diretion of the linear funtion. Their use has, hoever, its importane, partiularly in the output layer of neural netork here it is not neessary to transform the desired output values into the interval -, as it is often required for other transfer funtions that operate ithin this interval. The third and urrently most idely used type of funtions are nonlinear ontinuous monotoni funtions. To most ommonly used funtions of this group are the sigmoid and hyperboli tangent (Fig. 34 and 35). Suh transfer funtions allo the sensitivity of the neuron to be big for small signals hile for higher signal levels its sensitivity dereases. They are therefore highly resistant to failures. If an unduly large value omes to the input of the neuron, due to the urvature of these funtions, values lose to either or (-) appear at the output, depending on the argument sign. Slope is possible to be taken as the netork parameter hih signifiantly ontributes to the quality of its ativity. Slope also affets the speed of learning (or adaptation) of the netork. Therefore, algorithms have been developed hih, based on the dynamis of derease of the so alled netork error funtion, govern the slope values of transfer funtions of entire netork or of the partiular neurons. Y Z Fig. 34 Sigmoid transfer funtion of the neuron 75

76 Y Z Fig. 35 Hyperboli tangent transfer funtion Pereptron One of the most important models of the neuron is alled pereptron [9]; its potential is defined as a eighed sum of the signals entering. Pereptron inputs are real numbers; the output takes disrete values from the set {,}. If the inner potential of the neuron exeeds its threshold (defined by the symbol h), the neuron ill exite to. Otherise, the neuron is inhibited, hih is represented by a value of. Mathematially, this proedure an be expressed using the signum funtion as follos: y n z sgn x h sgn( z h ) i i i sgn( x ) x x here z is given by the equation: n z i x i i h By introduing a permanent neuron in the input ith a state of exitation of x = and a link to our neuron = - h, the preeding equation an be simplified as follos: y sgn n i x i i The transfer funtion of the pereptron is the so alled sharp nonlinearity (nonlinear jump funtion) Fig

77 n If e analyze the expression i x i so that e put it equal to zero, e obtain a hyperplane i equation (in to-dimensional ase represented by a straight line). Thus, using the vetor reord: W X This hyperplane divides the entrane area into to half-spaes. If the input vetor is loated in one half-spae bounded as follos, the output is exited to the value of ; similarly, the vetor of the seond half-spae auses a zero output. Pereptron is, thus, able to distinguish to ategories of inputs. The question no is ho to determine the values of neuron eights so that it is able to identify orretly (assoiate to lasses) the entries submitted. In order to ahieve this, it is neessary to adapt the pereptron, based on a training set, through an algorithm. One of the best knon priniples is the adaptation (learning) of the neuron aording to the Hebbs rule [9] hih is defined as follos: The eights are set randomly. If the output is orret, the eights do not hange. If the output is to be equal to, but it is, the eights are enlarged on ative inputs. If the output is to be equal to, but it is, the eights are redued on ative inputs. Yet inputs are ative only hen their value is above the threshold, that is non-zero. The amount of hanges in eights (inrement or derement) depends on the partiular seletion of one out of three possible options: To enlarge and redue the eights, solid gains are applied. Inreases vary depending on an error size. It is advantageous if they are bigger for a bigger error and vie versa. Thus ahieved aeleration of onvergene, hoever, an result in unstable learning. Variables and fixed inreases are ombined depending on the error size. Continuous pereptron Transfer funtion is ontinuous (and differentiable). The most ommonly used transfer funtions are standard sigmoid (Fig. 34) and hyperboli tangent (Fig. 35). Equation of a standard sigmoid has the folloing shape [9]: 77

78 n y σz z x h λz i i e here indiates the slope of the sigmoid and the inner potential is given by the relation already knon to us (). As follos from the ourse of standard sigmoid, the output of the neuron takes values from the interval <;>. From the fats given above, the folloing onlusions arise: i Exitation of the neuron varies beteen and here the value of means full neuron exitation in ontrast to the value hih orresponds to the ondition of inhibition. In ase the internal potential of the neuron is lose to +, there is a full exitation neurons, i.e. y=. On the other hand, in ase the internal potential of the neuron is lose to -, leads to a omplete inhibition of the neuron, thus y=. The tangent equation (Fig. 34) has the form of: y tanh(z) here z i xi h n i Artifiial neural netorks Definition of artifiial neural netork An artifiial neural netork generally refers to suh struture for distributed parallel data proessing, hih onsists of a ertain, usually very large number of interrelated performane elements. Eah of them an simultaneously reeive any finite number of different input data. It an pass an arbitrary finite number of idential information on the status of their single, hoever branhed, output on the other exeutive omponents. Eah exeutive element transforms input data to the output ones aording to ertain transfer funtion. Neural netork as a hole arries out some typial transformation funtion F (or a set of transformation funtions, even the adaptively hanging ones). A transformation funtion of the netork is understood as an assignment of the vetor of its outputs Y to the vetor of its inputs X, expressed by: Y = F(X) 78

79 Considerations referring to hih transformations of the funtion F an be arried out using neural netorks have been of interest sine the early development of their knoledge. Mathematial proof of the transformation properties of the funtion F of a generally strutured neural netork has not been available for many years. Only Heht-Nielsen and Hornik shoed the ay for more general understanding in this diretion. This leads to the Kolmogorov theorem, hih onerns the possibility of representation of ontinuous funtions of n variables using the final sum and superposition of ontinuous funtions of one variable. By applying the Kolmogorov theorem on the issue of neural netorks it as observed that in order to approximate any funtion f by the transformation funtion F, it is suffiient that the relevant neural netork has at least three layers of respetive numbers of neurons (exeutive elements) in eah layer. Essential harateristis of neural netorks Neural netorks use a distributed parallel proessing of the information hen performing the alulations. We an say that reording, proessing and transferring the information are arried out rather by means of the hole neural netork than by means of partiular memory plaes. It means that memory and information proessing ithin the neural netork are, in their natural substane, rather global than loal. Knoledge is reorded espeially through strength of linkages beteen partiular neurons. Linkages beteen neurons leading to a "orret anser" are being strengthened and linkages leading to a "rong anser" are being eakened by means of repeated exposure of examples desribing the problem area. Learning is a basi and essential feature of neural netorks. This fat expresses the basi differene beteen so far ommon usage of omputers and the usage of means on the basis of neural netorks. Main advantage and main differene at the same time, of neural netorks ompared to ommon means of the von Neumann arhiteture is exatly their learning ability. To reate a user programme e had to aim all our effort at reation of algorithms hih transform the input data set into the output data set. Neural netorks do not already need that diffiult stage. The ay of transforming the input data into the output data is determined by the learning stage based on the above mentioned exposure of samples (examples) desribing a given problem - training set. Therefore, there is no need to reate an algorithm. That need is substituted by submitting a training set to the neural netork and by its learning. In this sense, the neural netork reminds us of human intelligene, human ability 79

80 to learn from examples, human skills and knoledge that s/he is not able to solve algorithmially using ommon omputers beause of the absene of an analytial desription or beause the analysis is too diffiult. Importantly, neural netorks have the ability not only to remember all the model examples given by the training set, but they an generalize patterns and thus solve examples hih the netork has never met ith. This key apability in the terminology of neural netorks is alled generalization. Charateristis of neural netorks Arhiteture (topology) is determined by the number of neurons and their interonnetions. We distinguish beteen input, labour (hidden) and output neurons. The input neurons orrespond to reeptors, the output ones to effetors, and interonneted labour neurons reate paths beteen them to spread signals (impulses). Dissemination and proessing of information on its ay through the netork is enabled by the hange of the states of neurons lying on this road. State of a neural netork is given by the state of individual neurons lying on this ay. Configuration is determined by values of synapti eights of all onnetions beteen neurons. Aording to the diretion in hih the neural netork distributes the signals, neural netorks an be divided into to groups: Feedforard Neural Netork Feedbak Neural Netork For the feedforard neural netorks the outputs from one layer are direted to the inputs of the subsequent layer (therefore, they are alled multilayer neural netorks). Outputs from the last layer are the outputs from the entire netork (Fig. 36). 8

81 Fig. 36 Topology of multilayer feedforard netorks ith fully interonneted neurons [9] Fig. 37 Hopfields netork In ontrast, netorks ith feedbak (they are also alled reurrent) differ in the fat that the outputs from a layer are distributed not only to the subsequent layer, but are fed bak to the inputs of the given layer. An example might be a Hopfield netork (Fig. 37); it has one layer of neurons onneted to eah other ompletely. Response of the reurrent netork is timedependent due to the existene of an internal feedbak. After presenting the input data the appropriate output value is determined and it is simultaneously re-fed to the input hih it modifies. In the proess of learning of reurrent netorks e derive from the initial state of the netork and its gradual hanges take plae until a stable state is ahieved. These reurrent netorks an be unstable. For example, exitation an spread ontinuously, hih leads to divergene. Netorks ith feedbak are very interesting in terms of researh; hoever, in dealing ith real problems, to use feedforard neural netorks has proved to be advantageous. They are deployed in as many as 8% of appliations. 8

82 Modes of ork of neural netorks A neural netork keeps developing in time; the state of neurons and onnetions keep hanging; the eights are adapting. The ay these harateristis hange ith time is determined by three kinds of dynamis speifying the overall dynamis. They are the folloing dynamis: organizational (hange of topology), ative (hange of state), adaptive (hange of onfiguration). Work of artifiial neural netork thus alays takes plae in one of the three modes, orresponding to the dynamis mentioned (or the individual shemes may overlap in time). Organizational mode Organizational dynamis speifies the netork arhiteture and its possible hange. During the organizational mode the number of neurons, their arrangement and their interonnetions are determined. The layout is usually isolated from the other to modes; the netork designer has a onsiderable partiipation in the netork. It is usually determined heuristially based on the experiene and knoledge of the designer. Hoever, it is neessary to mention that there are adaptive algorithms, in hih a targeted gradual hange in geometry of netork forms a part or even the foundation of learning of the netork. Ative mode In ative mode, the neural netork is used either for the implementation of the requested transformation funtion, i.e. a reall ours (alulation), or it is used as one of the steps in the adaptive mode here it is neessary to alulate netork output and ompare it ith the desired output. In ative mode, states of input neurons are set at the beginning under the urrent model. All netork states onsist of the so alled state spae of the neural netork. Usually, development of the neural netork state in a disrete time is expeted. In eah suh time step, one or more neurons is seleted aording to the partiular rule of dynamis, the state of hih is updated on the basis of their inputs (states) represented by outputs of some neighbouring neurons that have updated their state in the previous step. 8

83 The ativation proess of most of the models is arried out in a one-time ay; for some models of neural netorks, the alulation is performed iteratively. Result of the alulation, and thus also the output of the neural netork, is the state of output neurons one the signal reahes their output. In the ative mode, thus, the netork reates reations to stimuli, i.e. it performs the transformation funtion. This funtion depends on the topology and netork onfiguration hih does not hange in the ative mode. Ative dynamis also speifies the transfer funtion of one neuron. If the regulation of this funtion is the same for all neurons in the netork, e talk about a homogeneous netork. Otherise, e speak of a heterogeneous netork inluding the ases here the funtions of neurons differ only in the values of slope. Adaptive mode In this mode, the netork learns in order to perform, in the ative mode, the funtion desired. There are to types of learning: learning ith a teaher learning ithout a teaher. When learning ith a teaher there is an outer riterion indiating hih output is orret. The eights in the netork are adjusted using a feedbak aording to ho lose the output is to the riterion. The netork is alays provided ith inputs as ell as ith the required outputs - the set of input-output ombinations is alled the training set. Differene beteen the desired and atual output is alulated. The eights are determined aording to an algorithm that ensures redution of errors beteen the real and desired output. Amount of hanges in eights is usually small. Then, the netork is given a ne output and the proess is repeated. After a large number of experiments, the netork learns to give a stable output in response to the inputs it reeives. The neural netork thus learns by omparing a real output ith the desired output, and by adjusting the eights of synapses in order to redue the differene beteen the real output and the desired one. The methodology of reduing the differene is determined by a learning algorithm. Methods of learning ith a teaher inlude for example bakpropagation, stohasti learning, geneti algorithm, learning ith error orretion and reinfored learning. Learning ithout a teaher does not have any external riterion of orretness. The algorithm of learning is designed in order to searh for speifi patterns ith ommon harateristis ithin the input data. The netork is provided ith inputs only and, using self- 83

84 organizing proess, it adapts in order to be able to distinguish beteen inputs adequately. Suh netorks are designed primarily for the lassifiation of input ombinations. Methods of unsupervised learning inlude the so alled Hebbs learning, differential Hebbs learning, minmax learning and ompetition. Adaptive dynamis speifies the initial netork onfiguration, and ho eights hange at a fixed topology in time. All possible onfigurations form the eight spae of the netork. In the adaptive mode, at the beginning, eights of all netork onnetions are determined for an initial onfiguration, for example using a random method. This is folloed by adaptation, mostly taking plae in disrete times (stages). The aim of adaptation is to find suh a netork onfiguration in the eight spae, hih ould, in the ative mode, implement the presribed funtion of netork ith minimal error. Netork error an be formulated in various ays, for example, square of the deviations of the desired output from the output of netork of all patterns from the training set is often used. Neural netorks an learn not only by hanging synapti eights, but also by adjusting the transfer funtion (slope) and the thresholds of neurons. Learning is usually a omplex nonlinear optimization problem the solution of hih an take a very long time on omputers. From the above one an derive that the ative mode is used to alulate the netork funtion hile the adaptive mode is used to find it. Individual modes an be linked one ith another. Advantages of blending the modes of neural netorks are obvious: for example, neuroregulator an regulate and also adapt to ne knoledge; in a longer learning proess it is possible to speify ation interventions in the ourse of regulation (ne input data into the training set). Nevertheless, blending the modes has indisputable drabaks, too. Perhaps the divergene of the adaptation proess is probably the most dangerous, espeially for industrial appliations, due to the hange of the learning set or of the netork topology. The hanges mentioned an even lead to netork osillation; in milder ases the netork response to ertain stimuli ill only get orse. It is, therefore, a matter of the expert, ho uses the netork, to determine if it is possible and effetive to apply blending modes. For operations ith enhaned seurity neither the hange of onfiguration nor the real-time adaptation is reommended. Blending the modes also inreases demands on tehnial implementation of neural netorks. It should be noted that the adaptation is mathematially (and hene omputationally) muh more diffiult proess than the alulation of responses in the ative mode. Moreover, hanging the topology of softare implementations of neural netorks usually represents 84

85 large alloations and dealloations of dynami variables (or objets). It is, therefore, neessary to adequately dimension the omputing tool that implements the neural netork. Despite the drabaks desribed it an be stated that blending operating modes of neural netorks represents their useful feature that an provide ertain appliations ith many benefits. Multilayer neural netorks Multilayer neural netorks are, due to their simpliity and flexibility of the most idely used models of neural netorks, deployed ithin a ide range of pratial appliations. They over the area of approximately 8% of all appliations arried out. Multilayer netorks have been named aording to their harateristi topology. Sometimes they also use the name of the feedforard netork, hih reflets the manner of signal distribution from input to output through the netork. Topology of multilayer netorks Figure 36 shos the arrangement of neurons and linkages harateristi for multilayer netorks. Basi element of this type of netorks is a ontinuous pereptron. The netork is omposed of at least three layers of neurons: input, output and at least one internal (hidden) layer. There is alays the so alled omplete linkage of neurons beteen to adjaent layers; it means that eah neuron of the loer layer is onneted to all neurons of the layer above. Any yles or interonnetions of neurons from the same layer are missing here. The input for this type of the netork is a vetor of numbers, hih is in the amount as the amount of the neurons of the input layer beause eah neuron of the input layer has only one input. Neurons in the input layer do not perform any mathematial operation ith the input values, the only purpose of these neurons is to distribute the input signals to the next layer so that eah neuron of the first hidden layer had an entire input vetor on its inputs. Signal distribution in this type of netork is the feedforard distribution [9]. Feedforard signal distribution in a multilayer netork Response of the neural netork is obtained by feedforard distribution of the input vetor through the netork toards the output neurons operating using this algorithm:. First, neurons of the input layer are exited to an appropriate level (mostly in the interval <,> or <-,>). 85

86 . These exitations are - by means of linkages - brought to a folloing higher layer and adjusted (strengthened or eakened) by synapti eights. 3. Eah neuron of this higher layer ill perform a total of adjusted signals from neurons of the loer layer and ill be exited to the level given by its transfer funtion. 4. This proess is arried out through all inner layers as far as the output layer here e ill obtain the exitations of all its neurons. In fat, e obtained a neural netork response to an input impulse given by the exitation of the input layer neurons. This is also the ay ho the signals in biologial system are transferred. The inner layer of biologial system an be reated for example by visual ells and, in the output layer of brain, partiular objets of monitoring are identified. Then the question still is ho the synapti eights leading to a orret response to the input signal are determined. The proess of determination of synapti eights is onneted ith a onept of learning - adaptation - of neural netork. [9]. Learning of multilayer neural netorks and the Bakpropagation method The aim of learning is to determine the transformation funtion using ertain riterion - the error funtion hih ould, ith some approximation, express a general relationship beteen input and output variables. Finding suh a relationship means solving a speifi problem area. Learning of the netork lies in the presentation of partiular patterns (vetors) from the training set the elements of hih form organized pairs of input-output. Eah training set pattern desribes ho neurons of input and output layer are exited. The netork response is obtained by submitting the input vetor and thanks to the knon signal propagation ithin the netork. At the beginning, it ill be quite different from the output given by the training set. Learning means an adaptation of eights in suh a ay to minimize the differenes beteen the netork responses from the desired outputs in the training set. The training set T an be formally defined as a set of elements (patterns) that are defined as ordered pairs in the folloing ay [9]: T I i I,O I,O I, p Op i i i i, k j 86

87 O i i i i o, n j here: p I i O i i j, o j number of patterns in the training set input layer exitation vetor onsisting of k neurons output layer exitation vetor onsisting of n neurons exitation of the j-th neuron of input or output layer Requirements for the learned netork outputs are not only to opy, ith minimal deviation, the output vetors of the training set but also to have a netork able to dedue the orret outputs for ne patterns. This property is alled generalization - generalization of the learned material, i.e. ho the neural netork is able to understand phenomena based on the learned matter that ere not part of the learning but that an someho be derived from the previous one. The training set is therefore divided into to subsets, one of hih is used to learn the netork (i.e. the adaptation of eights itself), the other one to verify hether the netork performs orret generalization. If the error of the learned netork response for the test set is too large in proportion to the error for the training set, it suggests that the netork generalization is poor. The phenomenon is knon as overtraining. This problem an be solved by dividing the group into three sets, and by setting apart a verifiation set. By traking the development of the verifiation set error, it is possible to determine the moment hen the netork begins to fix itself too muh on the individual patterns. At that moment, the learning stops or some of the learning parameters are hanged. The netork topology influenes orret generalization, in partiular the number of neurons in the hidden layer in proportion to the number of inputs and outputs. Currently used algorithms for the adaptation of pereptron layered neural netorks an, from a mathematial point of vie, be divided into to basi groups: Algorithms searhing for a minimum of the so alled eight or energy funtion in an n- dimensional spae (here n depends on the number of input and output neurons). The most ommonly used energy funtion is the sum of deviation quadrates of real outputs from the required ones, for all the training set elements. Hoever, it is also possible to use other types of funtions, suh as those based on hyperboli tangent, et. In ase of these 87

88 algorithms, the domain of mapping being sanned is not restrited, hih an lead to signifiantly slo global minimum searh. In addition to the speed of onvergene, another equally important aspet of quality of the learning algorithm is the generalization ability of the learned netork. Aording to theoretial onlusions, algorithms of this type an ahieve good results in only one of these areas, i.e. they either perform quik onvergene, or the learned netork has good generalization ability. Moreover, the generalization ability usually depends impliitly on the used algorithm, and it is not possible to make a request regarding its quality by, for example, means of a oeffiient. Another type of adaptation algorithms is represented by the ones that ill narro the domain of the sanned status spae. This results in faster onvergene of the algorithm to a minimum and, in addition, it is possible to use a suitable mathematial proedure to hoose suh a fration of the domain here the prerequisites for good generalization ability of the learned neural netork are present. In pratie, these algorithms are handled in order to searh for the eight funtion minimum again; hoever, at this point, it does not depend solely on the eights of links in the neural netork, but on some other variables as ell. This is mathematially equivalent to searh for a bound global extreme of an n-dimensional funtion. The best knon and most idely used method that allos the adaptation of multi-layer neural netork over the given training set is the Bakpropagation method. Unlike the already desribed forard movement of the neural netork signal propagation, this adaptation method is based on the opposite diretion of information propagation, from higher layers to the loer ones. First, e are going to desribe the priniple of this method [9]: First of all, e take the vetor of the I i i-th element from the training set and use it to exite the input layer neurons to a suitable level. We use the ell-knon method to perform the feedforard propagation of this signal up to the output layer of neurons. We ompare the desired status set by the vetor of the O i i-th element from the training set ith the real netork response of the urrent synapti eight setting. The differene beteen the real and the desired response defines the neural netork error. This error is then returned bak into the neural netork at ertain rate (learning rate) by 88

89 modifying the synapti eights beteen individual layers, from higher layers to the loer ones, so that the error in the folloing response is smaller. One the hole training set has been used, evaluate the total error for all the training set patterns and if it is higher than required, repeat the entire proess again. Seletion of patterns from the training set does not neessarily have to be sequential and, for example, random seletion is often used as ell. The hoie of strategy is not fixed and it rather belongs to the ategory of "ability to teah" neural netorks. Some patterns an be presented more often during the learning proess, others less often. Suitable strategy an greatly affet the speed and suess of learning. Total error in a learning algorithm means, for example, an average quadrati deviation alulated, taking into aount all the training set patterns. This error indiates the netork learning rate. Bakpropagation algorithm is based on minimizing the neural netork energy by means of the gradient adaptation method. The neural netork energy is a measure of netork learning, i.e. the deviation beteen the real and the desired output values of the neural netork for the given training set. It is basially the same as the total error, but to alulate the total error e prefer using statisti riteria. For the bakpropagation type of neural netork, the energy funtion E is defined as follos [9]: E p n y j o j i j i here: y j o j p n is the real response of the j-th output layer neuron required response of the j-th output layer neuron given by the training set pattern total number of training set patterns number of output layer neurons This gradient method allos for the alulation of the E error funtion derivatives, hih is hy it is neessary for the neuron transfer funtion to be differentiable. 89

90 The ay to ahieve the minimum error rests just in modifying the synapti eights beteen neurons i and j using the folloing formula (to make it lear, the seond index j ill not be used in the next part): i E i here: Δ i η is synapti eight hange is learning rate The above presented expression determines that the hange of synapti eight is given by a negative value of the partial derivative of error funtion aording to synapti eight, multiplied by the learning oeffiient. If the value of the derivative is high and positive, it means that the neural netork response error is steeply inreasing ith positive eight hange or, vie versa, the derivative is dereasing for negative values. It is therefore neessary to detrat part of the eight value or, vie versa, to inrease the eight. The oeffiient η refers to the extent of this hange. The higher it is, the greater hange in eights in the individual steps is as ell. Determination of the optimal value of this oeffiient is important for effetive adaptation of the neural netork. If the value is too high, the learning may beome unstable. If it is too small, the learning proess ill onverge to the minimum error funtion area too sloly. Hoever, setting this parameter is a matter of experimentation and searhing. Funtion E an be thought of as a urved surfae in hyperspae (Fig. 38). Immediate value of the eights then determines a point on this surfae. In order to ahieve the minimum neural netork energy, you must head from this point in the diretion of the steepest gradient. If e alulate the partial derivatives the energy spae gradient. E i for all eights, e reeive a vetor that determines 9

91 9 Fig. 38 Energy funtion E Energy funtion is a ompound one, hih is hy the partial derivative must be extended: i i z dz dy y E E If: n i i x i z z e z y the inner potential derivative an be quantified as: i i x z If e elaborate the output derivative: z z z z z z z e e e e e e e dz dy e an state that: y y z y No, e must only define the value of y E. Let us first onsider a situation hen the given neuron is a part of the neural netork output layer (Fig. 39).

92 Fig. 39 Adaptation of output layer neuron eights [9] It an be derived that for the expression and sample k E y y o j k The ase hen a neuron is found in one of the inner layers is illustrated in Fig. 4: The layer to hih the indexed expression belongs to is determined using a subsript and a supersript: E y n i E z i i z y here the sum is performed aross all neurons of the layer loated diretly above the neuron in question. Based on the validity of: y i z e an perform the folloing substitution: E y n i i E z i i 9

93 Fig. 4 Adaptation of inner layer neuron eights [9] The final formula for alulation of the neuron error gradient, inluding all the repeatedly used indies, is: E ij jk y j y j xi here δ jk means the differene beteen the real and required response of neuron j of pattern k of the outer or inner layer. For the outer layer: and for the inner one: jk y o j j k jk n ik i y i ji y i here ji is the eight of joint from the inner layer neuron j to the higher layer neuron i. The sum takes into aount all the neurons from the higher layer to the urrent one. It is obvious from the expressions that it is neessary to first set the error of neurons in higher layers, hih an subsequently be used to alulate the error of neurons in the loer layers [9]. 93

94 Aeleration of learning speed The Bakpropagation method is very often used for adaptation of neural netorks, but its disadvantage is the already mentioned slo onvergene of learning, ith the learning oeffiient set to exessively lo value or divergene for exessively high value. This problem an be solved in several ays:. Modifiation of the basi gradient method here the equation used for setting the synapti eights ill additionally inlude the so alled momentum (inertia) hih is taken into aount hen alulating the eight hanges from the previous step. The modified formula is: i E i i here: i is the eight hange in the previous step is the oeffiient of impat of eight hanges from the previous step (momentum, inertia) it indiates ho long it is neessary to keep ertain diretion before you turn aording to the ne hanges of gradient (this oeffiient as subsequently added to the algorithm to overome the so alled loal minima),, given by the previous steps., = means high netork inertia respeting the trend. Using parametri Bakpropagation PAB - it is an improvement of the regular method of Bakpropagation. It differs from the regular one by onverting the values of steepness and thresholds of neurons during the learning proess, in addition to synapti eights. 3. Using Netons method - it onverges ith suffiient speed in the viinity of the minimum error funtion. Hoever, it requires the alulation of seond derivatives, hih is hy it is diffiult to be alulated. 4. Using the onjugate gradient method - it, together ith Netons method, is another, more sophistiated and effiient method for nonlinear optimization. The method uses only the first derivatives in alulation, hih is hy it is more suitable for omputer use than the Netons method is. 94

95 5. Using other methods utilizing seond order derivation (Hessians matrix) or several of its approximations. This group of methods inludes, in addition to the Netons method, the quasi-netons method, the Gauss-Netons method, the pseudo-netons method and the Levenberg-Marquardts method. The seond order methods are reommended for their rapid onvergene and also beause they are haraterized by their high robustness. Fig. 4 Course of an error funtion during the neural netork learning. There is a derease of energy funtion E during the neural netork learning. A typial ourse of a ell-learned neural netork is illustrated in Figure 4. Fig. 4 Course of an error funtion hen deadloked in loal minimum 95