leitung: mathematical and computational modeling and simulation · mathematical and computational...

Mathematical and Computational Modeling and Simulation

Prof. Dr.-Ing. D.P.F.MöllerVAK 18.211

Sommersemester 2005

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Leitung: Prof. Dr.-Ing. D.P.F. Möller

2 Computational Modeling and Simulation Prof. Dr. Möller

Content

1. Modeling Continuous-Time and Discrete-Time Systems

2. Mathematical Description of Continuous-Time Systems

3. Mathematical Description of Discrete-Time Systems

4. Simulation Languages of Continuous-Time and Discrete-Time

Systems

5. Parameter Estimation of Dynamic Systems

6. Soft Computing in Simulation

7. Distributed Simulation

8. Virtual Reality in Simulation


Aims and Scopes

1. Modeling Continuous-Time and Discrete-Time Systemsintroductory material developing simulation models for real-world systems. Developed models usually take the form of a set of as-sumptions concerning the operation of real-world systems. Assump-tions are expressed in mathematical, logical, and symbolic relation-ships between the entities or objects of interest of real-world sys-tems. Once developed and verified, a model can be used to inves-tigate a wide variety of problems and questions about the real-world system, which is shown in the respective case study examples forthe several application domains such as

o biology, o business, o chemistry, o electrical engineering, o mechanical engineering, o medicine, o physics, o etc.


Aims and Scopes

2. Mathematical Description of Continuous-Time Systemsfocus on the most important mathematical methods in the

• time domain • frequency domain

that are used for mathematical description of real-world systems. Methods are based on

ordinary differential equations (ODEs) of nth order, sets of n first-order ordinary differential equations, partial differential equations (PDEs), superposition integral, convolution integral, Laplace transforms, etc.


Aims and Scopes

3. Mathematical Description of Discrete-Time Systemsfocus on the general principles modeling

queuing systems, discrete event concepts, Petri nets, statistical models, etc.


Aims and Scopes4. Simulation Languages for Computational M&S

contains the introductory material on the most interesting simulation systems at the language and logic level, such as

ACSL, AnyLogic, B2Spice A/D, CSMP, FEMLAB, GPSS, GPSS/H, MATLAB SIMULINKModelica, ModelMaker, SIDAS, SIMAN V, SIMSCRIPT, SLX,

and their application in the several case study examples.


Aims and Scopes

5. Parameter Identification of Dynamic Systemscontains a mathematical approach of ill-defined real-world systems for which the parameters of importance are not known or not measurable. Based on identification these unknown or un-measurable parameter can be estimated, using the several methods such as

gradient method, direct search method, least square method, etc.


Aims and Scopes

6. Soft-Computing Methodsfocus on fuzzy sets and neural networks in modeling and simulation to generate the basic insight that categories are not absolutely clear cut, they belong to a lesser or greater degree to the respectivecategory. Soft-computing breaks with the tradition that real-world systems can be precisely and unambiguously characterized, meaning divided into categories, for manipulation these formalizations according to precise and formal rules.


Aims and Scopes

7. Distributed Simulationcontains the introductory material of real-world systems that are distributed, and can be analyzed using the

tie-breaking method, critical time path method, High-Level-Architecture (HLA) concept.

The methods are introduced and used for real-world traffic problems.


Aims and Scopes

8. Virtual Reality contains the introductory material of computer-generated worlds that are based on

real-time computer graphics, color displays, advanced simulation software, etc.

Topics of virtual reality are used for real-world applications in the medical and geological domains.


Aims and ScopesCourse will provide a thorough foundation in • Modeling and simulation (M&S) methodology• Embedding M&S projects • Focusing on the several possible M&S application domains in

science and engineeringModeling methodology contains the respective mathematical descriptions of• Continuous-time dynamic systems• Discrete-time dynamic systems • Combined systemsDigital simulation contains• Continuous-time systems• Discrete-time systems• Simulation languages overview


Aims and Scopes

Course will also provide a thorough foundation in • Parameter estimation, necessary in case of ill defined systems• Distributed simulation

All topics are endowed by embedded examples and specific projects, which allow students getting their own experience in modeling and simulation in the several application domains in science and engineering, which will form the topics of the several projects


Web Material

Course material will be available from the web athttp://www.informatik.uni-hamburg.de/TISClickLehreclickSS2005click18.211 Mathematical and Computer Modeling and Simulation:

Methodologies and ApplicationsclickDownloadsTeil 1 (2,2 MB);Teil 2 ( 5.2 MB);


1. Modeling Continuous-Time and Discrete-Time Systems


1. Modeling Continuous-Time and Discrete-Time Systems1.1 Introduction1.2 Modeling Formalisms1.3 Basic Principles of Continuous-Time Systems1.3.1 Electrical RCL-Network1.3.2 Particle Dynamics1.3.3 Fluid Mechanics1.3.4 Thermal Dynamics1.3.4 Chemical Dynamics1.4 Block Diagram based Algebraic Representation of Dynamic Systems1.5 Basis Principles of Discrete-Time Systems1.5.1 Introduction1.5.2 Modeling Concept of Discrete-Time Systems1.5.3 Simulation Concepts1.6 Model Verification


1.1 Introduction

Why modeling?


1.1 Introduction

That´s why:

• Test is expensive – not achievable• For safety reasons• Non destructive testing • Colored ambiguous of the environment• Unstructured data• Phenomenological description• Very complex reality• Nonlinearities• Abstract system• ....


1.1 Introduction

Attempting to understand • Unknowns • Phenomena• ...in science and/or engineering, we are thinking in terms of models

Models are the most common possibility in science & engineering describing complex processes/systems and/or phenomema of realworld problems


1.1 Introduction

A model can be assumed as a reproduction of a real dynamic system/ process which with • simulation experiments can be done much more easier as with the

real system itself, and/or will be • the only possibility while it is not possible with the real object under

test.

Based on simulation (studies) deeper insight into the real dynamic sys-tem/process is available under • Best case operating conditions• Norm operating conditions• Worst case operating conditions


1.1 IntroductionM&S:powerful method when studying complex dynamic systems, remarkable advances in y systems theoryy computer science and engineeringy engineering

covers many areas in science and engineeringy automotive systemsy avionicsy biologyy chemistryy economyy electronicsy logisticy mechanicsy mediciney productiony sociologyy etc.


1.1 IntroductionM&S: iterative process, contains• (mathematical) model building • computer assisted simulation • approach to manipulate real complex dynamic systems in

accordance with the respective aims and scopes• changing model structure, and/or its parameters, and/or its inputs

and/or its outputs• match the real dynamic system

Derived model has achieved its purpose when an optimal match is obtained between the simulation results, based on the model, andthe data sets obtained from the real system measurements under test gathered through experimentation and measurements.


1.1 Introduction

Model building entails utilization of several types of information sources:• goals and purpose of modeling• determining boundaries• components of relevance• level of details• a priori knowledge of the real dynamic system being modeled• data sets gathered through experimentation and measurements on

the systems inputs and outputs• estimations of non-measurable data, and/or state space variab-les

of the real dynamic system


1.1 Introduction

AbstractModel

RealModel

Real System

ProgrammingPhysical ReproductionProgrammingProgrammingPhysical ReproductionPhysical Reproduction

Verification:ValidationFalsification

Verification:Verification:ValidationValidationFalsificationFalsification

Elements, Relations, AttributesElements, Relations, AttributesElements, Relations, Attributes

SimulationQualification

Rectification


1.1 Introduction

A part of our recognition is to be called a system, if all • Elements• Relations • Attributes are part of a whole structure, based on logical assumptions


1.1 Introduction

Elements may be• Components• Objects• Parts• ....Relations may be• Co-operations• Couplings• ...Attributes may be• Properties• Features• Signatures• ...


1.1 Introduction

Attributes also contain the connections between the system and the system environment

The attribute which describe the condition of the system is called the system state

Attributes which interact to each other describe the system related description


1.1 Introduction

Assuming Α will be a non empty amount of attributes α and Β will be the non empty amount of relations β, than a system description can be given as follows:

F: = (α∈Α, β∈Β)


1.1 IntroductionThe structural description of a system can be given by a matrix which may contain the• Input vector ui• Output vetcor yj• Operator kijas follows

yi = ∑ kij * uj

From this equation we will find that the ith equation of the above matrix equation will be

y = k*uwhile

ki1*u1 + ki2*u2 +, ..., kin*un = yi


1.1 Introduction

Systemu1

un

z1zn

y1

ym


1.1 Introduction

Model building of the human circulatory system through physical representation bypressure (P), volume (V), flow (Q). From left to right:

xQA = K*PVxCA = dV/dPAxCV = dV/dPVxRA = ∆P/Q = dPA/dQAxC = dV/dPxP = 1/C ∫dV = (1/C)(V + V0)xPA = (1/CA) * { VA(t=0) + ∫ (QA-Q)dt }xPV = (1/CV) * { VV(t=0) + ∫ (Q-QV)dt }xRA = (PA-PV)/QxQ = (PA-PV)/RA


1.1 Introduction


1.1 Introduction

Dynamic systems can be understood as systems which are decomposed to a certain level of detail, the• Behavior level• State structure level• Composite structure level


1.1 Introduction

Behavior Level:• Real systems may be described as black boxes and/or measure-

ments done at the real system in a chronological manner, the description of which is based on a set of trajectories, which reflect the behavior of the system under test

• Behavior level is of importance while experiments on the real sys-tem address this level, due to the input-output relationship, which may be expressed as

y(t) = F{u,x} with u(t) as input set, and y(t) as output set, and F as transfer function, and x as state space


1.1 Introduction

State Structure Level:• Real system may be described based on an internal system state

structure, which may generate, by iteration over time, • A set of trajectories, i.e. a behavior.• Internal state sets representing the state transition function, provi-

des rules for computing future state, given by the current state


1.1 Introduction

Composite Structure Level:• Real systems may be described by connecting together more

elementary black boxes, which may be introduced as network description

• Elementary black boxes are the components of which and each one must be given a system description state structure level

• Coupling specification determines interconnection of the components and interfacing of input and output variables


1.1 Introduction


1.1 Introduction

Deductive Method of Theoretical Axiomatic Modeling• Bottom up approach, starting with a high degree of well established

a priori knowledge of the system elements to build up a mathema-tical model which describes the system under test in a proper well defined way

• Problems may occur in assessing the range of applicability of these models, hence the deductive method has to be expanded by an experimental model validation technique

• Model verification by checking whether simulation results and data known from the real system match the error margin or not

• Model fit the assumed performance when the results, obtained from the model by simulation, compared with the results which may be data or measurables on the real system, are within the error margin

• If model is decided being unsatisfying, a modification is necessary at the different levels of the deductive modeling scheme


1.1 Introduction


1.1 Introduction

Empirical Method of Experimental Modeling:• Based on measurements available on the inputs and outputs of a

real system • Experimental model build up based on the measurements


1.1 Introduction


1.1 Introduction

Study of dynamic systems consist of four steps:

y Abstractiony Representation of the model, e.g. by mathematical notationy Analysis, e.g. by simulationy Optimization


1.1 Introduction

Abstraction• Means searching for a model that resembles the dynamic system

under test in its salient features but is easier to study• Real dynamic system is an objective which exist in the real world,

but its precise characteristics are often unknown• Applying test signals may determine the respective characteristics

Analytical solution• Means that a model that resembles the dynamic behavior has to be

determined, which can be based on the measured characteristics, obtained at least from the test signals, applied to the system inputs


1.1 IntroductionDeterministic test signals:

y Unit step 0 for t < t0

f(t) = u(t) for t0 < t < t1 0 for t > t1

y Ramp function u(t) = 0 for t < t0

f(t) = u(t) = a*[(t1 - t) / (t1 –t0)] for t < t < t u(t) = a for t > t1

The unit step and the ramp function prove to be particularly valuable in problemsin which successive switching characteristics may occur. As described in the equations above, it is possible to specify the order of the switching events. Thiswill be the same in the case that the given functions are delayed in time.


1.2 Modeling FormalismsModeling• Can be stated as a useful way solving complex problems in science,

technology, economy and other domains, because the success of systems analysis depends upon whether or not the model is properly chosen.

• Developing suitable models of real dynamic systems, a thorough understanding of the dynamic system and its operating range is of essential importance.

• From a more general point of view, three types of concepts may be stated as general systems concepts, which depend on a priori knowledge based on · knowledge of inputs,· knowledge of outputs,· knowledge of system states,for the decision of the unknown.


1.2 Modeling Formalisms



After a system model concept is found for the system under test, themodel formulation in terms of mathematical expressions is needed:• Linear equations• Non-linear equations• Integral equations• Difference equations• Differential equations• Petri net equations• Bond graph equations• Stochastic equations• Turing band equations• ...


1.2 Modeling Formalisms Mathematical modeling result in the relations of the system • which may be the variables of a biological, electrical, mechanical,

physical, etc. representation, • which may be described by ordinary differential equations (ODE´s),

representing the mathematical model of the system.A first order vector differential equation can be written as follows:

x´(t) = f( x(t), u(t), t); x(0) = x0

with the state vector x and the input vector u. This type of equation can be solved by evaluating the integral

x(t) = x0 + ∫ f( x(τ), u(τ), τ) dt

for successive values t-t1 within the simulation interval with the several numerical integration methods.



Time invariant, continuous-time model, based on ODEs

MCT: ( U, X, Y, f, g, T )with

u ∈ U: set of inputsx ∈ X: set of states

y ∈ Y: set of outputsf: rate of change function

g: output functionT: time domain

x´= f (x, u)y = g (x, u)


1.2 Modeling Formalisms The notation

MCT: ( U, X, Y, f, g, T )is a special case of a set structure. If M = Σ, we may use the three basic notations:• Input• Output• StateCorrespondingly we have a state X, a set U of input values, and a set Y of output values. A mathematical model of a system is called a dynamic one if it can be defined as set structure

Σ: = ( X, Y, U, v, t, a, b)

with X: state space, Y: set of output values, U: set of input values, v: set of admissible controls, T: time domain, a: state transition map, b: read out map b



Introducing the state vector x = [i, u1, u2]T for an electrical networksystem the respective model equations of which are

and

The state space equation of this linear continuous-time system isx´(t) = A*x(t)

whereby the (n,n)-system matrix A is given by

Ain iC

iC

iC

u111

1111

++=&

outiCi

Cu

222

11−=&

2111 uL

uL

i +=

−

−

−

=

22

1101

001

110

CRC

C

LLA

A



In many applications it may be necessary to have the option of specifying un-measurable and/or random inputs, which may be described by stochastic, time continuous models

Stochastic, continuous-time model

MSCT: ( U, V, W, X, f, g, t )

x´= f ( x, u, w, t )y = g ( x, v, t )

v, and w are random model disturbances


1.2 Modeling Formalisms Especially in management and operational research, the real complex dynamic systems can be thought being build up of a collection of events Even the state changes at specific time instantsDiscrete event model

MDE: ( V, S, Y, δ, λ, τ, T )with

V: set of external eventsS: set of sequential discrete event states

Y: set of outputsδ: transition functionλ: output function

τ: time function as approximation of the respective time step t as part of time domain T


1.2 Modeling Formalisms Many dynamic systems have properties that vary continuously in space which can be described using partial differential equations, which results in the following descriptionDistributed models, based on PDEs

MDM: ( U, Θ, Y, f, r, g, T )with

0 = f ( Θ, ϑΘ/ϑt, ϑΘ/ϑz, u, z, t); z ∈ Z

0 = r ( Θ, z, t); z ∈ Z

y = g ( Θ, z, t); z ∈ Z

t: independent variable, z: space coordinate z, Θ: vector of dependent variables which may vary in space and time. The equation hold in a spatial domain Z while conditions are provided on the boundary of the domain ϑZ. Inputs u, outputs y.



Dynamic systems may have many different specific modeling formalismexpressed by the mathematical equations, such like state space

State space model description

dx/dt = f [ x(t) ; u(t) ; t ]

y(t) = g [ x(t) ; u(t) ; t ]

with x(t): state space vector, dx/dt: derivative, y(t): output vector,u(t): control vector, f : nonlinear vector function, g: nonlinear vector function



Nonlinear system model

d/dt ( x0+ ∆x ) = f( x0, u0, t ) + (ϑf/ϑx)0 ∆x + ϑf/ϑu)0 ∆u

x´(t) = A x(t) + B u(t)

y(t) = C x(t) + D u(t)

with A: system matrix, which is a (n,n)-matrix; B: input- or control matrix, which is a (n,p)-matrix; C: observability, measure or output matrix, which is a (m,n)-matrix; D: as transition matrix, which is a (m,p)-matrix



Assuming that the influence of u(t) on y(t) is not direct through x(t), D≡0, which result in the structure diagram of a linear state space model


1.3 Basic Principles of Continuous-Time Systems

1.3.1 Electrical Elements1.3.2 Particle Dynamics1.3.3 Mechanical Elements1.3.4 Fluid Mechanics1.3.5 Diffusion Dynamics1.3.6 Thermodynamics1.3.7 Chemical Dynamics



1.3.1 Electrical Elements


1.3.1 Electrical ElementsAn important class of dynamic systems are those that can be described based on electrical networks consisting of • resistors, • capacitors, • inductors. Their important physical variables are • voltage, • current, • charge.



In an ideal resistor the voltage drop V across a resistor R is related to the current I through the resistor R, expressed by Ohm´s law, as

V = R*Iwhere R is a constant, called resistance, depending on the physical material constants χ and ρ, hence

• χ represent the conductivity of the material, • µ represent the mobility of electrons, • n*e represent the elementary charge, • ρ represent the density of electrons of the respective material, • A is the area of the conductive material the current I passes through

AAR 1*

*1 ρ

χ==

( )eneee ** −−=−= µρµχ



In an ideal capacitor the charge Q on a capacitor is related to the voltagedrop V across the capacitor C by

Q = C*Vwhere C is a proportional constant, called capacitance.In an ideal inductor the voltage drop V across the inductor is related to the rate of change of current through the inductor by

where L is a constant, called inductance. In any branch the current and the charge are related by

dtdILV *=

.dtdQI =

dtdI



Since resistors, capacitors, and inductors are typically connected in networks there are necessarily imposed relations between the physical variables; these relations are referred to as Kirchhoff´s laws: • Kirchhoff´s 1st law: The net current into any node of the network is

necessarily zero: ∑ν Iν = 0

• Kirchhoff`s 2nd law: The voltage drop V in any loop of a network is necessarily zero:

∑µ Vµ = ∑ν Iν*Rν

The signs associated with voltage, current, and charge depending on the convention used. In most cases electrical network includes voltage sources and current sources. The above relations form the basis forelementary electrical network analysis.



1.3.2 Particle Dynamics



Another important class of physical processes are those that can be de-scribed in terms of the motion of ideal particles. The important physical variables for each particle are its position, as measured from a fixed reference, its velocity and its acceleration. Also of importance is the force acting on the particle. The basic relation which characterizes the motion of a particle is based on Newton's law:

F = m*A,where the mass m of the particle is assumed constant. The acceleration of the particle is related to its velocity v by

and the velocity of the particle is related to its position H by

,dtdvA =

,dtdHv =



• Force, • Acceleration, • Velocity, • Position are considered as "vector“ variables.

In many situations it is desired to consider the motion of a collection of particles ⇒ convenient to examine free body diagrams that are used to indicate

the various forces on each particle in the collection.

Motion of rigid bodies consisting of an infinity of such particles, relevant equations for a rigid body can be obtained by using the relations given above, and integration over the body.



The resulting motion is described by equations as above in terms of the center of gravity of the body.

The rotational motion of the body is described in terms of the angular position of the body as measured from a fixed reference, its angular velocity, and its angular acceleration.

The torque acting on the rigid body is also important. It can be shown that the torque T acting on a rigid body and the angular acceleration of the rigid body are related by

T = I * αwhere moment of inertia I of the rigid body is assumed constant.



The angular acceleration of the rigid body is related to its angularvelocity by

and the angular velocity of the rigid body is related to its angularposition by

Torque, angular acceleration, angular velocity and angular position may also be considered as "vector" variables. The above relationships are essential ingredients in describing motion of particles and rigid bodies.

dtdωα =

dtdφω =



1.3.3 Mechanical Elements



1.3.4 Fluid Mechanics


1.3.3 Fluid MechanicsSome physical processes involve the motion of liquids and gases. Interest will focus on macroscopic motions rather than on microscopic motions of molecules. The two basic relationships which govern the macroscopic motions of fluids are • conservation of mass,• conservation of energy. It is usual to assume that liquids are incompressible so that the volume of a liquid is a constant. Gases are usually considered to be compressible; the ideal gas law, the so called Boyle-Mariotte law, is that the pressure P of a gas multiplied by its volume V, divided by its absolute temperature T, is always constant:

V*P = const.



Example: Application of water level control



Given is a water reservoir with the • cross section area A [m²], • water height in the water reservoir H [m]. Assuming that the water outlet qa [m³/s] will be proportional to the water inlet qz [m³/s], we can calculate the water volume V in the water reservoir

The water height in the water reservoir H will be influenced by the flow resistance R. The state space equation and the output equation of the systemshould be derived. Therefore the water reservoir volume can be expressed as:

V(t) = A*H(t).

az qqdtdV

−=



After differentiation we receive:

which result in the first order differential equation:

The term qa depends on the flow resistance expressed as follows:

Hence we receive the first order differential equation:

which result in the state space equation:

and in the output equation

dtdHA

dtdV *=

az qqdtdHA

dtdV

−== *

)(*1 tHR

q a =

)(*1* tHR

qdtdHA

dtdV

z −=−

)(**1**1 tHAR

qAdt

dHz=

)(*1)( tHR

ty =



1.3.5 Diffusion Dynamics



1.3.6 Thermodynamics


1.3.4 Diffusion Dynamics Another important class of physical processes are those that involve the transfer of heat. The basic principle of thermal dynamics if the principle of conservation of energy. Heat can be transferred between two bodies in such a way that the heat transfer rate Q to a body is related to the rate of change of the temperature of the body by the relation

where m is the mass of the body and Cp is the specific heat of the body. The heat transfer rate depends on a number of factors. If heat transfer depends on convection then the heat transfer rate is directly proportio-nal to the temperature difference between the body and its surroun-dings. As indicated previously the temperature of a gas also depends on its volume and pressure.

QdtdTCm p =**



1.3.7 Chemical Dynamics


1.3.5 Chemical Dynamics

Another class of interesting physical processes are those which are characterized by chemical reactions.The rate at which a chemical reaction occurs may be quite compli-cated since the rate generally depends on the amount of reactants as well as the particular nature of the reactions. The general principles of conservation of mass and conservation of energy are often useful as a means for describing certain aspects of the reactions.


1.4 Block Diagram based Algebraic RepresentationIn numerous scientific areas physical principles are well established and accepted while modeling. Such areas include

electromechanical energy conservation, nuclear reactions, operations research, physiology, population biology, economics, etc.


1.4 Block Diagram based Algebraic RepresentationCertain principles have even been suggested in the fields of

medicine, sociology, linguistics, anthropology, etc.,

which might serve as a basis for the use of the systems theory approach.


1.4 Block Diagram based Algebraic RepresentationOne of the most appropriate concept is the block diagram algebraic representation methodology for modeling and simulation of composite systems. From engineering we may consider a composite system which mayconsist of two or more sub-systems. There are many forms of composite systems, however. Mostly, they are build up on the following basic structures:

· - parallel· - sequential· - hybrid· - feedback


1.4 Block Diagram based Algebraic Representation

Assuming that the input output relation of composite systems can be considered as multivariable system (MVS), which is described by:

Yi(t) = Gi(t,τ), Ui(τ)dτ

with Ui and Yi as the input and the output, and Gi is the impulse response matrix of the system MVS, we can rewrite with

Gi(s) = ; i = 1,…,n

the algebraic notation of the input output relation of blocks, representing the sub-systems of a composite system.



From the composite connections top left side we find that in parallel connections U = U1 = U2, and Y = Y1 = Y2. In the feedback structure top right side we have U1 = U - Y2, and Y = Y1, and in the sequential system structure bottom side we have U = U1, Y1 = U2, and Y = Y2. Note that it was assumed that the systems G1 and G2 have compatible numbers of inputs and outputs. For top left side the impulse response equation of the parallel connection can be derived as

G(t,τ) = G1(t,τ) + G2(t,τ)



For the feedback connection, top right side, the impulse responsefunction is the solution of the integral

G(t,τ) = Gi(t,τ) - ∫ G1(t1,U) ∫ G2(U,V)G(V,τ)dudv

For the sequential solution, bottom side, we receive

G(t,τ) = ∫ G1(t1,U) G2(U,τ)dU



Combined parallel blocks

The output variables Y1 and Y2 are Y1 = U*G1 and Y2 = U*G2. Adding the

variables Y1and Y2 results in the relation Y = Y1 ± Y2 = U(G1 ± G2)



Combined parallel blocks

The output variable can be described as Y1 = (U ± Y2)*G = (U + Y1*H)*G.

The algebraic description is given by Y1 = [G/(1±G*H)]* U



Combined sequential blocks

The output variables are Y1 = U1*G1 and Y2 = G2*Y1 = G1*G2*U1



Permutation of blocks

The output variable Y2 = G1*G2*U = G2*G1*U



Moving a block before a summing point

The output variable is Y = (U1 ± U2)*G, and for the rearranged case Y =

U1*G ± U2*G



Moving a block behind a summing point

with Y = U1*G ± U2. The output variable after moving is Y = G*(U1 ± G-1*U2) which yields a multiplication with the inverse transfer function



Moving a block before a branch point

Y = G*U



Re-arranging summing points

Y = U1 ± U2 ± U3



Inversion

Y = G*U

and the inverse function U = G-1*Y


1.5 Principles of Discrete Time Systems

In distinction to the field of modeling and simulation of continuous time systems the treatment of time discrete systems follows a completely different modeling paradigm.

The difference depends on the appearance of the trajectories of the system variables under test.

For comparison the typical graph of continuous-time and discrete-time model variables may be inspected.

In both cases the x-axis represents the course of time, the y-axis marks the value of the model quantity.



The time continuous variables show permanent changes in value which

can mathematically be expressed by a differential equation.

In contrast the value of a time discrete variable may constant nearly all

the time. But there are only few points on the time scale the value

changes. At these points however the value changes abruptly and

without any interim value.



A characteristic example for a time discrete transient of a model

variable would be the number of persons waiting for service in front of

an information desk at the station. Changes in number are sudden: one

person enters or leaves the queue.

The process of joining the others who already are waiting there is not

differentiated in more detail: approaching, asking who is first and last.



The intention of time discrete models is to give a prognosis for the mean waiting time for the customers, the mean length of the queue, and so on Therefore the abstract process of model building reduces the dynamic behavior of the system on sudden changes in the number of peoplewaiting. The course of the number of people in queue is a classical time discrete model variablesWith this example in thought we may understand the two basic principles for building time discrete models: • definition of an event in the course of a model variable• conditioning its dynamic behavior between the events



Remark 1.5-1:

A time discrete event is an instantaneous occurrence that changes the

state of a system.

Remark 1.5-2:

The value of a time-discrete model quantity stays constant during the

time interval defined by two consecutive events, which may be stated as

condition for the course of a variable between events.



- Queuing Systems

- Manufacturing Systems

- High bay Warehouses

- Computer systems

- Network systems



Queuing systems

These systems distinguish stations which offer services and mo-

bile elements which request for services, being are able to move

from one service station to the other. The main task is how to or-

ganize the services to maximize their utilization and to minimize

the waiting time for the mobile elements.



Manufacturing systems

Manufacturing systems form an important application area of discrete

event modeling.

The stations are the highly automatisized machines of the plant, the

mobile elements are the raw materials, the semi-finished products, and

finally the assembled end product itself.

In addition highly automatisized transport systems from conveyors up to

intelligent automotive units complicate the systems behavior.



Manufacturing systems

Questions arises for simulation models for manufacturing systems are

quite similar to those of the queuing systems:

• minimization of production time

• maximization of utilization of the machines

But stations and strategies to move the mobile elements between them

by the transportation units are much more complex and specialized in

accordance to the technical realization of elements, stations, and

transportation system.



High bay Warehouses

Because highly automated the management of warehouse systems is a very reasonable application field for simulation. Because the goods stored are discrete elements and the places in the warehouse arediscrete as well, the model concentrates on discrete changes in state variables as are place free/ occupied and transitions by the autonomously guided vehicle system as are transport starts/ends. Main results of such type of models are: access time, optimal positioning of the goods, number of vehicles needed, …



Computer systemsHistorically this was the first application area for discrete simulation techniques. Main task is to optimize the architecture of a computer by simulation of its hardware components in relation to its operating system and observing the workload of•the CPU •the bus system•the storage•the peripheral devices •Typical parameters are the queuing strategies, strategies for sharing the processor and all the other parameters of the operating system. Discrete modeling unit for the simulation is the task with its needs concerning CPU time, storage space, external devices and so on.



Network systemsAll the parameters of interest within a single computer and its dynamic behavior can easily be transferred to a network of computers: workload of its elements, dimensioning of puffers, strategies for routing, and so on. Simulation of network systems with the data package as the unit which moves between the nodes of the system is a very up-to-date task for discrete event simulation


1.5.2 Modeling Concepts of Discrete Time Systems

With the definition of the event and the description of its semantic representation the way how to model time discrete event systems is obvious.

The description of the system dynamics consists of a chronological sorted list of events which occur between the start time and the end time of the observation. All knowledge about the system is represented in this list.

As in time continuous models an initial value for the model quantities influenced by the events must be given additionally.


Queuing System

In practice the modeler has to specify the events and to put them into the correct order. If we have a look on the events used to model the very simple system shown in Figure below we will find lots of very similar events: The example shows a single serving unit with a queue for the waiting customers. The customers may be created randomlyand receive a varying service time. After service, the customers leave the system. This system is the most simple example for discrete event simulation and is called a “single server system”.


1.5.2 Modeling Concepts for Discrete Time SystemsLets have a look at the events for the single server system shown in the previous Figure: element e1 enters queue at time t1, element e2 enters queue at time t3, element e3 enters queue at time t4, and so on. These are the events which describe arrivals of customers. On the other hand there are vents describing departures because the customers service time elapsed: element e1 finishes service at time t2, element e2 finishes service at time t6, element e3 finishes service at time t9, and so on. To simplify the task to specify all these events a more general specification schema is offered by model description languages and the correspon-ding simulation systems.

T


1.5.2 Modeling Concept of Discrete Time Systems



The idea is to build classes of events which describe the dynamics on a more abstracted level as the particular events introduced above. Main classes could be:

· Arrival of a customer· Customer enters queue· Start service· End of service· Customer leaves system



Using these more abstracted event classes • all arrivals • all enters in the queue • all service start up• …can be modeled by a single piece of model code. Therefore the general syntax of an event in a model description language consists of two defining parts:The event condition which specifies when the event will be executed.The body of the event which specifies what changes in the values of model quantities will happen. It is possible changing the values of a set of model quantities in thebody of one single event, e.g. an element is taken from the queue tothe service station the number of elements in the queue decreasesthe number of elements in service increases for the same amount.



With respect to the event condition a classification of events can be introduced as follows.• Time events

Whose event condition exclusively uses the simulation time T andwhose execution depends on the course of T exclusively• State events

whose condition is a free Boolean expression which can include any model variable and whose execution depends on the state of the model variables they change or even on the values of any other variables in the model



If the system dynamics follow some fixed rules such as iterations in time, dependencies in certain cases states of the model, the modeler has the possibility to formulate something like classes of events which represent more than one activity in real world by a single event in model description


1.5.3 Simulation Concepts

Running discrete time models, requires specific simulation algorithm.

The demands an algorithm has to fulfill are shown in the following

specification

• Execute the events which happen in the simulation period between

T_start and T_end completely

• Execute them exactly at the point of time their condition becomes

true

• Execute them in the right order

• Execute them without consumption of simulation time



Simulating discrete time systems, a very simple approach showing the advantages of the so called next-event-simulation best.

Assuming, the simulation interval is given by the starting time and the end time for the run. Furthermore the resolution ∆T of the time axis is determined, e.g. by the representation of numbers on a computer.

The most simple simulation algorithm could be as follows.



This algorithm executes the simulation correctly but consumes a huge amount lot CPU time while it check all event conditions at every time step ∆T. Caused by the characteristics of discrete event models nothing happens at the point of time under observation. Its typical for those systems to hold a given value constant a certain period of time until the next event may change it.Checking of event conditions at every point of time mostly will be dispensable and causes an enormous consumption of calculation time.



On the other hand the algorithm shown is very simple and demandsnothing else concerning the formulation of the eventsBecause of its run time behavior the algorithm is refined and the result is the so called next event algorithm, its data structure consists of two elements• current time• future event list, a list of events which are to be executed next



Each of the events mentioned before, has a time stamp showing the respective point of time where its condition becomes true. The respective next event list is ordered by growing time stamps, which with the event being executed next, top the list.

The advantage of this event list is that there will no other events in between two entries in the list. Hence the algorithm does not need to check all the conditions in between the two events, but knowing exactly when the next change in value of a model quantity will occur.

The algorithm is as follows


1.5.3 Simulation ConceptsDisadvantage of this approach is that it needs “the help” of the modeler: Somebody has to put new entries in the next event list. After initialization during start time it may be done by expanding the body of events: Within the event specification the modeler has to specify when the active event will be active again or if there is an other event which is triggered by the active event and when it is set up forexecution. These are the two types mentioned before: self- triggered events (e.g. by inter-arrival time) or condition triggered events at the same point of time (e.g. customer enters empty queue and is transmitted to the service unit at the same point of time).

All further much more sophisticated solutions for discrete eventsimulation algorithms are based on these two basic approaches. They modify the search for the next event in the list, they allow parallelism by distributing the event list, and they integrate continuous model elements in the processing of the simulation algorithm.

leitung: mathematical and computational modeling and simulation · mathematical and computational...

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