Introduction Mathematical modeling - cvut.cz · Introduction Mathematical modeling Examples Di...

Introduction Mathematical modeling Examples Difference equation iteration

IntroductionMathematical modelingModeling Systems and Processes

Bohumil Kovar

Department of Applied MahematicsCTU in Prague, Faculty of Transportation Sciences

1. lecture 11MSP2019

verze: 2019-02-25 16:18


Table of Contents

1 Introduction

Basic information

Literature

Exam

Prerequisites

Output knowledge

2 Mathematical modeling

3 Examples

4 Difference equation iteration


Basic information

Lecturer:

• Ing. Bohumil Kovar, Ph.D. ([email protected])Tu. 9:45 - 12:00 (reserve Th. 11:30 - 13:00)

Homepage:

http://zolotarev.fd.cvut.cz/msp

mailto:[email protected]

http://zolotarev.fd.cvut.cz/msp


Literature I

1 CARLSON, Gordon E. Signal and Linear System Analysis:with Matlab. 2. vyd. New York: John Wiley & Sons, 1998,768 s. ISBN 04-711-2465-6.

2 CHATURVEDI, Devendra K. Modeling and simulation ofsystems using MATLAB and Simulink. Boca Raton: CRCPress, 2009, 733 s. ISBN 978-143-9806-722.

3 OPPENHEIM, Alan V., Alan S. WILLSKY a Syed HamidNAWAB. Signals and Systems. 2. vyd. Upper Saddle River:Prentice Hall, 1997, 957 s. ISBN 01-381-4757-4.


Credits and final exam

The total number of points students can get from the tests duringthe semester is 40.

We require at least 25 points

Tests:

• 10 pt (2+4+4) – 3 tests,

• 4 pt – homeworks,

• 12 (6+6) – practical tests from Matlab and Simulink,

• 14pt – final credit test


Credits and final exam

Scoring ensures that if you get a credit (25 points and above), youcan automatically complete the course with the classification E orD.

In case you are looking for a better classification (A – C), you canacquire the remaining 20 points in the final exam.


Prerequisites

This is the knowledge that we assume you have.

1 basic operations with vectors and matrices

2 complex numbers

3 elementar functions

4 infinite series, derivatives and integrals of the function of onevariable

5 elementary math - fractions, algebraic calculations, equations,...

6 basic knowledge of Scilab/Matlab


Output knowledge

1 Laplace transformation for solving differential equationsdescribing continuous linear invariant systems

2 Z-transformation for solving differential equations describingdiscrete linear time-invariant systems

3 state-space description of the dynamic system

4 concept of stability of the dynamic system and the solutionverification methods

5 knowledge of MATLAB / SIMULINK environment formodeling of dynamic systems


Table of Contents

1 Introduction


Model of the system

External description

Internal description

3 Examples



Definition Study Testing Usage

This iterative process is typical for modeling projects and is one ofthe most useful aspects of modeling in terms of betterunderstanding how the system works.


System

Definition (System)

Characteristic features that are needed for modeling:

• the system is considered to be part of an environment thatcan be separated from its surroundings by physical or mentalboundaries,

• the system consists of subsystems, interconnected componentsthat communicate with each other.

It’s part of our world that somehow interacts with oursurroundings, for example through input and output.


What is modeling?

Model

The model can by considered as a replacment or simplification ofthe real world object in terms of its properties and functionality.

Modeling is only possible if we use some degree of abstractionand approximation.


Discrete and continuous model

input output?u(t) continuous system y(t)

u[n] discrete system y[n]


Why modeling systems?

Questions:

• How do we verify the accuracy of epidemic spread calculation?

• How do we verify the strength of the new bridge?

• How to verify the impact of the macroeconomic model beforeusing it?

If we can not pre-prove certain properties of our system, we areable to show the desired properties on its model!


Examples of real world modelsAntoni Gaudı

Spanish Catalan architect Antoni Gaudı disliked drawings andprefered to explore some of his designs — such as the SagradaFamılia — using scale models made of chains or weighted strings.

Gaudı’s upside-down physical models took him years to build butgave him more flexibility to explore organic designs, since everyadjustment would immediately trigger the ”physicalrecomputation” of optimal arches.


Examples of real world modelsAntoni Gaudı


Examples of real world modelsVW Polo crash test

The origins of industrial first principle computerized car crashsimulation lie in military defense and nuclear power plantapplications.

The sofware used for simulation of accidental crash of a militaryfighter plane into a nuclear power plant (1978) evolved totechnology for the simulation of destructive car crash tests (1986)

These simulation codes recreated a frontal impact of a fullpassenger car structure and they ran to completion on a computerovernight.

Engineers were able to make efficient and progressive improvementsof the crash behavior of the analyzed car body structure.


Examples of real world modelsVW Polo crash test


External description

The external description is based on the description of the systeminput u and the output y.

We understand the system as a black box, whose properties weobserve only if we examine its reaction to external events (signals,data).

The external model is described by the one differential equation forcontinuous time systems and the one difference equation fordiscrete time systems. The equations order are generally higherthan 1.


Internal description

The internal, so-called state-space, description uses the vector ofinternal states x to describe the dynamics of the system.

The input vector u and the output vector y are secondary variablesof the internal description. We describe state-based models by:

• a system of first order differential equations for continuoustime systems and a

• a system of first order difference equations for discrete timesystems.


The role of mathematics

Modeling is not self-sustaining:

• the outputs of the model must always be verified,

• possible errors are both in the model and in its calculation.

Verification: We calculate the correct model.

Validation: The model calculates correctly.


Table of Contents

1 Introduction


3 Examples



Flu epidemicSIR model (1/7)

SIR model equations

S ′(t) = −αI (t)S(t)

R ′(t) = βI (t)

I ′(t) = −S ′(t)− R ′(t) = αI (t)S(t)− βI (t)

S(t) for the number susceptibleI (t) for the number of infectiousR(t) for the number recovered (or immune)

S(t) + I (t) + R(t) = c

S ′(t) + I ′(t) + R ′(t) = 0


Flu epidemicSIR model - numerical solution (2/7)

What do these equations tell us? Suppose the S(0) = 100000healthy population, I (0) = 10 infected and R(0) = 10 immunewith infection rate α = 0.0001 and mortality β = 0.1. At timet = 0, today:

S ′(0) = −αI (0)S(0) = −100

R ′(0) = βI (0) = 1

I ′(0) = −S ′(0)− R ′(0) = αI (0)S(0)− βI (0) = 99

On the first day of the flu epidemic, the number of healthyindividuals will be reduced by 100, one person will die and thenumber of infected people will increase by 99.



So tomorrow, at t = 1 we can expect

S(1) ≈ S(0) + S ′(0) = 99900

R(1) ≈ R(0) + R ′(0) = 11

I (1) ≈ I (0) + I ′(0) = 109

a

S ′(1) = −αI (1)S(1) = −1088.91

R ′(1) = βI (1) = 10.9

I ′(1) = −S ′(1)− R ′(1) = αI (1)S(1)− βI (1) = 1078.01



Equations allow us to estimate changes in S , I ,R i in the past. Ifwe know the state of the epidemic today (in time t = 0), then wecan estimate the values yesterday (in time t = −1) as

S(−1) ≈ S(0)− S ′(0)

R(−1) ≈ R(0)− R ′(0)

I (−1) ≈ I (0)− I ′(0)

Thus, we can numerically analyze the changes of S , I and R intime, and predict how the epidemic will evolve. These arecalculations are recurrent and very easy to calculate usingcomputer.


Flu epidemicSIR model - Simulink (5/7)


Flu epidemicSIR model - Simulink (6/7)

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8

9

10 x 104

Time offset: 0 α = 0.00015, β = 0.11, S(0) = 100000, I (0) = R(0) = 10


Flu epidemicSIR model - Analytical solution (7/7)

Analysis of Equation for Infected people

I ′(t) = αS(t)I (t)− βI (t) = (αS(t)− β)I (t)

When

• S(t) > βα then I ′(t) > 0 and so the epidemic worsens and the

number of infected grows,

• S(t) < βα then I ′(t) < 0 the situation is getting better and

the number of infected decreases,

• ⇒ βα is a threshold.

The number of infected will therefore decrease if we can reduce thevalue of the α (and) or β coefficient.


Supply and DemandExample of price variation (1/2)

Offer (supply) Equation

The offer of today depends on the price of the yesterday, so thatthe offer rises with a rising price. For C > 0 supply is

s[k] = Cp[k − 1] +Au[k].

Demand Equation

The demand for today depends on the today price,so demand isfalling with a rising cost. For D > 0 demand is

d [k] = −Dp[k] + Bu[k].


Supply and DemandExample of price variation (1/2)

Equilibrium of supply and demand

s[k] = d [k]

then leads to the first-order difference equation

p[k] +CDp[k − 1] =

B −AD

u[k].


Supply and DemandExample of price variation - result

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Table of Contents

1 Introduction


3 Examples


The price equation iteration


Price equation iteration

Difference equation,

c[k] +CDc[k − 1] =

B −AD

u[k]

we can write in the canonical form1, where the outputs are locatedon the left side and the inputs on the right side of the equation,sorted by the time shift:

y [k] + αy [k − 1] = βu[k]

Now for k = 0, . . . , n, u[k] = 1[k] and initial condition y [−1] = 0we get

1writing in canonical form allows easier orientation.



For k = 0:

y [0] + γy [−1] = βu[0]

y [0] = β − γy [−1] = β

For k = 1:

y [1] + γy [0] = βu[1]

y [1] = β − γy [0] = β − βγ



For k = 2:

y [2] + γy [1] = βu[2]

y [2] = β − γy [1] = β − βγ + βγ2

. . . and generally for n:

y [n] + γy [n − 1] = βu[n]

y [n] = β − γy [n − 1] = β(1− γ + γ2 + · · ·+ (−γ)n

)



y [n] = β

n∑m=0

(−γ)m = β1− (−γ)n+1

1 + γ=

β

1 + γ+

βγ

1 + γ(−γ)n

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