System Modeling

Emam FathyDepartment of Electrical and Control Engineering

email: emfmz@yahoo.com

Lecture-2

1

Types of Systems

• Static System: If a system does not changewith time, it is called a static system.

• Dynamic System: If a system changes withtime, it is called a dynamic system.

2

Static Systems

• Following figure gives anexample of static systems, whichis a resistive circuit excited by aninput voltage u(t).

• Let the output be the voltageacross the resistance R3, andaccording to the circuittheory, we have

3

𝑦 𝑡 =𝑅2𝑅3

𝑅1 𝑅1 + 𝑅3 + 𝑅2𝑅3𝑢 𝑡

A system is said to be static if its output y(t) depends only on theinput u(t) at the present time t.

Dynamic Systems

• A system is said to be dynamic if its current output may depend onthe past history as well as the present values of the input variables.

• Mathematically,

Time Input, ::

]),([)(

tu

tuty 0

Example: A moving mass

M

y

u

Model: Force=Mass x Acceleration

𝑀 ሷ𝑦 = 𝑢

Dynamic Systems

examples: RC circuit, Bicycle, Car, Pendulum (in motion)

5

Ways to Study a System

6

System

Experiment with a model of the System

Experiment with actual System

Physical Model Mathematical Model

Analytical Solution

Simulation

Frequency Domain Time Domain Hybrid Domain

Model

• A model is a simplified representation or

abstraction of reality.

• Reality is generally too complex to copy

exactly.

• Much of the complexity is actually irrelevant

in problem solving.

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What is Mathematical Model?

A set of mathematical equations (e.g., differential eqs.) thatdescribes the input-output behavior of a system.

What is a model used for?

• Simulation

• Prediction/Forecasting

• Diagnostics

• Design/Performance Evaluation

• Control System Design

Black Box Model

• When only input and output are known.

• Internal dynamics are either too complex orunknown.

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Input Output

Grey Box Model

• When input and output and some informationabout the internal dynamics of the system isknown.

10

u(t) y(t)

y[u(t), t]

White Box Model

• When input and output and internal dynamicsof the system is known.

11

u(t) y(t)2

2

3dt

tyd

dt

tdu

dt

tdy )()()(

Transfer Function

• Transfer Function G(S) is the ratio of Laplace transformof the output to the Laplace transform of the input.Assuming all initial conditions are zero.

Plant y(t)u(t)

12

)(

)()(

SU

SYSG

Electrical Systems

Example: RC Circuit

• u is the input voltage applied at t=0

• y is the capacitor voltage

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Find out the transfer function of the RC network shown in figure.Assume that the capacitor is not initially charged.

𝑢 = 𝑅𝑖 +1

𝐶න 𝑖𝑑𝑡

𝑢

𝑅

+𝑦_

𝐶𝑖

Example

Find the transfer function relating the capacitor voltage, Vc(s), to the input voltage, V(s)

Example

Differential equation

0

( ) 1( ) ( ) ( )

tdi t

L Ri t i d v tdt C

ExampleRedraw the circuit using Laplace transform.

)(*

1*)( sI

scRsLsV

scsIsVC

*

1)()(

scsVsI C **)()(

)(*

1)( sI

ScRLssV

scsIsVC

*

1)()( scsVsI C **)()(

scsVsc

RLssV C **)(*

1)(

cLs

L

Rs

cL

RcscLs

scRsLsc

sV

sVC

*

1

*/1

1

1

*

1**

1

)(

)(

2

2

From (1) & (2)

…….. (1)

…….. (2)

Electric Network Transfer Functions

We can also present our answer in block diagram

Electric Network Transfer Functions

• Solution summary

laplace

Using mesh analysis

HW

• Find the transfer function, I2(s)/V(s)

Output I2(s)

Input V(s)

Mechanical Systems

Translational Mechanical System Transfer Function

• We are going to model translational mechanical system by a transfer function.

• In electrical we have three passive elements, resistor, capacitor and inductor. In mechanical we have spring, mass and viscous damper.

Example

• Consider the following system (friction is negligible)

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• Free Body Diagram

MF

kf

Mf

k

F

xM

• Where and are force applied by the spring and inertial force respectively.

kf Mf

Example

25

• Then the differential equation of the system is:

• Taking the Laplace Transform of both sides and ignoring initial conditions we get

MF

kf

Mf

Mk ffF

)()()( skXsXMssF 2

𝐹 = 𝑀 ሷ𝑥 + 𝑘𝑥

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)()()( skXsXMssF 2

• The transfer function of the system is

kMssF

sX

2

1

)(

)(

• if

12000

1000

Nmk

kgM

2

00102

ssF

sX .

)(

)(

Example

Example-2

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• Find the transfer function X2(s)/F(s) of the following system.

Free Body Diagram

M1

1kf

1Mf

Bf

M2

)(tF

1kf

2Mf

Bf2kf

2k

BMkk fffftF 221

)(

BMk fff 11

0

End Of Lec 2

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