SE201: Introduction to Systems Engineering

Mathematical Modeling

Mathematical Models

A mathematical model is a set of equations that relates different variables of the system

Mathematical models are essential for the analysis and design of control systems

Falling Ball Example A ball falling from a height of 100 meters

We need to determine a mathematical model that describes the behavior of the falling ball.

Objectives of the model: answer these questions:

1. When does the ball reach ground?2. What is the impact speed? Different assumptions results in different models

Falling Ball Example

Can you list some of the assumptions?

Falling Ball Example Assumptions for Model 1

1. Initial position = 100 x(0) = 1002. Initial speed = 0 v(0) = 03. Location: near sea level4. The only force acting on the ball is the

gravitational force (no air resistance)

ttvttx

Solution

8.9)()8.9(5.0100)(

:2

0)0(;100)0(

)(;8.9

:

vx

tvdtdx

dtdvModel

Falling Ball Example More models

Other mathematical models are possible. One such model includes the effect of air resistance. Here the drag force is assumed to be proportional to the square of the velocity.

0)0(;100)0(

)(;8.9

:2coeffient drag theis , resistanceair

2

2

vx

tvdtdxv

mc

dtdvModel

cwherecv

How far can this stunt driver jump?

List some assumptions for solving this problem

Stunt driver Assumptions:

Point mass Mass of car+driver =M Initial speed = v0

Angle of inclination =a No drag force

Model can be obtained to give the distance covered by the jump in terms of M,a, v0,…

Classification of Models

Static models: Models involve algebraic equations only

Dynamic models involves differential equations

Linear models: includes linear algebraic and/or differential equations

Other types of Models (nonlinear, distributed, discrete-time, hybrid,

…)

Transfer functions

A transfer function is used to describe the relationship between the input and output of a system or a subsystem

Transfer function is defined for systems described by linear algebraic and/or linear differential equations

Transfer function G

Input U Output Y

Y = G U

Static Models

The relationship between steady state values of input and output of a linear system is described by a linear static model

Uof valuestatesteady Yof valuestatesteady

G

Transfer function G

Input U Output Y

ExampleElectric Motor

Input (voltage supplied to motor) Volts Output (Motor speed ) revolution /min The transfer function represents the ratio of

steady state value of output over steady state value of input

Motor G

U (volts) Y (rev/min)

Uof valuestatesteady Yof valuestatesteady

G

ExampleElectric Motor

steady state value of input 12 Volts transfer function G= 500 revolution /min/volt What is the steady state value of output?

Motor G

U (volts) Y (rev/min)

/minrevolution 600012*500 Uof valuestatesteady Yof valuestatesteady

G

Block Diagrams Reductions

Objectives

• To understand the concept of Block diagrams and their assumptions

• To be able to reduce complex block diagrams to a simple (one block) diagram

Block diagrams

A block diagram is used to represent linear relationship between the input and the output of a subsystem.

Let the input and the output be U and Y, the relationship between U and Y is represented

by a rectangular block having a transfer function G

GU Y = G U

Block diagrams

A block diagram is pictorial representation of systems.

It shows different components (or subsystems) that make a system and shows their interactions

Each rectangular block represents a subsystem Inputs are represented by arrows entering the block Outputs are represented by arrows leaving the block

G

X Y

H

G

Voltage Divider

21

1

RRRG

eGeo

+e−

,21

1 eRR

Reo

1R

2R

oe

einput output

Two-input-two-output systems

2221

1211

gggg

G

11g21g

12g

22g

1u

1u

2u

2u

1y

1y

2y

2y

inputs outputs

Block diagram manipulation

HG H G

H

GH + G

H

G

G . 1+H G

_

Block diagram manipulation 2

G

X

Y G

X

Y

G

G

X

Y G

X

Y

G

V

Block Diagrams transformationsBlocks in cascade

GVX HG

YH

X Y

Two blocks in cascade are replaced by a single block whose transfer function is the product of transfer functions

Block Diagrams transformationsParallel Combination

G+H

YGX Y

H

X

Two blocks are in parallel they can be replaced by a single block whose transfer function is the sum of transfer functions

Block Diagrams transformationsMoving a summing point

G

X Y

GX

Y

G─1

UU

Block Diagrams transformationsMoving a summing point

G

X YG

XY

GUU

Block Diagrams transformationsMoving a pickoff point (#1)

G

V

Y G

V

Y

G

V

V

Block Diagrams transformationsMoving a pickoff point (#2)

G

X

Y G

X

Y

G─1

XX

Block Diagrams transformationsMoving a summing point

G

X Y

GX

Y

G─1

UU

Block Diagrams transformationsEliminating a Feedback Loop

XG

X Y

HGHG

1Y

Exercise

Find the transfer function relating X and Y

GX

Y

H

M

K

BD transformationsAn example

BD transformationsAn example

BD transformationsAn example

BD transformationsAn example

BD transformationsAn example

Keywords

• Summing point• Pickoff point• Transfer function• Cascade combination• Parallel combination• Feedback combination• Block diagram• Block diagram reduction

Example

Find the transfer function of the following system

G H K

?

Example

Find the transfer function of the following system

G H K

KHG

The transfer function of three blocks in series is the product of the three transfer functions

U VW Y

Y = KW = K (HV) = K H (GU) = KHG U

ExamplePressure Measurement System

The pressure measurement system consists of A pressure sensor Signal conditioner

Pressure SensorGenerate electric

current proportionalTo the pressure

G=0.1 mA/Pa

Signal Conditioner

)Amplifier(K=20

U Y

ExamplePressure Measurement System

Pressure SensorGenerate electric

current proportionalTo the pressure

G=0.1 mA/Pa

Signal Conditioner

)Amplifier(K=20

P V

T.F=20*0.1=2mA/PaVP

Example

Find the transfer function of the following system

G H

K

?

Example

Find the transfer function of the following system

G H

K

GHKGH

1

ExampleMotor with Speed Control

Find the transfer function of the following system

Amplifier + relay + motorG=600 rev/min/volt

Tachometer(speed sensor)K=3mvolt/rev/min

214600*003.01

6001

GKG