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7/27/2019 Chapter3 Note

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Chapter 3 Modeling of Physical Systems

3.1 Foundation of Modeling 3.2 Method in Dynamic Modeling 3.3 Classification of Dynamic Model

3.4 Mathematical Representation


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3.1 Foundation of Modeling (1) Role & Use of Model:

Eykhoff

A representation of the essential aspects of an existing

object (system) which presents knowledge of that object

(system) in a usable form.

Websters dictionary

A description or analogy used to help visualize something

(as a system) that cant be directly observed.


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3.1 Foundation of Modeling (2) Model ing is an art rather than a techn ique

Beh av io r Ob jec tiv e

Representat ion

Opt imal Model


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3.1 Foundation of Modeling (3) Principles of Model ing:

Principle of resemblance error tolerance

Principle of parsimony as simple as possible

Principle of objective use of information


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3.1 Foundation of Modeling (4) Method of Model ing:

Real

System

Model of Real

System

Information

UtilizationExperimental

Testing

Model

Building

System

Realization

Problem

Solving

Solution

Interpretation

Information

Processing

Behavior Representation Objective


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3.1 Foundation of Modeling (5) Types of Model:

Conceptual model

Analog model

Graphical model

Solid model

Physical model

Mathematical model


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3.2 Method in Dynamic Modeling (1) Method in Dynamic Modeling:

Actua l

System

Physical

Modeling

Mathematical

Modeling

Dynamic

Prediction & Analysis

Error

AcceptanceTest

No

Yes

Dynamic

Behavior


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3.2 Method in Dynamic Modeling (2) Phys ical Model:

An idealized physical system which resembles an actual system in its

salient features but which is more amenable to system analysis and synthesis.

Idealization Techniq ues:

Neglect small effect

Independent environment

Lumped characteristics

Linear relationships

Constant parameters

Neglect uncertainty and noise


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3.2 Method in Dynamic Modeling (3)Ex:

particle model

Newtonian particle

Fluid particle

Ideal gas

Photon

field model

Gravitational field

Flow field

Electromagnetic field


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3.2 Method in Dynamic Modeling (4) Mathematic al Model:

The description of object behavior by means of suitably chosenmathematical realizations.

Idealization:

Continuity

Directionality

Uniformity

Additivity

Constancy

Certainty


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3.2 Method in Dynamic Modeling (5) Model Verif ication:

Prediction error and acceptance criteria

Error Qualitative

Quantitative

Qualitative error phase plane portrait

Quantitative error e(t) (Well-controlled I.C.)

T

0

2

rmrm

dt(t)eT

1:RMS,)t(e:Ex

(t)eoffunctionNorm:(t)e

statereal:x,statemodel:x,)t(x)t(x)t(e


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3.3 Classification of Dynamic Model (1)

Model Ass umpt ions in th is course:

Deterministic, Lumped, Linear, Time-invariant (Constant coeff.), and Continuous.

Dynamic Model

Determinst ic Chao ti c Stochas ti c

Lumped parameters

(ODE)

Distributed parameters

(PDE)

Linear Non-linear

Costant coefficient Variable coeffic ient

Disc rete t ime Cont inuous t ime


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3.3 Classification of Dynamic Model (2) Various Phy sical and Mathematical Models of Real Pendulum :

Joint w i th clearance

Rod

g

m


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mg

q

2LA

L

3.3 Classification of Dynamic Model (3) Distr ibuted and Lumped Model :

Distributed Model Lumped Model

m

mg

qF

rF

q

l

0sing1

e..i

sinmgm

Fm

qq

qq

q q

l

l

l

A

0singL)3/2(

1e..i

)2

L(sinmg)mL

3

1(

MI

mL3

1I

2

AA

2

A

qq

qq

q


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3.3 Classification of Dynamic Model (4) Linear and Non-l inear Model:

Non-linear Model Linear Model0sin

gqq

l

qq

qqqq

q

qq

qq

q

q

q

qq

g

ge..i

0)(g

sin

pointoperating

pendulumInverted(2)

0g

sin

0pointoperating

pendulumRegular)1(

.......53

sin53

ll

l

l

m

m

g

q

qsin

0 2


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3.3 Classification of Dynamic Model (5) Time-invariant and Time-varying Model:

Time-invariant Model Time-varying Model

Non-linear Model

Linear Model

0g

qql

0sing

qql

Non-linear Model

Linear Model

0sin)tcosyg

1(g

tcosy)t(ya,mI

sinmgsinmaI

oo

2

oo

2

oA2

A

AA

q

q

qqq

l

l

ll

0)tcosyg

1(g

oo

2

q

ql

tcosy)t(y 0o

A

m

mg

qF

rF

q

l


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3.4 Mathematical Representation (1)Dynam ic Representat ion

Time Change

Disc rete Con t inuous Discrete Cont inuous

Dif ference

Equat ion

Differential

Equat ion

Fini te

state

machine

Discrete

event

mode l


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3.4 Mathematical Representation (2) Nonl inear i ty and Linear izat ion:

Nonlinearity in mechanical systems

x

FDry

Frict ion

x

FHard

Sof t

Nonl inear

Spr ing

x

FBacklashHysteres is s

e


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3.4 Mathematical Representation (3) Linearization:

Linear properties:Superposi t ion Output response of a system to the sum of inputs

is the sum of the responses to the individual inputs.

Input Output

If r1 (t) c1(t)

r2 (t) c2(t)r1(t) + r2(t) c1(t) + c2(t) Addi t ive Proper ty

Homogenei ty The response of a system to a multiplication of the

input by a scalar.

Input Output

If r1 (t) c1(t)

r1(t) c1(t) Scal ing Property


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3.4 Mathematical Representation (4) Mathematical Method :

Scalar functionTaylor series expansion

Vector function

xm)x(for

)xx(dx

df)x(f)x(fe..i

)x(0)xx(dx

df)x(f)x(f

o

o

o

xx

o

xx

o

2

o

xx

o

2

0,nn

xxn

0,22

xx2

0,11

xx1

n,02,00,1n21

)x(0)xx(x

f......)xx(

x

f

)xx(

x

f)x......,x,x(f)x......,x,x(fy

0,nn0,22

0,11


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3.4 Mathematical Representation (5)

1x

f

)x,(f 0,2b

)x,x(ff 0,20,10

)x,(f 0,2a

0,1x ab

g2x

h2x

0,22 xx

1x

f

),x(f 0,1 g

)x,x(f 0,20,1

),x(f 0,1 h

0,1x

g2x

h2x

0,22 xx

For f=f(x1,x

2)

ba

ba

)x,(f)x,(f

x

f 0,20,2

01

hghg

),x(f),x(f

x

f 0,10,1

02

)xx(x

f)xx(

x

f)x,x(ff 0,22

02

0,11

01

0,20,1

Exper imental Method :


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3.4 Mathematical Representation (6) Dynam ic Linear Model:

System

Dynamics

Linear

Model

Dynamic

Informat ion

Predict ion

Error


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3.4 Mathematical Representation (7) Classic al I/O Model:

Differential formEx: Vibration plant

Integral form

Key poin ts: Lin earity, Causality, Relaxation

x(t)y(t)-Output

f(t)-Input:I/O

(0)xx(0),I.C.

f(t)KxxCxM:System

x(t)y(t)-Output

f(t)-Input:I/O

)-g(t:System

d)(f)t(g)t(xt

0

y=xf(t)

Input

System

Output

C

K M

y=x

f(t)

f r ic t ionless


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3.4 Mathematical Representation (8) Modern State-space Model:

Define system states

State equation

)t(fM

1

xM

K

xM

C

x:Ex

21

1

xx

xx

)t(f

M

1

0

x

x

M

C

M

K

10

x

x

2

1

2

1

BfxAx

(0)x

I/O : Input

01C,xCy

i.e.

System:

I.C. :

Output

f(t)

)x(x1

)x(x2

statespace

x


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3.4 Mathematical Representation (9)In general

State equation :

Output equation :

I.C. :

: System states

: Control input

K: To be designed

xK)x(u

uDxCy

uBxAx

u

x

y

uDxCy

uBxAx

(0)x

u

x


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3.4 Mathematical Representation (10) State-sp ace and I/O Model:

State-space system model

I/O system model

ubxaxax

ubxaxax

22221212

12121111

u)baba(ubx)aaaa(x)aa(x

variablexu)baba(ubx)aaaa(x)aa(x

variablex

2111212221122211222112

2

1222121121122211122111

1

)dimensionone(uu,bbB,

aaaaA,

xxx.e.i

ub

b

x

x

aa

aa

x

xor

2

1

2221

1211

2

1

2

1

2

1

2221

1211

2

1

Output x1 and / or x2


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3.4 Mathematical Representation (11) Impu lse Respons e:

Unit impulse function

Impulse at t=a

Shifting property

0,1dt)t(

0t,0)t(

0e

e

)at(

f(b)

cba,dt)t(f)bt(c

a

e

e

1

t

ta

Area=1


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3.4 Mathematical Representation (13)r(t)

t

d

1 t

Arb itrary Input Respons e:

r(t) is composed of many impulses

Ex: Step response of 1st order system

r(t)

t

x(t)

)t(raxx

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