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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
q q
l
l
l
A
0singL)3/2(
1e..i
)2
L(sinmg)mL
3
1(
MI
mL3
1I
2
AA
2
A
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
qqqq
q
q
q
q
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