CHAPTER 2 Mathematical Modeling in Transfer Function Form
Transcript of CHAPTER 2 Mathematical Modeling in Transfer Function Form
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CHAPTER 2 Mathematical Modeling in
Transfer Function Form
DR. HERMAN WAHID | DR. SHAHDAN SUDIN DR. FATIMAH SHAM ISMAIL | DR. SHAFISHUHAZA SAHLAN
Department of Control and Mechatronics Engineering
Universiti Teknologi Malaysia
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Chapter Outline • Introduction to Laplace Transform and
Transfer Function • 2.1.1 Laplace Transform • 2.1.2 Transfer function
2.1
• Modeling of Electrical Systems 2.2
• Modeling of Mechanical Systems • 2.3.1 Translational system • 2.3.2 Rotational system • 2.3.3 Rotational system with gears
2.3
• Modeling of Electromechanical Systems 2.4 2
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2.1 Introduction to
Laplace Transform and Transfer
Function
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The Need for a Mathematical Model
Mathematical model of a dynamical system: May be obtained from the
schematics of the physical systems,
Based on physical laws of engineering
Newton’s Laws of motion Kirchoff’s Laws of electrical
network Ohm’s Law
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Design of controller
Mathematical modeling (of the plant to be
controlled)
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Modeling of Control System Plants
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Plant Model
Frequency Domain
C – T G(s)
D – T G(z-1)
Time Domain
C – T
D – T
)()()()()()(tuDtxCtytuBtxAtx
+=+=
)()()()()()1(
kuDkxCkykuBkxAkx
+=+=+
Transfer function
State-space equation
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We only cover this
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2.1.1 Laplace Transform
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Time-domain signals
Frequency-domain signals
Equations:
Laplace Transform:
Inverse Laplace Transform:
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Laplace Transform Table
Given f (t), what is F(s)? 7
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Laplace Transform Theorem
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Example 1: • Find the Laplace Transform of y(t), assuming zero
initial condition where u(t) is a unit step. • Solution:
• [Answer: ] 9
)(32)(32)(12)(2 sUsYssYsYs =++
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Inverse Laplace Transform • Recall:
• Therefore, for Inverse Laplace Transform,
• Refer to Laplace Transform Table on page 8.
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Given F(s), what is f (t)?
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Inverse Laplace Transform
• 3 situations:
i. Roots of D(s) are real & distinct, e.g.
ii. Roots of D(s) are real & repeated, e.g.
iii. Roots of D(s) are complex, e.g.
• Hint: Use ‘Partial Fraction Expansion’
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numerator
denominator
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2.1.2 Transfer Function
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2.1.2 Transfer Function, G(s)
• Definition:
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• Differential equation model:
• Laplace transform both sides (‘Differentiation Theorem’ from
p11) – assume zero initial condition:
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Transfer function
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Example 3: • Find the transfer function represented by:
• Use MATLAB to create the above transfer function.
• Find the response, c(t), to an input r(t) = u(t), a unit step input, assuming zero initial condition.
• [Answer: ] 16 2
1)()(
+=
ssRsC
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2.2 Modeling of
Electrical System
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Review on Electrical Circuit Analysis
• Ohm’s Law • Kirchoff’s Voltage Law • Kirchoff’s Current Law • Mesh & Nodal Analysis
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Kirchoff’s Current Law Ohm’s Law
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• Passive linear components i. Capacitor (C) – store energy ii. Resistor (R) – dissipate energy iii. Inductor (L) – store energy
• Relationships:
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Electrical Components
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Example 4: Single-loop network
• Find the transfer function of the circuit using – Differential Equation Method – Mesh Analysis (Laplace) – Nodal Analysis (Laplace)
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output input
Kirchoff’s Law
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Solution for example 4 1. Using differential equation
)()(1)()(0
tvdttiC
tRidt
tdiLt
=++ ∫
( ) )()(12 sVsVRCsLCs c =++
Current to charge
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)()(
2 ++=
RCsLCssVsVC
KVL
)()(1)()(2
2
tvtqcdt
tdqRdt
tqdL =++
dttdqti )()( =
)()( tCvtq c=From table
)()()()(2
2
tvtvdt
tdvRCdt
tvdLC ccc =++
Laplace transform
TRANSFER FUNCTION
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2. Solving in Laplace domain
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Time domain Laplace domain
KVL: )()(1)()( sVsICs
sLsIsRI =++
?)()(=
sVsI
?)()(=
sVsVC
?)(),(),( =sVsVsV CLR
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Example 5: Multiple-loop network
• Find the transfer function of the circuit using – Mesh Analysis – Nodal Analysis
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Find the transfer function, )()(2
sVsI
?)()(=
sVsVC
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Solution for Example 5
Next, find
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Generic equations – Two-loop Electrical System
• Depending on number of loops in the circuit, use the following rule to obtain simultaneous equations
I1(s)
- I2(s)
=
- I1(s)
+ I2(s)
=
Sum of impedances common to Mesh 1 and
Mesh 2
Sum of impedances
around Mesh 1
Sum of applied voltage around
Mesh 1
Sum of impedances common to Mesh 1 and
mesh 2
Sum of applied voltage around
Mesh 2
Sum of impedances
around Mesh 2
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2.3 Modeling of
Mechanical System Translational Rotational
Rotational with Gears
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2.3.1 Translational
• Newton’s Laws of Motion: i. First law: The velocity of a body remains constant
unless the body is acted upon by an external force. ii. Second law: The acceleration a of a body is parallel
and directly proportional to the net force F and inversely proportional to the mass m, i.e., F = ma.
iii. Third law: The mutual forces of action and reaction between two bodies are equal, opposite and collinear.
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Translational Mechanical System
3 passive and linear components in mechanical system: • Spring - energy storage element inductor
• Mass - energy storage element capacitor
• Viscous damper - energy-dissipative element resistor
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Translational Mechanical Components
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Spring, Mass & Damper in action
Applied force f(t) points to the right Mass is traveling toward the right All other forces impede the motion and act to opposite
direction Single input single output (SISO) system
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Example 6:
• Find the transfer function X(s)/F(s), for the following mechanical system.
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Find the transfer function X(s)/F(s) # Equation of motion: → positive direction to the right
)()()()()(
2
2
tftKxtdtdxf
dttxdM v =++
)()()()(2 sFsKXssXfsXMs v =++
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Differential equation of motion
Laplace equation of motion
Free body diagram
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• Find the transfer function X2(s)/F(s), for the following mechanical system.
• Answer:
Example 7:
33 )]()([)(
)()]()([
)()()()(
32322
223
2321312
1
232
KKsffsMKsfKsfKKsffsM
where
KsfsFsXsG
VVV
VVV
V
+++++−+−++++
=∆
∆+
==
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Generic equations – Two Translational Body System
X1(s)
- X2(s)
=
- X1(s)
+ X2(s)
=
Sum of impedances
connected to the motion at
x1
Sum of applied
forces at x2
Sum of applied
forces at x1
Sum of impedances
connected to the motion at
x2
Sum of impedances between x1
and x2
Sum of impedances between x1
and x2
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2.3.2 Rotational Mechanical System
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Example 8: Find the transfer function θ2(s) / T(s). The rod is supported by bearing and at either end is undergoing torsion. A torque is applied at the left and the displacement is measured at the right.
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Solution 8: Obtain the schematic from the physical system Assume:
o The torsion acts like a spring, concentrated at one particular point in the rod
o Inertia J1 to the left and J2 to the right o The damping inside the flexible shaft is negligible
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)()( 211
121
2
1 tTKdt
dDdt
dJ =−++ θθθθ
0)( 122
222
2
2 =−++ θθθθ Kdt
dDdt
dJ
(1)
(2)
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Equation of motion
Torques on J1 if J2 held still Torques on J1 when J2 is in motion
)()()()( 2112
1 sTsKsKsDsJ =−++ θθ
Final free body diagram for J1
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Torques on J2 if J2 in motion Torques on J2 if J1 in motion
Final free body diagram
for J2
Equation of motion
0)( K)()( 222
21 =+++− ssDsJsK θθ
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• Equations of motions:
• Hence giving the transfer function
• giving the block diagram
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0)()( )()()()(
22
21
2112
1
=+++−
=−++
KsDsJsKsTsKsKsDsJ
θθθ
)()(
)()(
22
2
12
1
2
KsDsJKKKsDsJ
where
KsTs
++−−++
=∆
∆=
θ
∆K )(2 sθT(s)
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Generic equations – Two Rotational Body System
θ1 (s)
- θ2 (s)
=
- θ1 (s)
+ θ2(s)
=
Sum of impedances connected
to the motion at θ1
Sum of applied
torques at θ2
Sum of applied
torques at θ1
Sum of impedances connected
to the motion at θ2
Sum of impedances between θ1
and θ2
Sum of impedances between θ1
and θ2
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2.3.3 Rotational with gears
• Gears o Used with rotational systems
(esp. those driven by motors). o Match driving systems with
loads. o E.g. Bicycles with gearing
systems - Uphill: shift gear for more
torque & less speed - Level road: shift gear for
more speed & less torque
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Input gear with radius r1 and N1 teeth rotated through angle θ1(t) due to torque T1(t)
Output gear with radius r2 and N2 teeth responds through angle θ2(t) and delivering a torque T2(t)
teethofnumber of ratio1 dispangular ratio ∝
radius of ratio teeth ofnumber ratio ∝
2211 θθ TT =
Just like translational motion, energy = force x disp.
2
1
2
1
rr
NN
=
1
2
2
1
1
2
NN
TT
==θθ
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Equivalent system at θ1 at input N1
Equivalent system at θ2 at output N2
)()()( 1
2
2
11
2
2
11
22
2
11 s
NNKss
NNDss
NNJT θθθ
+
+
=
)()()( 2222
1
21 sKsDssJs
NNT θθθ ++=
2
1
1
2
NN
=θθ
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• Generalizing the result – We can say that the rotational mechanical
impedances can be reflected through gear trains by multiplying the mechanical impedance by the ratio
2
shafton source ear teeth of gNumber of
tion shafton destinaear teeth of gNumber of
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Example 9 and solution:
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Find the transfer function θ2(s)/T1(s)
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2.4 Modeling of
Electromechanical System
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Electromechanical System • Electromechanical system: electrical + mechanical components
that generates a mechanical output by an electrical input (motor)
48 Schematic diagram
rotor
stator
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Develop magnetic field B by stationary permanent magnet Armature circuit with ia(t),
passes through magnetic field and produces a force F- (Fleming's left-hand rule )
The resulting torque turns the rotor (rotating member of motor)
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Since the current-carrying armature is rotating in a magnetic field, its voltage vb (back electromagnetic force emf) is proportional to angular valocity.
dttd
Ktv mbb
)()(
θ=
Taking the L.T: )()( ssKsV mbb θ=
(1)
(2)
KVL around the armature circuit
)()()()( sEsVssILsIR abaaaa =++ (3)
Torque developed by the motor (Tm ) is proportional to the armature current (ia).
)(1)()()( sTK
s IsIKsT mt
aatm == or (4) Substitute (2) and (4) into (3)
)()()()(
sEssKK
sTsLRamb
t
maa =++
θ (5)
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Equivalent mechanical loading on motor
)()()( 2 ssDsJsT mmmm θ+=Substitute (6) into (5) yields
)()()())(( 2
sEssKK
ssDsJsLRamb
t
mmmaa =+++
θθ
)(1)()(
αθ
+⇒
++
=ssK
RKK
DJ
ss
JRK
sEs
a
tbm
m
ma
t
a
m
Assume La <<<Ra , then (7) becomes
(6)
(7)
(8)
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DC motor driving a rotational mechanical load
2
2
12
2
1 )()(
+=
+=
NNDDsD
NNJJsJ LamLam ;
From (5), with La = 0
taking inverse Laplace transform
)()()(
tedt
tdK
KtTR
am
bt
ma =+θ
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)()()(
)( teRK
RKK
teRK
dttd
RKK
tT aa
tm
a
tba
a
tm
a
tbm +−=+−= ω
θ
⇒= 0ω aa
tm e
RK
T =
⇒= 0Tb
a
Ke
=ω a
stall
a
t
eT
RK
=
ωa
be
K =
aa
tm
a
tbm e
RK
RKK
T +−= ωAt steady state:
Electrical constants
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Example 10:
54
Find )()(
sEs
a
Lθ
Answer: )667.1(0417.0
)()(
+=
sssEs
a
Lθ
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Solution 10:
1) Find Mechanical Constants Jm and Dm
a. Total Inertia at the armature
b. Total damping at the armature
(1) 121017005
22
2
1 =
+=
+=
NNJJJ Lam
(2) 101018002
22
2
1 =
+=
+=
NNDDD Lam
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2) Find the electrical constants and
(4) 250
100
(3) 5100500
constants Electrical
===
===
∴
−loadno
ab
a
stall
a
t
eK
eT
RK
ω
100 50500
curve speed-Torque theFrom
===
−
a
loadno
stall
e
Tω
a
t
RK
bK
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3)
4) Find by establishing the ratio of
)667.1(
417.0 )(1
)/()(
:into (4) and (3) (2), (1), Substitute
+=
++
=ss
RKKD
Jss
JRKsE
(s)θ
a
bm
m
ma
a
m
τ
τ
)()(
sEs
a
Lθ
)667.1(0417.0
)667.1(10/417.0
)()(
)( 10)( ..
101001000
)()(
1
2
+=
+=∴
=
===
sssssEs
sseiNN
ss
a
L
Lm
L
m
θθθ
θθ
)()(
ss
L
m
θθ
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END OF CHAPTER 2
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