Automatic Control Systems -Lecture Note...
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Automatic Control Systems
Modeling of Physical Systems 3
Automatic Control Systems -Lecture Note 13-
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Automatic Control Systems
Equation of Motion
Two approaches to derive an equation of motion
i) Newtonian Mechanics : based on Newton’s 2nd law of motion
ii) Lagrangian Mechanics : analytic method based on energy
concept
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Automatic Control Systems
Equation of Motion
Newtonian Mechanics
describes rigid body motion using the balanced force relation
Linear motion
: vector sum of applied forces on a rigid body
: mass of rigid body
: vector of acceleration of rigid body
F
m
a
maF
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Automatic Control Systems
Equation of Motion
Rotational motion :
: sum of applied torques of rigid body
: mass moment of inertia of rigid body
: angular acceleration of rigid body
【Note】 Free body diagram : net description of forces exerted on
a rigid body convenient when deriving Newtonian equation of
motion
T J
T
J
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Automatic Control Systems
Equation of Motion
Largrangian Mechanics
derives equation of motion by using all the energy terms in a
rigid body such as kinetic, potential, and dissipating energies
Lagrange equation
: generalized coordinate, : kinetic energy
: potential energy, : dissipating energy
: non-conservative generalized force corresponding to
, 1,2, ,j
j jj j
d T T V DQ j n
dt q qq q
Tjq
V D
jQ jq
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Automatic Control Systems
Equation of Motion
【Note】 i) , , are functions of generalized variable
ii) Lagrangian :
iii) Lagrange equation
T V D jq
VTL
, 1,2, ,j
jj j
d L L DQ j n
dt qq q
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Automatic Control Systems
Equation of Motion
Kinetic energy
: mass and moment of inertia
: linear and angular velocity
【Note】 vector equation of kinetic energy
22
2
1
2
1JmvT
Jm ,
, v
1 1
2 2
T TT m J v v ω ω
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Automatic Control Systems
Equation of Motion
Dissipative friction energy
: viscous friction coefficient
: velocity
【Note】 Generalized force
1. an external force as function of generalized coordinate
variables
2. represents force for linear motion and torque for rotational
motion, respectively
2
2
1bvD
b
v
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Automatic Control Systems
Equation of Motion
Example : mass-spring-damper system
: mass, : spring constant,
: damping coefficient
: external force, : displacement
m
m
x
b
k
F
k
x
b
F
<Fig> mass-spring-damper system
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Automatic Control Systems
Equation of Motion
i) Newtonian mechanics
: F ma kx b x F m x
(1)m x b x kx F
<Fig> free body diagram
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Automatic Control Systems
Equation of Motion
ii) Largrangian mechanics :
1 dof system( )
1n xq 1
2 221 1 1
, , 2 2 2
, 0, ,
T m x V kx D b x
d T T V Dm x kx b x
dt x x xx
(1)m x b x kx F
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Automatic Control Systems
Equations of Mechanical Systems
【Example4】
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Automatic Control Systems
Equations of Mechanical Systems
【Example5】
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Automatic Control Systems
Equations of Mechanical Systems
【Example 6】Motor Coupling System
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Automatic Control Systems
Equations of Mechanical Systems
Torque equations
What is the order of the system? “ 4th ”
i) State space representation
State :
Output : Input :
(2)
(1)
2
2
2
2
dt
tdJttK
ttKdt
tdB
dt
tdJtT
LLLm
Lmm
mm
mm
1 2 1
3 4 3
, ,
,
m m
L L
x t t x t t x t
x t t x t t x t
tty L tTtu m
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Automatic Control Systems
Equations of Mechanical Systems
1 2
2
1 3 2
3 4
4 1 3
3
1
1
mm L m m
m m m
mm
m m m
m L
L L
L
x t x t
BKx t t t t T t
J J J
BKx t x t x t T t
J J J
x t x t
K Kx t t t x t x t
J J
y t t x t
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Automatic Control Systems
Equations of Mechanical Systems
【Note】 Alternative approach
State :
Output :
11
2 2
33
4
4
1
2
3
4
0 1 0 00
0 1
0 0 0 10
0 0 0
0 0 1 0
m
m m m
m
m L
x tx tBK K
x t J J J x tJ u t
x tx t
x tK Kx t J J
x t
x ty t
x t
x t
1 2 3- , , m L L mx t t t x t t x t t
1 3 1 : " "L my t t t x t x t dt x t impracital
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Automatic Control Systems
Equations of Mechanical Systems
ii) Transfer function representation
From (1), (2),
From (4),
(4), (5) → (3) :
Also
KBsJJKsJBsJJs
KsJ
sT
s
KBsJJKsJBsJJs
K
sBsJJsK
JBs
K
JJsT
ssG
ssK
Jss
ssJssK
ssKssBssJsT
mLmLmLm
L
m
m
mLmLmLm
mLmLmLmm
L
LL
Lm
LLLm
Lmmmmmm
23
2
23
234
2
2
2
1
(5)
(4)
(3)
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Automatic Control Systems
Sensors and Encoders
□ Sensors and Encoders
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Automatic Control Systems
Sensors and Encoders
automation
sensor
general sensor object detection touch
proximity
range sensor displacement
motor control
sensor
position
Speed/acceleration
force/torque/elastic
force
process control
sensor
temperature
Fluid/fluid speed/fluid
pressure
density/thickness
pH
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Automatic Control Systems
Sensors and Encoders
motor
control
sensor
analog
potentiometer
linear/rotary variable differential
transformer (LVDT/RVDT)
resolver
synchro
inductive
digital
optical encoder
absolute encoder
laser interferometer
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Automatic Control Systems
Sensors and Encoders
Potentiometer
position → proportion to electric voltage
tKte cs
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Automatic Control Systems
Sensors and Encoders
ttKte s 21
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Automatic Control Systems
Sensors and Encoders
【Example1】 Position control of DC motor
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Automatic Control Systems
Sensors and Encoders
【Example2】 Position control of AC motor
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Automatic Control Systems
Sensors and Encoders
Tachometers
Angular Velocity Electrical Voltage
; small generator
or Mechanical Energy Electrical Energy
Modeling
proportional to angular velocity
t t t t t
d te t K t K E s K s s
dt
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Automatic Control Systems
Sensors and Encoders
Velocity Control of DC motor
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Automatic Control Systems
Sensors and Encoders
Position-control system with tachometer feedback
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Automatic Control Systems
Sensors and Encoders
Encoder
Generate a coded reading of a measurement
Encoder Types
i) Incremental Encoder (optical encoder)
ii) Absolute Encoder
called “Shaft Encoder”
Shaft Encoder : digital transducer used for measuring angular
displacements and angular velocities
i) high resolution
ii) high accuracy
iii) suitable for digital control systems
iv) reduction in system cost
v) improvement of system reliability
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Automatic Control Systems
Sensors and Encoders
Incremental Encoder
Position or velocity detecting digital output
By counting the pulses or by timing the pulse width
Equally spaced and identical slit areas
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Automatic Control Systems
Sensors and Encoders
Incremental encoder (Single channel)
Single channel encoder no direction information
Dual channel encoder direction information detected
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Automatic Control Systems
Sensors and Encoders
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Automatic Control Systems
Sensors and Encoders
Absolute Encoder
Many pulse tracks for position indication
The pulse windows on the tracks can be organized into some
pattern (≡code)
i) Binary Code
ii) Gray Code : single bit continuous change
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Automatic Control Systems
Sensors and Encoders
Binary Code
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Automatic Control Systems
Sensors and Encoders
Gray Code
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Automatic Control Systems
Sensors and Encoders
LVDT (Linear Variable Differential Transformer)
Vref for direction information of movement
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Automatic Control Systems
Sensors and Encoders
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Automatic Control Systems
Sensors and Encoders
RVDT (Rotary Variable Differential Transformer)
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Automatic Control Systems
Sensors and Encoders
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Automatic Control Systems
Sensors and Encoders
Resolver
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Automatic Control Systems
Sensors and Encoders
Rotor : Primary coil ( )
Stator : Two sets of windings placed 90°apart
angular displacement( ), angular velocity( ), direction
information
Historically, resolvers were used to compute trigonometric
functions or to solve a vector into orthogonal components
useful for robotic control
01
02
cos
sin
ref
ref
v av
v av
refv
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Automatic Control Systems
Sensors and Encoders
Synchro-Transformer
Similar to the resolver
Consists of transmitter, receiver (or control transformer)
rtrefavv cos0