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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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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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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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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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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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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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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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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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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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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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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Equations of Mechanical Systems

【Example4】

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【Example5】

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【Example 6】Motor Coupling System

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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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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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【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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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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Sensors and Encoders

□ Sensors and Encoders

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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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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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Potentiometer

position → proportion to electric voltage

tKte cs

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ttKte s 21

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【Example1】 Position control of DC motor

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【Example2】 Position control of AC motor

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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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Velocity Control of DC motor

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Position-control system with tachometer feedback

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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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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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Incremental encoder (Single channel)

Single channel encoder no direction information

Dual channel encoder direction information detected

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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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Binary Code

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Gray Code

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LVDT (Linear Variable Differential Transformer)

Vref for direction information of movement

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RVDT (Rotary Variable Differential Transformer)

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Resolver

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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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Synchro-Transformer

Similar to the resolver

Consists of transmitter, receiver (or control transformer)

rtrefavv cos0

Automatic Control Systems -Lecture Note...

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