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