basic concepts

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Basic Concepts Block diagram representation of control systems Transfer functions Analysis of block diagrams P, PI and PID controllers ( Continuous and discrete forms) Stability of feedback control systems

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Basic Concepts. Block diagram representation of control systems Transfer functions Analysis of block diagrams P, PI and PID controllers ( Continuous and discrete forms) Stability of feedback control systems. Basic Block Diagram. PID:Process Instrumentation Diagram. - PowerPoint PPT Presentation

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Basic Concepts

Block diagram representation of control systems

Transfer functions Analysis of block diagrams P, PI and PID controllers ( Continuous

and discrete forms) Stability of feedback control systems

Basic Block Diagram

INPUT VARIABLE OUTPUT VARIABLESYSTEM

PID:Process Instrumentation Diagram

TT101

temperaturetransmitter

Thermocouple

TC101

temperaturecontroller

cooling waterinlet

cooling waterout

Feed

electronictransmissionline

ProductManualValve

sensor

pneumaticline

Automatic ControlValve

Block Diagram of Feedback Control System

CONTROLVALVE

TANKCONTROLLER

TRANSMITTER

WATERFLOW

VALVETset

SETPOINTERROR

ACTUATOR PROCESS

TEMPERATURE

Tm (MEASURED VARIABLE)

SIGNAL TO

CONTROLLEROUTPUT

MANIPULATEDVARIABLE

CONTROLLERVARIABLE

Laplace Transform

Common Signals

Name Function, f(t) Laplace transform, F(s)

Unit step function 0 0 ttu

0 1 ttu s

1

Unit impulse function(Diracdelta function)

0 , tt 0 0 tt

1

Ramp function 0 0 ttr

0 tkttr 2s

k

Sine function wttx sin 22

s

Properties of Laplace Transform

P r o p e r t yD e s c r i p t i o n

L i n e a r i t y P r o p e r t y

co n stan t w h ere

11

2121

K

tfLktfkL

tfLtfLtftfL

T i m e D e l a y sFe

tfLetfLs

s

D i f f e r e n t i a t i o n ofsFsd t

tfdL

I n t e g r a t i o n sFs

d ttfLt

o

1

F i n a l V a l u e T h e o r e m lim t

f t lim s 0

s F s

p r o v i d e d t h e l i m i t o n t h e l e f t

h a n d s i d e e x i s t s .

Transfer functions

0)0(),()()(

121 xtubtxadt

tdxa (BC.3)

subsxa0xsxsa 121

su1s

K

suasa

bsx

21

1

(BC.4)

1s

KsG

(BC.5)

Proportional Control

mteKtm c

seKsm c

tmeasuremen setpoint te

Proportional Integral Control

dt te teK tm

Ic

1

ses

KsmI

c

1

1

Proportional Integral Derivative (PID) Control

dt

de dt e te K tm D

Ic

1

)()1

1()( sess

Ksm DI

c

)()1

11()( se

s

s

sKsm D

Ic

Common Transfer Functions

0)0(',0)0(,22

22 xxKu x

dt

dx

dt

xd

Second-Order Transfer Functions

su s s

sx

1222

First-Order Plus Dead-Time (FOPDT)

1 s

Ke sy

s

Stability of Systems

sxps

Ksy

(BC.19)

1, sxttx (BC.20)

ps

Ksy

ptKety (BC.22)

ibap

Location of pole

22 bar

re

ibapi

a

br

R eal A xis

Imag

inar

y A

xis

φ

btibtKe

Keyat

pt

sincos

Generalization

n

n

n

ps

K

ps

K

ps

Kpspsps

sNsD

sNsG

.......

.......

2

2

1

1

21

tpn

tp neKeKty ...11

Stability and Pole Location

Imaginary Axis

Real Axis

Oscillatory growth

Exponential growth

Oscillatory growth

Expnential Decay

Oscillatory Decay

Oscillatory Decay

Sustained Oscillations

Sustained Oscillations

"For a transfer function to be stable, all its poles must lie to the left of the imaginary axis in the com-plex plane, i.e. in the left half plane (LHP)".

Stability of Closed Loop SystemsCONTROLLER ACTUATOR PROCESS OUTPUT

TRANSMITTERMEASURED VARIABLE

yset + e(s) m(s) q(s) y(s) sGc

G (s)

sGv sGp

sGT sym

-

01 Tvcp GGGG

setTvcp

pvcy

GGGG

GGGy

1

Example: Third order process

321

1

ssssGp

1, sGsGKsG Tvcc

c

c

set Ksss

K

y

y

321

0321 cKsss

Root Locus

Table BC.4 Table of roots of the character’s equation for various valves of cK

Kc root1 root2 root30.10.20.390.61.010.020.030.060.0100.0

-3.0467-3.0880-3.1564-3.2212-3.3247-4.3089-4.8371-5.2145-6.0000-6.7134

-1.8990-1.7909-1.4218 - 0.0542i-1.3894 - 0.3442i-1.3376 - 0.5623i-0.8455 - 1.7316i-0.5814 - 2.2443i-0.3928 - 2.5980i0 - 3.3166i0.3567 - 3.9575i

-1.0544-1.1211-1.4218 + 0.0542i-1.3894 + 0.3442i-1.3376 + 0.5623i-0.8455 + 1.7316i-0.5814 + 2.2443i-0.3928 + 2.5980i0 + 3.3166i0.3567 + 3.9575I

Root Locus Graph

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

Real Axis

Imag Axis

CONTROLLER TUNING)()

)1(

111()( ses

ssKsm D

Ic

0

0 .2

0 .4

0 .6

0 .8

1

1 .2

1 .4

0 1 2 3 4 5 6

N e w s e t p o i n t

O v e r s h o o t , AR i s e t i m e , T r

B

e ( t ) , E r r o r i n c o n t r o l

I S E e d t

D e c a y R a t i o B A

2

/

4PeriodTime Rise

Penod

eRatioDecay

eOvershoot

2

12

1

12

2

2

12

122

ss

sG

Ziegler Nichols Tuning

Table BC.5 Ziegler-Nichols tuning correlation

5.00.2K

2.1PID

33.3K

9.0PI

K

1P

K DII

Derivative-Integral-Prop

Integral alProportion

alProportion

1 s

Ke sG

s