PID Controllers: Theory, Design and...

PID

Con

trolle

rs:

Theo

ry, D

esig

n an

d Tu

ning

{rrGˀz ��G��

Y_UXUYWW`

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Lect

ure

cont

ent

•In

trodu

ctio

n•

Bas

ics

of P

ID c

ontro

llers

•Tu

ning

of P

ID c

ontro

llers

•O

ptim

izat

ion

in M

atla

b•

Aut

o tu

ning


Y_UXUYWW`

Z}Êz�ZT��V��

PID

-con

trolle

rs: i

ntro

duct

ion

•B

y fa

r the

mos

t pop

ular

con

trolle

r•

In p

roce

ss c

ontro

l >95

con

trolle

rs a

re o

f P

I(D)-

type

•G

ood

for l

inea

r pro

cess

con

trol

•R

elat

ivel

y ea

sy to

und

erst

and

(impo

rtant

re

ason

for w

ide

popu

larit

y)•

Stil

l man

y of

the

PID

-con

trol l

oops

are

po

orly

tune

d...


Y_UXUYWW`

[}Êz�ZT��V��

Typi

cal p

aper

mill

•O

ver 2

000-

500

cont

rol l

oops

•97

% P

I-con

trolle

rs•

Onl

y 20

% o

f PI-c

ontro

llers

wor

k w

ell

decr

easi

ng p

roce

ss v

aria

bilit

y•

Rea

son

for p

oor p

erfo

rman

ce:

–30

% p

oor t

unin

g–

30%

val

ve p

robl

ems

–20

% v

arie

ty o

f pro

blem

s (e

.g s

enso

r pr

oble

ms,

bad

cho

ice

of s

ampl

ing

rate

s...)


Y_UXUYWW`

\}Êz�ZT��V��

PID

-con

trolle

r•

Toda

y m

ost o

f the

PID

con

trolle

rs a

re

mic

ropr

oces

sor b

ased

•D

AM

ATR

OL

MC

100:

dig

ital s

ingl

e-lo

op u

nit

cont

rolle

r whi

ch is

use

d, fo

r exa

mpl

e, a

s P

ID

cont

rolle

r, ra

tio c

ontro

ller o

r man

ual c

ontro

l st

atio

n.

•O

ften

PID

con

trolle

rs a

rein

tegr

ated

dire

ctly

into

act

uato

rs(e

.g v

alve

s, s

ervo

s)


Y_UXUYWW`

]}Êz�ZT��V��

Sim

ple

idea

of f

eedb

ack.

..

•P

rinci

ple

of n

egat

ive

feed

back

:”In

crea

se th

e m

anip

ulat

ed v

aria

ble

whe

n pr

oces

s va

riabl

e is

sm

alle

r tha

n th

e se

tpoi

nt a

nd d

ecre

ase

the

man

ipul

ated

var

iabl

e w

hen

the

proc

ess

varia

ble

is la

rger

than

the

setp

oint

”

ey s

pu

yC

ontro

ller

Pro

cess

-166

6

l

x

n


Y_UXUYWW`

^}Êz�ZT��V��

Sim

ulin

k m

odel

ref

load

1

(s+1

)(s+1

)(s+1

)

Zero

-Pol

e

t

To

Wor

kspa

ce2

r

To

Wor

kspa

ce1

y

To

Wor

kspa

ce

Sco

pe

PID

PID

Con

trolle

r

Clo

ck

Ban

d-Li

mite

dW

hite

Noi

se


Y_UXUYWW`

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PID

con

trol

•”te

xtbo

ok” v

ersi

on o

f PID

•C

ontro

l var

iabl

e u

is a

sum

of:

–P

-term

(pro

porti

onal

to e

)–

I-ter

m (p

ropo

rtion

al to

inte

gral

of e

)–

D-te

rm (p

ropo

rtion

al to

der

ivat

ice

of e

)

•C

ontro

ller p

aram

eter

s:–

K =

pro

porti

onal

gai

n–

Ti =

inte

gral

tim

e–

Td=

deriv

ativ

e tim

e

1(

)(

)(

)(

)t

di

o

det

ut

Ke

te

dT

Tdt

WW

§·

�

�¨

¸©

¹³


Y_UXUYWW`

`}Êz�ZT��V��

Pro

porti

onal

act

ion

•H

igh

valu

e of

gai

n m

akes

the

syst

em

mor

e in

sens

itive

to

load

dis

turb

ance

•To

o la

rge

a ga

in

mak

es th

e sy

stem

m

ore

sens

itive

to

mea

sure

men

t noi

se•

Ste

ady-

stat

e er

ror

decr

ease

s w

hen

gain

in

crea

ses

•O

scill

atio

n ho

wev

er

ofte

n in

crea

ses

02

46

810

1214

1618

200

0.2

0.4

0.6

0.81

1.2

1.4

Clo

sed

loop

sys

tem

with

pro

porio

nal c

ontro

l

K =

5

K =

2

K =

1

��3

1 1G

s

�


Y_UXUYWW`

XW

}Êz�ZT��V��

Inte

gral

act

ion

•In

tegr

al te

rm

rem

oves

ste

ady

stat

e er

ror

•S

hort

inte

grat

ion

time

ofte

n le

ads

to

osci

llatio

n•

Long

inte

grat

ion

time

com

mon

in p

roce

ss

cont

rol

02

46

810

1214

1618

200

0.2

0.4

0.6

0.81

1.2

1.4

1.6

Clo

sed

loop

sys

tem

with

pro

porti

onal

and

inte

gral

con

trol

Ti =

1

Ti =

2

Ti =

5

Ti =

inf


Y_UXUYWW`

XX


Der

ivat

ive

actio

n•

Der

ivat

ive

term

can

pr

edic

t ou

tput

•Fa

st a

nd s

tabl

e re

spon

se•

Noi

se c

an m

ake

deriv

ativ

e co

ntro

l pr

oble

mat

ic•

Als

o lo

ng d

elay

s ar

epr

oble

mat

ic w

hen

usin

g de

rivat

ive

term

02

46

810

1214

1618

200

0.2

0.4

0.6

0.81

1.2

1.4

Der

ivativ

e co

ntro

l (K

=3, T

i=2)

Td =

0.7

Td

= 2

.5

Td =

0.1


Y_UXUYWW`

XY


Der

ivat

ive

actio

n

•Fa

st c

hang

es in

refe

renc

e si

gnal

resu

lt hi

gh d

eriv

ativ

esÆ

cont

rol s

igna

l sat

urat

es•

Fixe

s:–

com

putin

g de

rivat

ives

from

pro

cess

out

put

–us

ing

filte

red

deriv

ativ

e te

rm (t

his

is u

sed

ofte

n in

real

app

licat

ions

)


Y_UXUYWW`

XZ


Filte

red

deriv

ativ

e te

rm:

•B

enef

its:

–ea

sy to

impl

emen

t in

prac

tise

–m

ore

inse

nsiti

ve to

noi

se th

an n

orm

al d

eriv

ativ

e te

rm–

corr

espo

nds

with

der

ivat

ion

of lo

w p

ass

filte

red

sign

al–

by c

hoos

ing

tau

smal

lsy

stem

has

sam

e re

spon

se a

s by

usi

ng n

orm

al d

eriv

atio

n

^`

,1

,2,1

,2

()

,lim

()

()

()

1

s

ss

ss

Gs

sG

ss

Gs

sG

ss

W

°

®

°�

¯


Y_UXUYWW`

X[


Filte

red

deriv

ativ

e

02

46

810

1214

1618

200510152025303540

time(

s)

control signal u

Con

trol s

igna

l

norm

alfil

tere

din

finite

sig

nal p

eak

02

46

810

1214

1618

200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

output y

time

(s)

Ste

p re

spon

se (t

au =

0.1

)

e

PID

CO

NT

RO

LLE

R

uy

ul

ref

s

tau.

s+1

Tra

nsfe

r Fcn

uf

To

Wor

kspa

ce2

yf

To

Wor

kspa

ce

Sco

pe

G

Pro

cess

1 s

Inte

grat

or

1/T

i

Gai

n3

Kp

Gai

n2

Td

Gai

n1


Y_UXUYWW`

X\


Out

put d

eriv

atio

n –

Tach

omet

er fe

edba

ck

•U

se p

roce

ss o

utpu

t for

der

ivat

ion

and

gain

•N

o ze

ros

to c

ontro

lled

clos

ed-lo

op s

yste

m

(pre

vent

s ov

ersh

oots

)

G(s

)

Y ref

E

Ki/s

Kp+

Kds

Y+ -

+ -


Y_UXUYWW`

X]


Alte

rnat

ive

repr

esen

tatio

ns

1(

)1

di

Gs

KsT

sT§

·

��

¨¸

©¹

��

''

''

1(

)1

1d

i

Gs

KsT

sT§

·

��

¨¸

©¹

6

P I D

eu

66

PID

•Non

-inte

ract

ing

& in

tera

ctin

g

•Non

-inte

ract

ing

mor

e ge

nera

l

•Inte

ract

ing

com

mon

in c

omm

erci

al c

ontro

llers

(s

aid

to b

e ea

sier

to tu

ne m

anua

lly)


Y_UXUYWW`

X^


SIM

ULI

NK

PID

-con

trolle

r

•S

imul

ink

PID

-blo

ck is

of

form

:•

”text

boo

k” v

ersi

on is

:

PID

PID

Con

trolle

r1

pi

dK

KT

ss

��

e

PID

CO

NT

RO

LLE

R

uy

ref

Sco

pe

G

Pro

cess

1 s

Inte

grat

or

1/T

i

Gai

n3

Kp

Gai

n2

Td

Gai

n1

du/d

t

Der

ivat

ive


Y_UXUYWW`

X_


Inte

grat

or w

indu

p


Y_UXUYWW`

X`


Inte

grat

or w

indu

p•

All

actu

ator

s ha

ve li

mita

tions

:–

limite

d sp

eed

–va

lve

open

ing

•If

the

cont

rol v

aria

ble

reac

hes

actu

ator

lim

its Æ

feed

back

loop

is b

roke

n!•

Stil

l erro

r will

cont

inue

to in

tegr

ate Æ

very

larg

e in

tegr

al te

rm (”

win

d up

”)•

Larg

e tra

nsie

nts

whe

n th

e ac

tuat

or

satu

rate

s


Y_UXUYWW`

YW


Inte

grat

or w

indu

p

•In

tegr

ator

act

ion

mus

t be

stop

ped

whe

n ou

tput

sat

urat

es!

•S

olut

ions

:–

setp

oint

lim

itatio

n (li

mit

perfo

rman

ce,

win

dup

caus

ed b

y di

stur

banc

es?)

–in

crem

enta

l alg

orih

ms

–ba

ck c

alcu

latio

n an

d tra

ckin

g–

cond

ition

al in

tegr

atio

n


Y_UXUYWW`

YX


Whe

n is

PI c

ontro

l suf

ficie

nt?

•O

ften

deriv

ativ

e ac

tion

switc

hed

off

•D

omin

ant d

ynam

ics

are

of th

e 1.

ord

er•

For e

xam

ple:

–le

vel c

ontro

l in

sing

le ta

nk–

stirr

ed ta

nk w

ith p

erfe

ct m

ixin

g...

•W

hen

tight

con

trol n

ot n

eede

dÆ

PI-c

ontro

l ade

quat

e


Y_UXUYWW`

YY


Whe

n is

PID

con

trol s

uffic

ient

?

•D

omin

ant d

ynam

ics

are

of th

e 2.

ord

er•

PID

spe

eds

up th

e re

spon

se v

ersu

s P

I–

dam

ping

impr

oved

–hi

gher

gai

n ca

n be

use

d to

spe

ed u

p tra

nsie

nt re

spon

se•

For e

xam

ple:

–te

mpe

ratu

re c

ontro

l


Y_UXUYWW`

YZ


Whe

n P

ID c

ontro

l is

insu

ffici

ent?

•S

yste

m h

as h

igh

orde

r dyn

amic

s•

Sys

tem

is ti

me

varia

nt•

Long

del

ays

•N

on-li

near

pro

cess

•M

IMO

/MIS

O s

yste

m w

ith s

trong

cro

ss

depe

ncie

s


Y_UXUYWW`

Y[


Con

trolle

r des

ign

•P

robl

em: h

ow to

det

erm

ine

the

para

met

ers?

•Tu

ning

is a

trad

e-of

fs b

etw

een:

–lo

ad d

istu

rban

ce a

ttenu

atio

n–

effe

cts

of m

easu

rem

ent n

oise

–ro

bust

ness

to p

roce

ss v

aria

tions

–re

spon

se to

set

poin

t cha

nge

–m

odel

requ

irem

ents

–(c

ompu

tatio

nal r

equi

rem

ents

)


Y_UXUYWW`

Y\


Per

form

ance

crit

eria

02

46

810

1214

160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Step

Res

pons

e

Tim

e (s

ec)

Amplitude

rise

time

over

shoo

t

settl

ing

time

stea

dy s

tate


Y_UXUYWW`

Y]


Ope

n lo

op tu

ning

•O

pen

loop

•

Iden

tify

plan

t dyn

amic

s•

Let u

(t)be

a u

nit s

tep

(’goo

d’ te

st s

igna

l)•

Mea

sure

out

put

()

1()

ut

t

G(jZ

)(

)y

t


Y_UXUYWW`

Y^


Ope

n lo

op tu

ning

L

T

()

1sL

KG

se

sT�

�

K

From

figu

re

K=

1L=

1.3

sT=

4.4

sa=

0.4

a


Y_UXUYWW`

Y_


Zieg

ler-

Nic

hols

Con

trolle

rK

TiTd

Tp

P1/

a4L

PI

0.9/

a3L

5.7L

PID

1.2/

a2L

L/2

3.4L

a

L


Y_UXUYWW`

Y`


Zieg

ler-

Nic

hols

: res

pons

e

02

46

810

1214

1618

200

0.2

0.4

0.6

0.81

1.2

1.4

1.6

time(

s)

Zieg

ler-N

icho

ls re

spon

se


Y_UXUYWW`

ZW


Zieg

ler-

Nic

hols

ana

lysi

s

•D

ecay

ratio

is c

lose

to o

ne q

uarte

r•

Ove

rsho

ot is

qui

te la

rge

•S

impl

e an

d w

idel

y us

ed•

Ofte

n in

suffi

cien

tÆne

cess

ary

to h

ave

mor

e da

ta a

bout

pro

cess

dyn

amic

s•

Giv

es a

sta

rting

poi

nt fo

r fin

e tu

ning

•A

lso

frequ

ency

resp

onse

met

hod

can

be

used


Zieg

ler-

Nic

hols

Y_UXUYWW`

ZX



Y_UXUYWW`

ZY


Chi

en, H

rone

s &

Res

wic

k M

etho

d

•M

odifi

catio

n of

Zie

gler

-Nic

hols

met

hod

•B

ette

r dam

pled

clo

sed-

loop

ste

p re

spon

se•

Par

amet

er ta

bles

for:

–qu

icke

st re

spon

se w

ithou

t ove

rsho

ot–

quic

kest

resp

onse

with

20%

ove

rsho

ot•

Diff

eren

t par

amet

ers

for t

unin

g:–

setp

oint

resp

onse

–lo

ad d

istu

rban

ce


Y_UXUYWW`

ZZ


CH

R fo

r set

poin

t res

pons

e

•T

= tim

e co

nsta

nt

PID

Con

trolle

r par

amet

ers;

C

hien

, Rho

nes

Res

wic

k se

tpoi

nt re

spon

se m

etho

d0

%20

%ov

ersh

oot

Con

trolK

TiTd

KTi

Td

P0.

3/a

0.7/

aP

I0.

35 a

1.2T

0.6

aT

PID

0.6/

aT

0.5L

0.95

/a1.

4T0.

47L


Y_UXUYWW`

Z[


CH

R fo

r loa

d di

stur

banc

e re

sp.

PID

Contr

oller

par

amet

ers;

Chien

, Rho

nes R

eswi

ck lo

ad d

istur

banc

e re

spon

se m

etho

d0

%20

%ov

ersh

oot

Contr

olK

TiTd

KTi

Td

P0.

3/a

0.7/

aPI

0.6

a4L

0.7

a2.

3LPI

D0.

95/a

2.4L

0.42

L1.

2/a

2L0.

42L


Y_UXUYWW`

Z\


Z-N

vs.

CH

R

•O

vers

hoot

exi

st b

ut re

spon

se is

muc

h be

tter

whe

n C

HR

par

amet

ers

are

used

02

46

810

1214

1618

200

0.2

0.4

0.6

0.81

1.2

1.4

1.6

time(

s)

ZN CH

R 2

0%C

HR

0%

3

1(

)(

1)G

ss

�

väärin


Y_UXUYWW`

Z]


Ana

lytic

al tu

ning

met

hods

0

(pro

cess

tran

sfer

func

tion)

(con

trolle

r tra

nsfe

r fun

ctio

n)

G (c

lose

d lo

op tr

ansf

er fu

nctio

n)1

p c

pc

pc

G GG

G GG

o

�

0c

0

1G

(clo

sed

loop

tran

sfer

func

tion)

1p

GG

Go

��


Y_UXUYWW`

Z^


Clo

sed-

loop

tuni

ng

PID

G(s

)+ -

r(t)

= 0

e(t)

Line

ar sy

stem

c(t)

u(t)

1.Se

t Ti=

dan

d T d

= 0.

2.In

crea

se K

pun

til th

e sy

stem

osc

illat

es to

obt

ain

criti

cal

gain

Kcr

and

criti

cal p

erio

d T c

r(o

r fre

quen

cy)

0

1(

)(

)(

)t

pd

i

deu

tK

et

ed

TT

dtD

D§

·

��

¨¸

©¹

³


Y_UXUYWW`

Z_


Clo

sed-

loop

tuni

ng

PID

Con

trolle

r par

amet

ers;

Zi

egle

r-Nic

hols

freq

uenc

y re

spon

se m

etho

d

Con

trolK

TiTd

P0.

5Kcr

=2.3

PI

0.4K

cr=1

.90.

8Tcr

=1.8

PID

0.6K

cr=2

.80.

5Tcr

=1.1

0.12

5Tcr

=0.3

0

1(

)(

)(

)t

pd

i

deu

tK

et

ed

TT

dtD

D§

·

��

¨¸

©¹

³


Freq

uenc

y re

spon

se m

etho

dY_UXUYWW`

Z`



Freq

uenc

y re

spon

se in

terp

reta

tion

Y_UXUYWW`

[W


Freq

uenc

y re

spon

se –

phas

e m

argi

nY_UXUYWW`

[X


Com

paris

onY_UXUYWW`

[Y



Y_UXUYWW`

[Z


Rel

ay tu

ning

0

1(

)(

)(

)t

pd

i

deu

tK

et

ed

TT

dtD

D§

·

��

¨¸

©¹

³

PID

G(jZ

)+ -

r(t)

= 0

e(t)

Line

ar sy

stem

c(t)

u(t)

M

M�

Firs

t rel

ay o

n, to

obt

ain

criti

cal g

ain

Kcr

and

criti

cal p

erio

d T c

r


Rul

e-ba

sed

empi

rical

tuni

ng

•In

crea

sing

pro

porti

onal

gai

n de

crea

ses

stab

ility

•E

rror d

ecay

s m

ore

rapi

dly

if in

tegr

atio

n tim

e is

dec

reas

ed•

Dec

reas

ing

inte

grat

ion

time

decr

ease

s st

abilit

y•

Incr

easi

ng d

eriv

ativ

e tim

e im

prov

es s

tabi

lity

Y_UXUYWW`

[[



Tuni

ng m

aps

Y_UXUYWW`

[\



Tuni

ng m

apY_UXUYWW`

[]



Cou

nter

intu

itive

Y_UXUYWW`

[^


Y_UXUYWW`

[_


Pol

e P

lace

men

t

•P

roce

ss m

ust b

e m

odel

led

•D

efin

e th

e de

sire

d cl

osed

-loo

p po

les

•P

ID-c

ontro

ller g

ives

des

ired

clos

ed-lo

op

pole

s•

Pro

cess

par

amet

ers

max

. 3Æ

pole

s ca

n be

set

free

ly u

sing

PID

-con

trolle

r


Y_UXUYWW`

[`


Pol

e P

lace

men

t: ex

ampl

e•

Pro

cess

:

•C

ontro

ller (

PI):

•C

lose

d lo

op s

yste

m:

•C

hara

cter

istic

eq: Æ

(1)

pp

KG

sT

�

11(1

)c

GK

sT

�

()

1p

c

pc

GG

Gs

GG

�

10

cp

GG

�

21

0p

p

i

KK

KK

ss

TTT

��

�


Y_UXUYWW`

\W


Pol

e pl

acem

ent:

exam

ple

•D

esire

d cl

ose-

loop

pol

es

char

acte

rized

by

rela

tive

dam

ping

(ȗ) a

nd fr

eque

ncy

(Ȧ):

•B

y m

akin

g co

effic

ient

equ

al in

ch

arac

teris

tics

equa

tions

and

so

lvin

g co

ntro

ller p

aram

eter

s gi

ves:

22

00

20

ss

[ZZ

��

0 0 2 0

21

21

p

i

TK

K TT

T

[Z [Z Z

�

�


Gan

g of

six

•La

rges

t val

ue o

f tra

nsfe

r fun

ctio

n fro

m lo

ad

dist

urba

nce

to p

roce

ss o

utpu

t

Y_UXUYWW`

\X



Cho

osin

g Z

o

•M

ore

sens

itive

whe

n Z

oT is

sm

all

Y_UXUYWW`

\Y



App

roxi

mat

e m

odel

s•

Firs

t ord

er a

ppro

x

Y_UXUYWW`

\Z



App

roxi

mat

e m

odel

sY_UXUYWW`

\[



Y_UXUYWW`

\\


Opt

imiz

atio

n in

Mat

lab

•M

inim

ize

the

cost

func

tion

by s

imul

atin

g th

e co

ntro

lled

mod

el•

Use

fmin

sear

ch fu

nctio

n to

find

op

timiz

ed p

aram

eter

s fo

r the

PID

co

ntro

ller

•C

hoos

e ap

prop

riate

cos

t fun

ctio

n ca

refu

lly•

Don

’t fo

rget

loca

l min

ima


Y_UXUYWW`

\]


Ana

lytic

al tu

ning

met

hods

•Ȝ

–Tun

ing

•pr

oces

ses

with

long

dea

d tim

e•

desi

red

clos

ed-lo

op tf

:

•co

ntro

ller t

rans

fer f

unct

ion:

•Ȝ

< 1

fast

er re

spon

se (s

mal

ler

time

cons

tant

)

01

sLe

Gs

TO

�

� 1

(1)

csL

p

sTG

Ks

Te

O�

�

��


Y_UXUYWW`

\^


Opt

imiz

atio

n in

Mat

lab

1.G

uess

initi

al v

alue

s fo

r2.

Sol

ve y

(t)us

ing

sim

ulat

ion

(SIM

ULI

NK

) and

als

o th

e va

lue

of c

ost f

unct

ion

3.U

se o

ptim

izat

ion

algo

rithm

(FM

INS

EA

RC

H o

r FM

INU

NC

) to

upda

te th

e co

ntro

l par

amet

ers:

4.C

heck

if m

inim

um is

reac

hed

If no

t, se

t n =

n+1

and

go b

ack

to 2

.

��

,,

nn

np

id

JK

KK

00

0,

,,

Set

0p

id

KK

Kn

1 1 1

(dep

ends

on

optim

izat

ion

algo

rithm

)

(dep

ends

on

optim

izat

ion

algo

rithm

)

(dep

ends

on

optim

izat

ion

algo

rithm

)

nn

pp

nn

ii

nn

dd

KK

corr

ectio

n

KK

corr

ectio

nK

Kco

rrec

tion

� � �

�

�

�

1n

nJ

Jto

lera

nce

��

�


Y_UXUYWW`

\_


Opt

imiz

atio

n in

Mat

lab

-

+

Con

trolle

rSy

stem

PID

co

ntro

ller

Inpu

t��

��

��

��

��

,,

,,

tt

t

tt

tt

xf

xu

yh

xu

�

��

0

()

()

t

pi

dde

ut

Ke

tK

ed

Kdt

DD

�

�³

PID

con

trol

ler

Para

met

ers K

p, K

i,an

d K

d

Para

met

er v

ecto

r

ue

yy r

ef

,,

T

pi

dK

KK

ªº

¬

¼p

00

()

()

,,

()

()

tt

Tp

id

et

et

KK

Ke

de

d

dede

dtdt

DD

DD

ªº

ªº

«»

«»

«»

«»

«»

«»

ªº

¬¼ «

»«

»«

»«

»«

»«

»«

»«

»¬

¼¬

¼

³³

p


Y_UXUYWW`

\`


Cos

t crit

eria

0

()

ITAE

tet

dtf

³

0

()

ITE

tet

dtf

³

2

0

()

ITSE

tet

dtf

³

22

0

()

ISTE

te

tdt

f

³


Y_UXUYWW`

]W


Exa

mpl

e m

odel

2O

ut2

1O

ut1

ref

mea

sure

men

t noi

selo

ad d

istu

rban

ce

Tra

nspo

rtD

elay

ISE

To

Wor

kspa

ce3

t

To

Wor

kspa

ce2

r

To

Wor

kspa

ce1

y

To

Wor

kspa

ce

Sco

pe

PID

PID

Con

trolle

r

u2

Mat

hFu

nctio

n1

Gs

LTI S

yste

m

1 s

Inte

grat

orC

lock


Y_UXUYWW`

]X


Ope

n lo

op re

spon

se

05

1015

2025

3035

40-0

.20

0.2

0.4

0.6

0.81

1.2

1.4


Y_UXUYWW`

]Y


Cos

t fun

ctio

n in

Mat

lab

•%evaluate cost function from the

simulink model

•function J = cost(inputs)

•%paramaters

•Tsim=40; %simulation time

•model= 'pidmodel'; %simulink

model

•%assign new parameters to

workspace for simulation

•assignin('base','Kp',inputs(1));

•assignin('base','Ti',inputs(2));

•%simulate

•[t,x,y]=sim(model,Tsim);

•%y(2) corresponds with cost

•J=max(y(:,2));

IN MATLAB:

•fminsearch('cost',[1

10])

05

1015

2025

3035

40-0

.20

0.2

0.4

0.6

0.81

1.2

1.4 0

510

1520

2530

3540

0

0.51

1.52

2.53

3.54

4.5


Y_UXUYWW`

]Z


Gai

n sc

hedu

ling

Con

trolle

rP

roce

ss

Tabl

e

y sp

uyS

ched

ulin

gva

riabl

e

Con

trolle

r par

amet

ers


Y_UXUYWW`

][


Rul

e-B

ased

Met

hods

Spe

edS

tabi

lity

K in

crea

ses

incr

ease

sre

duce

s

T iin

crea

ses

redu

ces

incr

ease

s

T din

crea

ses

incr

ease

sin

crea

ses


Y_UXUYWW`

]\


MIM

O P

ID c

ontro

l-D

ecen

traliz

ed

•M

ultip

le In

put M

ultip

le O

utpu

t (M

IMO

)


Y_UXUYWW`

]]


MIM

O P

ID c

ontro

l-D

ecen

traliz

ed


Y_UXUYWW`

]^


MIM

O P

ID c

ontro

l-D

ecen

traliz

ed

Am

ount

of

inte

ract

ion

Lies

leht

oK

=0.

83, T

i=2,

I=0.

5U

nit s

tep

in r

ef1


Y_UXUYWW`

]_


MIM

O P

ID c

ontro

l-D

ecen

traliz

ed

Am

ount

of

inte

ract

ion

Lies

leht

oK

=1.

5, T

i=2,

I=0.

5U

nit s

tep

in r

ef2


Y_UXUYWW`

]`


MIM

O P

ID c

ontro

l -C

entra

lized

�

x

Ax

Bu

yC

x�St

ate-

spac

e eq

uatio

n

Tran

sfer

func

tion

1)

s�

G(s

)=C

(I-

AB


Y_UXUYWW`

^W


MIM

O P

ID c

ontro

l -C

entra

lized

G(s

)#

#G

(0)-1

Dec

oupl

ing

at st

eady

stat

e


Y_UXUYWW`

^X


MIM

O P

ID c

ontro

l –In

tegr

al g

ain

��

11

1.2

0.9

(0)

0.8

1

1.2

0.9

2.1

1.9

(0)

0.8

11.

72.

5i

G KG

HH

H�

�

ªº

«

»¬

¼

�ª

ºª

º

«»

«»

�¬

¼¬

¼


Y_UXUYWW`

^Y


MIM

O P

ID c

ontro

l –P

ropo

rtion

al g

ain

��

00

11

0

1.2

0.9

0.6

0.9

21

1lim

()

lim0.

81

0.2

0.3

41

31

0.6

0.9

18.3

50lim

()

0.2

0.33

1133

ss

ps

ss

sGs

s

ss

KsG

sG

GG

oo

��

o

ªº

«»

ªº

��

«»

«»

¬¼

«»

«»

��

¬¼

�ª

ºª

º

«»

«»

�¬

¼¬

¼


Y_UXUYWW`

^Z


()pidH

s

Sm

ith-p

redi

ctor (

)u

Hs

()

vH

s

()

tW�

()

dctW�

,(

)u

dcH

s

__

_

mSP

yu

ml

y

v

my

pe

Con

trol v

aria

ble

PID

con

trolle

r

Setp

oint

Proc

ess

Proc

ess m

odel

Proc

ess m

easu

rem

ent

Con

trolle

r with

dea

d-tim

e co

mpe

nsat

or


Y_UXUYWW`

^[


Ref

eren

ces

•K

. Åst

röm

and

T. H

äggl

und

PID

Con

trolle

rs: T

heor

y, D

esig

n, a

nd

Tuni

ng (3

rd e

d) (2

005)


Y_UXUYWW`

^\


PID

G(jZ

)+ -

e(t)

Line

ar sy

stem

Qou

t(t)

Qin

(t)r(

t)


Y_UXUYWW`

^]


/p

IK

Ks

�

Kp+

KI/s

e-sT

+ -

e(t)

Inpu

t

Qou

t(t)

Qin

(t)r(

t)

Out

put


Y_UXUYWW`

^^



Y_UXUYWW`

^_


/p

IK

Ks

�+ -

e(t)

Inpu

t

Qou

t(t)

Qin

(t)r(

t)sT

e�

Out

put


Y_UXUYWW`

^`


Gai

n sc

hedu

ling

Con

trolle

rP

roce

ss

Tabl

e

y sp

uyS

ched

ulin

gva

riabl

e

Con

trolle

r par

amet

ers

PID Controllers: Theory, Design and...

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Transcript of PID Controllers: Theory, Design and...