Compensation Using Frequency ResponseControl Systems Design

Dr. Hatem ElaydiElectrical Engineering Depatment

1

Islamic University of Gaza

Fall 2011

Compensators

The basic controller operation analysisActive compensation: PI D controllerPassive compensation: Phase-lead controller

Phase-lag controller Phase lag-lead controller

2

Phase lag-lead controller

Introduction

Compensation or correction of a control system

)1)(1()()( :Example

sTss

KsHsG

stablebecan systemloop-closed thismake

:getwecriterion,Hurwitz-RouthUsingSolution:

3

:getwecriterion,Hurwitz-RouthUsing

0)T 0(K

TT

TK

11

1

)()()( :if But

12

Tss

KsHsG

.or varyingonlystablebenot cansystem

loop-closedthisCriterion,Hurwitz-Routhtoding Accor

TK

Solution:

Definition of the compensation:

TτTss

sKsHsG

)1(

)1()()( :makeweIf

2

This closed-loop system can be stable.

We make the system stable by increasing a component.

Intro…

This procedureis called the compensation or correction.

Definition of the compensation:

4

increasing a component ,which makes the system’s performance to be improved, other than only varying the

system’s parameters, this procedure is called the compensation or correction of the system.

The compensator is an additional component or circuit that is inserted into a control system to compensate for a deficient performance.

component, 1sincreaseto ,)(

)()( :Example

12 Tss

KsHsG

Compensator

r.compensatoais 1s stable,becansystemthe

)(

1Tss

5

Types of the compensation

:typesseveralget canwesystem,theof

structurethein)(oflocationthetoaccordingand ),(

asdesignatedisr compensatotheoffunctiontransfer The

sGsG cc

Basic types of compensators

Active Compensators

PID controller

Transfer function:

DI

pc sτsτ

K(s)G )(1

1

6

DpDI

pIDIp

I

KKK

KsKs

KK

sτ

; 1

stability.improvingcontrolleraldifferenti--D

.clearingcontrollergintegratin-- I

y.sensitivitpromotingcontrolleralproportion --P

sse

PD controller )( :functiontransfer sKKsG Dpc

C(s)G(s)

R(s)－

＋pK

sK D

＋

＋

)( sGC

)2()(

:Assuming

2

n

nss

sG

dcompensatetheoffunctiontransfer loopopenThe

7

)2(

)()()( :issystem

dcompensatetheoffunctiontransfer loopopenThe 2

n

DPnc ss

sKKsGsG

D

P

K

Ks :at zeroloopopen

aaddingtoequivalentiscontrollerPD thethat showsIt

Effects of PD controller:

2) PD controller improve the system’s stability (to increase damping and reduce maximum overshoot);

3) PD controller reduce the rise time and settling time;

1) PD controller does not alter the system type;

8

3) PD controller reduce the rise time and settling time;

4) PD controller increase BW(Band Width) and improve GM & PM

－ bring in the noise !

PI controllers

KKsG Ipc1

)( :functionTransfer

)2()( :Assuming

2

n

n

sssG

C(s)G(s)

R(s)－

＋pK

sK I

1

＋

＋

)(sGC

:issystemdcompensatetheoffunctiontransfer loopopenThe

9

)2(

)(

)2(

)1

()()(

:issystemdcompensatetheoffunctiontransfer loopopenThe

2

22

n

IPn

n

IPnc

ss

KsK

sss

KKsGsG

0s :atpoleaand :at zero

loopopenaaddingtoequivalentiscontrollerPI thethat showsIt

P

IK

Ks

Effects of PI controller:

1) Increase the system’s type－clear the steady-state error ;

2) reduce BW(Band Width) and GM & PM

beneficial to the noise limiting ,

10

beneficial to the noise limiting ,not beneficial to the system’s stability.

G(s)R(s) C(s)

- +pK

sKI

1

PID controller Transfer function: sKs

KK(s)G DIpc 1

11

sI

sK D)(sGC

PID controller have advantages both of PI and PD.

Passive compensation

c

c

s1

(s)G controllerlag-phase2)

1 s1

s1(s)G controllerlead-phase)

1

1

Types of passive controller

12

ab

c

c

,

s1

s1

s1

s1(s)G controllerlead-lagphase3)

s1

(s)G controllerlag-phase2)

11

1

a

a

b

b

1 )( :functionTransfer

controller lead-Phase )

ps

zs

τs

τssGc

1

1

1

1

Passive controllers

13

τ p

τz

11

τj

jjωGc

1

1)(

responseFrequency

1

occurswhichat frequencythe

and,phase,theofvalueMaxmum

mm

m

zpm

:getwewhichfrom

tantan

)()(

11

jjGc

14

m

m

m

sin

sin

sin

:getwewhichfrom

1

1

1

11

Effects are similar to PD.Compensation ideal:

make ωm to be ωc !

Bode plotz pzp

1 1

1s

1s)( :functionTransfer

controllerlag-Phase )

ps

zssGc

2

Passive controllers

15

βτ p

τz

11

jω

jωjωGc

1

1)(

responseFrequency

Effects are similar to PI. 2

1

16

Compensation ideal:

Make 1/τto be in the lower frequency-band and far from ωc !

Bode plot

2

1

1 1

1

1

1

1)(

responseFrequency

b

b

a

ac j

j

j

jjωG

Effects are similar to PID.

Compensation ideal:

dBL( /)

dB/dec20

0

dB/dec20

b1

a1

b1

a1

)(

17

Compensation ideal:

First make the phase-lag compensation－to satisfy ess

and compensate a part of γc .

second make the phase-lead compensation－to satisfy the transitional requirements.

)(

090

090

Bode plot

Cascade compensation by frequency response methodCascade compensation by frequency response method

Phase-Lead Compensation using Bode diagram

d.compensate

becanangleLead-phasemaxmum,betomake :Ideal cm ).( is

systemteduncompensafunctiontransfer loopopen :Assume

sGHK 00

:oncompensati Active1. s)τ(K(s)G 1

18

)(

)()(desirethefromget becan :

desirethefromget becan :

:oncompensati Active1.

(s)(s)GGHK

sGsGHτ

KKeK

s)τ(K(s)G

c

c

jsC

jωsCCD

pssp

DpC

1

180

1

1

00

00

0

diagram.BodethefromandtheMeasure2)

.desiredtheand)functiontransfer loop-openthe

toaccordingsystemteduncompensatheofdiagramBodethePlot 1)

1)( τs1

τs1 :oncompensatiPassive2.

oo

coc

00

e(jωGHK

(s)G

ss

C

Cascade compensation by frequency responsemethod

19

.15~1060dB/dec,-;10~5

40dB/dec,-;5,at /20:)ofslopetheIf

.thetomadebetocascadethe

todueofincreasing theofbecauseangledcompensate:

.desired:

15~5: -Get 3)

oooo

oc0

00

c

cd

oococd

decdB(ωL

(s)GH K(s)G

φ

C

c

m

:fromget becanhere

1 :formulathefromget 5)

sin1

sin1get

1

1sinfrom4)

m

m

m1m

m

φ

φφ

Cascade compensation by frequency responsemethod

20

etc.theexamining

theofdiagramBodetheplot 6)

lglg

:fromget becanhere m

CC

c

ωω

,γω

(s) G(s)GHK

α(jw)GHKm

00

00 1020

Example:

thefor ionsspecificat,functiontransfer loop-Open 21

S

KGH(s)

))((. ; ; :aresystem cc20

2

1101045 ttress

Cascade compensation by frequency response method

solution: －－40dB/deccGlg20

21

10

cGlg20

).( 10101 sseK

1631000 . ;

Fig.6.4.1.inshownisdiagramBodeThe

c c

τs

τs(s) Gc

1

1 :oncompensatilead-phase theusemust We

10atangleaddand c 45

Cascade compensation by frequency response method

－－40dB/dec

10

cGlg20)180)()((

450045 o

oooo0m

jGH

φ ccd

17.0 sin1

sin1

om 45

m

m

24.0 101

cm

22

dBdBjGH 4.15log2020)(log20but

17.010

)7.14.1520lg20(

7.1and041.01

24.01 makeweSo

cc

ccc

KK

Ks

sKG

Phase-Lag Compensation using Bode diagram

)s.s

K

ss

K GH(jω v

1502

()()

:functiontransfer teduncompensaThe

.and,45 marginPhase

:aresystemthefor ionsSpecificat

c 20 vK

Example:

2

6.32－－20dB/dec

－－40dB/dec－－900

－－20lgβ

23

r.compensatotheDtermine

1 ,1

1(s):on compensatilag-phase theusecan We

6.32at 20

shown issystemteduncompensa theofdiagramBodeThe

cc

s

sGc

020c

－－900

－－1800

solution:

oo130)(where1.5frequencythelocateWe 50 cc φ

1020log20dB

20dBisat nattenuatiorequiredThe

.20)(lg 1.5, At

c

.c

dBjGH51

20

Phase-Lag Compensation using Bode diagram

2

6.32－－20dB/dec

－－40dB/dec－－900

－－20lgβ

24

s

ssG

c

7.661

67.61)(67.6

r)compensatolag-phasethefor at error 5Allowing(Consider

15.010

1 :Make

c

020c

90

－－1800

Validate……

Phase-Lag Compensation using Bode diagram

Steps of the phase-lag compensation:

1

)(log200log2 :fromGet 3)

t.requirementhesatisfyhichFind2)

.desiredtheand)functiontransfer loop-openthe

toaccordingsystemteduncompensatheofdiagramBodethePlot 1)

cc

00

jGH

w

e(jωGHK ss

c

25

etc.andionsspecificatthe

validatetosystemdcompensatetheofdiagramBodethePlot )5

.get to)105:(1

Make)4

cc

～kk

c

Phase-Lag-lead Compensation using Bode diagram

Compensation according to the desired frequency response

Example

First: make the phase-lag compensation－to satisfy ess

and compensate a part of γc .

Second: make the phase-lead compensation－to satisfy the requirementsγc and ωc etc.

26

Example

device.on compensatiseries thedetermine

)1(

1

:issystemteduncompensa theoffunction transfer loop-open theIf

shownisresponsefrequency desiredThe

ssGH(s)

solution

10

100－－2020dB/dec

－－4040dB/dec

Compensation according to the desired frequency responseCompensation according to the desired frequency response

In terms of the desired frequency response we have:

)101.0(

10)()()(

sssGHsGsGH cdesire

)()(

1

1

sssGH

27

10

100－－2020dB/dec

－－4040dB/dec).(

)(

)().()(

)()(

1010

110

11

101010

s

s

sssssGH

sGHsG desire

c

1

andsystem)phase

(minimumshown issystemaof

responsefrequency loop-opendesiredThe

GH(s) 100－－2020dB/dec

－－4040dB/dec

Exercise:

28

system.

teduncompensa theandsystemdcompensate

ebetween theperformanc thecompareand

deviceon compensaticascade theDetermine

)1(

ssGH(s)

10

100

－－4040dB/dec

1