feedback properties · 2008. 4. 17. · good feedback loop design ⇒ high loop (pk ) gain and high...

2008 Spring ME854 - GGZ Performance Specification and Limitations

Performance Specification and LimitationsPerformance Specification and Limitations

Feedback PropertiesFeedback Properties

systemfeedback standard following heConsider t

++ K P +

+

r -

u

di

up

dy

n

).()()()()( :Note

)()()( ,))(()(

to fromy sensitivitinput the Definely.respective

outputplant at the loop thebreaking ),()()(

and plant, theinput to at the loop thebreaking ),()()(

functions transfer loopoutput andinput Define

i

1

i

susPsKsdsu

sdsSsusLIsS

ud

sKsPsL

sPsKsL

pip

ipi

pi

o

i

−=

=+=

=

=

−

ies?uncertaint of face theinfeedback of benefits theachieve How to



++K P

+

+

r-

udi

up

dy

n

ly.respective matrics, difference returnoutput andinput

called are )( and )( . since ofary complementa is :Note

))()(()()(

))()(()()(

are matricesy sensiticitary complementoutput andinput The

)()()( ,))(()(

to fromy sensitivitoutput theDefine

1

1

1

sLIsLISITST

sLIsLsSIsT

sLIsLsSIsT

sdsSsysLIsS

yd

oi

oooo

iiii

ooo

++−=

+=−=

+=−=

=+=

−

−

−

iiooP

iioo

iooo

oioo

dSdKSnrKSu

dTdKSnrKSu

PdSnTdrSyr

dSPdSnrTy

+−−=

−−−=

−+−=−

++−=

)(

)(

)(

)(

thenstable, internally is system loop closed theIf




For plant output y++

K P+

+

r-

udi

up

dy

n

iiooP

iioo

iooo

oioo

dSdKSnrKSu

dTdKSnrKSu

PdSnTdrSyr

dSPdSnrTy

+−−=

−−−=

−+−=−

++−=

)(

)(

)(

)(

directly.output plant effacts which reference theimpact todirect has noiseSensor :Note

)())(()(

)(

1))(()(

reduce torequiresoutput plant at the rejection edisturbanc Good

)( reduce rejection (s) edisturbancoutput Plant

)()( reduce rejection (s) edisturbancinput Plant

1

1

rn

dPSPPKIPS

dPKI

PKIS

sSd

sPsSd

iio

o

o

oi

↓⇒=+=

↓⇒+

=+=

⇒

⇒

−

−

σσσ

σσσ

directly.input plant effacts which reference theimpact todirect has noiseSensor :Note

)(

1))(()(

)())(()(

reduce torequiresinput plant at therejection edisturbanc Good

)()( reducerejection (s) edisturbancoutput Plant

)( reduce rejection (s) edisturbancinput Plant

1

1

rn

dKPI

KPIS

dKSKKPIKS

sKsSd

sSd

ii

oi

i

ii

↓⇒+

=+=

↓⇒=+=

⇒

⇒

−

−

σσσ

σσσ

For plant input up




++K P

+

+

r-

udi

up

dy

n

)(

1)())(()( 1)(or 1)(

)(

1)())(()( 1)(or 1)(

invertible are and Suppose

1)( 1 )(

1)( 1 )(

imply that equations These

1)( if ,1)(

1)(

1)(

1

1)( if ,1)(

1)(

1)(

1 Then

1)()(1)(

1)()(1)( Note

1-1

1-1

PPPKIKKSKPPK

KKPPKIPSKPPK

KP

KPS

PKS

KPσKPσ

SKPσ

PKσPKσ

SPKσ

KPKPIKPσ

PKPKIPKσ

o

o

i

o

i

o

σσσσσσ

σσσσσσ

σσ

σσ

σ

σ

σσ

σσ

=≈+=⇒>>>>

=≈+=⇒>>>>

>>⇔<<

>>⇔<<

>−

≤≤+

>−

≤≤+

+≤+≤−

+≤+≤−

−

−




++K P

+

+

r-

udi

up

dy

n

)(reject 1 )(gain controller Large

)(reject 1)()(gain loopoutput Large of eperformanc Good

dominated is frequency eBetween th


4444444444 34444444444 21

4444444444 84444444444 76

d

d

ii

p

i

dP

dKPLu

>>

>>=

σ

σσperformance

)(reject 1 )(gain controller Large

)(reject 1)()(gain loopoutput Large of eperformanc Good

:Summary



4444444444 34444444444 21

4444444444 84444444444 76

id

d

i

o

dK

dPKLy

>>

>>=

σ

σσperformance

input.plant at rejection edisturbanc goodimply not doesoutput at rejection

edisturbanc good s,other word in ;)( meany mecessarilnot does )(

samll Hence,diagnoal. are and both unless general, in that Note

io

io

SS

KPSS

σσ

≠

disturbance

disturbance

gain)(controller high and gain)(loop Highdesign loopfeedback Good KPK⇒




++K P

+

+

r-

udi

up

dy

n

dominated is frequency at small )( gain loop small )(or

requires general in whichpoles, plan halfrighyt no has

)det()det()))(det(())(det(

if stable is system perturbed The ).0 whenstable is system theis,(that stable

nomial is system loop closed theand stable with)( toperturbed is model

plant that theAssuming y.uncertaint model todue high

yarbitraril made benot can gain loop ereality th inBut

∆⇒∆∆

∆++=+

∆++=∆++

=∆

∆∆+

PKTT

TIPKIPKI

PKIPKIPKII

PI

oo

o

To

σσ

48476

frequency highat gain loop Low

requiresit gSuppressin frequency. highat t significan is noisesensor Typically,

gain loop highat ,)(

that Recall. noisesensor consider Now

o

oioo

T

nryPKdSPdSnrTy

n

−≈++−=




++K P

+

+

r-

udi

up

dy

n

frequencyhigh at low relatively be shall ))((or ))(( gains loop Therefore,

1)]([

1)]([

1))(( if have We.invertible is Assuming

)()()(

since )saturationactuator (leading controlhigh ly unreasonab causemay

)1))(( while1))((or 1))(((

ofbandwidth theoutside gains loop large hand,other In the

1

1

jωLjωL

jPjP

jPP

ddnrPdTdnrKSdTdnrKSu

u

jP jωLjωLP

io

iiiiiio

io

σσ

ωσωσ

ωσ

ωσσσ

>>=

<<

−−−≈−−−=−−−=

<<>>>>

−

−

small is gain loop the whenlarge not too )( keep todesirable isit Therefore,

)()(

sincelow is gain loop wherefrequency at the small be shall gain

controller the,saturationactuator avoid To ).( gain controller heConsider t

K

dnrKdTdnrKSu

K

iio

σ

σ

−−≈−−−=




++K P

+

+

r-

udi

up

dy

n

1)( ),( ),( >>KKPPK σσσ

MK

KPPK

≤

<<

)(

1)( ),(

σ

σσ




ionsspecificat definefor used are )( functions ghtingdesign wei control futureFor

,1

,1)(

where

1)()(

:ionspecificat thedescribe

ofunction t weightinga usecan oneOr .))()(1()( where

,)(

,)(

becan ion specificat thesystemscalar afor example,For functions.

transferloop closed some of in termsor functionsy sensitivitry complement

and/or functionsy sensitivit on the specified becan

systemfeedback a of objectives eperformanc that theRecall

0

0

1

0

0

e

e

oe

o

o

o

W

/M

/εjW

jSjW

jKjPjS

MjS

jS

>∀

≤∀=

≤

+=

>∀≤

≤∀≤

−

ωω

ωωω

ωω

ωωω

ωωω

ωωεω

Weighted Weighted HH22 and and HH∞∞ PerformancePerformance

++K P

+

+

r-

udi

up

dy

n




++ K P +

+

-

n~

nW

eWey

n

dW

d~

d

uW

id~

id

pu

u

u~

iW

rWr

Standard feedback configuration with weights

Weighting matrices are used for following purposes:

1. Sensor/actuator unit conversion, signal scaling over channels and frequencies

2. Wi and Wd may be chosen to reflect frequency contents of di and d

3. Wn may be used to model frequency content of sensor noise n

4. We can be use to scale the output requirements (shape of output sensitivity)

5. Wu may be used for providing control and actuator requirements

6. Wr is an option to be used with non-unity feedback systems




K P-

n~

nW

eWey

n

dW

d~

d

uW

id~

id

pu

u

u~

iW

rWr

It is very essential that some appropriate weighting

matrices be used in order to utilize the optimalcontrol theory to be discussed in this class (i.e., H2 and H∞ theory).

As part of the controller design process, a very important step is to select weighting

matrices We, Wd, and Wu, and possibly Wn, Wi, and Wr. Note that selection of these

weighting matrices is not trivial and during the design process, these weighting matrices are

often used as controller design “tuning parameters”.

Control design is a process of selecting a controller K(s) such that some certain

weighted signals are made small.




K P-

n~

nW

eWey

n

dW

d~

d

uW

id~

id

pu

u

u~

iW

rWr

HHHH2222 Performance

2

2

2

2

22

2

2

2

2

20

22

2

}~{

functioncost

following theusemay design control realisticA ).actuator(s saturatemay that

gain controlhigh very toleadsfunction cost above ofon minimizati controllerA

norm an }{}{

~ edisturbanc the todue

error theofenergy expected theminimize weIf n.expectatio thedenotes where

)(

and

)()(~

i.e., direction, random with impulsean as modeled

elyapproximat becan ~

edisturbanc that theAssume

=+

⇐==

=

=

∫∞

∗

dou

doe

doe

WKSW

WSWueE

H

HWSWdteEeE

de

E

IE

ttd

d

ρρ

ηη

ηδ




K P-

n~

nW

eWey

n

dW

d~

d

uW

id~

id

pu

u

u~

iW

rWr

HHHH2222 Performance – a constrained optimization problem

.matrix weightingspecial a with controller

an is problem thisosolution t The t.requiremen eperformancoutput given a to

subjectenergy expected control theminimize todinterprete becan problem This

}){(

constraintenergy expectedoutput an subject to

norm an }{}{

matrix eightingconstant w a with control

weighteda ofenergy expected theminimize weIf n.expectatio thedenotes where

)~~

(

by definedmatrix covariance its with process random

mean zero a is ~

edisturbanc that theassume Now

2

22

2

2

2

20

22

2

2

2

e

ydo

dou

u

d

W

H

WSyyE

HWKSWdtuEuE

Wu

E

WddE

d

σσ ≤=

⇐==

=

∗

∞

∗

∫




K P-

n~

nW

eWey

n

dW

d~

d

uW

id~

id

pu

u

u~

iW

rWr

HHHH∞∞∞∞ Performance

yuncertaint modeling of toleranceii)

error sensor and edisturbanc of tradeoffi)

:control on sLimitation 2H

2

2

2

22

21

~

21

~

21

~

}~{sup

criterion mixeda or with

~sup

constraintenergy control subject to

sup

:functioncost

following theg Minimizinproblem. control following heconsider tNow

2

2

2

2

2

2

∞≤

∞≤

∞≤

∞∞

=+

≤=

=

dou

doe

d

dou

d

doe

d

WKSW

WSWue

WKSWu

WSWe

HH

ρρ

γ




K P-

n~

nW

eWey

n

dW

d~

d

uW

id~

id

pu

u

u~

iW

rWr

HHHH∞∞∞∞ Performance

problem.y sensitivit-mixed thecalledusually is problem design

This matrices. scalingy uncertaintdependent -frequency theare and where

}{sup

functioncost a weighted minimize can Or we

21

2

21

2

2

2

21

2

2

∞

∞≤

=+

H

WW

WKSW

WSWue

o

doe

d ρρ

Note: for a scalar system, the H∞ norm minimization problem can be viewed as

minimizing the maximum magnitude of the system’s steady-state response

under the worst case sinusoidal inputs.

This leads to the next optimization problem …..




K P-

n~

nW

eWey

n

dW

d~

d

uW

id~

id

pu

u

u~

iW

rWr

LLLL2222 to LLLL∞∞∞∞ Performance

et.given targ a than less is

magnitudeoutput case worst thesuch that size)actuator (minimum magnitude

controlmaximun theminimize 1, than less norm with edisturbanc allfor

:follows as dinterprete becan problem This

)(sup

constraintoutput an subject to

over norm an )(sup

norm weightedfollowing theMinimizing

2

22

1

2

1

2

2

2

2

L

ty

LtuW

L

y

d

u

d

σ≤

ℜ⇐

∞≤

∞∞≤

∞

Fortunately, this control problem is equivalent to the constrained H2 optimal

control design problem shown before.



Selection of Weighting FunctionsSelection of Weighting Functions

An Ad Hoc Process

10 , :shootover

4

: timesettling

8.03.0 , 16.26.0

: timerise

input step afor that know we theory,control classic From

)2(

systemorder second standard aconsider

,function weightinga choose To t.significan are and wheresfrequencie low

typicallys,frequencie of rang aover small keep want to webefore, discussed As

)1( ,)(

iserror tracking thebefore, discussed

as left.on shown systemfeedback SISOan Consider

21

2

1

<<=

≈

≤≤+

≈

+==

−=−+−=−=

−−

−

ξ

ξω

ξω

ξ

ξωω

ξ

πξ

eM

t

t

ssPK L

Wdr

S

PKSSPdTndrSyr e

p

n

s

n

r

n

n

e

i

++K P

+

+

r-

udi

up

dy

n




22

n

2

)2(

1

1

consider function,y sensitivit the torelated is eperformanc theSince

along. of functiona isshoot over b)

and response system a) :pointskey Two

nn

n

ss

ss

L S

ωξωξω

ξ

ω

++

+=

+=

∝++

K P+

+

r-

udi

up

dy

n

n

s

s

nb

M

jSSM

ω

αωω

ξα

αξα

ξαα

ω

ω

=

++=

+−

+=

==

=

∞

max

2

2222

22

max

815.05.0

4)1(

4

)(:let and

2/

bandwidth loop Closed

05.0=ξ

10.0=ξ

20.0=ξ

80.0=ξ

00.1=ξ

50.0=ξ




⇒

−=⇒= −

−

maximal findcan one , Using

)ln(

)ln(

and between

iprelationsh theusing minimal desired calculatecan We

),( tsrequiremendomain given timefor Now

max_min

1

min

,

2

s

P

Pp

P

psr

M

M

MeM

M

Mtt

ξ

πξ

ξ

ξ

ξ

πξ

++K P

+

+

r-

udi

up

dy

n

s

MsW

ωjsMs

sS

tt

bse

bs

r

ω

ωω

ω

ξ

+=

⇒

∀=+

≤

/

lyequivalentOr

responese. ry timesatisfacto provide will

,/

(s)

function y sensitivit

a Therefore, . requiredbandwidth minimal the

calculatecan one , and desired , using Then,

b

smin




k

kb

bM

s

e

e

b

bMs

e

e

e

e

e

sW

W

sW

W

WeSW

SW

SWS

ks

s

+

+=

⇒+

+=

≤≤

∞<=

=

∞

∞

εω

ω

ε

εω

ω

ω

ε

follows as modified becan

required isansition steeper tr a If t.requiremen

eperformanc esatisfy th to suitable a choose

alwayscan wepurpose, practicalfor Therefore,

becan of choice possible One

1 s.t. )0(

such that matrix weightinga choose togoing are weperfectly,

input step the trackingof Instead hniques.design tec optimalour by handled benot

canit and (unstable) axisimaginary on pole a has function y transfersensitivit

weighted that theNote . and 0)0(that

implieshich infinity w is 0at weighting theSince

1

++K P

+

+

r-

udi

up

dy

n




⇒+

+=

+=

⇒

−−−=

bcu

ubcu

u

bc

ubcu

u

bc

u

i

e

u

s

MsW

W

MsW

W

M

KS

TddnrKSu

W

W

ωεω

ε

ωω

ω

/

ghtingproper wei afor introduced is

an system stable CL of limitation toDue

/

be would candidate aHence,

sfrequencie

noisesensor and )(bandwidth controllerby limited isgain frequency High ii)

frequency lowin gain largefairly actuator theoflimit saturation and

andeffort control ofcost by the limited is sfrequencie lowin of Magnitude i)

:signal controlin slimitation following heConsider t

)(

equation signal control thegconsiderinby ofthat

similar to is weightingcontrol ofselection The

u

++K P

+

+

r-

udi

up

dy

n

uε



BodyBody’’s Gain and Phase Relations Gain and Phase Relation

++K P

+

+

r-

udi

up

dy

n

small toobenot shall 2

)(sin4)(1)(1

differencereturn hand,other On the

Margin Phase )( )(ln

slope crossBody

Therefore,

time)ofmost frequency crossing @ (negativeplot

Body of slop theis )(ln

and from deviates

en quickly wh decreases 2

cothln that Note

ln2

cothln and )ln(: where

2cothln

ln1)(

henfunction t transfer loop phase minimal stable a be )(Let

2

11

0

2/2/

2/2/

0

0

ccc

c

c

jLjLjL

jLd

jLd

d

jLd

ee

ee

dd

LdjL

sL

ωπωω

ωπυ

ω

υ

ωω

ωυ

υωωυ

υυ

υπω

υυ

υυ

∠+=+=+

∠+∝

−

+==

=∠

−−

−

−

∞

∞−∫

υ-3 -2 -1 0 1 2 3

0

1

2

3

4

5

6

7

8

2cothln

υ




++K P

+

+

r-

udi

up

dy

n

20db).-(or 1- angreater th be toslopover cross theto

wanteNormally w small). (or too deep toobenot should

)( of slopover cross thesummary,a as Therefore, ωjL

υ

(-2) 40db- slop

12 margin phase with1060

10)( o

42

5

≈

≈++

=ss

sL

-60

-40

-20

0

20

40

Mag

nitud

e (

dB

)

100

101

102

103

104

-180

-135

-90

-45

0

Pha

se

(d

eg

)

Bode Diagram

Frequency (rad/sec)

-30

-20

-10

0

10

20

30

Mag

nitud

e (

dB

)

101

102

103

104

-135

-90

-45

0

Pha

se

(d

eg

)

Bode Diagram

Frequency (rad/sec )

(-1) db 20- slop

78 margin phase with1060

10500)( o

42

5

≈

≈++

+=

ss

ssL




++K P

+

+

r-

udi

up

dy

n

Effect of non-minimal

phase transfer function

( )

lag pahse additional scontribute function trasfer loop of zeros phase minimal-Non

28,53,90:/

Define

2cothln

)(ln1

2cothln

)(ln1

)()(

Therefore, .)()( and phase minimal and stable is where

)()(

as factorized be can )(

then),( of zeros plan halfright be ,Let

4/,2/,0

001

1 0

00

1 0

00

1 0

000

2

2

1

1

21

0

⇒

−−−=+

+−∠=

+

+−∠+=

+

+−∠+=

+

+−∠+∠=∠

=

+

+−

+

+−

+

+−=

=

∞

∞−=

∞

∞−=

=

∫ ∑

∫ ∑

∏

ooo

L

L

zzz

k

i i

i

k

i i

imp

k

i i

imp

mpmp

mp

k

k

k

zj

zjz

zj

zjd

d

jLd

zj

zjd

d

jLd

zj

zjjLjL

sLsLL

sLzs

zs

zs

zs

zs

zssL

sL

sLz,zz

ωωω

ωφ

ωω

υυ

υ

ω

π

ωω

υυ

υ

ω

π

ωω

ωω



BodyBody’’s Sensitivity Integrals Sensitivity Integral

++K P

+

+

r-

udi

up

dy

n

effect. bedWater frequeniesother at onebelow kept isit if one

exceeds functiony sensitivit theof magnitude the

over which rangefrequency a is thereTherefore,

0)(ln

tosimplifies integral thestable, is )( If

)Re()(ln

holds integraly sensitivit s Bode'following the then),( of poles plane half-right

open be ,let and zeros thanpoles more two

leastat withfunction transfer loop-open thebe )(Let

0

10

21

⇒

=

=

∫

∑∫

∞

=

∞

ωω

ωω

djS

sL

pdjS

sL

p,pp

sL

m

i

i

mL

)( ωjS

+

−

1

ω

constants. given some are 0 and 0, ,0 where

),[ ,1~)(

example, For s.frequencie specific

a above small be function transfer loop open requires often sconstraint Bandwidth

h

1

>>>

∞∈∀<≤≤+

βω

ωωεω

ω β

h

hh

M

MjL



BodyBody’’s Sensitivity Integrals Sensitivity Integral

++K P

+

+

r-

udi

up

dy

n

[ ]( )

[ ]( )

.)(over t significan be couldy Sensitivit

~11

)(max

have We 0. stable, is )( that whenNote

.)Re(

where

~11

)(max

Then,

left. theonplot see low, tohigh from variesgain

loop whensfrequencie tworepresent and Let

)(

,

1

)(

,

hl

lh

lh

m

i i

lh

hlh

l

hl

lh

hlh

l

hl

jS

sL

pπ

ejS

ωω

εε

ω

α

ωωα

εε

ω

ωω

ωωβωωω

ω

ωωω

ωωβωωω

ω

α

ωωω

−⇒

−

≥

=

−=

−

≥

−−

∈

=

−−

∈

∑

-60

-40

-20

0

20

40

Ma

gn

itud

e (

dB

)

100

101

102

103

104

-180

-135

-90

-45

0

Ph

as

e (

de

g)

Bode Diagram

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feedback properties · 2008. 4. 17. · good feedback loop design ⇒ high loop (pk ) gain and high...

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