Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order...

77
1.1 Mario Garcia-Sanz Mario Garcia-Sanz Automatic Control & Computer Science Department Public University of Navarra 31006 Pamplona, Spain (Email: [email protected]) NATO. RTO-LS-SCI-195, May-June 2008 1 st Session Beyond the Classical Performance Limitations Controlling Uncertain MIMO Systems: UAV Applications

Transcript of Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order...

Page 1: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.1

Mar

io G

arci

a-Sa

nz

Mar

io G

arci

a-Sa

nzA

utom

atic

Con

trol &

Com

pute

r Sci

ence

Dep

artm

ent

Publ

ic U

nive

rsity

of N

avar

ra31

006

Pam

plon

a, S

pain

(Em

ail:

mgs

anz@

unav

arra

.es)

NA

TO

. R

TO

-LS-

SCI-

195,

May

-Jun

e 20

08

1stSe

ssio

n

Bey

ond

the

Cla

ssic

al P

erfo

rman

ce

Lim

itatio

ns C

ontr

ollin

g U

ncer

tain

MIM

O

Syst

ems:

UA

V A

pplic

atio

ns

Page 2: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

Report Documentation Page Form ApprovedOMB No. 0704-0188

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4. TITLE AND SUBTITLE Beyond the Classical Performance Limitations Controlling UncertainMIMO Systems: UAV Applications

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13. SUPPLEMENTARY NOTES See also ADM002223. Presented at the NATO/RTO Systems Concepts and Integration Panel LectureSeries SCI-195 on Advanced Autonomous Formation Control and Trajectory Management Techniques forMultiple Micro UAV Applications held in Glasgow, United Kingdom on 19-21 May 2008.

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15. SUBJECT TERMS

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT Same as

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Page 3: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.2

Mar

io G

arci

a-Sa

nz

Out

line

1.-

QFT

Con

trolle

r Des

ign

Tech

niqu

e Fu

ndam

enta

ls

2.-

Rea

l-wor

ld Q

FT c

ontro

l app

licat

ions

and

exa

mpl

es

3.-

Non

-dia

gona

l MIM

O Q

FT c

ontro

ller d

esig

n m

etho

dolo

gies

4.-A

pplic

atio

n: R

obus

t QFT

con

trol f

or a

MIM

O S

pace

craf

t with

flex

ible

suns

hiel

d

5.-S

witc

hing

robu

st c

ontro

l: B

eyon

d th

e lin

ear l

imita

tions

.

6.-E

xam

ple:

Sw

itchi

ng c

ontro

l for

Unm

anne

d V

ehic

les

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1.3

Mar

io G

arci

a-Sa

nz

1.-Q

FT C

ontr

olle

r D

esig

n T

echn

ique

Fun

dam

enta

ls

1.1.

-Int

rodu

ctio

n

Ach

ieve

s rea

sona

bly

low

loop

gai

ns, i

.e.,

avoi

ds o

r min

imiz

es: S

enso

r noi

se

ampl

ifica

tion,

Sat

urat

ion,

Hig

h Fr

eque

ncy

unce

rtain

ties.

Look

s for

a d

esig

n th

at c

ombi

nes:

Mod

el +

Par

amet

er U

ncer

tain

ty. (

Rob

ustn

ess)

.Pe

rfor

man

ce S

peci

ficat

ions

.M

inim

um O

rder

Con

trolle

r. Tr

ansp

aren

cy o

f the

Tec

hniq

ue.

Freq

uenc

y D

omai

n Te

chni

que.

Use

s the

Nic

hols

Cha

rt (N

C).

The

QFT

desi

gn o

bjec

tive

is to

des

ign

and

impl

emen

t a ro

bust

con

trol f

or a

sy

stem

with

Unc

erta

inty

that

satis

fies t

he d

esire

d Pe

rfor

man

ce S

peci

ficat

ions

.

Qua

ntita

tive

Feed

back

The

ory

(Q.F

.T.).

Intro

duce

d by

Pro

f. Is

aac

Hor

owitz

.19

59. F

irst i

deas

.19

72. T

he n

ame.

1973

. Hor

owitz

at t

he A

ir Fo

rce

Off

ice

of S

cien

tific

Res

earc

h (f

irst g

rant

).19

92. P

rof.

Hou

piso

rgan

izes

the

Firs

t Int

erna

tiona

l Sym

posi

um o

n Q

FT.

Unt

il th

en, I

nt. S

ympo

sia

ever

y tw

o ye

ars:

199

5, 1

997,

199

9, 2

001,

200

3, 2

005,

200

7.

a re

liabl

e co

ntro

l des

ign

met

hodo

logy

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1.4

Mar

io G

arci

a-Sa

nz

Perf

orm

ance

Spec

ifica

tions

(P.S

.)Pl

antM

odel

+ U

ncer

tain

ty QFT

Con

trol

ler

Des

ign

Per

form

ance

Rob

ustn

ess

F ,

GF

, G

Min

imum

Ord

erCo

ntro

ller

Tran

spar

ency

ofth

ete

chni

que

Min

imum

Cost

ofFe

edba

ck

Bou

nds

Loop

Shap

ing

L 0

PD

1D

2

GF

RY

_+

++

++

QFT

. A su

cces

sful

robu

st c

ontro

l th

eory

for r

eal-w

orld

ap

plic

atio

ns:

Stab

le a

nd U

nsta

ble

Syst

ems,

SISO

and

MIM

O P

roce

sses

, A

nalo

g an

d D

iscr

ete

Syst

ems,

Line

ar a

nd N

on-li

near

Pla

nts,

Proc

esse

s with

con

stan

t and

va

riabl

e pa

ram

eter

s, M

inim

um a

nd N

on-m

inim

um

Phas

e Sy

stem

s, C

asca

de C

ontro

l Sys

tem

s, et

c.

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1.5

Mar

io G

arci

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Step

1: C

ontro

l Spe

cific

atio

ns: S

tabi

lity

Step

2: C

ontro

l Spe

cific

atio

ns: P

erfo

rman

ceSt

ep 3

: Spe

cify

Pla

nt m

odel

s + U

ncer

tain

tySt

ep 4

: Obt

ain

tem

plat

es a

t spe

cifie

d ω i

(des

crib

es u

ncer

tain

ty)

Step

5: S

elec

t nom

inal

pla

nt P

o(s)

Step

6: D

eter

min

e st

abili

ty c

onto

ur (U

-con

tour

) on

N.C

.St

eps 7

-9: D

eter

min

e tra

ckin

g, d

istur

banc

e, &

opt

imal

bou

nds

Step

10:

Syn

thes

ize

nom

inal

Lo(

s) =

G(s

)Po(s

)--

Satis

fies a

ll bo

unds

& st

abili

ty c

onto

ur--

Obt

ain

G(s

) = L

o(s)/P

o(s)

Step

11:

Syn

thes

ize

pref

ilter

F(s)

.St

ep 1

2: S

imul

ate

linea

r sys

tem

(J ti

me

resp

onse

s)St

ep 1

3: S

imul

ate

with

nonl

inea

ritie

s

QFT

Des

ign

Proc

edur

eQ

FT D

esig

n Pr

oced

ure

Con

trol

Sp

ecifi

catio

ns

Mod

el +

un

cert

aint

y

Bou

nds

Loo

psha

ping

Pref

ilter

1.2.

-MIS

O a

nalo

g co

ntro

l sys

tem

des

ign

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1.6

Mar

io G

arci

a-Sa

nz

Step

1, 2

: Con

trol

Spe

cific

atio

ns:

Step

1, 2

: Con

trol

Spe

cific

atio

ns:

Stab

ility

and

Per

form

ance

Stab

ility

and

Per

form

ance

RR

efer

ence

WC

ontro

llerI

nput

Dis

turb

ance

VPl

antI

nput

Dis

turb

ance

DPl

antO

utpu

t Dis

turb

ance

NN

oise

FU

EY

N

DV

WR

Pref

ilter

Sens

or

Plan

tC

ontro

ller

-G

P

H

)VD

N(1

)RW(

1

N1

R1

1W

1V

1D

1

N1

)RW(

1V

1D

11

PH

GPHG

FH

GPG

U

HG

PHF

HG

PH

GP

HG

PH

GPHP

HG

PHE

HG

PH

GP

FH

GPGP

HG

PPH

GP

Y

++

+−

++

=

+−

++

++

++

+−

=

+−

++

++

++

=

Spec

ifica

tions

in te

rms o

f T.F

.Fo

r exa

mpl

e:

)(

11

sW

HG

P≤

+

(1)

(2)

(3)

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1.7

Mar

io G

arci

a-Sa

nz

Gai

n M

argi

n:G

M ≥

1 +

1/μ

(mag

nitu

de)

Phas

e M

argi

n:PM

≥18

0º-θ

(deg

)

whe

re:

μis

the

circ

le M

spec

ifica

tion

in m

agni

tude

: M

m=

20 lo

g 10(

μ)θ

= 2

cos-1

(0.5

/μ) ∈

[0, 1

80º]

YP(

s)G

(s)

R

Plan

tC

ontro

ller

-

1po

int

at18

018

0po

int

at1

1

1

=+

=−

==

==

=≤

+=

=

)j

(Lº

PM

ºM/

GM

eM

)j

(G

)j

(P

)j

(L

W)

j(

T

)j

(G

)j

(P

)j

(G

)j

(P

)j

(T

)j

(R

)j

(Y

j

s

ωψ

ψωω

ω

μω

ωω

ωω

ωωω

ψ

The

Stab

ility

(Gai

n an

d Ph

ase

Mar

gins

) is

rel

ated

with

the

Max

imum

clo

sed-

loop

R

eson

ance

Mm

spec

ifica

tion.

PGH

PGH

Ws

11

+≤

Ws1

GM

PMO

vers

hoot

1.1

(0.8

dB

)1.

99 (5

.9 d

B)

55º

∼11

%1.

2 (1

.58

dB)

1.83

(5.2

dB

)49

º∼

18 %

1.3

(2.2

8 dB

)1.

77 (5

.0 d

B)

45º

∼22

%

1.4

(2.9

dB

)1.

71 (4

.7 d

B)

41.8

º∼

27 %

Stab

ility

-350

-300

-250

-200

-150

-100

-50

0-4

0

-30

-20

-10010203040

6 db3 db1

db0.5

db0.25

db

0 db

-1

db -3 d

b

-6 d

b

-12

db

-20

db

-40

db

Phas

e. O

pen

loop

(deg

)

Magnitude. Openloop(dB)-1

80

-5 º

-10

º

-2 º

-330

º-3

0 º

-300

º-6

0 º

-90

º-2

70 º

Circ

leM

m=

2 db

2 db

L =

G P

-350

-300

-250

-200

-150

-100

-50

0-4

0

-30

-20

-10010203040

6 db3 db1

db0.5

db0.25

db

0 db

-1

db -3 d

b

-6 d

b

-12

db

-20

db

-40

db

Phas

e. O

pen

loop

(deg

)

Magnitude. Openloop(dB)-1

80

-5 º

-10

º

-2 º

-330

º-3

0 º

-300

º-6

0 º

-90

º-2

70 º

Circ

leM

m=

2 db

2 db

L =

G P

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1.8

Mar

io G

arci

a-Sa

nz

Tim

e-D

omai

n Sp

ecifi

catio

ns:

Des

ire s

yste

m o

utpu

t y(t)

to li

e be

twee

n sp

ecifi

ed u

pper

and

low

er b

ound

s, y(

t) Uan

d y(

t) L, r

espe

ctiv

ely.

Figu

res o

f mer

it(F

OM

), ba

sed

upon

a st

ep in

put s

igna

l r(t)

= R

0u -

1(t),

Mp

peak

ove

rsho

ot; t

rris

e tim

e; t p

peak

tim

e; a

nd t s

settl

ing

time.

Dep

ends

on

requ

irem

ents

that

the

desi

gner

wan

ts fo

r the

spec

ific

Pla

nt:

Airp

lane

, Hea

ting

syst

em, M

achi

nery

, Win

d Tu

rbin

e, e

tc...

Des

ired

syst

em p

erfo

rman

ce

spec

ifica

tions

: tim

e do

mai

n re

spon

se sp

ecifi

catio

ns;

t r L

t r U

MP

t Pt

y(t) L

y(t) U

r(t)

1.0

0.9

0.1

Acc

epta

ble

Perf

orm

ance

Are

a

t s

Settl

ing

Tim

eTo

lera

nce

Trac

king

Perf

orm

ance

WPG PG

HW

sa

sb

77

1≤

+≤

Page 10: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.9

Mar

io G

arci

a-Sa

nz

Tran

slate

d in

toth

e fr

eque

ncy

dom

ain

are,

BU

and

B L, t

he u

pper

and

low

er

boun

ds re

spec

tivel

y: P

eak

over

shoo

t Lm

Mm

& fr

eque

ncy

band

wid

th ω

h.

(Not

e: in

crea

sing

δ δδδR(j

ω ωωωi)

abov

e 0

dB c

ross

ing)

Des

ired

syst

em p

erfo

rman

ce

spec

ifica

tions

: fr

eque

ncy

dom

ain

resp

onse

sp

ecifi

catio

ns.

+ 0 - -12

dB

LmT R

ωω

i

LmM

m

BU

= Lm

T RU

= Lm

BR

U

δ hf

BL

= Lm

T R L =

LmB

R L

δ δδδ R(jω ωωω

)

Ban

dwid

th

Trac

king

WPG PG

HW

sa

sb

77

1≤

+≤

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1.10

Mar

io G

arci

a-Sa

nz

22

2

72

)()

/(

nn

nU

Rb

ss

s

as

aT

ωξω

++

+=

=

)s()

s()s(

TW

LR

as

32

17

σσ

++

+=

=

Upp

erfu

nctio

n

Low

erfu

nctio

n

01

23

45

67

89

1000.2

0.4

0.6

0.811.2

1.4

Tim

e (s

)

t r(L)

t r(U

)

Mp

y(t) U

y(t) L

ω(r

ad/s

)

BL(

jω)

BU(jω

)

10-1

100

101

-40

-35

-30

-25

-20

-15

-10-505

δ R(jω

)

Mm

2 db

ω(ra

d/s)

10-1

100

101

05101520dB

δ R(jω

)

δ δδδ R(j

ω ωωω) =

BU

(jω ωωω

) -B

L(jω ωωω

)cl

osed

loop

Trac

king

WPG PG

HW

sa

sb

77

1≤

+≤

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1.11

Mar

io G

arci

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nz

Freq

uenc

y-D

omai

n Sp

ecifi

catio

ns:

Des

ire T

F sy

stem

Y(ω

)/U(ω

) to

lie

un

der a

spec

ific

boun

d, B

(ω)

Figu

res o

f mer

it(F

OM

), R

eson

ance

; Ban

dwid

th; R

oll-o

ff; L

ow F

requ

ency

, etc

.

Dep

ends

on

requ

irem

ents

that

the

desi

gner

wan

ts fo

r the

spec

ific

Pla

nt:

Stru

ctur

e re

sona

nce,

Noi

se m

easu

rem

ent,

Dis

turb

ance

s, St

eady

Sta

te E

rror

s, et

c...

+ 0 - -12

dB

Lm(Y

(ω)/U

(ω))

ωω

r

Res

onan

ceB

(ω)

Ban

dwid

th

Rol

l-Off

Dist

urba

nce

reje

ctio

n1

12

+≤

PG

HW

s

Page 13: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.12

Mar

io G

arci

a-Sa

nz

22

2

22

21

1

nn

nd

ss

s

ss

HG

PT

ωξωξ

++

+≤

+=

=

-30

-20

-10010

Magnitude (dB)

10-1

100

101

102

04590Phase (deg)

Bode

Dia

gram

Freq

uenc

y (r

ad/s

ec)

Dis

turb

ance

reje

ctio

n

Dist

urba

nce

reje

ctio

n1

12

+≤

PG

HW

s

Page 14: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.13

Mar

io G

arci

a-Sa

nz

Tran

sfer

func

tions

and

spec

ifica

tions

Eq

.No.

{} 1

11

1),

()

()

(1

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

ωω

ωδ

ωω

ωω

ωωωω

ωω

ωω

∈≤

⋅+

⋅=

==

⋅=

jj

Gj

Pj

Gj

Pj

Nj

Yj

Dj

Uj

Fj

Rj

Yj

T(T

1)

{}

22

22

),(

)(

)(

11

)(

)(

)(

ωω

ωδ

ωω

ωωω

∈≤

⋅+

==

jG

jP

jD

jY

jT

(T2)

{}

33

13

),(

)(

)(

1)

()

()

()

ωω

δω

ωω

ωωω

∈≤

⋅+

==

jG

jP

jP

jD

jY

jT

(T3)

{}

44

24

),(

)(

)(

1)

()

()

()

()

()

()

()

()

ωω

δω

ωω

ωω

ωωω

ωωω

∈≤

⋅+

=⋅

==

=j

Gj

Pj

Gj

Fj

Rj

Uj

Nj

Uj

Dj

Uj

T(T

4)

{}

5su

p5

5in

f5

),(

)(

)(

1)

()

()

()

()

()

()

ωω

δω

ωω

ωω

ωωω

ωδ

∈≤

⋅+

⋅=

=≤

jG

jP

jG

jP

jF

jR

jY

jT

(T5)

Clo

sed

loop

spec

ifica

tions

are

usu

ally

des

crib

ed in

term

s of f

requ

ency

func

tions

δk(

ω)

that

are

impo

sed

on th

e m

agni

tude

of t

he sy

stem

tran

sfer

func

tions

|Tk|

, k =

1, .

.. 5

(1) r

obus

t sta

bilit

y, c

ontro

l eff

ort l

imit

in th

e in

put d

istu

rban

ce re

ject

ion,

sen

sor n

oise

atte

nuat

ion

(2) o

utpu

t sys

tem

dis

turb

ance

reje

ctio

n

(3) i

nput

syst

em d

istu

rban

ce re

ject

ion

(4) c

ontro

l eff

ort l

imit

in th

e ou

tput

dis

turb

ance

reje

ctio

n, n

oise

atte

nuat

ion,

and

trac

king

(5) s

igna

l tra

ckin

g

Page 15: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.14

Mar

io G

arci

a-Sa

nz

Why

Unc

erta

inty

?W

hy U

ncer

tain

ty?

Step

3: P

lant

Mod

el +

Unc

erta

inty

Step

3: P

lant

Mod

el +

Unc

erta

inty

Tes

t-be

d fo

r L

arge

Mul

tipol

eG

ener

ator

s

M.T

orre

s

Up

to fo

ur

500

kW

Mot

ors

driv

es a

30

00 k

W

Gen

erat

or

Page 16: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.15

Mar

io G

arci

a-Sa

nz

Shad

ed r

egio

nre

pres

ents

th

e re

gion

of p

lant

un

certa

inty

.

Mot

or r

epre

sent

edby

6

LTI t

rans

fer f

unct

ions

(ι=

1,2,

…,J)

.

23

4

16

5

a

K 0

Reg

ion

of p

lant

para

met

er u

ncer

tain

tyK

max

Reg

ion

of p

lant

par

amet

er u

ncer

tain

ty.

Km

in

a max

a min

)(

)(

)(

)(

as

sK

as

Vs

sP

fm+

A Si

mpl

e M

athe

mat

ical

Des

crip

tion

Mot

ortra

nsfe

r fun

ctio

n is

:

Para

met

ers K

and

ava

ry:

K∈

(Km

in, K

max

) an

d a

∈(a

min

, am

ax)

Vf

Ra

Rf

i fω

Lf

40 20 0

-20

-40

-60

-80

-100

Magnitude (dB)

110

100

Plan

t 1Pl

ant 2

Plan

t 3Pl

ant 4

Plan

t 5Pl

ant 6

Freq

uenc

y (ω

)

Freq

uenc

y (ω

)1

1010

0-1

80

-170

-160

-150

-140

-130

-120

-110

-100-9

0

Plan

t 1,2

Plan

t 3,6

Plan

t 4,5

Phase (degrees)

Page 17: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.16

Mar

io G

arci

a-Sa

nz

Step

4: T

empl

ates

Step

4: T

empl

ates

Plan

t Tem

plat

eob

tain

ed b

y m

appi

ngPo

ints

of u

ncer

tain

ty re

gion

into

poi

nts o

n to

the

N.C

.C

urve

dra

wn

thro

ugh

poin

ts –

Shad

ed a

rea

labe

led

ℑP(

j1)

ℑP (

j1)

CB A

D

K=1

0

a=10

a=1

K=1

20 17 dB 0 -3-1

80°

-90°

-135

°-0

.04

dB

Phas

e

dB

N.C

. cha

ract

eriz

ing

P(s)

ove

r the

regi

on

of u

ncer

tain

ty.

a)s(

sKa

(s)

P+

ω =

1

rad/

s

23

4

16

5

a

K 0

Reg

ion

of

plan

t par

amet

erun

certa

inty

Km

ax=1

0

Km

in=1

a max

=10

a min

=1

Page 18: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.17

Mar

io G

arci

a-Sa

nz

Tem

plat

es fo

r oth

er v

alue

s of ω

iar

e ob

tain

ed

Cha

ract

eris

tic o

f tem

plat

es:

Star

ting

from

low

val

ues o

f ωi,

(nar

row

wid

th),t

he a

ngul

ar w

idth

be

com

es la

rger

(med

ium

freq

.)

For i

ncre

asin

g va

lues

of ω

i

tem

plat

es b

ecom

e na

rrow

er a

gain

.

Even

tual

ly a

ppro

ach

stra

ight

lin

e: h

eigh

t V d

BPh

ase

Magnitude

ω 1 ω 2 ω 3

ω 4

ω 5

ω 6

ω 6> ω

5 >

…>

ω 1

dBV

Klo

g20

Klo

g20

]P

log

20P

log

[20

limm

inm

axm

inm

ax10

1010

10=

−=

−=

Δ∞

→ω

Page 19: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.18

Mar

io G

arci

a-Sa

nz

-140

-120

-100

-80

-60

-40

-30

-28

-26

-24

-22

-20

-18

-16

-14

5

Pha

se(d

egre

es)

Magnitude(dB)

-140

-120

-100

-80

-60

-40

-30

-28

-26

-24

-22

-20

-18

-16

-14

5

Pha

se(d

egre

es)

Magnitude(dB)

128

punt

os

-140

-120

-100

-80

-60

-40

-30

-28

-26

-24

-22

-20

-18

-16

-14

5

Pha

se(d

egre

es)

Magnitude(dB)

Who

leTe

mpl

ate

Edge

sTem

plat

e

Con

tour

Tem

plat

e

[]

[]

[]

[]

∈τ∈

∈∈

τ−+

=

=2.0,1.0

,2,1

,12,

10,

5,1

)ex

p()

( 1

1

kb

a

sb

sa

ks

P

PExam

ple:

4-di

men

sion

al p

aram

eter

spac

e

forω

= 5

rad/

sec

Ω4

Ω1

Page 20: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.19

Mar

io G

arci

a-Sa

nz

Cho

se a

ny p

lant

Kee

p th

e sa

me

plan

t(s

et o

f par

amet

ers)

as th

e no

min

al fo

r all

freq

uenc

ies

ℑP (

j1)

C

B A

D

K=1

0

a=10

a=1

K=1

20 17 dB 0 -3-1

80°

-90°

-135

°-0

.04

dB

Phas

e

dB

Step

5: N

omin

al P

lant

Step

5: N

omin

al P

lant

Page 21: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.20

Mar

io G

arci

a-Sa

nz

The

Forb

idde

n R

egio

n is

ex

tend

ed b

y V

dB

.

-350

-300

-250

-200

-150

-100

-50

0-4

0

-30

-20

-10010203040

6 db3

db1 db0.

5 db0.

25 d

b0

db

-1 d

b

-3 d

b

-6 d

b

-12

db

-20

db

-40

db

Pha

se. O

pen

loop

(deg

)

Magnitude. Openloop(dB)

-180

-5 º

-10

º

-2 º

-330

º-3

0 º

-300

º

-60

º

-90

º-2

70 º

2 db

V

L o(jω

) -no

min

alL(

jω) -

max

imum

due

to u

ncer

tain

ty

Tem

plat

eV

Ps

Ks

z

sp

Ks

iim

iin

nm

()

()

()

=+

+∏

⎯→

⎯⎯⎯

= =

→∞

−1 1

ωU

nive

rsal

Hig

hFr

eque

ncy

Bou

nd(U

HFB

)

Step

6: U

Step

6: U

-- Con

tour

(Sta

bilit

y bo

unds

)C

onto

ur (S

tabi

lity

boun

ds)

Page 22: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.21

Mar

io G

arci

a-Sa

nz

22

2 2)

()/

(

nn

nU

Rs

s

as

aT

ωωξ

ω+

++

=

)()

()(

k3

21

σσ

σ+

++

=s

ss

TL

R

Upp

erfu

nctio

n

Step

7: T

rack

ing

Bou

nds o

n L

Step

7: T

rack

ing

Bou

nds o

n L

oo

Low

erfu

nctio

n

01

23

45

67

89

1000.2

0.4

0.6

0.811.2

1.4

Tim

e (s

)

t r(L)

t r(U

)

Mp

y(t) U

y(t) L

y(t) U

. ( ω

n=

1; a

= 1

; ξ=

0.6

)

y(t) L.

( σ1

= 0.

5; σ

2=

1; σ

3=

2; k

= 1

)

dB

ω(r

ad/s

)

BL(

jω)

BU(jω

)

10-1

100

101

-40

-35

-30

-25

-20

-15

-10-505

δ R(jω

)

Mm

2 db

ω(ra

d/s)

10-1

100

101

05101520dB

δ R(jω

)

δ R(jω

) = B

U(jω

) -B

L(jω

)cl

osed

loop

YF

PG

UE

D

R

Pre-

filte

rPl

ant

Con

trolle

r-

Page 23: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.22

Mar

io G

arci

a-Sa

nz

Solu

tion

forB

R(jω

i) re

quire

s:

be sa

tisfie

d fo

r all

L ι(jω

i).

And

ar

e be

twee

n B

Uan

d B

L

dB)

(jL

B-)(j

UB

)(j

)(j

Ti

ii

Ri

ωω

δω

=≤

Δ

+ 0 -

dBLm

ω

BU

δ L(jω

1)δ L(

jω2)

δ L(jω

3)δ L(

jω4)

BL

δ R(jω

4)

δ R(jω

3)

δ R(jω

2)

δ R(jω

1)

TL ι

Clo

sed-

loop

resp

onse

s:

LTI p

lant

s with

onl

y G

(s)

Clo

sed-

loop

resp

onse

s:

LTI p

lant

s with

G(s

) and

F(s

)

+ 0 -

dBLm

ω

BU

BL

T Rι

)(j

Ti

Δ

YF

PG

UE

D

R

Pre-

filte

rPl

ant

Con

trolle

r-

Page 24: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.23

Mar

io G

arci

a-Sa

nz

Bas

ed o

nth

eno

min

al p

lant

At s

peci

fied

ωi

By

use

of te

mpl

ates

Alo

ng e

very

NC

ph

ase

grid

line

-350

-300

-250

-200

-150

-100

-50

0-4

0

-30

-20

-10010203040

6db3

db1db0.

5db0.

25db

0db

-1db

-3db

-6db

-12

db

-20d

b

-40

db

Pha

se. O

pen

loop

(deg

)

Magnitude. Open loop(dB)

-180

-5 º

-10

º

-2 º

-330

º-3

0 º

-300

º-6

0 º

-90

º-2

70 º

Rep

eat

proc

edur

eon

su

ffic

ient

ph

ase

grid

lines

to p

rovi

de

enou

gh p

oint

s to

dra

w B

R(ω

i)

BR(jω

i)fo

r suf

ficie

nt

phas

e gr

id li

nes

The

cla

ssic

gra

phic

al

proc

edur

e

Page 25: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.24

Mar

io G

arci

a-Sa

nz

Cas

e 1

Dis

turb

ance

at P

lant

Out

put [

d 2(t)

= D

ou-1

,(t),

d 1(t)

= 0

]th

e di

stur

banc

e co

ntro

l rat

io fo

r inp

ut d

2(t) i

s,

Subs

titut

ing

L =

G P

= 1

/lyi

elds

whi

ch h

as th

e m

athe

mat

ical

form

at re

quire

d to

use

the

N.C

.

L1

1(s

)DY

(s)

(s)

D2

T2

+=

=

+=

=1

(s)

DY(s

)(s

)T

2D

2

PD

1D

2

GF

RY

_++

++

+

A 2

DO

F fe

edba

ck st

ruct

ure.

Step

8: D

istu

rban

ce R

ejec

tion

Bou

nds

Step

8: D

istu

rban

ce R

ejec

tion

Bou

nds

The

cla

ssic

gra

phic

al

proc

edur

e

Page 26: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.25

Mar

io G

arci

a-Sa

nz

0° °°°-2

0° °°°-4

0° °°°-6

0° °°°-8

0° °°°-1

00° °°°

-120

° °°°-1

40° °°°

-160

° °°°-1

80° °°°

-200

° °°°-2

20° °°°

-240

° °°°-2

60° °°°

-280

° °°°-3

00° °°°

12 8 6 5 4 3 2 1dB

0.5

0.35

-24 -18

-12

-9 -6 -5 -4 -3 -2dB -1

0dB

-0.2

5

-0.5

AB

C D

β CD

ω2

ωb

ω1ωa

Lm l

280° or -80°

240° or

-120

°

(Not

e: U

se th

e ne

gativ

e an

gle

for

L si

nce

n >

w)

L =

1/l

24 d

B

18 d

B

12 d

B

For -

Lm

TD o

r BD(jω

i) ev

alua

tion

24 1620 12 8 4 0 -4 -8 -12

-16

-20

-24

-28

-24

-16

-20

-12 -8 -4 0 4 8 12 16 20 24 28

Lm L

Lm l

Attenuation, dB

Phas

e an

gle,

φ, d

eg

Rot

ated

Nic

hols

cha

rt.

∠L

∠l

Tem

plat

e of

[1/P

(jωi)]

BD

2(jω

i)

A N

.C.

is ro

tate

d 18

0o

Cha

nge

of si

gn

of th

e ve

rtica

l ax

isin

dB

, and

ho

rizon

tal a

xis

in d

eg.

Sim

ilar t

hat

used

for t

he

track

ing

boun

ds

Now

look

ing

for:

()

()

dB)(j

L/11

L/1lo

g20

L1

1lo

g20

)(j

)(j

log

20

)(j

Tlo

g20

iD

210

1010

iD

210

ωδ

ωωω

≤+

=

=+

==

=

2DY

Page 27: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.26

Mar

io G

arci

a-Sa

nz

Ineq

ualit

ies B

ound

s Exp

ress

ions

(S

teps

6 to

8)

G(s

)F(

s)P(

s)

D1(

s)D

2(s)

R(s

)

N(s

)

Y(s

)

U(s

)

E(s)

+ -

++

++

+

+

Pre-

filte

rC

ontro

ller

Plan

t with

unce

rtain

ty

H(s

)

Let’s

con

side

r the

two-

degr

ees-

of-f

reed

om fe

edba

ck sy

stem

.

In a

gen

eral

real

-wor

ld p

robl

em P

(s)w

ill p

rese

nt u

ncer

tain

ty{P

}.

The

com

pens

ator

G(s

)and

a th

e pr

e-fil

ter F

(s)w

ill b

e de

sign

ed to

mee

t rob

ust s

tabi

lity

and

robu

st p

erfo

rman

ce sp

ecifi

catio

ns,

and

to d

eal w

ithre

fere

nces

R(s

), di

stur

banc

es D

1,2(

s), s

igna

l noi

se N

(s)a

nd sa

tura

ble

cont

rol e

ffor

t U(s

),

min

imiz

ing

the

‘cos

t of t

he fe

edba

ck’(

exce

ssiv

e ba

ndw

idth

)

The

mod

ern

proc

edur

e

Page 28: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.27

Mar

io G

arci

a-Sa

nz

Tran

sfer

func

tions

and

spec

ifica

tions

Eq.N

o.

{} 1

11

1),

()

()

(1

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

ωω

ωδ

ωω

ωω

ωωωω

ωω

ωω

∈≤

⋅+

⋅=

==

⋅=

jj

Gj

Pj

Gj

Pj

Nj

Yj

Dj

Uj

Fj

Rj

Yj

T(1

)

{}

22

22

),(

)(

)(

11

)(

)(

)(

ωω

ωδ

ωω

ωωω

∈≤

⋅+

==

jG

jP

jD

jY

jT

(2)

{}

33

13

),(

)(

)(

1)

()

()

()

ωω

δω

ωω

ωωω

∈≤

⋅+

==

jG

jP

jP

jD

jY

jT

(3)

{}

44

24

),(

)(

)(

1)

()

()

()

()

()

()

()

()

ωω

δω

ωω

ωω

ωωω

ωωω

∈≤

⋅+

=⋅

==

=j

Gj

Pj

Gj

Fj

Rj

Uj

Nj

Uj

Dj

Uj

T(4

)

{}

5su

p5

5in

f5

),(

)(

)(

1)

()

()

()

()

()

()

ωω

δω

ωω

ωω

ωωω

ωδ

∈≤

⋅+

⋅=

=≤

jG

jP

jG

jP

jF

jR

jY

jT

(5)

Clo

sed

loop

spec

ifica

tions

are

usu

ally

des

crib

ed in

term

s of f

requ

ency

func

tions

δk(

ω)

that

are

impo

sed

on th

e m

agni

tude

of t

he sy

stem

tran

sfer

func

tions

|Tk|

, k =

1, .

.. 5

(1) r

obus

t sta

bilit

y, c

ontro

l eff

ort l

imit

in th

e in

put d

istu

rban

ce re

ject

ion,

sen

sor n

oise

atte

nuat

ion

(2) o

utpu

t sys

tem

dis

turb

ance

reje

ctio

n

(3) i

nput

syst

em d

istu

rban

ce re

ject

ion

(4) c

ontro

l eff

ort l

imit

in th

e ou

tput

dis

turb

ance

reje

ctio

n, n

oise

atte

nuat

ion,

and

trac

king

(5) s

igna

l tra

ckin

g

Tabl

e 1

Page 29: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.28

Mar

io G

arci

a-Sa

nz

k-pr

oble

mB

ound

Qua

drat

ic In

equa

lity

10

1)

cos(

21

12

2 1

2≥

+⋅

+⋅

⋅+

⋅−

⋅g

pg

φδ

20

11

)co

s(2

2 2

22

≥−

+⋅

+⋅

⋅+

⋅δ

θφ

gp

gp

30

1)

cos(

22 32

22

≥−

+⋅

+⋅

⋅+

⋅δ

θφ

pg

pg

p

40

1)

cos(

21

22 4

2≥

+⋅

+⋅

⋅+

⋅−

gp

gp

θφ

δ

50

)co

s()

cos(

21

12 52

22 5

22 5

22

≥−

+⋅

+−

++

⋅−

δθ

φδ

θφ

δd

ee

dd

ed

ed

e

pp

gp

pp

pg

pp δ 5=δ

5sup

/δ5i

nfφ∠

g)

j(G

i{} 1

,...,

0,

)(

)(

−=

∠=

=m

rp

jP

jP

ir

ωω

Plan

t

Con

trolle

r

Each

pla

nt in

the

ω i-tem

plat

e an

d th

e co

ntro

ller c

an b

e ex

pres

sed

in it

s pol

ar fo

rm:

Then

, sub

stitu

ting

and

rear

rang

ing

the

ineq

ualit

ies -

Eq. (

1) to

(5) i

n Ta

ble

1-, t

hey

can

be re

duce

d to

the

quad

ratic

ineq

ualit

ies –

k-pr

oble

m(1

) to

(5) i

n Ta

ble

2-.

Solv

ing

equa

litie

s suc

h as

ag2 +

bg+c

= 0

the

set o

f ω ωωωi-b

ound

sfor

{δ k=

1,..,

5} is

com

pute

d.

Tabl

e 2

Page 30: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.29

Mar

io G

arci

a-Sa

nz

Alg

orith

mto

com

pute

th

ebo

unds

1.D

iscr

etiz

eth

edo

mai

n{

}k

ωin

toa

finite

set

{} k

ik

ni

,...,

1,

==

ωΩ

.2.

Esta

blis

hth

eun

certa

inLT

I pla

ntm

odel

s{

} )(

ω jP

=℘

and

map

itsbo

unda

ryfo

reac

hfr

eque

ncy

ki

Ωω

∈on

the

Nic

hols

char

t. A

seto

fn

tem

plat

es {

} )(

ij

, n

i,..

.,1

=is

obta

ined

. Eac

hte

mpl

ate

{} 1

,...,

0,

)(

)(

−=

∠=

=m

rp

jP

jP

ir

ωω

cont

ains

mpo

ints

orpl

ants

. Sel

ecto

neof

them

as th

eno

min

al p

lant

00

0)

ω∠

=p

jP

i.

3.N

ow, t

heco

nditi

onst

om

eetb

y th

eco

ntro

ller

φω

∠=

gj

Gi)

(ha

veto

be c

ompu

ted.

4.D

efin

e a

rang

e, Φ

, for

the

com

pens

ator

’sph

aseφ

, and

disc

retiz

eit;

fore

xam

ple

[] º0:º5:º

360

−=

∈Φ

φ.

5.C

hoos

ea

sing

le fr

eque

ncy

ki

Ωω

∈.

6.C

hoos

ea

sing

le c

ontro

ller’

spha

seΦ

φ∈

.7.

Cho

ose

a si

ngle

pla

ntin

theω

i-tem

plat

e:

θω

∠=

pj

Pi

r)

(.

8.A

tthi

sste

p, th

ek

feed

back

pro

blem

isre

duce

dto

solv

ea

kqu

adra

ticin

equa

lity

with

outu

ncer

tain

ty. T

hefe

edba

ck

prob

lem

sin

equa

tions

(1) t

o(5

) in

Tabl

eI a

re re

duce

dto

ineq

ualit

iesi

n Ta

ble

II.9.

Com

pute

the

max

imum

)(

rm

axm

axP

gg

=an

dth

em

inim

um)

(r

min

min

Pg

g=

ofth

etw

oro

ots

g 1an

dg 2

that

solv

eth

ek

quad

ratic

ineq

ualit

y,.

10.R

epea

tSte

ps6

and

7 fo

rthe

mpl

ants

1,..

.0

),(

−=

mr

jP

ir

ωin

the

tem

plat

e)

(i

jP

ω.

11.C

hoos

eth

em

ostr

estri

ctiv

eof

the

m)

(r

max

Pg

and

the

m)

(r

min

Pg

. Thu

s, )

(Pg m

axan

d)

(Pg m

inar

e ob

tain

ed. T

hey

are

the

max

imum

and

min

imum

boun

dpo

ints

fort

heco

ntro

llerm

agni

tude

g at

a ph

ase

φ .12

.Rep

eatS

tep

5 ov

erth

era

nge

Φ. T

heun

ion

of)

(Pg m

axan

d)

(Pg m

info

reac

φ∈

give

∠m

axg

and

φ∠

min

g,

resp

ectiv

ely.

13.N

owth

ebo

unds

fort

heop

enlo

optra

nsm

issio

n0

00

)(

ψω

∠=

lj

Li

are

com

pute

d. S

etφ

ψ∠

⋅=

∠m

axm

axg

pl

00

0an

ψ∠

⋅=

∠m

inm

ing

pl

00

0, b

eing

[] 0:º5:º

360

, 00

−=

+=

φθ

φψ

. The

sebo

unds

will

be la

belle

d)

(i

kj

.14

.Rep

eatS

tep

4 ov

erth

era

nge

. The

seto

fbou

ndsf

orth

ek

con

trolp

robl

em{

}k

ii

kj

ωω

∈),

(ha

s jus

tbee

nco

mpu

ted.

Y. C

hait,

and

O. Y

aniv

, “M

ulti-

inpu

t/sin

gle-

outp

ut

com

pute

r-ai

ded

cont

rol d

esig

n us

ing

the

Qua

ntita

tive

Feed

back

The

ory,

”In

t. J.

Rob

ust &

N

on-l

inea

r C

ontr

ol, v

ol.3

, pp.

47-

54, 1

993.

Page 31: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.30

Mar

io G

arci

a-Sa

nz

-350

-300

-250

-200

-150

-100

-50

0-5

0

-40

-30

-20

-1001020

1

1 11

2 2 23 3 3

Pha

se (d

egre

es)

Magnitude (dB)

All B

ound

s

The

set o

f ω ωωωi-b

ound

sPe

rfor

man

ceSp

ecifi

catio

ns(P

.S.)

Plan

tMod

el+

Unc

erta

inty

+

Page 32: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.31

Mar

io G

arci

a-Sa

nz

Step

9: O

ptim

al B

ound

s St

ep 9

: Opt

imal

Bou

nds

BBoo(

j(jω ωωωω ωωω

ii))

-350

-300

-250

-200

-150

-100

-50

0

-30

-20

-1001020

Pha

se (d

egre

es)

Magnitude (dB)

Inte

rsec

tion

of B

ound

s

-350

-300

-250

-200

-150

-100

-50

0-5

0

-40

-30

-20

-1001020

1

1 11

2 2 23 3 3

Pha

se (d

egre

es)

Magnitude (dB)

All B

ound

s

Inte

rsec

tion

of ω ωωω

i-bou

nds

One

line

for

each

fr

eque

ncy

ω ωωω iTh

e m

ost d

eman

ding

one

Page 33: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.32

Mar

io G

arci

a-Sa

nz

Step

10:

Syn

thes

izin

g G

(s)

Step

10:

Syn

thes

izin

g G

(s)

or L

oop

Shap

ing

Lor

Loo

p Sh

apin

g L

oo(s)(s)

Shap

ing

ofL o

(jω) =

Po(

jω) G

(jω)

Onl

y on

e L

= L o

to b

e sh

ape!

!

Lo(

jωi)

mus

t be

at e

very

ωi:

-out

side

the

U-c

onto

ur-a

bove

the

cont

inuo

us b

ound

s B0(

jω)

-bel

ow th

e di

scon

t. B

ound

s B0(

jω)

Synt

hesi

ze ra

tiona

l fun

ctio

nL o

(s)

Bui

ld u

p G

(jω) t

erm

-by-

term

add

ing

som

e el

emen

ts li

ke: g

ain,

real

pol

es a

nd z

eros

, co

mpl

ex p

oles

and

zer

os, i

nteg

rato

rs,

diff

eren

tiato

rs, l

ead/

lag

netw

orks

, not

ch

filte

rs, s

econ

d or

der T

F, e

tc.

Com

pens

ator

: G(s

) = L

o(s)

/Po(

s)

Prob

ably

one

of t

he m

ost d

iffic

ult s

teps

of

the

met

hodo

logy

for t

he b

egin

ner.

-180

°

8db

16db

24db 0db

-8db

-16d

b

-24d

b

-140

°-1

00°

-60°

Bo(j

1)

Bo(j

2) Bo(j

3)

Bo(j

4)

Bo(j

6)

Bo(j

10)

Bo(j

20)

ω=1

ω=2 ω=3

ω=5 ω

=10

ω=2

0

ω=3

0

ω=4

=60

ω=1

00

U-c

onto

ur(S

tabi

lity

Bou

nd)

LmL

o(jω

)

LmL

o(jω

)

A

ω=5

0

[B0(j

ω)]

ω≥ω

h=40

Page 34: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.33

Mar

io G

arci

a-Sa

nz

1.

Gai

n

2.

Rea

l Pol

e

3.

Rea

l Zer

o

4.

Com

plex

Pol

e

5.

Com

plex

Zer

o)1

(1

s2

s

)1(

1s

2s

1

1zs

1ps1

k

n

2 n2

n

2 n2

+ωζ

+ωζ

++

Proc

edur

e:B

uild

up

ter

m-b

y-te

rm a

ddin

g so

me

elem

ents

like

:

1s

2s

1s

2s

1ps

1zsss1

1s

bs

b1

sa

sa

n22

n2n1

2

n2

nn

22

1

22

1

+ωζ

+ωζ

++

++

++

6.

2ºor

der /

2ºo

rder

7.

Inte

grat

or

8.

Diff

eren

ciat

or

9.

Lead

/Lag

N

etw

ork

10. N

otch

Filt

eret

c...

∏ ==

n 0i

)](j i

[GK

)(j

P)

(jL

oo

ωω

ω

Page 35: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.34

Mar

io G

arci

a-Sa

nz

The

Inte

ract

ive

Des

ign

Env

iron

men

t(ID

E):

(Ter

asof

t, ve

rsio

n2)

Inte

ract

ive

tool

tode

sign

the

cont

rolle

rG(s

)Fu

nctio

n:

lpsh

ape(

...)

Onl

y on

e L

= L o

to b

e sh

ape!

!

& the

cont

rolle

ris

for t

hew

hole

set o

fun

certa

inpl

ants

Page 36: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.35

Mar

io G

arci

a-Sa

nz

Step

11:

St

ep 1

1: P

refil

ter

Pref

ilter

Des

ign

Des

ign

F(s)

F(s)

+ 0 -

dBLm

ω

BU

δ L(jω

1)δ L(

jω2)

δ L(jω

3)δ L(

jω4)

BL

δ R(jω

4)

δ R(jω

3)

δ R(jω

2)

δ R(jω

1)

TL

ι

Fig.

9 C

lose

d-lo

op re

spon

ses:

LT

I pla

nts w

ith o

nly

G(s

)

Fig.

10

Clo

sed-

loop

resp

onse

s:

LTI p

lant

s with

G(s

) and

F(s

)

+ 0 -

dBLm

ω

BU

BL

TR

ι

to li

ebe

twee

n B U

&B L

for

all J

pla

nts

A P

refil

terF

(s) i

s nee

ded

YF

PG

UE

D

R

Pre-

filte

rPl

ant

Con

trolle

r-

Page 37: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.36

Mar

io G

arci

a-Sa

nz

ω(r

ad/s

)

LmT R

L

LmT R

U

db

101

0

δ R(jω

i)

LmT m

ax

LmT m

in

ωi

100

10-1

δ R(jω

i)

LmT

LmT

BLm

TLm

F

LmT

LmT

BLm

TLm

F

RU

max

Um

axm

ax

RL

min

Lm

inm

in

−=

−=

−=

−=

Lm F

max

Lm F

min

+ 0 -

dBLm

ω

LmT R

L-L

mT m

in

0db/

dec

-20d

b/de

c

-40d

b/de

c

LmT R

U-L

mT m

ax

Freq

uenc

y bo

unds

on

the

pref

ilter

F(s)

.

F(s)

is sy

nthe

size

d,

in d

ashe

s, th

at li

es w

ithin

th

e up

per &

low

er

plot

s

Lm F

jj

jm

axm

in(

) <

Lm

F(

) <

Lm

F(

ωω

limF

ss→

=0

1(

)

We

have

tom

ove

dow

nm

ore

than

F max

but

less

than

F min

.

Page 38: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.37

Mar

io G

arci

a-Sa

nz

(Ter

asof

t, ve

rsio

n2)

Page 39: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.38

Mar

io G

arci

a-Sa

nz

Step

12:

Sim

ulat

e lin

ear

syst

em (J

tim

e re

spon

ses)

Step

12:

Sim

ulat

e lin

ear

syst

em (J

tim

e re

spon

ses)

Step

13:

Sim

ulat

e w

ithSt

ep 1

3: S

imul

ate

with

nonl

inea

ritie

sno

nlin

eari

ties

10-2

100

102

104

-70

-60

-50

-40

-30

-20

-100

Wei

ght:

--

Magnitude (dB)

Freq

uenc

y(ra

d/se

c)

10-2

10-1

100

101

-40

-35

-30

-25

-20

-15

-10-505

Wei

ght:

--

Magnitude (dB)

Freq

uenc

y(ra

d/se

c)

Tim

e (s

ec.)

Amplitude

Ste

p R

espo

nse

00.

20.

40.

60.

81

1.2

1.4

1.6

1.8

20

0.2

0.4

0.6

0.81

Freq

uenc

y D

omai

n

Tim

e D

omai

n

Page 40: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.39

Mar

io G

arci

a-Sa

nz

Des

ign

proc

edur

e:

Pref

ilter

desi

gnF(

s)

Satis

fact

ory

sim

ulat

ion

?

Rea

djus

tmen

t

Indu

stri

al

Indu

stri

al

Impl

emen

tatio

nIm

plem

enta

tion

Exp

erim

enta

l Ver

ifica

tion

?

Ver

ifica

tion

In th

efr

eq. &

tim

edo

mai

n?

Ver

ifica

tion

In th

efr

eque

ncy

dom

ain?

Boun

dsCo

mpu

tatio

n

Yes

Spec

ifica

tions

disc

ussio

nM

athe

mat

ical

mod

elre

visio

n

Non

linea

rm

odel

s

Spec

ifica

tions

Tem

plat

es

Line

ariza

tion

Mat

hem

atic

alm

odel

Doe

saSo

lutio

nex

ist?

Loop

-sha

ping

G(s

)

Rea

djus

tmen

t

Rea

djus

tmen

t

No

No

No

Star

t

End

Page 41: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.40

Mar

io G

arci

a-Sa

nz

•B

ooks

:

–H

OR

OW

ITZ,

I. M

., 19

93, Q

uant

itativ

e F

eedb

ack

Des

ign

Theo

ry (Q

FT)

.Q

FT P

ub.,

660

Sout

h M

onac

o Pa

rkw

ay, D

enve

r, C

olor

ado

8022

4-12

29.

–Y

AN

IV, O

., 19

99, Q

uant

itativ

e F

eedb

ack

Des

ign

of L

inea

r an

d N

on-l

inea

r C

ontr

ol S

yste

ms.

Klu

verA

cade

mic

Pub

., IS

BN

: 0-7

923-

8529

-2.

–SI

DI,

M..,

200

2, D

esig

n of

Rob

ust C

ontr

ol S

yste

ms:

Fro

m c

lass

ical

to

mod

ern

prac

tical

app

roac

hes.

Krie

gerP

ublis

hing

.

–H

OU

PIS,

CH

., R

ASM

USS

EN, S

J., G

AR

CIA

-SA

NZ,

M.,

2006

, Q

uant

itativ

e F

eedb

ack

Theo

ry. F

unda

men

tals

and

App

licat

ions

. 2nd

Editi

on. A

CR

C b

ook,

Tay

lor a

nd F

ranc

is.

1.3.

-Ref

eren

ces

Page 42: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.41

Mar

io G

arci

a-Sa

nz

C.H

. Hou

pis,

S.J.

Ras

mus

sen

and

M. G

arcí

a-Sa

nz"Q

uant

itativ

e Fe

edba

ck T

heor

y.

Fund

amen

tals

and

appl

icat

ions

”.

2nded

ition

, 624

pag

es, a

CR

C P

ress

boo

k,

Tayl

or &

Fra

ncis,

Boc

a R

atón

, Flo

rida,

U

SA, I

SBN

: 084

9333

709,

Janu

ary

2006

. C.H

. Hou

pis,

S.J.

Ras

mus

sen

and

M. G

arcí

a-Sa

nz"S

olut

ions

Man

ual t

o Q

uant

itativ

e Fe

edba

ck

Theo

ry. F

unda

men

tals

and

appl

icat

ions

”. 2

nd

editi

on, 9

0 pa

ges,

Tayl

or &

Fra

ncis,

Boc

a R

atón

, Flo

rida,

USA

, Jan

uary

2006

.

Air

Forc

eIn

stitu

teof

Tech

nolo

gyW

right

-Pat

ters

onA

FB,

Day

ton,

Ohi

o, U

SA, 2

003

Page 43: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.42

Mar

io G

arci

a-Sa

nz

•In

tern

atio

nal S

ympo

sia

on Q

uant

itativ

e Fe

edba

ck T

heor

y an

d R

obus

t Fr

eque

ncy

Dom

ain

Met

hods

–U

p to

now

ther

e ha

ve b

een

eigh

t Int

. Sym

p. o

n Q

FT:

1.-H

oupi

s, C

.H.,

Cha

nder

, P. (

Edito

rs).

Writ

ghtP

atte

rson

Airf

orce

Bas

e, D

ayto

n,

Ohi

o, U

SA, A

ugus

t 199

2.

2.-N

wok

ah, O

.D.I.

, Cha

nder

, P.(

Edito

rs).

Purd

ue U

nive

rsity

, Wes

t Laf

ayet

te, I

ndia

na,

USA

, Aug

ust 1

995.

3.-P

etro

poul

akis

, L.,

Lei

thea

d, W

.E.(E

dito

rs).

Uni

vers

ity o

f Stra

thcl

yde,

Gla

sgow

, Sc

otla

nd, U

K, A

ugus

t 199

7.

4.-B

oje,

E.,

and

Eite

lber

g, E

.(Ed

itors

). U

nive

rsity

of N

atal

, Dur

ban,

Sou

th A

fric

a,

Aug

ust 1

999.

5.-G

arcí

a-Sa

nz, M

.(Ed

itor)

. Pub

lic U

nive

rsity

of N

avar

ra, P

ampl

ona,

Spa

in, A

ugus

t20

01.

6.-B

oje,

E.,

and

Eite

lber

g, E

.(Ed

itors

). U

nive

rsity

of C

ape

Tow

n, C

ape

Tow

n, S

outh

A

fric

a, D

ecem

ber 2

003.

7.-C

olgr

en, R

.(Ed

itor)

. Uni

vers

ity o

f Kan

sas,

Law

renc

e, K

ansa

s, U

SA, A

ugus

t 200

5.

8.-G

utm

an, P

-O.(

Edito

r). T

echn

ion,

Hai

fa, I

srae

l, Ju

ly 2

007.

Page 44: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.43

Mar

io G

arci

a-Sa

nz

•Sp

ecia

lIss

ues:

1.-N

wok

ah, O

.D.I.

(Gue

st E

dito

r). H

orow

itz a

nd Q

FT D

esig

n M

etho

ds. S

peci

al Is

sue.

In

tern

atio

nal J

ourn

al o

f Rob

ust a

nd N

onlin

ear

Con

trol

. Vol

. 4, N

um 1

, Jan

uary

-Fe

brua

ry 1

994.

Wile

y.

2.-H

oupi

s, C

.H. (

Gue

st E

dito

r). Q

uant

itativ

e Fe

edba

ck T

heor

y. S

peci

al Is

sue.

In

tern

atio

nal J

ourn

al o

f Rob

ust a

nd N

onlin

ear

Con

trol

. Vol

. 7, N

um 6

, Jun

e 19

97.

Wile

y.

3.-E

itelb

erg,

Edu

ard

(Gue

st E

dito

r). I

saac

Hor

owitz

. Spe

cial

Issu

e. In

tern

atio

nal J

ourn

al

of R

obus

t and

Non

linea

r C

ontr

ol. P

art 1

, Vol

. 11,

Num

10,

Aug

ust 2

001

and

Part

2,

Vol

. 12,

Num

4, A

pril

2002

. Wile

y.

4.-G

arci

a-Sa

nz, M

ario

(Gue

st E

dito

r). R

obus

t Fre

quen

cy D

omai

n. S

peci

al Is

sue.

In

tern

atio

nal J

ourn

al o

f Rob

ust a

nd N

onlin

ear

Con

trol

. Vol

. 13,

Num

7, J

une

2003

. W

iley.

5.-G

arci

a-Sa

nz, M

ario

and

Hou

pis,

Con

stan

tine

(Gue

st E

dito

rs).

Qua

ntita

tive

Feed

back

T

heor

y. In

Mem

oria

m o

f Isa

ac H

orow

itz. S

peci

al Is

sue.

Inte

rnat

iona

l Jou

rnal

of

Rob

ust a

nd N

onlin

ear

Con

trol

. Vol

. 17,

Num

2-3

, Jan

uary

200

7. W

iley.

Page 45: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.44

Mar

io G

arci

a-Sa

nz

•BO

RG

HES

AN

I, C

., C

HA

IT, Y

., Y

AN

IV, O

.,20

02,

Qua

ntita

tive

Fee

dbac

k Th

eory

Tool

box

-For

use

with

MA

TLA

B, 2

nd E

d.U

SA

ht

tp://

ww

w.te

raso

ft.co

m/p

rodu

cts/

qft/

•G

UTM

AN

, P.O

., Q

syn.

H

aifa

, Isr

ael.

•H

OU

PIS,

C.H

., R

ASM

USS

EN

, S.,

GA

RC

IA-S

AN

Z, M

., 20

06Q

FT

CA

D T

oolf

orM

ISO

and

MIM

O s

yste

ms.

With

the

book

, Qua

ntita

tive

Fee

dbac

k Th

eory

. Fun

dam

enta

ls a

nd a

pplic

atio

nsTa

ylor

and

Fran

cis,

2nd

editi

onU

SA

1.4.

-QFT

Sof

twar

e to

ols

9.50

Page 46: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.45

Mar

io G

arci

a-Sa

nz

Out

line

1.-

QFT

Con

trolle

r Des

ign

Tech

niqu

e Fu

ndam

enta

ls

2.-

Rea

l-wor

ld Q

FT c

ontro

l app

licat

ions

and

exa

mpl

es

3.-

Non

-dia

gona

l MIM

O Q

FT c

ontro

ller d

esig

n m

etho

dolo

gies

4.-A

pplic

atio

n: R

obus

t QFT

con

trol f

or a

MIM

O S

pace

craf

t with

flex

ible

suns

hiel

d

5.-S

witc

hing

robu

st c

ontro

l: B

eyon

d th

e lin

ear l

imita

tions

.

6.-E

xam

ple:

Sw

itchi

ng c

ontro

l for

Unm

anne

d V

ehic

les

Page 47: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.46

Mar

io G

arci

a-Sa

nz

Mul

tipol

e,

Var

iabl

e Sp

eed,

Dire

ct D

rive

1650

kW

Win

d Tu

rbin

e.TW

T165

0. M

.Tor

res(

Spai

n).

•M

ore

than

20

cont

rol l

oops

•V

ery

Non

-line

ar m

odel

s•

MIM

O p

lant

•Pa

ram

eter

unc

erta

inty

•H

igh

relia

bilit

y ne

eded

•O

ptim

um e

ffic

ienc

y

E. T

orre

s, M

. Gar

cía-

Sanz

"Exp

erim

enta

l Res

ults

of t

he V

aria

ble

Spee

d, D

irec

t Dri

ve

Mul

tipol

eSy

nchr

onou

s W

ind

Turb

ine:

TW

T165

0". W

ind

Ene

rgy,

200

4.

2.-R

eal-w

orld

QFT

con

trol

app

licat

ions

: C

ontr

ol o

fa L

arge

Win

dTu

rbin

e

Page 48: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.47

Mar

io G

arci

a-Sa

nz

TWT1

650.

Mec

hani

cal D

iagr

am

Pate

nts b

y M

.Tor

res

Win

d

Tran

sf

Grid

Gea

rbox

Indu

ctio

nG

ener

atorC

lass

ical

Sys

tem

Win

d

Dire

ctD

rive

Sync

hron

ous

gene

rato

r

Mul

tipol

eSy

stem

Page 49: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.48

Mar

io G

arci

a-Sa

nz

TWT1

650.

Ele

ctri

cal B

lock

Dia

gram

Vca

pM

ultip

ole

Sync

hron

ous

Gen

erat

or

AC

-DC

con

v

I exc

I gen

Con

verte

rs

Win

d

Pitc

hM

otor

s

Tran

sf

DC

-AC

con

v

Win

d

Line

Yaw

Mot

ors

Con

trol S

yste

m

Safe

ty R

edun

dant

Sys

tem

Exci

tatio

n

IGB

TsIG

BTs

Var

iabl

e fr

eque

ncy

from

2.5

Hz

to 1

2.5

Hz

Con

stan

t fr

eque

ncy

50 H

z

DC

link

Var

iabl

e Sp

eed

M.T

orre

s

Page 50: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.49

Mar

io G

arci

a-Sa

nz

TWT-

1650

Page 51: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.50

Mar

io G

arci

a-Sa

nz

TWT1

650.

Fir

st P

roto

type

bui

lt in

May

200

1

Firs

t Pro

toty

pe a

t Cab

anill

asW

ind

Farm

(Spa

in)

M.T

orre

s

Page 52: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.51

Mar

io G

arci

a-Sa

nz

TWT1

650.

Fir

st P

roto

type

bui

lt in

May

200

1

Tow

er: 7

0 m

; B

lade

: 40

mR

otor

: 82

m ;

Iner

tia ro

tor:

5,00

0,00

0 K

g m

2

M.T

orre

s

Page 53: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.52

Mar

io G

arci

a-Sa

nz

Con

trol L

oop

1:In

put:

Win

d Tu

rbin

e R

otor

Spe

ed

Ref

. = 2

0 rp

mO

utpu

t:Pi

tch

Ang

le R

efer

ence

.

Con

trol L

oop

2:In

put:

Pitc

h A

ngle

Ref

eren

ce.

Out

put:

Pitc

h Sp

eed

Ref

eren

ce.

Con

trol L

oop

3:In

put

Pitc

h Sp

eed

Ref

eren

ce.

Out

put:

Mot

or C

urre

nt R

efer

ence

.

P 3(s

)P 2

(s)

C2(

s)

Pitc

hA

ngle

Ref

I(s)

--

Pitc

han

gle

cont

rolle

rPitc

hSp

eed

Ref

C3(

s)

Pitc

hsp

eed

cont

rolle

r

P 1(s

)

Pitc

hSp

eed

Pitc

hA

ngle

Rot

or

Spee

dC

1(s)

Rot

or

Spee

dR

ef

-R

otor

spee

dco

ntro

ller

Con

trol S

yste

m

TWT1

650.

Act

ual R

esul

ts.

Exam

ple

of th

ree

of m

ore

than

20

loop

s.M

.Tor

res

Targ

et: T

o co

ntro

l the

Win

d Tu

rbin

e R

otor

Spe

ed .

Act

uato

rs: P

itch

angl

e m

ovem

ent.

3 In

depe

nden

t dr

iven

bla

des.

Page 54: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.53

Mar

io G

arci

a-Sa

nz

06.A

pril.

2003

. Cab

anill

asW

ind

Farm

(Spa

in).

Ver

y H

igh

Win

d:A

ver.

24

m/s

TWT1

650.

Act

ual R

esul

ts.

Exam

ple

of th

ree

of m

ore

than

20

loop

s.

Targ

et: c

ontro

l Rot

or S

peed

(r.p

.m.).

Set-p

oint

: 20

rpm

Act

uato

rs: P

itch

angl

e m

ovem

ent.

3 In

depe

nden

t driv

en b

lade

s.

010

020

030

040

050

060

00510152025

Tim

e (s

ec)

Rot

or S

peed

-rpm

-

010

020

030

040

050

060

01415161718192021222324

Tim

e (s

ec)

Con

trol P

itch

Ang

le: r

ef(k

),1(r)

,2(b

),3(g

) (de

g bl

ade)

010

020

030

040

050

060

01618202224262830

Tim

e (s

ec)

Win

d S

peed

-m/s

-

M.T

orre

s

1.7

MW

800

kNm

4.5

MW

2140

kN

m

Win

dG

ust

24

m/s

Test

at v

ery

high

Win

d Sp

eed.

Ver

y N

on-li

near

mod

el.

()

3p

2ai

rV

CR

5.0Po

wer

πρ

=

Cp =

f(V

,Ω,β

)

Page 55: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.54

Mar

io G

arci

a-Sa

nz

TWT1

650.

Act

ual R

esul

ts.

Exam

ple

of th

ree

of m

ore

than

20

loop

s.

Targ

et: c

ontro

l Rot

or S

peed

(r.p

.m.).

Set-p

oint

: 20

rpm

Act

uato

rs: P

itch

angl

e m

ovem

ent.

3 In

depe

nden

t driv

en b

lade

s.

M.T

orre

s15

m

/s

410

420

430

440

450

460

470

480

490

500

510

1213141516171819

Tim

e (s

ec)

Win

d S

peed

-m/s

-

410

420

430

440

450

460

470

480

490

500

510

024681012141618202224

Tim

e (s

ec)

Rot

or S

peed

-rpm

-

410

420

430

440

450

460

470

480

490

500

510

456789101112

Tim

e (s

ec)

Con

trol P

itch

Ang

le: r

ef(k

),1(r)

,2(b

),3(g

) (de

g bl

ade)

Pitc

h A

ngle

Ref

eren

ce

Pitc

h A

ngle

04.F

ebru

ary.

2003

. Cab

anill

asW

ind

Farm

(Spa

in).

Med

ium

Win

d:A

vera

ge 1

5 m

/s

Test

at v

ery

high

Win

d Sp

eed.

Ver

y N

on-li

near

mod

el.

()

3p

2ai

rV

CR

5.0Po

wer

πρ

=

Cp =

f(V

,Ω,β

)

Page 56: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.55

Mar

io G

arci

a-Sa

nz

M.T

orre

s

Vid

eo:

TWT1

650

Page 57: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.56

Mar

io G

arci

a-Sa

nz

Out

line

1.-

QFT

Con

trolle

r Des

ign

Tech

niqu

e Fu

ndam

enta

ls

2.-

Rea

l-wor

ld Q

FT c

ontro

l app

licat

ions

and

exa

mpl

es

3.-

Non

-dia

gona

l MIM

O Q

FT c

ontro

ller d

esig

n m

etho

dolo

gies

4.-A

pplic

atio

n: R

obus

t QFT

con

trol f

or a

MIM

O S

pace

craf

t with

flex

ible

suns

hiel

d

5.-S

witc

hing

robu

st c

ontro

l: B

eyon

d th

e lin

ear l

imita

tions

.

6.-E

xam

ple:

Sw

itchi

ng c

ontro

l for

Unm

anne

d V

ehic

les

Page 58: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.57

Mar

io G

arci

a-Sa

nz

This

sect

ion

disc

usse

s how

the

QFT

tech

niqu

e ca

n be

ap

plie

d to

the

desi

gn o

f MIM

O c

ontro

l sys

tem

s.

2x2

Exam

ple

of a

MIM

O s

yste

m

T 1N

1

T 2Q

p 11 p 12 p 21 p 22

g 11 g 22

+ +-

-e 1

e 2

r 1 r 2

2x2

Plan

t

N1

N2

T 1 T 2

QH

OT

WA

TER

TAN

K

HEA

TEX

CH

AN

GER

PRO

DU

CT

HO

TW

ATE

R

New

Pro

blem

s: In

tera

ctio

nbe

twee

n co

ntro

l lo

ops,

Inpu

t and

out

put D

irect

ions

, Inp

ut-

outp

ut P

airin

g, T

rans

mis

sion

Zer

os(R

HP)

.

New

Too

ls: R

GA

, SV

D,

Smith

-McM

illan

3.N

on-d

iago

nal M

IMO

QFT

C

ontr

olle

r D

esig

n M

etho

dolo

gies

g 21 g 12

(non

-dia

gona

l con

trol

ler)

Page 59: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.58

Mar

io G

arci

a-Sa

nz

A fu

lly p

opul

ated

(non

-dia

gona

l) m

atrix

com

pens

ator

allo

ws t

he d

esig

ner m

uch

mor

e de

sign

flex

ibili

tyto

gov

ern

MIM

O sy

stem

s tha

n th

e cl

assi

cal d

iago

nal c

ontro

ller.

This

sess

ion

exte

ndst

he c

lass

ical

dia

gona

l QFT

com

pens

ator

desi

gn to

a fu

lly

popu

late

d m

atrix

com

pens

ator

desi

gn.

In th

is se

ssio

n th

ree

case

sare

stud

ied:

-t

he re

fere

nce

track

ing,

-t

he e

xter

nal d

istu

rban

ce re

ject

ion

at th

e pl

ant i

nput

and

-t

he e

xter

nal d

istu

rban

ce re

ject

ion

at p

lant

out

put.

The

defin

ition

of t

hree

cou

plin

g m

atri

ces(

C1,

C2,

C3)

of t

he n

on-d

iago

nal e

lem

ents

ar

e us

ed to

qua

ntify

the

amou

nt o

f loo

p in

tera

ctio

n an

d to

des

ign

the

non-

diag

onal

co

mpe

nsat

ors r

espe

ctiv

ely.

This

yie

lds a

crit

erio

n to

pro

pose

a se

quen

tial d

esig

n m

etho

dolo

gyof

the

fully

po

pula

ted

mat

rix c

ompe

nsat

orin

the

QFT

robu

st c

ontro

l fra

me.

3.1.

-Int

rodu

ctio

n

Gar

cía-

Sanz

M.,

Egañ

aI.

(200

2). Q

uant

itativ

e N

on-d

iago

nal

Con

trol

ler

Des

ign

for

Mul

tivar

iabl

e Sy

stem

s with

Unc

erta

inty

. In

t. J.

Rob

ust N

onlin

ear

Con

trol

, Vol

. 12,

No.

4, p

p. 3

21-3

33.

Gar

cía-

Sanz

M.,

Egañ

aI.,

Bar

rera

s M. (

2005

). D

esig

n of

qu

antit

ativ

e fe

edba

ck th

eory

non

-dia

gona

l con

trol

lers

for

use

in

unce

rtai

n m

ultip

le-in

put m

ultip

le-o

utpu

t sys

tem

s. IE

EC

ontr

ol

Theo

ry a

nd A

pplic

atio

ns. V

ol. 1

52, N

. 02,

pp.

177

-187

.

Page 60: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.59

Mar

io G

arci

a-Sa

nz

Con

side

r an

nxn

linea

r mul

tivar

iabl

e sy

stem

(see

Fig

ure)

, com

pose

d of

a

plan

t P, a

fully

pop

ulat

ed m

atrix

com

pens

ator

G, a

nd a

pre

filte

rF:

MIM

O S

yste

m

=

==

nnn

n

nn

nnn

n

nn

nnn

n

nn

ff

f

ff

f

ff

f

gg

g

gg

g

gg

g

pp

p

pp

p

pp

p

.........

;

.........

;

.........

21

222

21

112

11

21

222

21

112

11

21

222

21

112

11

F

GP

F(s)

G(s

)P(

s)u

r’r

y

T(s)

-

P do(

s) d od o’

P di(s

) d id i’

whe

re P

∈,

and

is th

e se

t of

poss

ible

pla

nts d

ue to

unc

erta

inty

.

Page 61: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.60

Mar

io G

arci

a-Sa

nz

The

plan

t inv

erse

, den

oted

by

P*

, is p

rese

nted

in th

e fo

llow

ing

form

at:

[]

+=

+=

==

0...

p...

0...

p...

0

p0

00

00

0p

** 1n

* n1

* nn

* 11* ij

1-B

PP

p

and

whe

re th

e co

mpe

nsat

or m

atrix

is b

roke

n up

into

two

parts

asf

ollo

ws:

+=

+=

0...

g...

0...

g...

0

g0

00

...0

00

g

n1

1n

nn

11

bd

GG

G

The

follo

win

g in

trodu

ces a

mea

sure

men

t in

dex

to q

uant

ify th

e lo

op in

tera

ctio

n in

the

thre

e cl

assi

cal c

ases

: ref

eren

ce tr

acki

ng,

exte

rnal

dis

turb

ance

s at t

he p

lant

inpu

t, an

d th

e ex

tern

al d

istu

rban

ces a

t the

pla

nt o

utpu

t.

F(s)

G(s

)P(

s)u

r’r

y

T(s)

-

P do(

s) d od o’

P di(s

) d id i’

Page 62: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.61

Mar

io G

arci

a-Sa

nz

Ref

eren

ce T

rack

ing.

The

trans

fer f

unct

ion

mat

rix o

f the

con

trol s

yste

m fo

r the

refe

renc

e tra

ckin

g pr

oble

m, w

ithou

t any

ext

erna

l dis

turb

ance

, is w

ritte

n as

follo

ws:

()

'/

/1

rr

rr

yr

yF

TT

GP

GP

Iy

==

+=

()

()

()

[] r

rr

r-

-y/

ry/

rT

GG

GI

GG

IT

bb

1-1

d1

d1-

1d

1+

−+

++

=−

−B

and

appl

ying

the

defin

ition

s of :

P* =

Λ ΛΛΛ+

Ban

d

G=

Gd

+ G

b

A d

iago

nal t

erm

A n

on-d

iago

nal t

erm

F(s)

G(s

)P(

s)u

r’r

y

T(s)

-

P do(

s) d od o’

P di(s

) d id i’

Page 63: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.62

Mar

io G

arci

a-Sa

nz

A d

iago

nal t

erm

:

A d

iago

nal

term(

)d

1-1

d1

y/r_

dG

GI

T−

+=

-

g ii

r iy i

t i(s)

-

u i*1 iip

A n

on-d

iago

nal t

erm

: (

)(

)[

](

)1

1-1

d1-

bb

1-1

d1

y/r_

bC

GI

TG

GG

IT

y/r

B−

−=

+=

+−

+-

A n

on-d

iago

nal

term

g ii

u i0

y i

t i(s)

-

=

=n 1

kj

ij1j

rc

d

*1 iip

()

=+

−=

mm

mm

mm

bb

cc

c

cc

c

cc

c

12

11

1

21221

211

11121

111

1y/

rT

GB

GC

()

()

()

−+

−−

==n 1

kik

kjik

* ikij

ijij1

1g

p1

gc

t

C1

repr

esen

ts th

e co

uplin

g m

atri

x C

of t

he e

quiv

alen

t sys

tem

for r

efer

ence

trac

king

pro

blem

s

≠⇔

==

⇔=

=i

k0

ik

1kiki

ki

Page 64: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.63

Mar

io G

arci

a-Sa

nz

The

trans

fer f

unct

ion

mat

rix o

f the

con

trol s

yste

m fo

r the

ext

erna

l dis

turb

ance

re

ject

ion

at p

lant

inpu

tpro

blem

, with

out a

ny e

xter

nal d

istu

rban

ce, i

s writ

ten

as,

and

appl

ying

the

defin

ition

s of

P*

= Λ ΛΛΛ

+ B

and

G

= G

d+

Gb

A d

iago

nal

term

A n

on-d

iago

nal

term

Ext

erna

l dist

urba

nce

reje

ctio

n at

pla

nt in

put.

()

'/

/1

idi

diy

idi

yi

dP

dd

yT

TP

GP

I=

=+

=−

()

()

()

[] i

-i

-i

y/di

dd

dy/

dib

1-1

d1

1-1

d1

TG

GI

GI

T+

+−

+=

−−

B

F(s)

G(s

)P(

s)u

r’r

y

T(s)

-

P do(

s) d od o’

P di(s

) d id i’

Page 65: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.64

Mar

io G

arci

a-Sa

nz

A d

iago

nal t

erm

:

A d

iago

nal

term(

)1-

1d

1y/

di_d

−+

=G

IT

-

g ii

0y i

t i(s)

-

u i

dii

*1 iip

A n

on-d

iago

nal t

erm

:

A n

on-d

iago

nal

term

()

()

()

21-

1d

1-b

1-1

d1

y/di

_bB

CG

IT

GG

IT

y/di

−−

+=

++

=-

g ii

u i0

y i

t i(s)

-

=

=n 1

kj

ij2j

dic

d

*1 iip

C2

repr

esen

ts th

e co

uplin

g m

atri

x of

the

equi

vale

nt sy

stem

for e

xter

nal d

istu

rban

ce re

ject

ion

at th

e pl

ant i

nput

pro

blem

s

≠⇔

==

⇔=

=i

k0

ik

1kiki

ki

()

y/di

TG

Cb

+=

B2

)1(

t)g

p(c

ikkj

ikn 1

k* ik

ij2δ

−+

==

Page 66: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.65

Mar

io G

arci

a-Sa

nz

The

trans

fer f

unct

ion

mat

rix o

f the

con

trol s

yste

m fo

r the

ext

erna

l dis

turb

ance

re

ject

ion

at p

lant

out

putp

robl

em, w

ithou

t any

ext

erna

l dis

turb

ance

, is w

ritte

n as

,

and

appl

ying

the

defin

ition

s of

P*

= Λ ΛΛΛ

+ B

and

G

= G

d+

Gb

A d

iago

nal

term

A n

on-d

iago

nal

term

Ext

erna

l dist

urba

nce

reje

ctio

n at

pla

nt o

utpu

t.

()

'/

/1

odo

doy

odo

yo

dd

dy

PT

TG

PI

==

+=

()

()

()

[] o

-o

-o

dd

dy/

doy/

doT

GG

IG

IT

b1-

1d

11

d1

+−

++

+=

−−

BB

F(s)

G(s

)P(

s)u

r’r

y

T(s)

-

P do(

s) d od o’

P di(s

) d id i’

Page 67: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.66

Mar

io G

arci

a-Sa

nz

≠⇔

==

⇔=

=i

k0

ik

1kiki

ki

()

y/do

TG

BB

Cb

+−

=3

)1(

)(

)1(

ikkj

ik1

* ikij

* ij3i

δ−

+−

−=

=

tg

pp

cn k

C3

repr

esen

ts th

e co

uplin

g m

atri

x of

the

equi

vale

nt sy

stem

for e

xter

nal d

istu

rban

ce re

ject

ion

at th

e pl

ant o

utpu

t pro

blem

s

A d

iago

nal t

erm

:

A d

iago

nal

term(

)1d

1y/

do_d

−+

=G

IT

-

g ii

0y i

t i(s)

-

u i

doi

*1 iip

A n

on-d

iago

nal t

erm

:

A n

on-d

iago

nal

term

()

()

[]

()

31-

1d

1-b

1-1

d1

y/do

_bC

GI

TG

GI

Ty/

doB

B−

−+

=+

−+

=-

g ii

u i0

y i

t i(s)

-

=

=n 1

kj

ij3j

doc

d

*1 iip

Page 68: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.67

Mar

io G

arci

a-Sa

nz

The

Cou

plin

g el

emen

ts

To d

esig

n a

MIM

O c

ompe

nsat

or w

ith a

low

cou

plin

g le

vel,

it is

nec

essa

ry to

stud

y th

e in

fluen

ce o

f eve

ry n

on-d

iago

nal e

lem

ent g

ijon

the

coup

ling

elem

ents

c1i

j, c 2i

jan

d c 3i

j.

Hyp

othe

sis

Thus

,

()

()

jjkj

ik* ik

ij* ij

tof

band

wid

th

in th

ean

dj,

kfo

r,

tg

pjjt

gp

≠+

>>+

jjkj

jj t

ofba

ndw

idth

in

the

and

j,k

for

,≠

>>t

t

Page 69: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.68

Mar

io G

arci

a-Sa

nz

Due

to h

ypot

hesi

s, th

e co

uplin

g ef

fect

s c1i

j,c 2

ij,c 3

ijar

e co

mpu

ted

as,

track

ing

dist

urba

nce

reje

ctio

n at

the

plan

t inp

ut

dist

urba

nce

reje

ctio

n at

pla

nt o

utpu

t

()

()

ji

;g

p

gp

gg

cjj

* jj

ij* ij

jjij

ij1≠

+

+−

=

()

()

ji

;g

p

gp

cjj

* jj

ij* ij

ij2≠

++=

()

()

ji

;g

p

gp

pp

cjj

* jj

ij* ij

* jj* ij

ij3≠

+

+−

=

Page 70: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.69

Mar

io G

arci

a-Sa

nz

The

Opt

imum

non

-dia

gona

l con

trol

ler

The

optim

um n

on-d

iago

nal c

ompe

nsat

ors f

or th

e th

ree

case

s (tra

ckin

g an

d di

stur

banc

e re

ject

ion

at p

lant

inpu

t and

out

put)

are

obta

ined

mak

ing

last

thre

e Eq

s. eq

ual t

o ze

ro.

whe

re th

e fu

nctio

n F

pd(A

) mea

ns in

eve

ry c

ase

a st

able

pro

per f

unct

ion

mad

e fr

om

the

dom

inan

t pol

es a

nd z

eros

of t

he e

xpre

ssio

n A

.

ji

for

,pp

gg

* jj* ijjj

opt

ij≠

=NN

pdF

()

ji

for

,p

gN

* ijop

tij

≠−

=pd

F

ji

for

,pp

gg

N* jjN* ij

jjop

tij

≠=

pdF

track

ing

dist

urba

nce

reje

ctio

n at

the

plan

t inp

ut

dist

urba

nce

reje

ctio

n at

pla

nt o

utpu

t

Page 71: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.70

Mar

io G

arci

a-Sa

nz

Ever

y un

certa

in p

lant

can

be

any

plan

t rep

rese

nted

by

the

fam

ily:

whe

re

i

s the

nom

inal

pla

nt (≠

P o), a

nd

is th

e m

axim

um o

f the

non

-pa

ram

etric

unc

erta

inty

radi

i

{}

()

n1,

...,

ji,

for

,p

0,

1p

p* ij

ijij

N* ij

* ij=

≤≤

+=

N* ijp

* ijpΔij

The

nom

inal

pla

nts

min

imis

e th

e m

axim

um o

f the

non

-par

amet

ric

unce

rtain

ty ra

dii

and

th

at

com

pris

e th

e pl

ant t

empl

ates

.

N* ijp * ijpΔ

* jjpΔ

* ijpΔ

N * ijp* ijpΔ

Imag

ReN * ijp

ij

ij

Page 72: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.71

Mar

io G

arci

a-Sa

nz

Des

ign

Met

hodo

logy

Step

A.C

ontr

olle

r St

ruct

ure,

C

oupl

ing

Ana

lysi

s, In

put/o

utpu

t P

airi

ng a

nd lo

op o

rder

ing.

Firs

t, th

e m

etho

dolo

gy b

egin

s pai

ring

the

plan

t inp

uts a

nd o

utpu

ts a

nd se

lect

ing

the

cont

rolle

rst

ruct

ure

with

the

Rel

ativ

e G

ain

Ana

lysi

s (R

GA

-Bris

tol)

tech

niqu

e.

This

is fo

llow

ed b

y ar

rang

ing

the

mat

rix

P* so

that

ha

s the

smal

lest

pha

se

mar

gin

freq

uenc

y,

the

next

smal

lest

ph

ase

mar

gin

freq

uenc

y, a

nd so

on.

The

sequ

entia

ltec

hniq

ue, c

ompo

sed

of n

stag

es (n

loop

s), r

epea

ts st

eps (

B a

nd C

) fo

r eve

ry c

olum

n k

= 1

to n

.

The

com

pens

ator

des

ign

met

hod

is a

sequ

entia

l pr

oced

ure

by c

losin

g lo

ops.

()1

* 11p−

()1

* 22p−

nnnk

2n1n

knkk

2k1k

n2k2

2221

n1k1

1211

g...

g...

gg

......

...

g...

g...

gg

......

...

g...

g...

gg

g...

g...

gg

Step

n

...

=

2n1n

2k1k

2221

1211

1n1k2111

...

0...

0...

gg

......

...

0...

0...

gg

......

...

0...

0...

gg

0...

0...

gg

0...

0...

0g

......

...

0...

0...

0g

......

...

0...

0...

0g

0...

0...

0g

G

Step

1St

ep 2

Page 73: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.72

Mar

io G

arci

a-Sa

nz

Step

B.D

esig

n of

the

diag

onal

co

mpe

nsat

or e

lem

ents

gkk

.

This

des

ign

of th

e el

emen

t gkk

is

calc

ulat

ed u

sing

the

stan

dard

QFT

lo

op-s

hapi

ng te

chni

que

for t

he

inve

rse

of th

e eq

uiva

lent

pla

nt

in

ord

er to

ach

ieve

robu

st st

abili

ty

and

robu

st p

erfo

rman

ce

spec

ifica

tions

.

nnnk

2n1n

knkk

2k1k

n2k2

2221

n1k1

1211

g...

g...

gg

......

...

g...

g...

gg

......

...

g...

g...

gg

g...

g...

gg

Step

n

...

=

2n1n

2k1k

2221

1211

1n1k2111

...

0...

0...

gg

......

...

0...

0...

gg

......

...

0...

0...

gg

0...

0...

gg

0...

0...

0g

......

...

0...

0...

0g

......

...

0...

0...

0g

0...

0...

0g

G

Step

1St

ep 2

()1

p*e− k

kk

[]

[]

[]

() [

](

)[

](

)[

]1

1*

1,1

1*

1,1

,11

*,1

1,

1*

1,

1*

*

−−

−−

−−

−−

−−

−−

=

+

++

−=

Pp

gp

gp

gp

pp

e ik

ii

ie

ii

ki

ie

ki

ii

ie i

i

ie ik

ie ik

k

e kkk

kp

pp

pp

p

pp

p

pp

p

*1

* 21

* 1

1* 33

1* 32

1* 31

1* 23

1* 22

1* 21

1* 13

1* 12

1* 11 fu

nctio

nof

Page 74: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.73

Mar

io G

arci

a-Sa

nz

Step

C.D

esig

n of

the

non-

diag

onal

co

mpe

nsat

or e

lem

ents

gij

The

(n-1

) non

-dia

gona

l ele

men

ts g

ik

(i ≠

k, i

= 1,

2,...

n) o

f the

k-th

com

pens

ator

col

umn

are

desi

gned

to,

min

imis

e th

e cr

oss-

coup

ling

term

s cik

.nn

nk2n

1n

knkk

2k1k

n2k2

2221

n1k1

1211

g...

g...

gg

......

...

g...

g...

gg

......

...

g...

g...

gg

g...

g...

gg

Step

n

...

=

2n1n

2k1k

2221

1211

1n1k2111

...

0...

0...

gg

......

...

0...

0...

gg

......

...

0...

0...

gg

0...

0...

gg

0...

0...

0g

......

...

0...

0...

0g

......

...

0...

0...

0g

0...

0...

0g

G

Step

1St

ep 2

Step

D.T

he d

esig

n of

the

pref

ilter

Fdo

es n

ot p

rese

nt a

ny d

iffic

ulty

be

caus

e th

e fin

al T

y/rfu

nctio

n sh

ows

less

loop

inte

ract

ion.

The

refo

re, t

he

pref

ilter

Fca

n be

dia

gona

l.

()

()

0g

p

gp

gg

cjj

* jj

ij* ij

jjij

ij1=

+

+−

=

()

()

0g

p

gp

cjj

* jj

ij* ij

ij2=

++=

()

()

0g

p

gp

pp

cjj

* jj

ij* ij

* jj* ij

ij3=

+

+−

=

Page 75: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.74

Mar

io G

arci

a-Sa

nz

The

sequ

entia

l non

-dia

gona

l MIM

O Q

FT te

chni

que

intro

duce

d he

re a

rriv

es a

t a

robu

st st

able

clos

ed-lo

op sy

stem

if ,

for e

ach

P∈

TP

a)ea

ch L

i(s)

= g ii(

s) (

p ii*e)−1

, i=

1, .

.., n

, sa

tisfie

s th

e N

yqui

sten

circ

lem

ent

cond

ition

,

b)no

RH

P po

le-z

ero

canc

ella

tions

occ

ur b

etw

een

g ii(s)

and

(pii*e

)−1, i

=1, .

.., n

,

c)no

Sm

ith-M

cMill

an p

ole-

zero

can

cella

tions

occ

ur b

etw

een

P(s)

and

G(s

), an

d

d)no

Sm

ith-M

cMill

an p

ole-

zero

can

cella

tions

occ

ur in

⏐P*

(s) +

G(s

)⏐

Rob

ust S

tabi

lity

of th

e M

IMO

syst

em

Che

cked

at

eac

h lo

op

Che

cked

at

the

end

Page 76: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.75

Mar

io G

arci

a-Sa

nz

Des

ign

of th

e co

ntro

ller

mat

rix

G(s

)

Non

-dia

gona

l MIM

O Q

FT m

etho

d

Cal

cula

tion

of tr

ansm

issi

on z

eros

Smith

-McM

illan

form

Mod

ifica

tion

of R

HP

tran

smis

sion

zer

os

Non

-dia

gona

l ele

men

ts g

ij(i

≠j) p

lace

d in

the

last

col

umn

of th

e m

atrix

con

trolle

r G(s

)

Are

ther

e R

HP

trans

mis

sion

zer

os?

Yes

No E

ND

nnnk

2n1n

knkk

2k1k

n2k2

2221

n1k1

1211

g...

g...

gg

......

...

g...

g...

gg

......

...

g...

g...

gg

g...

g...

gg

To re

mov

e th

e R

HP

tra

nsm

issi

on z

eros

th

at c

ould

be

intro

duce

d in

the

prev

ious

con

trolle

r de

sign

RH

P tr

ansm

issio

n ze

ros o

f the

M

IMO

syst

em

Page 77: Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order Controller. Transparency of the Technique. z Frequency Domain Technique. Uses the

1.76

Mar

io G

arci

a-Sa

nz

Any

ques

tions

?

Than

ks

End

first

part