406 MUL TIV ARIABLE FEEDBA CK CONTR OL... minimize control system complexity subject to the...
28
solution, yet has to be done in ignorance of this solution, which can then turn out to be unsuitable in ways that were not foreseen. Therefore, control system design usually proceeds iteratively through the steps of modelling, control structure design, controllability analysis, performance and robustness weights selection, controller synthesis, control system analysis and nonlinear simulation. Rosenbrock (1974) makes the following observation: Solutions are constrained by so many requirements that it is virtually impossible to list them all. The designer finds himself threading a maze of such requirements, attempting to reconcile conflicting demands of cost, performance, easy maintenance, and so on. A good design usually has strong aesthetic appeal to those who are competent in the subject. Most (if not all) available control theories assume that a control structure is given at the outset. They therefore fail to answer some basic questions which a control engineer regularly meets in practice. Which variables should be controlled, which variables should be measured, which inputs should be manipulated, and which links should be made between them? The objective of this chapter is to describe the main issues involved in control structure design and to present some of the available quantitative methods, for example, for decentralized control. 10.1 Introduction Control structure design was considered by Foss (1973) in his paper entitled “Critique of process control theory” where he concluded by challenging the control theoreticians of the day to close the gap between theory and applications in this important area. Later Morari et al. (1980) presented an overview of control structure design, hierarchical control and multilevel optimization in their paper “Studies in the synthesis of control structure for chemical processes”, but the gap still remained, and still does to some extent today. Control structure design is clearly important in the chemical process industry because of the complexity of these plants, but the same issues are relevant in most other areas of control where we have large-scale systems. For example, in the late 1980s Carl Nett (Nett, 1989; Nett and Minto, 1989) gave a number of lectures based on his experience on aero-engine control at General Electric, under the title “A quantitative approach to the selection and partitioning of measurements and manipulations for the control of complex systems”. He noted that increases in controller complexity unnecessarily outpaces increases in plant complexity, and that the objective should be to
Transcript of 406 MUL TIV ARIABLE FEEDBA CK CONTR OL... minimize control system complexity subject to the...
404
MULTIVARIABLEFEEDBACKCONTROL
solu
tion,
yeth
asto
bedo
nein
igno
ranc
eof
this
solu
tion,
whi
chca
nth
entu
rnou
tto
beun
suita
ble
inw
ays
that
wer
eno
tfor
esee
n.
The
refo
re,
cont
rol
syst
emde
sign
usua
llypr
ocee
dsite
rativ
ely
thro
ugh
the
step
sof
mod
ellin
g,co
ntro
lst
ruct
ure
desi
gn,
cont
rolla
bilit
yan
alys
is,
perf
orm
ance
and
robu
stne
ssw
eigh
tsse
lect
ion,
cont
rolle
rsy
nthe
sis,
cont
rol
syst
eman
alys
isan
dno
nlin
ears
imul
atio
n.R
osen
broc
k(1
974)
mak
esth
efo
llow
ing
obse
rvat
ion:
Solu
tions
are
cons
trai
ned
byso
man
yre
quir
emen
tsth
atit
isvi
rtua
llyim
poss
ible
tolis
tthe
mal
l.T
hede
sign
erfin
dshi
mse
lfth
read
ing
am
aze
ofsu
chre
quir
emen
ts,a
ttem
ptin
gto
reco
ncile
confl
ictin
gde
man
dsof
cost
,pe
rfor
man
ce,
easy
mai
nten
ance
,an
dso
on.
Ago
odde
sign
usua
llyha
sst
rong
aest
hetic
appe
alto
thos
ew
hoar
eco
mpe
tent
inth
esu
bjec
t.
10
CONTROL
STRUCTURE
DESIGN
Mos
t(if
nota
ll)av
aila
ble
cont
rolt
heor
ies
assu
me
that
aco
ntro
lstr
uctu
reis
give
nat
the
outs
et.
The
yth
eref
ore
fail
toan
swer
som
eba
sic
ques
tions
whi
cha
cont
rol
engi
neer
regu
larl
ym
eets
inpr
actic
e.W
hich
vari
able
ssh
ould
beco
ntro
lled,
whi
chva
riab
les
shou
ldbe
mea
sure
d,w
hich
inpu
tssh
ould
bem
anip
ulat
ed,a
ndw
hich
links
shou
ldbe
mad
ebe
twee
nth
em?
The
obje
ctiv
eof
this
chap
ter
isto
desc
ribe
the
mai
nis
sues
invo
lved
inco
ntro
lstr
uctu
rede
sign
and
topr
esen
tso
me
ofth
eav
aila
ble
quan
titat
ive
met
hods
,for
exam
ple,
for
dece
ntra
lized
cont
rol.
10.1
Intr
oduc
tion
Con
trol
stru
ctur
ede
sign
was
cons
ider
edby
Foss
(197
3)in
his
pape
ren
title
d“C
ritiq
ueof
proc
ess
cont
rolt
heor
y”w
here
heco
nclu
ded
bych
alle
ngin
gth
eco
ntro
lth
eore
ticia
nsof
the
day
tocl
ose
the
gap
betw
een
theo
ryan
dap
plic
atio
nsin
this
impo
rtan
tare
a.L
ater
Mor
arie
tal.
(198
0)pr
esen
ted
anov
ervi
ewof
cont
rols
truc
ture
desi
gn,h
iera
rchi
calc
ontr
olan
dm
ultil
evel
optim
izat
ion
inth
eirp
aper
“Stu
dies
inth
esy
nthe
sis
ofco
ntro
lstr
uctu
refo
rche
mic
alpr
oces
ses”
,but
the
gap
still
rem
aine
d,an
dst
illdo
esto
som
eex
tent
toda
y.
Con
trol
stru
ctur
ede
sign
iscl
earl
yim
port
anti
nth
ech
emic
alpr
oces
sin
dust
rybe
caus
eof
the
com
plex
ityof
thes
epl
ants
,but
the
sam
eis
sues
are
rele
vant
inm
osto
ther
area
sof
cont
rol
whe
rew
eha
vela
rge-
scal
esy
stem
s.Fo
rex
ampl
e,in
the
late
1980
sC
arl
Net
t(N
ett,
1989
;N
ett
and
Min
to,
1989
)ga
vea
num
ber
ofle
ctur
esba
sed
onhi
sex
peri
ence
onae
ro-e
ngin
eco
ntro
latG
ener
alE
lect
ric,
unde
rthe
title
“Aqu
antit
ativ
eap
proa
chto
the
sele
ctio
nan
dpa
rtiti
onin
gof
mea
sure
men
tsan
dm
anip
ulat
ions
for
the
cont
rol
ofco
mpl
exsy
stem
s”.
He
note
dth
atin
crea
ses
inco
ntro
ller
com
plex
ityun
nece
ssar
ilyou
tpac
esin
crea
ses
inpl
ant
com
plex
ity,a
ndth
atth
eob
ject
ive
shou
ldbe
to
406
MULTIVARIABLEFEEDBACKCONTROL
...m
inim
ize
cont
rol
syst
emco
mpl
exity
subj
ect
toth
eac
hiev
emen
tof
accu
racy
spec
ifica
tions
inth
efa
ceof
unce
rtai
nty.
�
-
-
-
KP
sens
edou
tput
sco
ntro
lsig
nals
exog
enou
sin
puts
(wei
ghte
d)ex
ogen
ous
outp
uts
(wei
ghte
d)
u
vz
w
Fig
ure
10.1
:Gen
eral
cont
rolc
onfig
urat
ion
InC
hapt
er3.
8w
eco
nsid
ered
the
gene
ralc
ontr
olpr
oble
mfo
rmul
atio
nin
Figu
re10
.1,
and
stat
edth
atth
eco
ntro
llerd
esig
npr
oble
mis
to
�
Find
aco
ntro
llerK
whi
chba
sed
onth
ein
form
atio
nin
v
,gen
erat
esa
cont
rols
igna
l
u
whi
chco
unte
ract
sth
ein
fluen
ceof
w
on
z
,the
reby
min
imiz
ing
the
clos
ed-l
oop
norm
from
w
to
z
.
How
ever
,if
we
goba
ckto
Cha
pter
1(p
age
1),t
hen
we
see
that
this
ison
lySt
ep7
inth
eov
eral
lpro
cess
ofde
sign
ing
aco
ntro
lsys
tem
.In
this
chap
ter
we
are
conc
erne
dw
ithth
est
ruct
ural
deci
sion
s(S
teps
4,5,
6an
d7)
asso
ciat
edw
ithth
efo
llow
ing
task
sof
cont
rols
truc
ture
desi
gn:
1.T
hese
lect
ion
ofco
ntro
lled
outp
uts
(ase
tof
vari
able
sw
hich
are
tobe
cont
rolle
dto
achi
eve
ase
tof
spec
ific
obje
ctiv
es;s
eese
ctio
ns10
.2an
d10
.3):
Wha
tar
eth
eva
riab
les
z
inF
igur
e10
.1?
2.T
hese
lect
ion
ofm
anip
ulat
ions
and
mea
sure
men
ts(s
ets
ofva
riab
les
whi
chca
nbe
man
ipul
ated
and
mea
sure
dfo
rco
ntro
lpur
pose
s;se
ese
ctio
n10
.4):
Wha
tar
eth
eva
riab
lese
ts
u
and
v
inF
igur
e10
.1?
3.T
hese
lect
ion
ofa
cont
rol
confi
gura
tion
(ast
ruct
ure
inte
rcon
nect
ing
mea
sure
-m
ents
/com
man
dsan
dm
anip
ulat
edva
riab
les;
see
sect
ions
10.6
,10
.7an
d10
.8):
Wha
tis
the
stru
ctur
eof
K
inF
igur
e10
.1,t
hati
s,ho
wsh
ould
we
“pa
ir”
the
vari
-ab
lese
ts
u
and
v
?4.
The
sele
ctio
nof
aco
ntro
ller
type
(con
trol
law
spec
ifica
tion,
e.g.
PID
-con
trol
ler,
deco
uple
r,L
QG
,etc
):W
hata
lgor
ithm
isus
edfo
r
K
inF
igur
e10
.1?
The
dist
inct
ion
betw
een
the
wor
dsco
ntro
lstr
uctu
rean
dco
ntro
lcon
figur
atio
nm
ayse
emm
inor
,bu
tno
teth
atit
issi
gnifi
cant
with
inth
eco
ntex
tof
this
book
.T
heco
ntro
lst
ruct
ure
(or
cont
rol
stra
tegy
)re
fers
toal
lst
ruct
ural
deci
sion
sin
clud
edin
the
desi
gnof
aco
ntro
lsy
stem
.O
nth
eot
her
hand
,th
eco
ntro
lco
nfigu
rati
on
CONTROLSTRUCTUREDESIGN
407
refe
rson
lyto
the
stru
ctur
ing
(dec
ompo
sitio
n)of
the
cont
rolle
r
K
itsel
f(a
lso
calle
dth
em
easu
rem
ent/m
anip
ulat
ion
part
ition
ing
orin
put/o
utpu
tpa
irin
g).C
ontr
olco
nfigu
ratio
nis
sues
are
disc
usse
din
mor
ede
tail
inSe
ctio
n10
.6.
The
sele
ctio
nof
cont
rolle
dou
tput
s,m
anip
ulat
ions
and
mea
sure
men
ts(t
asks
1an
d2
com
bine
d)is
som
etim
esca
lled
inpu
t/ou
tput
sele
ctio
n.
Idea
lly,
the
task
sin
volv
edin
desi
gnin
ga
com
plet
eco
ntro
lsy
stem
are
perf
orm
edse
quen
tially
;fir
sta
“top
-dow
n”se
lect
ion
ofco
ntro
lled
outp
uts,
mea
sure
men
tsan
din
puts
(with
little
rega
rdto
the
confi
gura
tion
ofth
eco
ntro
llerK
)and
then
a“b
otto
m-
up”
desi
gnof
the
cont
rols
yste
m(i
nw
hich
the
sele
ctio
nof
the
cont
rolc
onfig
urat
ion
isth
em
ost
impo
rtan
tde
cisi
on).
How
ever
,in
prac
tice
the
task
sar
ecl
osel
yre
late
din
that
one
deci
sion
dire
ctly
influ
ence
sth
eot
hers
,so
the
proc
edur
em
ayin
volv
eite
ratio
n.
One
impo
rtan
tre
ason
for
deco
mpo
sing
the
cont
rol
syst
emin
toa
spec
ific
cont
rol
confi
gura
tion
isth
atit
may
allo
wfo
rsi
mpl
etu
ning
ofth
esu
bcon
trol
lers
with
out
the
need
for
ade
taile
dpl
ant
mod
elde
scri
bing
the
dyna
mic
san
din
tera
ctio
nsin
the
proc
ess.
Mul
tivar
iabl
ece
ntra
lized
cont
rolle
rsm
ayal
way
sou
tper
form
deco
mpo
sed
(dec
entr
aliz
ed)
cont
rolle
rs,b
utth
ispe
rfor
man
cega
inm
ustb
etr
aded
off
agai
nstt
heco
stof
obta
inin
gan
dm
aint
aini
nga
suffi
cien
tlyde
taile
dpl
antm
odel
.
The
num
ber
ofpo
ssib
leco
ntro
lst
ruct
ures
show
sa
com
bina
tori
algr
owth
,so
for
mos
tsys
tem
sa
care
fule
valu
atio
nof
alla
ltern
ativ
eco
ntro
lstr
uctu
res
isim
prac
tical
.Fo
rtun
atel
y,w
eca
nof
ten
from
phys
ical
insi
ght
obta
ina
reas
onab
lech
oice
ofco
ntro
lled
outp
uts,
mea
sure
men
tsan
dm
anip
ulat
edin
puts
.In
othe
rca
ses,
sim
ple
cont
rolla
bilit
ym
easu
res
aspr
esen
ted
inC
hapt
ers
5an
d6
may
beus
edfo
rqu
ickl
yev
alua
ting
orsc
reen
ing
alte
rnat
ive
cont
rols
truc
ture
s.
Som
edi
scus
sion
onco
ntro
lst
ruct
ure
desi
gnin
the
proc
ess
indu
stry
isgi
ven
byM
orar
i(1
982)
,Shi
nske
y(1
988)
,Ste
phan
opou
los
(198
4)an
dB
alch
enan
dM
umm
e(1
988)
.Asu
rvey
onco
ntro
lst
ruct
ure
desi
gnis
give
nby
van
deW
alan
dde
Jage
r(1
995)
.A
revi
ewof
cont
rol
stru
ctur
ede
sign
inth
ech
emic
alpr
oces
sin
dust
ry(p
lant
wid
eco
ntro
l)is
give
nby
Lar
sson
and
Skog
esta
d(2
000)
.T
here
ader
isre
ferr
edto
Cha
pter
5(p
age
160)
for
anov
ervi
ewof
the
liter
atur
eon
inpu
t-ou
tput
cont
rolla
bilit
yan
alys
is.
10.2
Opt
imiz
atio
nan
dco
ntro
l
InSe
ctio
ns10
.2an
d10
.3w
ear
eco
ncer
ned
with
the
sele
ctio
nof
cont
rolle
dva
riab
les
(out
puts
).T
hese
are
the
vari
able
s
z
inFi
gure
10.1
,but
we
will
inth
ese
two
sect
ions
call
them
y
.
The
sele
ctio
nof
cont
rolle
dou
tput
sin
volv
esse
lect
ing
the
vari
able
s
y
tobe
cont
rolle
dat
give
nre
fere
nce
valu
es,y�r.
Her
eth
ere
fere
nce
valu
e
r
isse
tats
ome
high
erla
yer
408
MULTIVARIABLEFEEDBACKCONTROL
inth
eco
ntro
lhi
erar
chy.
Thu
s,th
ese
lect
ion
ofco
ntro
lled
outp
uts
(for
the
cont
rol
laye
r)is
usua
llyin
timat
ely
rela
ted
toth
ehi
erar
chic
alst
ruct
urin
gof
the
cont
rol
syst
emw
hich
isof
ten
divi
ded
into
two
laye
rs:
�
opti
miz
atio
nla
yer
—co
mpu
tes
the
desi
red
refe
renc
eco
mm
ands
r
(out
side
the
scop
eof
this
book
)
�
cont
rol
laye
r—
impl
emen
tsth
ese
com
man
dsto
achi
eve
y�r
(the
focu
sof
this
book
).
Add
ition
alla
yers
are
poss
ible
,as
isill
ustr
ated
inFi
gure
10.2
whi
chsh
ows
aty
pica
l
?
Sche
dulin
g(w
eeks
)
Site
-wid
eop
timiz
atio
n(d
ay)
A AA A AU
Loc
alop
timiz
atio
n(h
our)
?�� �
?
Supe
rvis
ory
cont
rol
(min
utes
)
�
���� ��
C C CC CW
Reg
ulat
ory
cont
rol
(sec
onds
)
� � � � ��� �+
���� �9
Con
trol
laye
r
Fig
ure
10.2
:Typ
ical
cont
rols
yste
mhi
erar
chy
ina
chem
ical
plan
t
cont
rolh
iera
rchy
for
aco
mpl
ete
chem
ical
plan
t.H
ere
the
cont
roll
ayer
issu
bdiv
ided
into
two
laye
rs:
supe
rvis
ory
cont
rol
(“ad
vanc
edco
ntro
l”)
and
regu
lato
ryco
ntro
l(“
base
cont
rol”
).W
eha
veal
soin
clud
eda
sche
dulin
gla
yera
bove
the
optim
izat
ion.
Inge
nera
l,th
ein
form
atio
nflo
win
such
aco
ntro
lhie
rarc
hyis
base
don
the
high
erla
yer
CONTROLSTRUCTUREDESIGN
409
send
ing
refe
renc
eva
lues
(set
poin
ts)t
oth
ela
yer
belo
w,a
ndth
elo
wer
laye
rrep
ortin
gba
ckan
ypr
oble
ms
inac
hiev
ing
this
,see
Figu
re10
.3(b
).
The
optim
izat
ion
tend
sto
bepe
rfor
med
open
-loo
pw
ithlim
ited
use
offe
edba
ck.
On
the
othe
rha
nd,t
heco
ntro
lla
yer
ism
ainl
yba
sed
onfe
edba
ckin
form
atio
n.T
heop
timiz
atio
nis
ofte
nba
sed
onno
nlin
ear
stea
dy-s
tate
mod
els,
whe
reas
we
ofte
nus
elin
ear
dyna
mic
mod
els
inth
eco
ntro
llay
er(a
sw
edo
thro
ugho
utth
ebo
ok).
The
reis
usua
llya
time
scal
ese
para
tion
with
fast
erlo
wer
laye
rsas
indi
cate
din
Figu
re10
.2.
Thi
sm
eans
that
the
setp
oint
s,as
view
edfr
oma
give
nla
yer
inth
ehi
erar
chy,
are
upda
ted
only
peri
odic
ally
.Bet
wee
nth
ese
upda
tes,
whe
nth
ese
tpoi
nts
are
cons
tant
,iti
sim
port
antt
hatt
hesy
stem
rem
ains
reas
onab
lycl
ose
toits
optim
um.
Thi
sob
serv
atio
nis
the
basi
sfo
rSe
ctio
n10
.3w
hich
deal
sw
ithse
lect
ing
outp
uts
for
the
cont
roll
ayer
.
From
ath
eore
tical
poin
tof
view
,th
eop
timal
coor
dina
tion
ofth
ein
puts
and
thus
the
optim
alpe
rfor
man
ceis
obta
ined
with
ace
ntra
lize
dop
tim
izin
gco
ntro
ller
,whi
chco
mbi
nes
the
two
laye
rsof
optim
izat
ion
and
cont
rol;
see
Figu
re10
.3(c
).A
llco
ntro
lac
tions
insu
chan
idea
lco
ntro
lsy
stem
wou
ldbe
perf
ectly
coor
dina
ted
and
the
cont
rol
syst
emw
ould
use
on-l
ine
dyna
mic
optim
izat
ion
base
don
ano
nlin
ear
dyna
mic
mod
elof
the
com
plet
epl
anti
nste
adof
,for
exam
ple,
infr
eque
ntst
eady
-sta
teop
timiz
atio
n.H
owev
er,t
his
solu
tion
isno
rmal
lyno
tus
edfo
ra
num
ber
ofre
ason
s;in
clud
ing
the
cost
ofm
odel
ling,
the
diffi
culty
ofco
ntro
ller
desi
gn,m
aint
enan
cean
dm
odifi
catio
n,ro
bust
ness
prob
lem
s,op
erat
orac
cept
ance
,and
the
lack
ofco
mpu
ting
pow
er.
As
note
dab
ove
we
may
also
deco
mpo
seth
eco
ntro
llay
er,a
ndfr
omno
won
whe
nw
eta
lkab
out
cont
rol
confi
gura
tions
,hie
rarc
hica
lde
com
posi
tion
and
dece
ntra
lizat
ion,
we
gene
rally
refe
rto
the
cont
roll
ayer
.
Mes
arov
ic(1
970)
revi
ews
som
eid
eas
rela
ted
toon
-lin
em
ulti-
laye
rst
ruct
ures
appl
ied
tola
rge-
scal
ein
dust
rial
com
plex
es.
How
ever
,ac
cord
ing
toL
unze
(199
2),
mul
tilay
erst
ruct
ures
,al
thou
ghof
ten
used
inpr
actic
e,la
cka
form
alan
alyt
ical
trea
tmen
t.N
ever
thel
ess,
inth
ene
xtse
ctio
nw
epr
ovid
eso
me
idea
son
how
tose
lect
obje
ctiv
es(c
ontr
olle
dou
tput
s)fo
rth
eco
ntro
lla
yer,
such
that
the
over
all
goal
issa
tisfie
d.
Rem
ark
1In
acco
rdan
cew
ithL
unze
(199
2)w
eha
vepu
rpos
ely
used
the
wor
dla
yer
rath
erth
anle
vel
for
the
hier
arch
ical
deco
mpo
sitio
nof
the
cont
rol
syst
em.
The
diff
eren
ceis
that
ina
mul
tile
vel
syst
emal
lun
itsco
ntri
bute
tosa
tisfy
ing
the
sam
ego
al,
whe
reas
ina
mul
tila
yer
syst
emth
edi
ffer
entu
nits
have
diff
eren
tobj
ectiv
es(w
hich
pref
erab
lyco
ntri
bute
toth
eov
eral
lgo
al).
Mul
tilev
elsy
stem
sha
vebe
enst
udie
din
conn
ectio
nw
ithth
eso
lutio
nof
optim
izat
ion
prob
lem
s.
Rem
ark
2T
heta
sks
with
inan
yla
yerc
anbe
perf
orm
edby
hum
ans
(e.g
.man
ualc
ontr
ol),
and
the
inte
ract
ion
and
task
shar
ing
betw
een
the
auto
mat
icco
ntro
lsys
tem
and
the
hum
anop
erat
ors
are
very
impo
rtan
tin
mos
tca
ses,
e.g.
anai
rcra
ftpi
lot.
How
ever
,the
seis
sues
are
outs
ide
the
scop
eof
this
book
.
410
MULTIVARIABLEFEEDBACKCONTROL
d
? ? ?
? ? ? ?�
? ?
�
Opt
imiz
er
Obj
ectiv
e
Gu
y (a)
Con
trol
ler
Opt
imiz
er
Obj
ectiv
e
Gr
y
-+ u (b
)
Obj
ectiv
e
y
u (c)
Opt
imiz
ing
Con
trol
ler
G
Fig
ure
10.3
:Alte
rnat
ive
stru
ctur
esfo
rop
timiz
atio
nan
dco
ntro
l.(a
)O
pen-
loop
optim
izat
ion.
(b)
Clo
sed-
loop
impl
emen
tatio
nw
ithse
para
teco
ntro
lla
yer.
(c)
Inte
grat
edop
timiz
atio
nan
dco
ntro
l.
10.3
Sele
ctio
nof
cont
rolle
dou
tput
s
Aco
ntro
lled
outp
utis
anou
tput
vari
able
(usu
ally
mea
sure
d)w
ithan
asso
ciat
edco
ntro
lobj
ectiv
e(u
sual
lya
refe
renc
eva
lue)
.In
man
yca
ses,
itis
clea
rfro
ma
phys
ical
unde
rsta
ndin
gof
the
proc
ess
wha
tthe
cont
rolle
dou
tput
ssh
ould
be.F
orex
ampl
e,if
we
cons
ider
heat
ing
orco
olin
ga
room
,the
nw
esh
ould
sele
ctro
omte
mpe
ratu
reas
the
cont
rolle
dou
tput
y
.In
othe
rca
ses
itis
less
obvi
ous
beca
use
each
cont
rolo
bjec
tive
may
notb
eas
soci
ated
with
am
easu
red
outp
utva
riab
le.T
hen
the
cont
rolle
dou
tput
s
y
are
sele
cted
toac
hiev
eth
eov
eral
lsys
tem
goal
,and
may
nota
ppea
rto
beim
port
ant
vari
able
sin
them
selv
es.
Exa
mpl
e10
.1C
ake
baki
ng.T
oge
tan
idea
ofth
eis
sues
invo
lved
inou
tput
sele
ctio
nle
tus
cons
ider
the
proc
ess
ofba
king
aca
ke.
The
over
all
goal
isto
mak
ea
cake
whi
chis
wel
lba
ked
insi
dean
dw
ith
ani
ceex
teri
or.
The
man
ipul
ated
inpu
tfo
rac
hiev
ing
this
isth
ehe
atin
put,
u=Q
,(an
dw
ew
illa
ssum
eth
atth
edu
rati
onof
the
baki
ngis
fixed
,e.g
.at1
5m
inut
es).
Now
,ifw
eha
dne
ver
bake
da
cake
befo
re,a
ndif
we
wer
eto
cons
truc
tthe
stov
eou
rsel
ves,
we
mig
htco
nsid
erdi
rect
lym
anip
ulat
ing
the
heat
inpu
tto
the
stov
e,po
ssib
lyw
ith
aw
att-
met
erm
easu
rem
ent.
How
ever
,th
isop
en-l
oop
impl
emen
tati
onw
ould
not
wor
kw
ell,
asth
eop
tim
alhe
atin
putd
epen
dsst
rong
lyon
the
part
icul
arov
enw
eus
e,an
dth
eop
erat
ion
isal
sose
nsit
ive
todi
stur
banc
es,
for
exam
ple,
from
open
ing
the
oven
door
orw
hate
ver
else
mig
htbe
inth
eov
en.I
nsh
ortm
the
open
-loo
pim
plem
enta
tion
isse
nsit
ive
toun
cert
aint
y.A
nef
fect
ive
way
ofre
duci
ngth
eun
cert
aint
yis
tous
efe
edba
ck.T
here
fore
,in
prac
tice
we
look
upth
eop
tim
alov
ente
mpe
ratu
rein
aco
okbo
ok,a
ndus
ea
clos
ed-l
oop
impl
emen
tati
onw
here
ath
erm
osta
tis
used
CONTROLSTRUCTUREDESIGN
411
toke
epth
ete
mpe
ratu
re
y
atit
spr
edet
erm
ined
valu
e
T
.
The
(a)
open
-loo
pan
d(b
)cl
osed
-loo
pim
plem
enta
tion
sof
the
cake
baki
ngpr
oces
sar
eil
lust
rate
din
Fig
ure
10.3
.In
(b)
the
“op
tim
izer
”is
the
cook
book
whi
chha
sa
pre-
com
pute
dta
ble
ofth
eop
tim
alte
mpe
ratu
repr
ofile
.T
here
fere
nce
valu
e
r
for
tem
pera
ture
isth
ense
ntdo
wn
toth
eco
ntro
llay
erw
hich
cons
ists
ofa
sim
ple
feed
back
cont
roll
er(t
heth
erm
osta
t).
Rec
all
that
the
title
ofth
isse
ctio
nis
sele
ctio
nof
cont
rolle
dou
tput
s.In
the
cake
baki
ngpr
oces
sw
ese
lect
oven
tem
pera
ture
asth
eco
ntro
lled
outp
ut
y
inth
eco
ntro
lla
yer.
Itis
inte
rest
ing
tono
teth
atco
ntro
lling
the
oven
tem
pera
ture
inits
elf
has
nodi
rect
rela
tion
toth
eov
eral
lgoa
lof
mak
ing
aw
ell-
bake
dca
ke.S
ow
hydo
we
sele
ctth
eov
ente
mpe
ratu
reas
aco
ntro
lled
outp
ut?
We
now
wan
tto
outli
nean
appr
oach
for
answ
erin
gqu
estio
nsof
this
kind
.
Inth
efo
llow
ing,
we
lety
deno
teth
ese
lect
edco
ntro
lled
outp
uts
inth
eco
ntro
llay
er.
Not
eth
atth
ism
ayal
soin
clud
edi
rect
lyus
ing
the
inpu
ts(o
pen-
loop
impl
emen
tatio
n)by
sele
ctin
g
y=u
.Tw
odi
stin
ctqu
estio
nsar
ise:
1.W
hatv
aria
bles
y
shou
ldbe
sele
cted
asth
eco
ntro
lled
vari
able
s?2.
Wha
tis
the
optim
alre
fere
nce
valu
e(y
opt)
for
thes
eva
riab
les?
The
seco
ndpr
oble
mis
one
ofop
timiz
atio
nan
dis
exte
nsiv
ely
stud
ied
(but
not
inth
isbo
ok).
Her
ew
ew
ant
toga
inso
me
insi
ght
into
the
first
prob
lem
.We
mak
eth
efo
llow
ing
assu
mpt
ions
:
(a)
The
over
allg
oal
can
bequ
antifi
edin
term
sof
asc
alar
cost
func
tion
J
whi
chw
ew
antt
om
inim
ize.
(b)
For
agi
ven
dist
urba
nce
d
,th
ere
exis
tsan
optim
alva
lue
uopt(d)
and
corr
espo
ndin
gva
lue
yopt(d)
whi
chm
inim
izes
the
cost
func
tion
J
.(c
)T
here
fere
nce
valu
es
r
for
the
cont
rolle
dou
tput
s
y
shou
ldbe
cons
tant
,i.e
.
r
shou
ldbe
inde
pend
ento
fth
edi
stur
banc
es
d
.Typ
ical
ly,s
ome
aver
age
valu
eis
sele
cted
,e.g
.r=yopt(� d)
For
exam
ple,
inth
eca
keba
king
proc
ess
we
may
assi
gnto
each
cake
anu
mbe
r
P
ona
scal
efr
om0
to10
,ba
sed
onca
kequ
ality
.A
perf
ectly
bake
dca
keac
hiev
es
P=10
,and
anac
cept
ably
bake
dca
keac
hiev
es
P>6
(aco
mpl
etel
ybu
rned
cake
may
corr
espo
ndto
P=1
).In
anot
her
case
P
coul
dbe
the
oper
atin
gpr
ofit.
Inbo
thca
ses
we
can
sele
ct
J=�P
,and
the
over
all
goal
ofth
eco
ntro
lsy
stem
isth
ento
min
imiz
e
J
.
The
syst
embe
havi
ouri
sa
func
tion
ofth
ein
depe
nden
tvar
iabl
es
u
and
d
,so
we
may
wri
te
J=J(u;d).
For
agi
ven
dist
urba
nce
d
the
optim
alva
lue
ofth
eco
stfu
nctio
nis
Jopt(d),J(uopt;d)=min uJ(u;d)
(10.
1)
Idea
lly,w
ew
ant
u=uopt
.How
ever
,thi
sw
illno
tbe
achi
eved
inpr
actic
e,an
dw
ese
lect
cont
rolle
dou
tput
s
y
such
that
:
412
MULTIVARIABLEFEEDBACKCONTROL
�
The
inpu
t
u
(gen
erat
edby
feed
back
toac
hiev
e
y�r)
shou
ldbe
clos
eto
the
opti
mal
inpu
tuopt(d).
Not
eth
atw
eha
veas
sum
edth
at
r
isin
depe
nden
tof
d
.
Wha
thap
pens
if
u6=uopt,
e.g.
due
toa
dist
urba
nce?
Obv
ious
ly,w
eth
enha
vea
loss
whi
chca
nbe
quan
tified
by
L=J�Jopt
,and
are
ason
able
obje
ctiv
efo
rse
lect
ing
cont
rolle
dou
tput
s
y
isto
min
imiz
eso
me
norm
ofth
elo
ss,f
orex
ampl
e,th
ew
orst
-ca
selo
ss
Worst�caseloss:
�,maxd2DjJ(u;d)�J(uopt;d)
|{z}
L
j
(10.
2)
Her
e
D
isth
ese
tof
poss
ible
dist
urba
nces
.As
“dis
turb
ance
s”w
esh
ould
also
incl
ude
chan
ges
inop
erat
ing
poin
tand
mod
elun
cert
aint
y.
10.3
.1Se
lect
ing
cont
rolle
dou
tput
s:D
irec
tev
alua
tion
ofco
st
The
“bru
tefo
rce”
appr
oach
for
sele
ctin
gco
ntro
lled
vari
able
sis
toev
alua
teth
elo
ssfo
ral
tern
ativ
ese
tsof
cont
rolle
dva
riab
les.
Spec
ical
ly,b
yso
lvin
gth
eno
nlin
ear
equa
tions
,w
eev
alua
tedi
rect
lyth
eco
stfu
nctio
n
J
for
vari
ous
dist
urba
nces
d
and
cont
role
rror
s
e,as
sum
ing
y=r+e
whe
re
r
iske
ptco
nsta
nt.T
hese
tof
cont
rolle
dou
tput
sw
ithsm
alle
stw
orst
-cas
eor
aver
age
valu
eof
J
isth
enpr
efer
red.
Thi
sap
proa
chis
may
betim
eco
nsum
ing
beca
use
the
solu
tion
ofth
eno
nlin
ear
equa
tions
mus
tbe
repe
ated
for
each
cand
idat
ese
tof
cont
rolle
dou
tput
s.
Ifw
ew
ithco
nsta
ntre
fere
nces
(set
poin
ts)
r
can
achi
eve
anac
cept
able
loss
,the
nth
isse
tof
cont
rolle
dva
riab
les
issa
idto
bese
lf-o
ptim
izin
g.H
ere
r
isus
ually
sele
cted
asth
eop
timal
valu
efo
rth
eno
min
aldi
stur
banc
e,bu
tth
ism
ayno
tbe
the
best
choi
cean
dits
valu
em
ayal
sobe
foun
dby
optim
izat
ion
(“op
timal
back
-off
”).
The
spec
ial
case
ofm
easu
rem
ent
sele
ctio
nfo
rin
dire
ctco
ntro
lis
cove
red
onpa
ge43
9.
10.3
.2Se
lect
ing
cont
rolle
dou
tput
s:L
inea
ran
alys
is
We
here
use
alin
eara
naly
sis
ofth
elo
ssfu
nctio
n.T
his
resu
ltsin
the
usef
ulm
inim
umsi
ngul
arva
lue
rule
.H
owev
er,
note
that
this
isa
loca
lan
alys
is,
whi
chm
aybe
mis
lead
ing,
for
exam
ple,
ifth
eop
timum
poin
tof
oper
atio
nis
clos
eto
infe
asib
ility
.
Con
side
rth
elo
ss
L=J(u;d)�Jopt(d)
(10.
2),w
here
d
isa
fixed
(gen
eral
lyno
n-ze
ro)
dist
urba
nce.
We
mak
eth
efo
llow
ing
addi
tiona
lass
umpt
ions
:
(d)
The
cost
func
tion
J
issm
ooth
,or
mor
epr
ecis
ely
twic
edi
ffer
entia
ble.
CONTROLSTRUCTUREDESIGN
413
(e)
The
optim
izat
ion
prob
lem
isun
cons
trai
ned.
Ifit
isop
timal
toke
epso
me
vari
able
ata
cons
trai
nt,t
hen
we
assu
me
that
this
isim
plem
ente
dan
dco
nsid
erth
ere
mai
ning
unco
nstr
aine
dpr
oble
m.
(f)
The
dyna
mic
sof
the
prob
lem
can
bene
glec
ted,
that
is,w
eco
nsid
erth
est
eady
-st
ate
cont
rola
ndop
timiz
atio
n.
For
afix
ed
d
we
may
then
expr
ess
J(u;d)
inte
rms
ofa
Tayl
orse
ries
expa
nsio
nin
u
arou
ndth
eop
timal
poin
t.W
ege
t
J(u;d)
=
Jopt(d)+
� @J @u
� T opt
|{z}
=0
(u�uopt(d))+
1 2(u�uopt(d))T
� @2J
@u2
� opt
(u�uopt(d))+
���
(10.
3)
We
will
negl
ect
term
sof
thir
dor
der
and
high
er(w
hich
assu
mes
that
we
are
reas
onab
lycl
ose
toth
eop
timum
).T
hese
cond
term
onth
eri
ghth
and
side
in(1
0.3)
isze
roat
the
optim
alpo
intf
oran
unco
nstr
aine
dpr
oble
m.
Equ
atio
n(1
0.3)
quan
tifies
how
u�uopt
affe
cts
the
cost
func
tion.
Nex
t,to
stud
yho
wth
isre
late
sto
outp
utse
lect
ion
we
use
alin
eari
zed
mod
elof
the
plan
t,w
hich
for
afix
ed
d
beco
mes
y�yopt=G(u�uopt)
whe
re
G
isth
est
eady
-sta
tega
inm
atri
x.If
G
isin
vert
ible
we
then
get
u�uopt=G�1(y�y opt)
(10.
4)
(IfG
isno
tinv
ertib
lew
em
ayus
eth
eps
eudo
-inv
erse
Gy
whi
chre
sults
inth
esm
alle
stpo
ssib
le
ku�uoptk 2
for
agi
ven
y�yopt.
)W
ege
t
J�Jopt�
1 2� G�1(y�y opt)� T�
@2J
@u2
� opt
G�1(y�y opt)
(10.
5)
whe
reth
ete
rm
(@2J=@u2) opt
isin
depe
nden
tofy
.Obv
ious
ly,w
ew
ould
like
tose
lect
the
cont
rolle
dou
tput
ssu
chth
at
y�yopt
isze
ro.
How
ever
,th
isis
not
poss
ible
inpr
actic
e.To
see
this
,wri
te
y�y opt=y�r+r�y opt=e+e opt
(10.
6)
Firs
t,w
eha
vean
optim
izat
ion
erro
r
eopt(d),
r�y opt(d),
beca
use
the
algo
rith
m(e
.g.
aco
okbo
ok)
pre-
com
pute
sa
desi
red
r
whi
chis
diff
eren
tfr
omth
eop
timal
y opt(d).
Inad
ditio
n,w
eha
vea
cont
rol
erro
r
e=
y�r
beca
use
the
cont
rol
laye
ris
not
perf
ect,
for
exam
ple
due
topo
orco
ntro
lpe
rfor
man
ceor
anin
corr
ect
mea
sure
men
tor
estim
ate
(ste
ady-
stat
ebi
as)
of
y
.If
the
cont
roli
tsel
fis
perf
ectt
hen
e=n
(mea
sure
men
tnoi
se).
Inm
ostc
ases
the
erro
rs
e
and
eopt(d)
can
beas
sum
edin
depe
nden
t.
414
MULTIVARIABLEFEEDBACKCONTROL
Exa
mpl
e10
.1C
ake
baki
ng,c
onti
nued
.Let
usre
turn
toou
rin
itia
lqu
esti
on:
Why
sele
ctth
eov
ente
mpe
ratu
reas
aco
ntro
lled
outp
ut?
We
have
two
alte
rnat
ives
:a
clos
ed-l
oop
impl
emen
tati
onw
ith
y=T
(the
oven
tem
pera
ture
)an
dan
open
-loo
pim
plem
enta
tion
wit
h
y=u=Q
(the
heat
inpu
t).
Fro
mex
peri
ence
,w
ekn
owth
atth
eop
tim
alov
ente
mpe
ratu
re
Topt
isla
rgel
yin
depe
nden
tof
dist
urba
nces
and
isal
mos
tth
esa
me
for
any
oven
.Thi
sm
eans
that
we
may
alw
ays
spec
ify
the
sam
eov
ente
mpe
ratu
re,s
ay
Tr=190Æ
C,a
sob
tain
edfr
omth
eco
okbo
ok.O
nth
eot
her
hand
,the
opti
mal
heat
inpu
tQopt
depe
nds
stro
ngly
onth
ehe
atlo
ss,
the
size
ofth
eov
en,
etc,
and
may
vary
betw
een,
say
100
Wan
d
5000
W.
Aco
okbo
okw
ould
then
need
toli
sta
diffe
rent
valu
eof
Qr
for
each
kind
ofov
enan
dw
ould
inad
diti
onne
edso
me
corr
ecti
onfa
ctor
depe
ndin
gon
the
room
tem
pera
ture
,how
ofte
nth
eov
endo
oris
open
ed,e
tc.
The
refo
re,
we
find
that
itis
muc
hea
sier
toke
ep
e opt=T�Topt
[Æ
C]
smal
lth
anto
keep
Qr�Qopt
[W]
smal
l.In
sum
mar
y,th
em
ain
reas
onfo
rco
ntro
llin
gth
eov
ente
mpe
ratu
reis
tom
inim
ize
the
opti
miz
atio
ner
ror.
From
(10.
5)an
d(1
0.6)
,we
conc
lude
that
we
shou
ldse
lect
the
cont
rolle
dou
tput
s
y
such
that
:
1.
G�1
issm
all(
i.e.G
isla
rge)
;th
ech
oice
of
y
shou
ldbe
such
that
the
inpu
tsha
vea
larg
eef
fect
on
y
.2.
e opt(d)=r�y opt(d)
issm
all;
the
choi
ceof
y
shou
ldbe
such
that
its
opti
mal
valu
e
yopt(d)
depe
nds
only
wea
kly
onth
edi
stur
banc
esan
dot
her
chan
ges.
3.
e=y�r
issm
all;
the
choi
ceof
y
shou
ldbe
such
that
itis
easy
toke
epth
eco
ntro
ler
ror
e
smal
l.
Not
eth
at
��(G�1)=
1=�(G),
and
sow
ew
ant
the
smal
lest
sing
ular
valu
eof
the
stea
dy-s
tate
gain
mat
rix
tobe
larg
e(b
utre
call
that
sing
ular
valu
esde
pend
onsc
alin
gas
isdi
scus
sed
belo
w).
The
desi
reto
have
�(G)
larg
eis
cons
iste
ntw
ithou
rin
tuiti
onth
atw
esh
ould
ensu
reth
atth
eco
ntro
lled
outp
uts
are
inde
pend
ento
fea
chot
her.
Tous
e
�(G)
tose
lect
cont
rolle
dou
tput
s,w
ese
efr
om(1
0.5)
that
we
shou
ldfir
stsc
ale
the
outp
uts
such
that
the
expe
cted
mag
nitu
deof
yi�y iopt
issi
mila
r(e
.g.1
)in
mag
nitu
defo
rea
chou
tput
,an
dsc
ale
the
inpu
tssu
chth
atth
eef
fect
ofa
give
nde
viat
ion
uj�uj opt
onth
eco
stfu
nctio
n
J
issi
mila
rfo
rea
chin
put
(suc
hth
at
� @2 J=@u2� o
pt
iscl
ose
toa
cons
tant
times
aun
itary
mat
rix)
.We
mus
tal
soas
sum
eth
atth
eva
riat
ions
in
yi�y iopt
are
unco
rrel
ated
,orm
ore
prec
isel
y,w
em
usta
ssum
e:
(g)
The
“wor
st-c
ase”
com
bina
tion
ofou
tput
devi
atio
ns, y
i�y iopt
,cor
resp
ondi
ngto
the
dire
ctio
nof
�(G),
can
occu
rin
prac
tice.
Pro
cedu
re.T
heus
eof
the
min
imum
sing
ular
valu
eto
sele
ctco
ntro
lled
outp
uts
may
besu
mm
ariz
edin
the
follo
win
gpr
oced
ure:
1.Fr
oma
(non
linea
r)m
odel
com
pute
the
optim
alpa
ram
eter
s(i
nput
san
dou
tput
s)fo
rva
riou
sco
nditi
ons
(dis
turb
ance
s,op
erat
ing
poin
ts).
(Thi
syi
elds
a“l
ook-
up”
tabl
eof
optim
alpa
ram
eter
valu
esas
afu
nctio
nof
the
oper
atin
gco
nditi
ons.
)2.
From
this
data
obta
info
rea
chca
ndid
ate
outp
utth
eva
riat
ion
inits
optim
alva
lue,
v i=(yi opt;m
ax
�y iopt;m
in)=2
.
CONTROLSTRUCTUREDESIGN
415
3.Sc
ale
the
cand
idat
eou
tput
ssu
chth
atfo
rea
chou
tput
the
sum
ofth
em
agni
tude
sof
v i
and
the
cont
rol
erro
r(e
i,in
clud
ing
mea
sure
men
tno
ise
ni)
issi
mila
r(e
.g.
jv ij+je ij=1
).4.
Scal
eth
ein
puts
such
that
aun
itde
viat
ion
inea
chin
putf
rom
itsop
timal
valu
eha
sth
esa
me
effe
cton
the
cost
func
tion
J
(i.e
.suc
hth
at
� @2 J=@u2� o
pt
iscl
ose
toa
cons
tant
times
aun
itary
mat
rix)
.5.
Sele
ctas
cand
idat
esth
ose
sets
ofco
ntro
lled
outp
uts
whi
chco
rres
pond
toa
larg
eva
lue
of
�(G).
G
isth
etr
ansf
erfu
nctio
nfo
rth
eef
fect
ofth
esc
aled
inpu
tson
the
scal
edou
tput
s.
Not
eth
atth
edi
stur
banc
esan
dm
easu
rem
ent
nois
een
ter
indi
rect
lyth
roug
hth
esc
alin
gof
the
outp
uts
(!).
Exa
mpl
e.T
heae
ro-e
ngin
eap
plic
atio
nin
Cha
pter
12pr
ovid
esa
nice
illu
stra
tion
ofou
tput
sele
ctio
n.T
here
the
over
all
goal
isto
oper
ate
the
engi
neop
tim
ally
inte
rms
offu
elco
nsum
ptio
n,w
hile
atth
esa
me
tim
est
ayin
gsa
fely
away
from
inst
abil
ity.
The
opti
miz
atio
nla
yer
isa
look
-up
tabl
e,w
hich
give
sth
eop
tim
alpa
ram
eter
sfo
rth
een
gine
atva
riou
sop
erat
ing
poin
ts.S
ince
the
engi
neat
stea
dy-s
tate
has
thre
ede
gree
s-of
-fre
edom
we
need
tosp
ecif
yth
ree
vari
able
sto
keep
the
engi
neap
prox
imat
ely
atth
eop
tim
alpo
int,
and
five
alte
rnat
ive
sets
ofth
ree
outp
uts
are
give
n.T
heou
tput
sar
esc
aled
asou
tlin
edab
ove,
and
ago
odou
tput
set
isth
enon
ew
ith
ala
rge
valu
eof
�(G),
prov
ided
we
can
also
achi
eve
good
dyna
mic
cont
rol
perf
orm
ance
.
Rem
ark.
Not
eth
atou
rde
sire
toha
ve
�(G)
larg
efo
rou
tput
sele
ctio
nis
not
rela
ted
toth
ede
sire
toha
ve
�(G)
larg
eto
avoi
din
putc
onst
rain
tsas
disc
usse
din
Sect
ion
6.9.
Inpa
rtic
ular
,th
esc
alin
gs,a
ndth
usth
em
atri
x
G
,are
diff
eren
tfor
the
two
case
s.
10.3
.3Se
lect
ion
ofco
ntro
lled
vari
able
s:Su
mm
ary
Gen
eral
ly,t
heop
timal
valu
esof
allv
aria
bles
will
chan
gew
ithtim
edu
ring
oper
atio
n(d
ueto
dist
urba
nces
and
othe
rch
ange
s).F
orpr
actic
alre
ason
s,w
eha
veco
nsid
ered
ahi
erar
chic
alst
rate
gyw
here
the
optim
izat
ion
ispe
rfor
med
only
peri
odic
ally
.The
ques
tion
isth
en:
Whi
chva
riab
les
(con
trol
led
outp
uts)
shou
ldbe
kept
cons
tant
(bet
wee
nea
chop
timiz
atio
n)?
Ess
entia
lly,w
efo
und
that
we
shou
ldse
lect
vari
able
s
y
for
whi
chth
eva
riat
ion
inop
timal
valu
ean
dco
ntro
lerr
oris
smal
lcom
pare
dto
thei
rco
ntro
llabl
era
nge
(the
rang
e
y
may
reac
hby
vary
ing
the
inpu
t
u
).W
eco
nsid
ered
two
appr
oach
esfo
rse
lect
ing
cont
rolle
dou
tput
s:
1.“B
rute
forc
e”ev
alua
tion
tofin
dth
ese
tw
ithth
esm
alle
stlo
ssim
pose
dby
usin
gco
nsta
ntva
lues
for
the
setp
oint
s
r.2.
Max
imiz
atio
nof
�(G)
whe
re
G
isap
prop
riat
ely
scal
ed(s
eeth
eab
ove
proc
edur
e).
Ifth
elo
ssim
pose
dby
keep
ing
cons
tant
setp
oint
sis
acce
ptab
leth
enw
eha
vese
lf-
optim
izin
gco
ntro
l.T
heob
ject
ive
ofth
eco
ntro
llay
eris
then
toke
epth
eco
ntro
lled
416
MULTIVARIABLEFEEDBACKCONTROL
outp
uts
atth
eirr
efer
ence
valu
es(w
hich
are
com
pute
dby
the
optim
izat
ion
laye
r).T
heco
ntro
lled
outp
uts
are
ofte
nm
easu
red,
butw
em
ayal
soes
timat
eth
eir
valu
esba
sed
onot
her
mea
sure
dva
riab
les.
We
may
also
use
othe
rm
easu
rem
ents
toim
prov
eth
eco
ntro
lof
the
cont
rolle
dou
tput
s,fo
rex
ampl
e,by
use
ofca
scad
eco
ntro
l.T
hus,
the
sele
ctio
nof
cont
rolle
dan
dm
easu
red
outp
uts
are
two
sepa
rate
issu
es,
alth
ough
the
two
deci
sion
sar
eob
viou
sly
clos
ely
rela
ted.
The
mea
sure
men
tse
lect
ion
prob
lem
isbr
iefly
disc
usse
din
the
next
sect
ion.
The
nin
sect
ion
10.5
we
disc
uss
the
rela
tive
gain
arra
yof
the
“big
”tr
ansf
erm
atri
x(w
ithal
lca
ndid
ate
outp
uts
incl
uded
),as
aus
eful
scre
enin
gto
olfo
rse
lect
ing
cont
rolle
dou
tput
s.
10.4
Sele
ctio
nof
man
ipul
atio
nsan
dm
easu
rem
ents
We
are
here
conc
erne
dw
ithth
eva
riab
lese
ts
u
and
v
inFi
gure
10.1
.Not
eth
atth
em
easu
rem
ents
used
byth
eco
ntro
ller
(v
)ar
ein
gene
rald
iffe
rent
from
the
cont
rolle
dva
riab
les
(z
),be
caus
e1)
we
may
notb
eab
leto
mea
sure
allt
heco
ntro
lled
vari
able
s,an
d2)
we
may
wan
tto
mea
sure
and
cont
rola
dditi
onal
vari
able
sin
orde
rto
�
stab
ilize
the
plan
t(or
mor
ege
nera
llych
ange
itsdy
nam
ics)
�
impr
ove
loca
ldis
turb
ance
reje
ctio
n
Stab
iliza
tion
.W
eus
ually
star
tth
eco
ntro
ller
desi
gnby
desi
gnin
ga
(low
er-l
ayer
)co
ntro
ller
tost
abili
zeth
epl
ant.
The
issu
eis
then
:W
hich
outp
uts
(mea
sure
men
ts)
and
inpu
ts(m
anip
ulat
ons)
shou
ldbe
used
for
stab
iliza
tion?
We
shou
ldcl
earl
yav
oid
satu
ratio
nof
the
inpu
ts,
beca
use
this
mak
esth
esy
stem
effe
ctiv
eop
en-l
oop
and
stab
iliza
tion
isim
poss
ible
.A
reas
onab
leob
ject
ive
isth
eref
ore
tom
inim
ize
the
requ
ired
inpu
tus
age
ofth
est
abili
zing
cont
rol
syst
em.
Ittu
rns
out
that
this
isac
hiev
ed,
for
asi
ngle
unst
able
mod
e,by
sele
ctin
gth
eou
tput
(mea
sure
men
t)an
din
put
(man
ipul
atio
n)co
rres
pond
ing
toth
ela
rges
tel
emen
tsin
the
outp
utan
din
put
pole
vect
ors
(yp
and
up
),re
spec
tivel
y(s
eere
mar
kon
page
2)(H
avre
,199
8)(H
avre
and
Skog
esta
d,19
98b)
.T
his
choi
cem
axim
izes
the
(sta
te)
cont
rolla
bilit
yan
dob
serv
abili
tyof
the
unst
able
mod
e.
Loc
aldi
stur
banc
ere
ject
ion.
Form
easu
rem
ents
,the
rule
isge
nera
llyto
sele
ctth
ose
whi
chha
vea
stro
ngre
latio
nshi
pw
ithth
eco
ntro
lled
outp
uts,
orw
hich
may
quic
kly
dete
cta
maj
ordi
stur
banc
ean
dw
hich
toge
ther
with
man
ipul
atio
nsca
nbe
used
for
loca
ldis
turb
ance
reje
ctio
n.
The
sele
cted
man
ipul
atio
nssh
ould
have
ala
rge
effe
cton
the
cont
rolle
dou
tput
s,an
dsh
ould
belo
cate
d“c
lose
”(i
nte
rms
ofdy
nam
icre
spon
se)
toth
eou
tput
san
dm
easu
rem
ents
.
For
am
ore
form
alan
alys
isw
em
ayco
nsid
erth
em
odel
yall=Galluall+Gdalld
.H
ere
CONTROLSTRUCTUREDESIGN
417
�y all=
allc
andi
date
outp
uts
(mea
sure
men
ts)
�uall=
allc
andi
date
inpu
ts(m
anip
ulat
ions
)
The
mod
elfo
ra
part
icul
arco
mbi
natio
nof
inpu
tsan
dou
tput
sis
then
y=Gu+Gdd
whe
re
G=SOGallSI;
Gd=SOGdall
(10.
7)
Her
e
SO
isa
non-
squa
rein
put
“sel
ectio
n”m
atri
xw
itha
1
and
othe
rwis
e
0
’sin
each
row
,and
SI
isa
non-
squa
reou
tput
“sel
ectio
n”m
atri
xw
itha
1
and
othe
rwis
e
0
’sin
each
colu
mn.
For
exam
ple,
with
SO
=I
all
outp
uts
are
sele
cted
,an
dw
ith
SO
=[0
I]
outp
ut1
has
notb
een
sele
cted
.
Toev
alua
teth
eal
tern
ativ
eco
mbi
natio
ns,o
nem
ay,b
ased
on
G
and
Gd
,per
form
anin
put-
outp
utco
ntro
llabi
lity
anal
ysis
asou
tline
din
Cha
pter
6fo
rea
chco
mbi
natio
n(e
.g,c
onsi
dert
hem
inim
umsi
ngul
arva
lue,
RH
P-ze
ros,
inte
ract
ions
,etc
).A
tlea
stth
ism
aybe
usef
ulfo
rel
imin
atin
gso
me
alte
rnat
ives
.Am
ore
invo
lved
appr
oach
,bas
edon
anal
yzin
gac
hiev
able
robu
stpe
rfor
man
ceby
negl
ectin
gca
usal
ity,i
sou
tline
dby
Lee
etal
.(19
95).
Thi
sap
proa
chis
mor
ein
volv
edbo
thin
term
sof
com
puta
tion
time
and
inth
eef
fort
requ
ired
tode
fine
the
robu
stpe
rfor
man
ceob
ject
ive.
An
even
mor
ein
volv
ed(a
ndex
act)
appr
oach
wou
ldbe
tosy
nthe
size
cont
rolle
rsfo
rop
timal
robu
stpe
rfor
man
cefo
rea
chca
ndid
ate
com
bina
tion.
How
ever
,the
num
ber
ofco
mbi
natio
nsha
sa
com
bina
tori
algr
owth
,so
even
asi
mpl
ein
put-
outp
utco
ntro
llabi
lity
anal
ysis
beco
mes
very
time-
cons
umin
gif
ther
ear
em
any
alte
rnat
ives
.For
apl
antw
here
we
wan
tto
sele
ct
m
from
M
cand
idat
em
anip
ulat
ions
,an
dl
from
L
cand
idat
em
easu
rem
ents
,the
num
bero
fpo
ssib
ilitie
sis
� L l�� M m� =
L!
l!(L�l)!
M!
m!(M
�m)!
(10.
8)
Afe
wex
ampl
es:f
or
m=l=1
and
M
=L=2
the
num
bero
fpos
sibi
litie
sis
4;fo
r
m=l=2
and
M
=L=4
itis
36;f
or
m=l=5
and
M
=L=10
itis
6350
4;an
dfo
r
m=M
,l=5
and
L=100
(sel
ectin
g5
mea
sure
men
tsou
tof
100
poss
ible
)th
ere
are
7528
7520
poss
ible
com
bina
tions
.
Rem
ark.
The
num
ber
ofpo
ssib
ilitie
sis
muc
hla
rger
ifw
eco
nsid
eral
lpos
sibl
eco
mbi
natio
nsw
ith1
to
M
inpu
tsan
d1
to
L
outp
uts.
The
num
ber
is(N
ett,
1989
):
P M m=1
P L l=1
� L l�� M m� .F
orex
ampl
e,w
ith
M
=L=2
ther
ear
e4+
2+2+
1=9
cand
idat
es(4
stru
ctur
esw
ithon
ein
put
and
one
outp
ut,2
stru
ctur
esw
ithtw
oin
puts
and
one
outp
ut,2
stru
ctur
esw
ithon
ein
put
and
two
outp
uts,
and
1st
ruct
ure
with
two
inpu
tsan
dtw
oou
tput
s).
One
way
ofav
oidi
ngth
isco
mbi
nato
rial
prob
lem
isto
base
the
sele
ctio
ndi
rect
lyon
the
“big
”m
odel
s
Gall
and
Gdall
.For
exam
ple,
one
may
cons
ider
the
sing
ular
valu
ede
com
posi
tion
and
rela
tive
gain
arra
yof
Gall
asdi
scus
sed
inth
ene
xtse
ctio
n.T
his
rath
ercr
ude
anal
ysis
may
beus
ed,
toge
ther
with
phys
ical
insi
ght,
rule
sof
thum
ban
dsi
mpl
eco
ntro
llabi
lity
mea
sure
s,to
perf
orm
apr
e-sc
reen
ing
inor
der
tore
duce
the
poss
ibili
ties
toa
man
agea
ble
num
ber.
The
seca
ndid
ate
com
bina
tions
can
then
bean
alyz
edm
ore
care
fully
.
418
MULTIVARIABLEFEEDBACKCONTROL
10.5
RG
Afo
rno
n-sq
uare
plan
t
Asi
mpl
ebu
teff
ectiv
esc
reen
ing
tool
for
sele
ctin
gin
puts
and
outp
uts,
whi
chav
oids
the
com
bina
tori
alpr
oble
mju
stm
entio
ned,
isth
ere
lativ
ega
inar
ray
(RG
A)
ofth
e“b
ig”
tran
sfer
mat
rix
Gall
with
all
cand
idat
ein
puts
and
outp
uts
incl
uded
,
�
=
Gall�GyT all
.
Ess
entia
lly,f
orth
eca
seof
man
yca
ndid
ate
man
ipul
atio
ns(i
nput
s)on
em
ayco
nsid
erno
tusi
ngth
ose
man
ipul
atio
nsco
rres
pond
ing
toco
lum
nsin
the
RG
Aw
here
the
sum
ofth
eel
emen
tsis
muc
hsm
alle
rth
an1
(Cao
,199
5).S
imila
rly,
for
the
case
ofm
any
cand
idat
em
easu
red
outp
uts
(orc
ontr
olle
dou
tput
s)on
em
ayco
nsid
erno
tusi
ngth
ose
outp
uts
corr
espo
ndin
gto
row
sin
the
RG
Aw
here
the
sum
ofth
eel
emen
tsis
muc
hsm
alle
rth
an1.
Tose
eth
is,w
rite
the
sing
ular
valu
ede
com
posi
tion
of
Gall
as
Gall=U�VH
=Ur�rVH r
(10.
9)
whe
re
�r
cons
ists
only
ofth
e
r=rank(G)
non-
zero
sing
ular
valu
es,
Ur
cons
ists
ofth
e
r
first
colu
mns
of
U
,an
d
Vr
cons
ists
ofth
e
r
first
colu
mns
of
V
.T
hus,
Vr
cons
ists
ofth
ein
putd
irec
tions
with
ano
n-ze
roef
fect
onth
eou
tput
s,an
d
Ur
cons
ists
ofth
eou
tput
dire
ctio
nsw
eca
naf
fect
(rea
ch)
byus
eof
the
inpu
ts.
Let
e j=[0
���0
1
0
���0]T
bea
unit
vect
orw
itha
1in
posi
tion
j
and
0’s
else
whe
re.T
hen
the
j’th
inpu
tis
uj=eT ju
.Defi
ne
e i
ina
sim
ilar
way
such
that
the
i’th
outp
utis
yi=eT iy
.We
then
have
that
eT jVr
yiel
dsth
epr
ojec
tion
ofa
unit
inpu
tuj
onto
the
effe
ctiv
ein
puts
pace
of
G
,and
we
follo
wC
ao(1
995)
and
defin
e
Projectionforinputj=keT jVrk 2
(10.
10)
whi
chis
anu
mbe
rbe
twee
n0
and
1.Si
mila
rly,
eT iUr
yiel
dsth
epr
ojec
tion
ofa
unit
outp
ut
yi
onto
the
effe
ctiv
e(r
each
able
)ou
tput
spac
eof
G
,and
we
defin
e
Projectionforoutputi=keT iUrk 2
(10.
11)
whi
chis
anu
mbe
rbet
wee
n0
and
1.T
hefo
llow
ing
theo
rem
links
the
inpu
tand
outp
ut(m
easu
rem
ent)
proj
ectio
nto
the
colu
mn
and
row
sum
sof
the
RG
A.
The
orem
10.1
(RG
Aan
din
put
and
outp
utpr
ojec
tion
s.)
The
i’th
row
sum
ofth
eR
GA
iseq
ualt
oth
esq
uare
ofth
e
i’th
outp
utpr
ojec
tion
,and
the
j’th
colu
mn
sum
ofth
eR
GA
iseq
ualt
oth
esq
uare
ofth
e
j’th
inpu
tpro
ject
ion,
i.e.
m X j=1
�ij=keT iUrk2 2;
l X i=1
�ij=keT jVrk2 2
(10.
12)
For
asq
uare
non-
sing
ular
mat
rix
both
the
row
and
colu
mn
sum
sin
(10.
12)a
re1.
CONTROLSTRUCTUREDESIGN
419
Pro
of:
See
App
endi
xA
.4.2
.
2
The
RG
Ais
aus
eful
scre
enin
gto
olbe
caus
eit
need
only
beco
mpu
ted
once
.It
incl
udes
all
the
alte
rnat
ive
inpu
tsan
d/or
outp
uts
and
thus
avoi
dsth
eco
mbi
nato
rial
prob
lem
.Fro
m(1
0.12
)w
ese
eth
atth
ero
wan
dco
lum
nsu
ms
ofth
eR
GA
prov
ide
aus
eful
way
ofin
terp
retin
gth
ein
form
atio
nav
aila
ble
inth
esi
ngul
arve
ctor
s.Fo
rth
eca
seof
extr
ain
puts
the
RG
A-v
alue
sde
pend
onth
ein
put
scal
ing,
and
for
extr
aou
tput
son
the
outp
utsc
alin
g.T
heva
riab
les
mus
tth
eref
ore
besc
aled
prio
rto
the
anal
ysis
.
Exa
mpl
e10
.2C
onsi
der
apl
antw
ith
2
inpu
tsan
d
6
cand
idat
eou
tput
sof
whi
chw
ew
antt
ose
lect
2
.The
plan
tand
its
RG
A-m
atri
xar
e
Gall=
2 6 6 6 6 410
10
10
9
2
1
2
�1
2
2
0
23 7 7 7 7 5;
�=
2 6 6 6 6 4�0:1050
0:6303
0:5742
�0:1008
0:1317
�0:0616
0:4034
0:2101
�0:0042
0:0252
0
0:2969
3 7 7 7 7 5
The
reex
ist
� 6 2� =15
com
bina
tion
sw
ith
2
inpu
tsan
d
2
outp
uts.
The
RG
Am
aybe
usef
ulin
prov
idin
gan
init
ial
scre
enin
g.T
hesi
xro
wsu
ms
ofth
eR
GA
-mat
rix
are
0:5252;0:4734;0:0700;0:6134;0:0210
and
0:2969
.To
max
imiz
eth
eou
tput
proj
ecti
onw
esh
ould
sele
ctou
tput
s1
and
4.Fo
rth
isse
lect
ion
�(G)=2:12
whe
reas
�(Gall)=�2(Gall)=
2:69
wit
hal
lou
tput
sin
clud
ed.
Thi
ssh
ows
that
we
have
not
lost
muc
hga
inin
the
low
-gai
ndi
rect
ion
byus
ing
only
2
ofth
e
6
outp
uts.
How
ever
,the
rear
ea
larg
enu
mbe
rof
othe
rfa
ctor
sth
atde
term
ine
cont
roll
abil
ity,
such
asR
HP
-zer
os,s
ensi
tivi
tyto
unce
rtai
nty,
and
thes
em
ustb
eta
ken
into
acco
untw
hen
mak
ing
the
final
sele
ctio
n.
The
follo
win
gex
ampl
esh
ows
that
alth
ough
the
RG
Ais
anef
ficie
ntsc
reen
ing
tool
,it
mus
tbe
used
with
som
eca
utio
n.
Exa
mpl
e10
.3C
onsi
der
apl
antw
ith
2
inpu
tsan
d
4
cand
idat
eou
tput
sof
whi
chw
ew
antt
ose
lect
2
.We
have
:
Gall=
2 6 410
10
10
9
2
1
2
13 7 5;
�=
2 6 4�2:57
3:27
1:96
�1:43
0:80
�0:42
0:80
�0:42
3 7 5
The
four
row
sum
sof
the
RG
A-m
atri
xar
e
0:70
,
0:53
,
0:38
and
0:38
.T
hus,
tom
axim
ize
the
outp
utpr
ojec
tion
we
shou
ldse
lect
outp
uts
1
and
2
.H
owev
er,
this
yiel
dsa
plan
t
G1
=
� 1010
10
9� w
hich
isil
l-co
ndit
ione
dw
ith
larg
eR
GA
-ele
men
ts,
�(G1)=
� �910
10
�9� ,
and
isli
kely
tobe
diffi
cult
toco
ntro
l.O
nth
eot
her
hand
,se
lect
ing
outp
uts
1
and
3
yiel
ds
G2
=
� 1010
2
1� w
hich
isw
ell-
cond
itio
ned
wit
h
�(G2)=
� �12
2
�1� .
For
com
pari
son,
the
min
imum
sing
ular
valu
esar
e:
�(Gall)=1:05
,�(G1)=0:51
,and
�(G2)=0:70
.
We
disc
uss
onpa
ge43
5th
ese
lect
ion
ofex
tra
mea
sure
men
tsfo
rus
ein
aca
scad
eco
ntro
lsys
tem
.
420
MULTIVARIABLEFEEDBACKCONTROL
10.6
Con
trol
confi
gura
tion
elem
ents
We
now
assu
me
that
the
mea
sure
men
ts,
man
ipul
atio
nsan
dco
ntro
lled
outp
uts
are
fixed
.The
avai
labl
esy
nthe
sis
theo
ries
pres
ente
din
this
book
resu
ltin
am
ultiv
aria
ble
cont
rolle
r
K
whi
chco
nnec
tsal
lav
aila
ble
mea
sure
men
ts/c
omm
ands
(v
)w
ithal
lav
aila
ble
man
ipul
atio
ns(u
),
u=Kv
(10.
13)
(the
vari
able
s
v
will
mos
tlybe
deno
ted
y
inth
efo
llow
ing)
.How
ever
,suc
ha
“big
”(f
ull)
cont
rolle
rm
ayno
tbe
desi
rabl
e.B
yco
ntro
lco
nfigu
ratio
nse
lect
ion
we
mea
nth
epa
rtiti
onin
gof
mea
sure
men
ts/c
omm
ands
and
man
ipul
atio
nsw
ithin
the
cont
rol
laye
r.M
ore
spec
ifica
lly,w
ede
fine
Con
trol
confi
gura
tion
.T
here
stri
ctio
nsim
pose
don
the
over
all
cont
roll
er
K
byde
com
posi
ngit
into
ase
tof
loca
lco
ntro
ller
s(s
ubco
ntro
ller
s,un
its,
elem
ents
,bl
ocks
)w
ith
pred
eter
min
edli
nks
and
wit
ha
poss
ibly
pred
eter
min
edde
sign
sequ
ence
whe
resu
bcon
trol
lers
are
desi
gned
loca
lly.
Ina
conv
entio
nalf
eedb
ack
syst
ema
typi
calr
estr
ictio
non
K
isto
use
aon
ede
gree
-of
-fre
edom
cont
rolle
r(so
that
we
have
the
sam
eco
ntro
llerf
or
r
and
�y
).O
bvio
usly
,th
islim
itsth
eac
hiev
able
perf
orm
ance
com
pare
dto
that
ofa
two
degr
ees
offr
eedo
mco
ntro
ller.
Inot
her
case
sw
em
ayus
ea
two
degr
ees-
of-f
reed
omco
ntro
ller,
but
we
may
impo
seth
ere
stri
ctio
nth
atth
efe
edba
ckpa
rtof
the
cont
rolle
r(K
y
)is
first
desi
gned
loca
llyfo
rdi
stur
banc
ere
ject
ion,
and
then
the
prefi
lter
(Kr
)is
desi
gned
for
com
man
dtr
acki
ng.I
nge
nera
lthi
sw
illlim
itth
eac
hiev
able
perf
orm
ance
com
pare
dto
asi
mul
tane
ous
desi
gn(s
eeal
soth
ere
mar
kon
page
105)
.Sim
ilara
rgum
ents
appl
yto
othe
rca
scad
esc
hem
es.
Som
eel
emen
tsus
edto
build
upa
spec
ific
cont
rolc
onfig
urat
ion
are:
�
Cas
cade
cont
rolle
rs
�
Dec
entr
aliz
edco
ntro
llers
�
Feed
forw
ard
elem
ents
�
Dec
oupl
ing
elem
ents
�
Sele
ctor
s
The
sear
edi
scus
sed
inm
ore
deta
ilbe
low
,and
inth
eco
ntex
tof
the
proc
ess
indu
stry
inSh
insk
ey(1
988)
and
Bal
chen
and
Mum
me
(198
8).F
irst
,som
ede
finiti
ons:
Dec
entr
aliz
edco
ntro
lis
whe
nth
eco
ntro
lsys
tem
cons
ists
ofin
depe
nden
tfee
dbac
kco
ntro
ller
sw
hich
inte
rcon
nect
asu
bset
ofth
eou
tput
mea
sure
men
ts/c
omm
ands
wit
ha
subs
etof
the
man
ipul
ated
inpu
ts.
The
sesu
bset
ssh
ould
not
beus
edby
any
othe
rco
ntro
ller
.
Thi
sde
finiti
onof
dece
ntra
lized
cont
rol
isco
nsis
tent
with
itsus
eby
the
cont
rol
com
mun
ity.
Inde
cent
raliz
edco
ntro
lw
em
ayre
arra
nge
the
orde
ring
of
CONTROLSTRUCTUREDESIGN
421
mea
sure
men
ts/c
omm
ands
and
man
ipul
ated
inpu
tssu
chth
atth
efe
edba
ckpa
rtof
the
over
allc
ontr
olle
rK
in(1
0.13
)has
afix
edbl
ock-
diag
onal
stru
ctur
e.
Cas
cade
cont
rol
isw
hen
the
outp
utfr
omon
eco
ntro
ller
isth
ein
putt
oan
othe
r.T
his
isbr
oade
rth
anth
eco
nven
tiona
ldefi
nitio
nof
casc
ade
cont
rolw
hich
isth
atth
eou
tput
from
one
cont
rolle
ris
the
refe
renc
eco
mm
and
(set
poin
t)to
anot
her.
Fee
dfor
war
del
emen
tsli
nkm
easu
red
dist
urba
nces
and
man
ipul
ated
inpu
ts.
Dec
oupl
ing
elem
ents
link
one
set
ofm
anip
ulat
edin
puts
(“m
easu
rem
ents
”)
wit
han
othe
rse
tof
man
ipul
ated
inpu
ts.T
hey
are
used
toim
prov
eth
epe
rfor
man
ceof
dece
ntra
lize
dco
ntro
lsys
tem
s,an
dar
eof
ten
view
edas
feed
forw
ard
elem
ents
(alt
houg
hth
isis
notc
orre
ctw
hen
we
view
the
cont
rols
yste
mas
aw
hole
)whe
reth
e“
mea
sure
ddi
stur
banc
e”is
the
man
ipul
ated
inpu
tco
mpu
ted
byan
othe
rde
cent
rali
zed
cont
roll
er.
Sele
ctor
sar
eus
edto
sele
ctfo
rco
ntro
l,de
pend
ing
onth
eco
ndit
ions
ofth
esy
stem
,a
subs
etof
the
man
ipul
ated
inpu
tsor
asu
bset
ofth
eou
tput
s.
Inad
ditio
nto
rest
rict
ions
onth
est
ruct
ure
of
K
,w
em
ayim
pose
rest
rict
ions
onth
ew
ay,
orra
ther
inw
hich
sequ
ence
,th
esu
bcon
trol
lers
are
desi
gned
.Fo
rm
ost
deco
mpo
sed
cont
rols
yste
ms
we
desi
gnth
eco
ntro
llers
sequ
entia
lly,s
tart
ing
with
the
“fas
t”or
“inn
er”
or“l
ower
-lay
er”
cont
roll
oops
inth
eco
ntro
lhie
rarc
hy.I
npa
rtic
ular
,th
isis
rele
vant
for
casc
ade
cont
rol
syst
ems,
and
itis
som
etim
esal
sous
edin
the
desi
gnof
dece
ntra
lized
cont
rols
yste
ms.
The
choi
ceof
cont
rol
confi
gura
tion
lead
sto
two
diff
eren
tw
ays
ofpa
rtiti
onin
gth
eco
ntro
lsys
tem
:�
Vert
ical
deco
mpo
siti
on.
Thi
sus
ually
resu
ltsfr
oma
sequ
entia
lde
sign
ofth
eco
ntro
lsy
stem
,e.
g.ba
sed
onca
scad
ing
(ser
ies
inte
rcon
nect
ing)
the
cont
rolle
rsin
ahi
erar
chic
alm
anne
r.
�
Hor
izon
tal
deco
mpo
siti
on.T
his
usua
llyin
volv
esa
set
ofin
depe
nden
tde
cent
ral-
ized
cont
rolle
rs.
Rem
ark
1Se
quen
tial
desi
gnof
ade
cent
raliz
edco
ntro
ller
resu
ltsin
aco
ntro
lsy
stem
whi
chis
deco
mpo
sed
both
hori
zont
ally
(sin
ce
K
isdi
agon
al)
asw
ella
sve
rtic
ally
(sin
ceco
ntro
llers
athi
gher
laye
rsar
etu
ned
with
low
er-l
ayer
cont
rolle
rsin
plac
e).
Rem
ark
2O
fco
urse
,a
perf
orm
ance
loss
isin
evita
ble
ifw
ede
com
pose
the
cont
rol
syst
em.
For
exam
ple,
for
ahi
erar
chic
alde
cent
raliz
edco
ntro
lsys
tem
,if
we
sele
cta
poor
confi
gura
tion
atth
elo
wer
(bas
e)co
ntro
llay
er,t
hen
this
may
pose
fund
amen
tall
imita
tions
onth
eac
hiev
able
perf
orm
ance
whi
chca
nnot
beov
erco
me
byad
vanc
edco
ntro
ller
desi
gns
athi
gher
laye
rs.
The
selim
itatio
nsim
pose
dby
the
low
er-l
ayer
cont
rolle
rsm
ayin
clud
eR
HP-
zero
s(s
eeth
eae
ro-e
ngin
eca
sest
udy
inC
hapt
er12
)or
stro
ngin
tera
ctio
ns(s
eeth
edi
still
atio
nca
sest
udy
inC
hapt
er12
whe
reth
e
LV
-con
figur
atio
nyi
elds
larg
eR
GA
-ele
men
tsat
low
freq
uenc
ies)
.
Inth
isse
ctio
n,w
edi
scus
sca
scad
eco
ntro
llers
and
sele
ctor
s,an
dgi
veso
me
just
ifica
tion
for
usin
gsu
ch“s
ubop
timal
”co
nfigu
ratio
nsra
ther
than
dire
ctly
422
MULTIVARIABLEFEEDBACKCONTROL
desi
gnin
gth
eov
eral
lco
ntro
ller
K
.L
ater
,in
Sect
ion
10.7
,w
edi
scus
sin
mor
ede
tail
the
hier
arch
ical
deco
mpo
sitio
n,in
clud
ing
casc
ade
cont
rol,
part
ially
cont
rolle
dsy
stem
san
dse
quen
tial
cont
rolle
rde
sign
.Fi
nally
,in
Sect
ion
10.8
we
cons
ider
dece
ntra
lized
diag
onal
cont
rol.
10.6
.1C
asca
deco
ntro
lsys
tem
s
We
wan
tto
illus
trat
eho
wa
cont
rols
yste
mw
hich
isde
com
pose
din
tosu
bcon
trol
lers
can
beus
edto
solv
em
ultiv
aria
ble
cont
rol
prob
lem
s.Fo
rsi
mpl
icity
,w
ehe
reus
esi
ngle
-inp
utsi
ngle
-out
put(
SISO
)co
ntro
llers
ofth
efo
rm
ui=Ki(s)(ri�y i)
(10.
14)
whe
re
Ki(s)
isa
scal
ar.N
ote
that
whe
neve
rwe
clos
ea
SISO
cont
roll
oop
we
lose
the
corr
espo
ndin
gin
put,
ui,
asa
degr
eeof
free
dom
,but
atth
esa
me
time
the
refe
renc
e,
r i
,bec
omes
ane
wde
gree
offr
eedo
m.
Itm
aylo
oklik
eit
isno
tpos
sibl
eto
hand
leno
n-sq
uare
syst
ems
with
SISO
cont
rolle
rs.
How
ever
,si
nce
the
inpu
tto
the
cont
rolle
rin
(10.
14)
isa
refe
renc
em
inus
am
easu
rem
ent,
we
can
casc
ade
cont
rolle
rsto
mak
eus
eof
extr
am
easu
rem
ents
orex
tra
inpu
ts.
Aca
scad
eco
ntro
lst
ruct
ure
resu
ltsw
hen
eith
erof
the
follo
win
gtw
osi
tuat
ions
aris
e:
�
The
refe
renc
e
r i
isan
outp
utfr
oman
othe
rcon
trol
ler(
typi
cally
used
fort
heca
seof
anex
tra
mea
sure
men
tyi)
,see
Figu
re10
.4(a
).T
his
isco
nven
tion
alca
scad
eco
ntro
l.
�
The
“mea
sure
men
t”
yi
isan
outp
utfr
oman
othe
rco
ntro
ller
(typ
ical
lyus
edfo
rth
eca
seof
anex
tra
man
ipul
ated
inpu
t
uj
;e.
g.in
Figu
re10
.4(b
)w
here
u2
isth
e“m
easu
rem
ent”
for
cont
rolle
r
K1
).T
his
casc
ade
sche
me
isre
ferr
edto
asin
put
rese
ttin
g.
10.6
.2C
asca
deco
ntro
l:E
xtra
mea
sure
men
ts
Inm
any
case
sw
em
ake
use
ofex
tra
mea
sure
men
ts
y2
(sec
onda
ryou
tput
s)to
prov
ide
loca
ldis
turb
ance
reje
ctio
nan
dlin
eari
zatio
n,or
tore
duce
the
effe
ctof
mea
sure
men
tno
ise.
For
exam
ple,
velo
city
feed
back
isfr
eque
ntly
used
inm
echa
nica
lsys
tem
s,an
dlo
cal
flow
casc
ades
are
used
inpr
oces
ssy
stem
s.L
et
u
beth
em
anip
ulat
edin
put,
y 1
the
cont
rolle
dou
tput
(with
anas
soci
ated
cont
rol
obje
ctiv
e
r1
)an
dy2
the
extr
am
easu
rem
ent.
Cen
tral
ized
(par
alle
l)im
plem
enta
tion
.Ace
ntra
lized
impl
emen
tatio
n
u=K(r�
y) ,
whe
re
K
isa
2-in
put-
1-ou
tput
cont
rolle
r,m
aybe
wri
tten
u=K11(s)(r 1�y 1)+K12(s)(r 2�y 2)
(10.
15)
CONTROLSTRUCTUREDESIGN
423
r 1-+ -
c-K1
-cr 2
+ -
-
K2
-u2
Plan
t-
y 2
6
y 1
q
6
(a)
Ext
ram
easu
rem
ents
y 2
(con
vent
iona
lca
scad
eco
ntro
l)
r u2
-+ -
c-
K1
-u 1
r-+ -
c-
K2
-u 2 q
6
Plan
t
-y
q
6
(b)
Ext
rain
puts
u2
(inp
utre
setti
ng)
Fig
ure
10.4
:Cas
cade
impl
emen
tatio
ns
whe
rein
mos
tca
ses
r 2=0
(sin
cew
edo
not
have
ade
gree
offr
eedo
mto
cont
rol
y 2
).
Cas
cade
impl
emen
tati
on(c
onve
ntio
nalc
asca
deco
ntro
l).T
oob
tain
anim
plem
en-
tatio
nw
ithtw
oSI
SOco
ntro
llers
we
may
casc
ade
the
cont
rolle
rsas
illus
trat
edin
Figu
re10
.4(a
):
r 2=K1(s)(r 1�y 1);
(10.
16)
u2=K2(s)(r 2�y 2);r 2=bu 1
(10.
17)
Not
eth
atth
eou
tput
r 2
from
the
slow
erpr
imar
yco
ntro
ller
K1
isno
tam
anip
ulat
edpl
anti
nput
,but
rath
erth
ere
fere
nce
inpu
tto
the
fast
erse
cond
ary
(ors
lave
)co
ntro
ller
K2
.For
exam
ple,
casc
ades
base
don
mea
suri
ngth
eac
tual
man
ipul
ated
vari
able
(in
whi
chca
se
y2=um
)ar
eco
mm
only
used
tore
duce
unce
rtai
nty
and
nonl
inea
rity
atth
epl
anti
nput
.
With
r 2=
0
in(1
0.15
)th
ere
latio
nshi
pbe
twee
nth
ece
ntra
lized
and
casc
ade
impl
emen
tatio
nis
K11=K2K1
and
K12=K2
.
An
adva
ntag
ew
ithth
eca
scad
eim
plem
enta
tion
isth
atit
mor
ecl
earl
yde
coup
les
the
desi
gnof
the
two
cont
rolle
rs.I
tal
sosh
ows
mor
ecl
earl
yth
at
r2
isno
ta
degr
ee-o
f-fr
eedo
mat
high
erla
yers
inth
eco
ntro
lsys
tem
.Fin
ally
,it
allo
ws
for
inte
gral
actio
nin
both
loop
s(w
here
asus
ually
only
K11
shou
ldha
vein
tegr
alac
tion
in(1
0.15
)).
On
the
othe
rha
nd,
ace
ntra
lized
impl
emen
tatio
nis
bette
rsu
ited
for
dire
ctm
ultiv
aria
ble
synt
hesi
s;se
eth
eve
loci
tyfe
edba
ckfo
rth
ehe
licop
ter
case
stud
yin
Sect
ion
12.2
.
Rem
ark.
Con
side
rco
nven
tiona
lca
scad
eco
ntro
lin
Figu
re10
.4(a
).In
the
gene
ralc
ase
y 1
and
y2
are
notd
irec
tlyre
late
dto
each
othe
r,an
dth
isis
som
etim
esre
ferr
edto
aspa
rall
elca
scad
e
424
MULTIVARIABLEFEEDBACKCONTROL
r 1-+ -
c-
K1
-c
+ -
-
K2
-u2
G2
-c?d 2
++
-q y 2
6
G1
-c?d 1
++
-y 1
q
6
Fig
ure
10.5
:C
omm
onca
seof
casc
ade
cont
rol
whe
reth
epr
imar
you
tput
y 1
depe
nds
dire
ctly
onth
eex
tra
mea
sure
men
t y2
.
cont
rol.
How
ever
,it
isco
mm
onto
enco
unte
rth
esi
tuat
ion
inFi
gure
10.5
whe
re
y 1
depe
nds
dire
ctly
on
y 2
.T
his
isa
spec
ial
case
ofFi
gure
10.4
(a)
with
“Pla
nt”=
� G 1G2
G2
� ,an
dit
is
cons
ider
edfu
rthe
rin
Exa
mpl
e10
.1.
Exe
rcis
e10
.1C
onve
ntio
nalc
asca
deco
ntro
l.W
ith
refe
renc
eto
the
spec
ial
(but
com
mon
)ca
seof
conv
enti
onal
casc
ade
cont
rol
show
nin
Fig
ure
10.5
,M
orar
ian
dZ
afiri
ou(1
989)
conc
lude
that
the
use
ofex
tra
mea
sure
men
tsis
usef
ulun
der
the
foll
owin
gci
rcum
stan
ces:
(a)
The
dist
urba
nce
d2
issi
gnifi
cant
and
G1
isno
n-m
inim
umph
ase.
(b)
The
plan
t
G2
has
cons
ider
able
unce
rtai
nty
asso
ciat
edw
ith
it–
e.g.
apo
orly
know
nno
nlin
ear
beha
viou
r–
and
the
inne
rlo
opse
rves
tore
mov
eth
eun
cert
aint
y.
Inte
rms
ofde
sign
they
reco
mm
ende
dth
at
K2
isfir
stde
sign
edto
min
imiz
eth
eef
fect
of
d2
on
y 1
(wit
h
K1
=0
)an
dth
en
K1
isde
sign
edto
min
imiz
eth
eef
fect
of
d1
on
y 1
.W
ew
ant
tode
rive
conc
lusi
ons
(a)
and
(b)
from
anin
put-
outp
utco
ntro
llab
ilit
yan
alys
is,
and
also
,(c
)ex
plai
nw
hyw
em
aych
oose
tous
eca
scad
eco
ntro
lif
we
wan
tto
use
sim
ple
cont
roll
ers
(eve
nw
ith
d2=0
).
Out
line
ofso
lutio
n:(a
)Not
eth
atif
G1
ism
inim
umph
ase,
then
the
inpu
t-ou
tput
cont
roll
abil
ity
of
G2
and
G1G2
are
inth
eory
the
sam
e,an
dfo
rre
ject
ing
d 2
ther
eis
nofu
ndam
enta
lad
vant
age
inm
easu
ring
y 1
rath
erth
an
y 2
.(b
)T
hein
ner
loop
L2
=
G2K2
rem
oves
the
unce
rtai
nty
ifit
issu
ffici
entl
yfa
st(h
igh
gain
feed
back
)an
dyi
elds
atr
ansf
erfu
ncti
on
(I+L2)�1L2
clos
eto
I
atfr
eque
ncie
sw
here
K1
isac
tive
.(c)
Inm
ost
case
s,su
chas
whe
nP
IDco
ntro
ller
sar
eus
ed,
the
prac
tica
lba
ndw
idth
isli
mit
edby
the
freq
uenc
y
wu
whe
reth
eph
ase
ofth
epl
ant
is
�180Æ
(see
sect
ion
5.12
),so
anin
ner
casc
ade
loop
may
yiel
dfa
ster
cont
rol(
for
reje
ctin
g
d1
and
trac
king
r 1
)if
the
phas
eof
G2
isle
ssth
anth
atof
G1G2
.
Exe
rcis
e10
.2To
illu
stra
teth
ebe
nefit
ofus
ing
inne
rca
scad
esfo
rhi
gh-o
rder
plan
ts,
case
(c)
inth
eab
ove
exam
ple,
cons
ider
Fig
ure
10.5
and
let
G1=
1
(s+1)2
;
G2=
1s+1
We
use
afa
stpr
opor
tion
alco
ntro
ller
K2=25
inth
ein
ner
loop
,whe
reas
aso
mew
hats
low
erP
ID-c
ontr
olle
ris
used
inth
eou
ter
loop
,
K1(s)=Kc
(s+1)2
s(0:1s+1)
;
Kc=5
Sket
chth
ecl
osed
-loo
pre
spon
se.W
hati
sth
eba
ndw
idth
for
each
ofth
etw
olo
ops?
CONTROLSTRUCTUREDESIGN
425
Com
pare
this
wit
hth
eca
sew
here
we
only
mea
sure
y 1
,so
G
=
G1G2
,an
dus
ea
PID
-co
ntro
ller
K(s)
wit
hth
esa
me
dyna
mic
sas
K1(s)
butw
ith
asm
alle
rva
lue
ofKc
.Wha
tis
the
achi
evab
leba
ndw
idth
?F
ind
are
ason
able
valu
efo
r
Kc
(sta
rtin
gw
ith
Kc=1
)an
dsk
etch
the
clos
ed-l
oop
resp
onse
(you
wil
lsee
that
itis
abou
tafa
ctor
5sl
ower
wit
hout
the
inne
rca
scad
e).
10.6
.3C
asca
deco
ntro
l:E
xtra
inpu
ts
Inso
me
case
sw
eha
vem
ore
man
ipul
ated
inpu
tsth
anco
ntro
lled
outp
uts.
The
sem
aybe
used
toim
prov
eco
ntro
lpe
rfor
man
ce.C
onsi
der
apl
ant
with
asi
ngle
cont
rolle
dou
tput
y
and
two
man
ipul
ated
inpu
ts
u1
and
u2
.So
met
imes
u2
isan
extr
ain
put
whi
chca
nbe
used
toim
prov
eth
efa
st(t
rans
ient
)co
ntro
lof
y
,but
ifit
does
noth
ave
suffi
cien
tpow
eror
isto
oco
stly
tous
efo
rlo
ng-t
erm
cont
rol,
then
afte
ra
whi
leit
isre
sett
oso
me
desi
red
valu
e(“
idea
lres
ting
valu
e”).
Cen
tral
ized
(par
alle
l)im
plem
enta
tion
.Ace
ntra
lized
impl
emen
tatio
n
u=K(r�
y)
whe
re
K
isa
1-in
put2
-out
putc
ontr
olle
r,m
aybe
wri
tten
u1=K11(s)(r�y);
u2=K21(s)(r�y)
(10.
18)
Her
etw
oin
puts
are
used
toco
ntro
lone
outp
ut,s
oto
geta
uniq
uest
eady
-sta
tefo
rthe
inpu
ts
u1
and
u2
.W
eus
ually
let
K11
have
inte
gral
cont
rol
whe
reas
K21
does
not.
The
n
u2(t)
will
only
beus
edfo
rtr
ansi
ent
(fas
t)co
ntro
lan
dw
illre
turn
toze
ro(o
rm
ore
prec
isel
yto
itsde
sire
dva
lue
ru2
)as
t!1
.
Cas
cade
impl
emen
tati
on(i
nput
rese
ttin
g).T
oob
tain
anim
plem
enta
tion
with
two
SISO
cont
rolle
rsw
em
ayca
scad
eth
eco
ntro
llers
assh
own
inFi
gure
10.4
(b).
We
agai
nle
tin
putu
2
take
care
ofth
efa
stco
ntro
land
u1
ofth
elo
ng-t
erm
cont
rol.
The
fast
cont
roll
oop
isth
en
u2=K2(s)(r�y)
(10.
19)
The
obje
ctiv
eof
the
othe
rsl
ower
cont
rolle
ris
then
tous
ein
put u
1
tore
seti
nput
u2
toits
desi
red
valu
e
r u2
:
u1=K1(s)(r u2
�y 1);
y 1=u2
(10.
20)
and
we
see
that
the
outp
utfr
omth
efa
stco
ntro
ller
K2
isth
e“m
easu
rem
ent”
for
the
slow
cont
rolle
rK1
.
With
r u2
=0
the
rela
tions
hip
betw
een
the
cent
raliz
edan
dca
scad
eim
plem
enta
tion
is
K11=�K1K2
and
K21=K2
.
The
casc
ade
impl
emen
tatio
nag
ain
has
the
adva
ntag
eof
deco
uplin
gth
ede
sign
ofth
etw
oco
ntro
llers
.Ita
lso
show
sm
ore
clea
rly
that
ru2
,the
refe
renc
efo
ru2
,may
beus
edas
ade
gree
-of-
free
dom
athi
gher
laye
rsin
the
cont
rols
yste
m.F
inal
ly,w
eca
nha
vein
tegr
alac
tion
inbo
th
K1
and
K2
,but
note
that
the
gain
of
K1
shou
ldbe
nega
tive
(if
effe
cts
of
u1
and
u2
on
y
are
both
posi
tive)
.
426
MULTIVARIABLEFEEDBACKCONTROL
Rem
ark
1Ty
pica
lly,t
heco
ntro
llers
ina
casc
ade
syst
emar
etu
ned
one
ata
time
star
ting
with
the
fast
loop
.For
exam
ple,
for
the
cont
rol
syst
emin
Figu
re10
.6w
ew
ould
prob
ably
tune
the
thre
eco
ntro
llers
inth
eor
der
K2
(inn
erca
scad
eus
ing
fast
inpu
t),
K3
(inp
utre
setti
ngus
ing
slow
erin
put)
,and
K1
(fina
ladj
ustm
ento
f
y 1
).
Rem
ark
2In
proc
ess
cont
rol,
the
casc
ade
impl
emen
tatio
nof
inpu
tre
setti
ngis
som
etim
esre
ferr
edto
asva
lve
posi
tion
cont
rol,
beca
use
the
extr
ain
put
u2
,usu
ally
ava
lve,
isre
set
toa
desi
red
posi
tion
byth
eou
ter
casc
ade.
Exe
rcis
e10
.3D
raw
the
bloc
kdi
agra
ms
for
the
two
cent
rali
zed
(par
alle
l)im
plem
enta
tion
sco
rres
pond
ing
toF
igur
e10
.4.
Exe
rcis
e10
.4D
eriv
eth
ecl
osed
-loo
ptr
ansf
erfu
ncti
ons
for
the
effe
ctof
r
on
y
,
u1
and
u2
for
the
casc
ade
inpu
tre
sett
ing
sche
me
inF
igur
e10
.4(b
).A
san
exam
ple
use
G
=
[G11
G12]=
[1
1]
and
use
inte
gral
acti
onin
both
cont
roll
ers,
K1
=
�1=s
and
K2=10=s.
Show
that
inpu
tu2
isre
seta
tste
ady-
stat
e.
Exa
mpl
e10
.4T
wo
laye
rsof
casc
ade
cont
rol.
Con
side
rth
esy
stem
inF
igur
e10
.6w
ith
two
man
ipul
ated
inpu
ts(u
2
and
u3
),on
eco
ntro
lled
outp
ut(y
1
whi
chsh
ould
becl
ose
to
r 1
)an
dtw
om
easu
red
vari
able
s(y
1
and
y 2
).In
put
u2
has
am
ore
dire
ctef
fect
on
y 1
than
does
inpu
t
u3
(the
reis
ala
rge
dela
yin
G3(s))
.Inp
ut
u2
shou
ldon
lybe
used
for
tran
sien
tcon
trol
asit
isde
sira
ble
that
itre
mai
nscl
ose
to
r 3=r u2
.The
extr
am
easu
rem
enty
2
iscl
oser
than
y 1
toth
ein
putu
2
and
may
beus
eful
for
dete
ctin
gdi
stur
banc
es(n
otsh
own)
affe
ctin
g
G1
.
e e e
q- -
-
-
?-
-
-
-
??
r 1r 2
r 3
K1
K2
K3
u2 u
3
G1
G2
G3
-
+
-
+ +
-
qy 1
-
+
+ 6e-
-y 2q
Fig
ure
10.6
:Con
trol
confi
gura
tion
with
two
laye
rsof
casc
ade
cont
rol.
InF
igur
e10
.6co
ntro
ller
s
K1
and
K2
are
casc
aded
ina
conv
enti
onal
man
ner,
whe
reas
cont
roll
ers
K2
and
K3
are
casc
aded
toac
hiev
ein
putr
eset
ting
.The
corr
espo
ndin
geq
uati
ons
are
bu 1=K1(s)(r 1�y1)
(10.
21)
u2=K2(s)(r 2�y2);
r 2=bu 1
(10.
22)
CONTROLSTRUCTUREDESIGN
427
u3=K3(s)(r 3�y3);
y3=u2
(10.
23)
Con
trol
ler
K1
cont
rols
the
prim
ary
outp
ut
y 1
atit
sre
fere
nce
r 1
byad
just
ing
the
“in
put”
bu 1,w
hich
isth
ere
fere
nce
valu
efo
r
y 2
.C
ontr
olle
r
K2
cont
rols
the
seco
ndar
you
tput
y 2
usin
gin
putu
2
.Fin
ally
,con
trol
ler
K3
man
ipul
ates
u3
slow
lyin
orde
rto
rese
tinp
ut
u2
toit
sde
sire
dva
lue
r 3
.
Exe
rcis
e10
.5P
roce
ssco
ntro
lap
plic
atio
n.A
prac
tica
lca
seof
aco
ntro
lsy
stem
like
the
one
inF
igur
e10
.6is
inth
eus
eof
apr
e-he
ater
toke
epth
ere
acto
rte
mpe
ratu
re
y 1
ata
give
nva
lue
r 1
.In
this
case
y 2
may
beth
eou
tlet
tem
pera
ture
from
the
pre-
heat
er,u
2
the
bypa
ssflo
w(w
hich
shou
ldbe
rese
tto
r 3
,say
10
%of
the
tota
lflo
w),
and
u3
the
flow
ofhe
atin
gm
ediu
m(s
team
).M
ake
apr
oces
sflo
wsh
eet
wit
hin
stru
men
tati
onli
nes
(not
abl
ock
diag
ram
)fo
rth
ishe
ater
/rea
ctor
proc
ess.
10.6
.4E
xtra
inpu
tsan
dou
tput
s(l
ocal
feed
back
)
Inm
any
case
spe
rfor
man
cem
aybe
impr
oved
with
loca
lfe
edba
cklo
ops
invo
lvin
gex
tra
man
ipul
ated
inpu
tsan
dex
tra
mea
sure
men
ts.H
owev
er,t
heim
prov
emen
tmus
tbe
trad
edof
fag
ains
tthe
cost
ofth
eex
tra
actu
ator
,mea
sure
men
tand
cont
rols
yste
m.
An
exam
ple
whe
relo
cal
feed
back
isre
quir
edto
coun
tera
ctth
eef
fect
ofhi
gh-o
rder
lags
isgi
ven
fora
neut
raliz
atio
npr
oces
sin
Figu
re5.
24on
page
208.
The
use
oflo
cal
feed
back
isal
sodi
scus
sed
byH
orow
itz(1
991)
.
10.6
.5Se
lect
ors
Split
-ran
geco
ntro
lfor
extr
ain
puts
.We
assu
med
abov
eth
atth
eex
tra
inpu
tis
used
toim
prov
edy
nam
icpe
rfor
man
ce.A
noth
ersi
tuat
ion
isw
hen
inpu
tcon
stra
ints
mak
eit
nece
ssar
yto
add
am
anip
ulat
edin
put.
Inth
isca
seth
eco
ntro
lran
geis
ofte
nsp
litsu
chth
at,
for
exam
ple,
u1
isus
edfo
rco
ntro
lw
hen
y2[ymin;y1],
and
u2
isus
edw
hen
y2[y1;ymax].
Sele
ctor
sfo
rto
ofe
win
puts
.A
com
plet
ely
diff
eren
tsi
tuat
ion
occu
rsif
ther
ear
eto
ofe
win
puts
.Con
side
rthe
case
with
one
inpu
t(
u
)and
seve
ralo
utpu
ts(y
1;y2;:::
).In
this
case
,w
eca
nnot
cont
rol
all
the
outp
uts
inde
pend
ently
,so
we
eith
erne
edto
cont
rola
llth
eou
tput
sin
som
eav
erag
em
anne
r,or
we
need
tom
ake
ach
oice
abou
tw
hich
outp
uts
are
the
mos
tim
port
antt
oco
ntro
l.Se
lect
ors
orlo
gic
switc
hes
are
ofte
nus
edfo
rthe
latte
r.A
ucti
onee
ring
sele
ctor
sar
eus
edto
deci
deto
cont
rolo
neof
seve
ral
sim
ilar
outp
uts.
For
exam
ple,
this
may
beus
edto
adju
stth
ehe
atin
put
(u
)to
keep
the
max
imum
tem
pera
ture
(maxiy i
)in
afir
edhe
ater
belo
wso
me
valu
e.O
verr
ide
sele
ctor
sar
eus
edw
hen
seve
ral
cont
rolle
rsco
mpu
teth
ein
put
valu
e,an
dw
ese
lect
the
smal
lest
(or
larg
est)
asth
ein
put.
For
exam
ple,
this
isus
edin
ahe
ater
whe
reth
ehe
atin
put(
u
)no
rmal
lyco
ntro
lste
mpe
ratu
re(y
1
),ex
cept
whe
nth
epr
essu
re(y
2)
isto
ola
rge
and
pres
sure
cont
rolt
akes
over
.
428
MULTIVARIABLEFEEDBACKCONTROL
10.6
.6W
hyus
eca
scad
ean
dde
cent
raliz
edco
ntro
l?
As
isev
iden
tfr
omFi
gure
10.6
(a),
deco
mpo
sed
cont
rol
confi
gura
tions
can
easi
lybe
com
equ
iteco
mpl
exan
ddi
fficu
ltto
mai
ntai
nan
dun
ders
tand
.Itm
ayth
eref
ore
bebo
thsi
mpl
eran
dbe
tteri
nte
rms
ofco
ntro
lper
form
ance
tose
tup
the
cont
rolle
rdes
ign
prob
lem
asan
optim
izat
ion
prob
lem
and
let
the
com
pute
rdo
the
job,
resu
lting
ina
cent
raliz
edm
ultiv
aria
ble
cont
rolle
ras
used
inot
her
chap
ters
ofth
isbo
ok.
Ifth
isis
the
case
,why
isca
scad
ean
dde
cent
raliz
edco
ntro
luse
din
prac
tice?
The
rear
ea
num
ber
ofre
ason
s,bu
tthe
mos
tim
port
anto
neis
prob
ably
the
cost
asso
ciat
edw
ithob
tain
ing
good
plan
tmod
els,
whi
char
ea
prer
equi
site
fora
pply
ing
mul
tivar
iabl
eco
ntro
l.O
nth
eot
her
hand
,with
casc
ade
and
dece
ntra
lized
cont
role
ach
cont
rolle
ris
usua
llytu
ned
one
ata
time
with
am
inim
umof
mod
ellin
gef
fort
,som
etim
esev
enon
-li
neby
sele
ctin
gon
lya
few
para
met
ers
(e.g
,the
gain
and
inte
gral
time
cons
tant
ofa
PI-c
ontr
olle
r).A
fund
amen
talr
easo
nfo
rap
plyi
ngca
scad
ean
dde
cent
rali
zed
cont
rol
isth
usto
save
onm
odel
ling
effo
rt.S
ince
casc
ade
and
dece
ntra
lized
cont
rols
yste
ms
depe
ndm
ore
stro
ngly
onfe
edba
ckra
ther
than
mod
els
asth
eirs
ourc
eof
info
rmat
ion,
itis
usua
llym
ore
impo
rtan
t(r
elat
ive
toce
ntra
lized
mul
tivar
iabl
eco
ntro
l)th
atth
efa
stco
ntro
lloo
psbe
tune
dto
resp
ond
quic
kly.
Oth
erad
vant
ages
ofca
scad
ean
dde
cent
raliz
edco
ntro
linc
lude
the
follo
win
g:th
eyar
eof
ten
easi
erto
unde
rsta
ndby
oper
ator
s,th
eyre
duce
the
need
for
cont
rol
links
and
allo
wfo
rde
cent
raliz
edim
plem
enta
tion,
thei
rtu
ning
para
met
ers
have
adi
rect
and
“loc
aliz
ed”
effe
ct,a
ndth
eyte
ndto
bele
ssse
nsiti
veto
unce
rtai
nty,
for
exam
ple,
inth
ein
putc
hann
els.
The
issu
eof
sim
plifi
edim
plem
enta
tion
and
redu
ced
com
puta
tion
load
isal
soim
port
anti
nm
any
appl
icat
ions
,but
isbe
com
ing
less
rele
vant
asth
eco
stof
com
putin
gpo
wer
isre
duce
d.
Bas
edon
the
abov
edi
scus
sion
,the
mai
nch
alle
nge
isto
find
aco
ntro
lcon
figur
atio
nw
hich
allo
ws
the
(sub
)con
trol
lers
tobe
tune
din
depe
nden
tlyba
sed
ona
min
imum
ofm
odel
info
rmat
ion
(the
pair
ing
prob
lem
).Fo
rin
dust
rial
prob
lem
s,th
enu
mbe
rof
poss
ible
pair
ings
isus
ually
very
high
,but
inm
ostc
ases
phys
ical
insi
ghta
ndsi
mpl
eto
ols,
such
asth
eR
GA
,ca
nbe
help
ful
inre
duci
ngth
enu
mbe
rof
alte
rnat
ives
toa
man
agea
ble
num
ber.
Tobe
able
totu
neth
eco
ntro
llers
inde
pend
ently
,w
em
ust
requ
ire
that
the
loop
sin
tera
cton
lyto
alim
ited
exte
nt.
For
exam
ple,
one
desi
rabl
epr
oper
tyis
that
the
stea
dy-s
tate
gain
from
ui
to
y i
inan
“inn
er”
loop
(whi
chha
sal
read
ybe
entu
ned)
,do
esno
tch
ange
too
muc
has
oute
rlo
ops
are
clos
ed.
For
dece
ntra
lized
diag
onal
cont
rol
the
RG
Ais
aus
eful
tool
for
addr
essi
ngth
ispa
irin
gpr
oble
m.
Why
dow
ene
eda
theo
ryfo
rca
scad
ean
dde
cent
raliz
edco
ntro
l?W
eju
star
gued
that
the
mai
nad
vant
age
ofde
cent
raliz
edco
ntro
lw
asits
savi
ngon
the
mod
ellin
gef
fort
,but
any
theo
retic
altr
eatm
ento
fde
cent
raliz
edco
ntro
lreq
uire
sa
plan
tmod
el.
Thi
sse
ems
tobe
aco
ntra
dict
ion.
How
ever
,ev
enth
ough
we
may
not
wan
tto
use
am
odel
totu
neth
eco
ntro
llers
,w
em
ayst
illw
ant
tous
ea
mod
elto
deci
deon
aco
ntro
lst
ruct
ure
and
tode
cide
onw
heth
erac
cept
able
cont
rol
with
ade
cent
raliz
ed
CONTROLSTRUCTUREDESIGN
429
confi
gura
tion
ispo
ssib
le.T
hem
odel
ling
effo
rtin
this
case
isle
ss,b
ecau
seth
em
odel
may
beof
am
ore
“gen
eric
”na
ture
and
does
not
need
tobe
mod
ified
for
each
part
icul
arap
plic
atio
n.
10.7
Hie
rarc
hica
land
part
ialc
ontr
ol
Ahi
erar
chic
alco
ntro
lsy
stem
resu
ltsw
hen
we
desi
gnth
esu
bcon
trol
lers
ina
sequ
entia
lm
anne
r,us
ually
star
ting
with
the
fast
loop
s(“
botto
m-u
p”).
Thi
sm
eans
that
the
cont
rolle
rat
som
ehi
gher
laye
rin
the
hier
arch
yis
desi
gned
base
don
apa
rtia
llyco
ntro
lled
plan
t.In
this
sect
ion
we
deri
vetr
ansf
erfu
nctio
nsfo
rpa
rtia
lco
ntro
l,an
dpr
ovid
eso
me
guid
elin
esfo
rde
sign
ing
hier
arch
ical
cont
rols
yste
ms.
10.7
.1P
arti
alco
ntro
l
Part
ialc
ontr
olin
volv
esco
ntro
lling
only
asu
bset
ofth
eou
tput
sfo
rw
hich
ther
eis
aco
ntro
lobj
ectiv
e.W
edi
vide
the
outp
uts
into
two
clas
ses:
�y 1
–(t
empo
rari
ly)
unco
ntro
lled
outp
ut(f
orw
hich
ther
eis
anas
soci
ated
cont
rol
obje
ctiv
e)
�y 2
–(l
ocal
ly)
mea
sure
dan
dco
ntro
lled
outp
ut
We
also
subd
ivid
eth
eav
aila
ble
man
ipul
ated
inpu
tsin
asi
mila
rm
anne
r:�u2
–in
puts
used
for
cont
rolli
ng
y2
�u1
–re
mai
ning
inpu
ts(w
hich
may
beus
edfo
rco
ntro
lling
y1
)
We
have
inse
rted
the
wor
dte
mpo
rari
lyab
ove,
sinc
e
y1
isno
rmal
lya
cont
rolle
dou
tput
atso
me
high
erla
yeri
nth
ehi
erar
chy.
How
ever
,we
here
cons
ider
the
part
ially
cont
rolle
dsy
stem
asit
appe
ars
afte
rhav
ing
impl
emen
ted
only
alo
calc
ontr
olsy
stem
whe
re
u2
isus
edto
cont
roly
2
.In
mos
tof
the
deve
lopm
entt
hatf
ollo
ws
we
assu
me
that
the
outp
uts
y2
are
tight
lyco
ntro
lled.
Four
appl
icat
ions
ofpa
rtia
lcon
trol
are:
1.Se
quen
tial
desi
gnof
dece
ntra
lize
dco
ntro
ller
s.T
heou
tput
s
y
(whi
chin
clud
e
y 1
and
y2
)al
lha
vean
asso
ciat
edco
ntro
lob
ject
ive,
and
we
use
ahi
erar
chic
alco
ntro
lsy
stem
.W
efir
stde
sign
aco
ntro
ller
K2
toco
ntro
lth
esu
bset
y2
.W
ithth
isco
ntro
ller
K2
inpl
ace
(apa
rtia
llyco
ntro
lled
syst
em),
we
may
then
desi
gna
cont
rolle
r K1
for
the
rem
aini
ngou
tput
s.2.
Sequ
enti
alde
sign
ofco
nven
tion
alca
scad
eco
ntro
l.T
heou
tput
s
y2
are
addi
tiona
lm
easu
red
(“se
cond
ary”
)va
riab
les
whi
char
eno
tim
port
ant
vari
able
sin
them
selv
es.T
here
ason
for
cont
rolli
ng
y2
isto
impr
ove
the
cont
rolo
f
y1
.The
430
MULTIVARIABLEFEEDBACKCONTROL
refe
renc
es
r 2
are
used
asde
gree
sof
free
dom
for
cont
rolli
ng
y1
soth
ese
t
u1
isof
ten
empt
y.3.
“Tr
ue”
part
ial
cont
rol.
The
outp
uts
y
(whi
chin
clud
e
y1
and
y2
)al
lha
vean
asso
ciat
edco
ntro
lob
ject
ive,
and
we
cons
ider
whe
ther
byco
ntro
lling
only
the
subs
et
y 2
;we
can
indi
rect
lyac
hiev
eac
cept
able
cont
rolo
f
y1
,tha
tis,
the
outp
uts
y 1
rem
ain
unco
ntro
lled
and
the
setu
1
rem
ains
unus
ed.
4.In
dire
ctco
ntro
l.T
heou
tput
s
y1
have
anas
soci
ated
cont
rol
obje
ctiv
e,bu
tth
eyar
eno
tm
easu
red.
Inst
ead,
we
aim
atin
dire
ctly
cont
rolli
ng
y1
byco
ntro
lling
the
“sec
onda
ry”
mea
sure
dva
riab
les
y2
(whi
chha
veno
asso
ciat
edco
ntro
lobj
ectiv
e).
The
refe
renc
es
r 2
are
used
asde
gree
sof
free
dom
and
the
set
u1
isem
pty.
Thi
sis
sim
ilar
toca
scad
eco
ntro
l,bu
tth
ere
isno
“out
er”
loop
invo
lvin
g
y1
.In
dire
ctco
ntro
lwas
disc
usse
din
Sect
ion
10.7
.4.
The
follo
win
gta
ble
show
sm
ore
clea
rly
the
diff
eren
cebe
twee
nth
efo
urap
plic
atio
nsof
part
ial
cont
rol.
Inal
lca
ses
ther
eis
aco
ntro
lob
ject
ive
asso
ciat
edw
ith
y1
and
afe
edba
cklo
opin
volv
ing
mea
sure
men
tand
cont
rolo
f
y2
.
Mea
sure
men
tand
Con
trol
obje
ctiv
eco
ntro
lof
y1
?fo
r
y2
?
Sequ
entia
ldec
entr
aliz
edco
ntro
lY
esY
esSe
quen
tialc
asca
deco
ntro
lY
esN
o“T
rue”
part
ialc
ontr
olN
oY
esIn
dire
ctco
ntro
lN
oN
o
The
four
prob
lem
sar
ecl
osel
yre
late
d,an
din
all
case
sw
e(1
)w
antt
heef
fect
ofth
edi
stur
banc
eson
y1
tobe
smal
l(w
hen
y2
isco
ntro
lled)
,and
(2)
wan
titt
obe
easy
toco
ntro
ly2
usin
g
u2
(dyn
amic
ally
)..L
etus
deri
veth
etr
ansf
erfu
nctio
nsfo
r
y1
whe
n
y 2
isco
ntro
lled.
One
diffi
culty
isth
atth
isre
quir
esa
sepa
rate
anal
ysis
fore
ach
choi
ceof
y2
and
u2
,and
the
num
bero
falte
rnat
ives
has
aco
mbi
nato
rial
grow
thas
illus
trat
edby
(10.
8).
By
part
ition
ing
the
inpu
tsan
dou
tput
s,th
eov
eral
lmod
el
y=Gu
may
bew
ritte
n
y 1=G11u1+G12u2+Gd1d
(10.
24)
y 2=G21u1+G22u2+Gd2d
(10.
25)
Ass
ume
now
that
feed
back
cont
rolu
2=K2(r2�y2�n2)
isus
edfo
rthe
“sec
onda
ry”
subs
yste
min
volv
ing
u2
and
y2
,see
Figu
re10
.7.B
yel
imin
atin
g
u2
and
y2
,we
then
gett
hefo
llow
ing
mod
elfo
rth
ere
sulti
ngpa
rtia
llyco
ntro
lled
syst
em:
y 1
=
� G 11�G12K2(I+G22K2)�1G21
� u 1+
� G d1�G12K2(I+G22K2)�1Gd2
� d+
G12K2(I+G22K2)�1(r2�n2)
(10.
26)
CONTROLSTRUCTUREDESIGN
431
dd d d
-
-
-
-?
?
-
�
?? ? ��
u1
u2
n2
r 2y 1
d
K2
G11
G12
G21
G22
Gd1
Gd2
+
+ y 2
+ -
+
+
+ +
Fig
ure
10.7
:Par
tialc
ontr
ol
Rem
ark.
(10.
26)
may
bere
wri
tten
inte
rms
oflin
ear
frac
tiona
ltra
nsfo
rmat
ions
.For
exam
ple,
the
tran
sfer
func
tion
from
u1
to
y1
is
Fl(G;�K2)=G11�G12K2(I+G22K2)�
1G21
(10.
27)
Tig
htco
ntro
lof
y2
.In
som
eca
ses
we
can
assu
me
that
the
cont
rol
of
y2
isfa
stco
mpa
red
toth
eco
ntro
lof
y1
.To
obta
inth
em
odel
we
may
form
ally
letK
2!1
in(1
0.26
),bu
titi
sbe
tter
toso
lve
for
u2
in(1
0.25
)to
get
u2=�G�122Gd2d�G�122G21u1+G�122y 2
We
have
here
assu
med
that
G22
issq
uare
and
inve
rtib
le,o
ther
wis
ew
eca
nge
tth
ele
ast-
squa
reso
lutio
nby
repl
acin
g
G�122
byth
eps
eudo
-inv
erse
,Gy 22
.On
subs
titut
ing
this
into
(10.
24)w
ege
t
y 1=(G11�G12G�122G21)
|{z}
,Pu
u1+(Gd1�G12G�122Gd2)
|{z}
,Pd
d+G12G�122
|{z},Pr
(r2�e 2
| {z}y2
)
(10.
28)
whe
re
Pd
isca
lled
the
part
ial
dist
urba
nce
gain
,w
hich
isth
edi
stur
banc
ega
info
ra
syst
emun
der
perf
ect
part
ial
cont
rol,
and
Pu
isth
eef
fect
of
u1
on
y1
with
y 2
perf
ectly
cont
rolle
d.In
man
yca
ses
the
set
u1
isem
pty
(the
rear
eno
extr
ain
puts
).T
head
vant
age
ofth
em
odel
(10.
28)
over
(10.
26)
isth
atit
isin
depe
nden
tof
K2
,but
we
stre
ssth
atit
only
appl
ies
atfr
eque
ncie
sw
here
y2
istig
htly
cont
rolle
d.Fo
rth
eca
seof
tight
cont
rolw
eha
ve
e2,y 2�r 2=n2
,i.e
.,th
eco
ntro
lerr
or
e2
equa
lsth
em
easu
rem
ente
rror
(noi
se)
n2
.
432
MULTIVARIABLEFEEDBACKCONTROL
Rem
ark.
Rel
atio
nshi
pssi
mila
rto
thos
egi
ven
in(1
0.28
)ha
vebe
ende
rive
dby
man
yau
thor
s,e.
g.se
eth
ew
ork
ofM
anou
siou
thak
iset
al.
(198
6)on
bloc
kre
lativ
ega
ins
and
the
wor
kof
Hag
gblo
man
dW
alle
r(1
988)
ondi
still
atio
nco
ntro
lcon
figur
atio
ns.
10.7
.2H
iera
rchi
calc
ontr
olan
dse
quen
tial
desi
gn
Ahi
erar
chic
alco
ntro
lsys
tem
aris
esw
hen
we
appl
ya
sequ
entia
ldes
ign
proc
edur
eto
aca
scad
eor
dece
ntra
lized
cont
rols
yste
m.
The
idea
isto
first
impl
emen
ta
loca
llo
wer
-lay
er(o
rin
ner)
cont
rol
syst
emfo
rco
ntro
lling
the
outp
uts
y2
.N
ext,
with
this
low
er-l
ayer
cont
rol
syst
emin
plac
e,w
ede
sign
aco
ntro
ller
K1
toco
ntro
l
y1
.T
heap
prop
riat
em
odel
for
desi
gnin
g
K1
isgi
ven
by(1
0.26
)(f
orth
ege
nera
lcas
e)or
(10.
28)
(for
the
case
whe
nw
eca
nas
sum
e
y 2
perf
ectly
cont
rolle
d).
The
obje
ctiv
esfo
rth
ishi
erar
chic
alde
com
posi
tion
may
vary
:
1.To
allo
wfo
rsim
ple
orev
enon
-lin
etu
ning
ofth
elo
wer
-lay
erco
ntro
lsys
tem
(K2
).2.
Toal
low
the
use
oflo
nger
sam
plin
gin
terv
als
for
the
high
erla
yers
(K1
).3.
Toal
low
sim
ple
mod
els
whe
nde
sign
ing
the
high
er-l
ayer
cont
rol
syst
em(K
1
).T
hehi
gh-f
requ
ency
dyna
mic
sof
the
mod
els
ofth
epa
rtia
llyco
ntro
lled
plan
t(e.
g.
Pu
and
Pr
)m
aybe
sim
plifi
edif
K1
ism
ainl
yef
fect
ive
atlo
wer
freq
uenc
ies.
4.To
“sta
biliz
e”1
the
plan
tus
ing
alo
wer
-lay
erco
ntro
lsy
stem
(K2
)su
chth
atit
isam
enab
leto
man
ualc
ontr
ol.
The
latte
ris
the
case
inm
any
proc
ess
cont
rol
appl
icat
ions
whe
rew
efir
stcl
ose
anu
mbe
rof
fast
er“r
egul
ator
y”lo
ops
inor
der
to“s
tabi
lize”
the
plan
t.T
hehi
gher
laye
rco
ntro
lsys
tem
(K1
)is
then
used
mai
nly
for
optim
izat
ion
purp
oses
,and
isno
tre
quir
edto
oper
ate
the
plan
t.
Bas
edon
thes
eob
ject
ives
,H
ovd
and
Skog
esta
d(1
993)
prop
osed
som
ecr
iteri
afo
rse
lect
ing
u2
and
y2
for
use
inth
elo
wer
-lay
erco
ntro
lsys
tem
:
1.T
helo
wer
laye
rm
ust
quic
kly
impl
emen
tth
ese
tpoi
nts
com
pute
dby
the
high
erla
yers
,tha
tis
,th
ein
put-
outp
utco
ntro
llabi
lity
ofth
esu
bsys
tem
invo
lvin
gus
eof
u2
toco
ntro
ly2
shou
ldbe
good
(con
side
r
G22
and
Gd2
).2.
The
cont
rol
of
y2
usin
g
u2
shou
ldpr
ovid
elo
cal
dist
urba
nce
reje
ctio
n,th
atis
,it
shou
ldm
inim
ize
the
effe
ctof
dist
urba
nces
on
y1
(con
side
r
Pd
for
y2
tight
lyco
ntro
lled)
.3.
The
cont
rolo
f
y2
usin
g
u2
shou
ldno
tim
pose
unne
cess
ary
cont
roll
imita
tions
onth
ere
mai
ning
cont
rolp
robl
emw
hich
invo
lves
usin
g
u1
and/
orr 2
toco
ntro
ly1
.By
“unn
eces
sary
”w
em
ean
limita
tions
(RH
P-ze
ros,
ill-c
ondi
tioni
ng,e
tc.)
that
did
not
1
The
term
s“s
tabi
lize”
and
“uns
tabl
e”as
used
bypr
oces
sop
erat
ors
may
notr
efer
toa
plan
ttha
tis
unst
able
ina
mat
hem
atic
alse
nse,
but
rath
erto
apl
ant
that
isse
nsit
ive
todi
stur
banc
esan
dw
hich
isdi
fficu
ltto
cont
rol
man
ually
.
CONTROLSTRUCTUREDESIGN
433
exis
tin
the
orig
inal
over
allp
robl
emin
volv
ing
u
and
y
.Con
side
rthe
cont
rolla
bilit
yof
Pu
for
y2
tight
lyco
ntro
lled,
whi
chsh
ould
notb
em
uch
wor
seth
anth
atof
G
.
The
seth
ree
crite
ria
are
impo
rtan
tfor
sele
ctin
gco
ntro
lcon
figur
atio
nsfo
rdi
still
atio
nco
lum
nsas
isdi
scus
sed
inth
ene
xtex
ampl
e.
Exa
mpl
e10
.5C
ontr
olco
nfigu
rati
ons
for
dist
illat
ion
colu
mns
.T
heov
eral
lco
ntro
lpr
oble
mfo
rth
edi
stil
lati
onco
lum
nin
Fig
ure
10.8
has
5in
puts
u=[L
V
D
B
VT]T
(the
sear
eal
lflow
s:re
flux
L
,boi
lup
V
,dis
till
ate
D,b
otto
mflo
w
B
,ove
rhea
dva
pour
VT
)and
5ou
tput
s
y=[yD
xB
MD
MB
p]T
(the
sear
eco
mpo
siti
ons
and
inve
ntor
ies:
top
com
posi
tion
y D
,bo
ttom
com
posi
tion
xB
,co
nden
ser
hold
up
MD
,re
boil
erho
ldup
MB
,pr
essu
re
p
)se
eF
igur
e10
.8.
Thi
spr
oble
mus
uall
yha
sno
inhe
rent
cont
rol
lim
itat
ions
caus
edby
RH
P-z
eros
,but
the
plan
tha
spo
les
finor
clos
eto
the
orig
inan
dne
eds
tobe
stab
iliz
ed.I
nad
diti
on,f
orhi
gh-p
urit
yse
para
tion
sth
e
5�5
RG
A-m
atri
xm
ayha
veso
me
larg
eel
emen
ts.A
noth
erco
mpl
icat
ion
isth
atco
mpo
siti
onm
easu
rem
ents
are
ofte
nex
pens
ive
and
unre
liab
le.
Inm
ost
case
s,th
edi
stil
lati
onco
lum
nis
first
stab
iliz
edby
clos
ing
thre
ede
cent
rali
zed
SISO
loop
sfo
rle
vela
ndpr
essu
reso
y2=[MD
MB
p]T
and
the
rem
aini
ngou
tput
sar
e
y1=[yD
xB
]T
The
thre
eSI
SOlo
ops
for
cont
roll
ing
y 2
usua
lly
inte
ract
wea
kly
and
may
betu
ned
inde
pend
entl
yof
each
othe
r.H
owev
er,
sinc
eea
chle
vel
(tan
k)ha
san
inle
tan
dtw
oou
tlet
flow
s,th
ere
exis
tsm
any
poss
ible
choi
ces
for
u2
(and
thus
for
u1
).B
yco
nven
tion
,eac
hch
oice
(“co
nfigu
rati
on”
)is
nam
edby
the
inpu
ts
u1
left
for
com
posi
tion
cont
rol.
For
exam
ple,
the
“
LV
-con
figur
atio
n”us
edin
man
yex
ampl
esin
this
book
refe
rsto
apa
rtia
lly
cont
roll
edsy
stem
whe
rew
eus
e
u1=[L
V]T
toco
ntro
l
y 1
(and
we
assu
me
that
ther
eis
aco
ntro
lsy
stem
inpl
ace
whi
chus
es
u2
=
[D
B
VT]T
toco
ntro
l
y 2
).T
he
LV
-con
figur
atio
nis
good
from
the
poin
tof
view
that
cont
rolo
fy1
usin
g
u1
isne
arly
inde
pend
ento
fthe
tuni
ngof
the
cont
roll
er
K2
invo
lvin
g
y 2
and
u2
.H
owev
er,
the
prob
lem
ofco
ntro
llin
g
y 1
by
u1
(“pl
ant”
Pu
)is
ofte
nst
rong
lyin
tera
ctiv
ew
ith
larg
est
eady
-sta
teR
GA
-ele
men
tsin
Pu
.
Ano
ther
confi
gura
tion
isth
e
DV
-con
figur
atio
nw
here
u1=[D
V]T
and
thus
u2
=[L
B
VT]T
.In
this
case
,th
est
eady
-sta
tein
tera
ctio
nsfr
om
u1
to
y1
are
gene
rall
ym
uch
less
,an
d
Pu
has
smal
lR
GA
-ele
men
ts.
But
the
mod
elin
(10.
26)
depe
nds
434
MULTIVARIABLEFEEDBACKCONTROL
BxB
LC
LC
MB
MD
V
Fz F
p
L
Dy D
PC
VT
Fig
ure
10.8
:Typ
ical
dist
illat
ion
colu
mn
cont
rolle
dw
ithth
e
LV
-con
figur
atio
n
stro
ngly
on
K2
(i.e
.on
the
tuni
ngof
the
leve
llo
ops)
,an
da
slow
leve
llo
opfo
r
MD
may
intr
oduc
eun
favo
urab
ledy
nam
ics
for
the
resp
onse
from
u 1
to
y1
.
The
rear
eal
som
any
othe
rpo
ssib
leco
nfigu
rati
ons
(cho
ices
for
the
two
inpu
tsin
u 1
);w
ith
five
inpu
tsth
ere
are
10
alte
rnat
ive
confi
gura
tion
s.F
urth
erm
ore,
one
ofte
nal
low
sfo
rth
epo
ssib
ilit
yof
usin
gra
tios
betw
een
flow
s,e.
g.
L=D
,as
poss
ible
degr
ees
offr
eedo
min
u1
,an
dth
issh
arpl
yin
crea
ses
the
num
ber
ofal
tern
ativ
es.
Exp
ress
ions
whi
chdi
rect
lyre
late
the
mod
els
for
vari
ous
confi
gura
tion
s,e.
g.re
lati
onsh
ips
betw
een
PLVu
;PLVd
and
PDV
u
;PDV
d
etc.
,ar
egi
ven
inH
aggb
lom
and
Wal
ler
(198
8)an
dSk
oges
tad
and
Mor
ari
(198
7a).
How
ever
,it
may
besi
mpl
erto
star
tfr
omth
eov
eral
l
5�5
mod
el
G
,and
deri
veth
em
odel
sfo
rth
eco
nfigu
rati
ons
usin
g(1
0.26
)or
(10.
28),
see
also
the
MA
TL
AB
file
onpa
ge50
1.
Tose
lect
ago
oddi
stil
lati
onco
ntro
lco
nfigu
rati
on,
one
shou
ldfir
stco
nsid
erth
epr
oble
mof
cont
roll
ing
leve
lsan
dpr
essu
re(y
2
).T
his
elim
inat
esa
few
alte
rnat
ives
,so
the
final
choi
ceis
base
don
the
2�2
com
posi
tion
cont
rol
prob
lem
(y1
).If
y2
isti
ghtl
yco
ntro
lled
then
none
ofth
eco
nfigu
rati
ons
seem
toyi
eld
RH
P-z
eros
in
Pu
.Im
port
ant
issu
esto
cons
ider
then
are
dist
urba
nce
sens
itiv
ity
(the
part
iald
istu
rban
cega
in
Pd
shou
ldbe
smal
l)an
dth
ein
tera
ctio
ns(t
heR
GA
-ele
men
tsof
Pu
).T
hese
issu
esar
edi
scus
sed
by,
for
exam
ple,
Wal
ler
etal
.(1
988)
and
Skog
esta
det
al.
(199
0).
Ano
ther
impo
rtan
tis
sue
isth
atit
isof
ten
not
desi
rabl
eto
have
tigh
tle
vel
loop
san
dso
me
confi
gura
tion
s,li
keth
e
DV
-con
figur
atio
nm
enti
oned
abov
e,ar
ese
nsit
ive
toth
etu
ning
of
K2
.T
hen
the
expr
essi
ons
for
Pu
and
Pd
,w
hich
are
used
inth
ere
fere
nces
men
tion
edab
ove,
may
nota
pply
.Thi
sis
furt
her
disc
usse
din
Skog
esta
d(1
997)
.
CONTROLSTRUCTUREDESIGN
435
Bec
ause
ofth
epr
oble
ms
ofin
tera
ctio
nsan
dth
ehi
ghco
stof
com
posi
tion
mea
sure
men
ts,
we
ofte
nfin
din
prac
tice
that
only
one
ofth
etw
opr
oduc
tco
mpo
siti
ons
isco
ntro
lled
(“tr
ue”
part
ialc
ontr
ol).
Thi
sis
disc
usse
din
deta
ilin
Exa
mpl
e10
.7be
low
.Ano
ther
com
mon
solu
tion
isto
mak
eus
eof
addi
tion
alte
mpe
ratu
rem
easu
rem
ents
from
the
colu
mn,
whe
reth
eir
refe
renc
eva
lues
are
setb
ya
com
posi
tion
cont
roll
erin
aca
scad
em
anne
r.
Insu
mm
ary,
the
over
all5
�5
dist
illat
ion
cont
rolp
robl
emis
solv
edby
first
desi
gnin
ga
3�3
cont
rolle
r
K2
for
leve
lsan
dpr
essu
re,a
ndth
ende
sign
ing
a
2�2
cont
rolle
r
K1
for
the
com
posi
tion
cont
rol.
Thi
sis
then
aca
seof
(blo
ck)
dece
ntra
lized
cont
rol
whe
reth
eco
ntro
ller
bloc
ks
K1
and
K2
are
desi
gned
sequ
entia
lly(i
nad
ditio
n,th
ebl
ocks
K1
and
K2
may
them
selv
esbe
dece
ntra
lized
).
Sequ
entia
lde
sign
isal
sous
edfo
rth
ede
sign
ofca
scad
eco
ntro
lsy
stem
s.T
his
isdi
scus
sed
next
.
Sequ
enti
alde
sign
ofca
scad
eco
ntro
lsys
tem
s
Con
side
rth
eco
nven
tiona
lcas
cade
cont
rols
yste
min
Figu
re10
.4(a
),w
here
we
have
addi
tiona
l“s
econ
dary
”m
easu
rem
ents
y2
with
noas
soci
ated
cont
rol
obje
ctiv
e,an
dth
eob
ject
ive
isto
impr
ove
the
cont
rolo
fthe
prim
ary
outp
uts
y1
bylo
cally
cont
rolli
ng
y 2
.The
idea
isth
atth
issh
ould
redu
ceth
eef
fect
ofdi
stur
banc
esan
dun
cert
aint
yon
y 1
.
From
(10.
28),
itfo
llow
sth
atw
esh
ould
sele
ctse
cond
ary
mea
sure
men
ts
y2
(and
inpu
ts
u2
)suc
hth
at
kPdki
ssm
alla
ndat
leas
tsm
alle
rtha
n
kGd1k.
Inpa
rtic
ular
,the
sear
gum
ents
appl
yat
high
erfr
eque
ncie
s.Fu
rthe
rmor
e,it
shou
ldbe
easy
toco
ntro
ly1
byus
ing
asde
gree
sof
free
dom
the
refe
renc
es
r2
(for
the
seco
ndar
you
tput
s)or
the
unus
edin
puts
u1
.M
ore
prec
isel
y,w
ew
ant
the
inpu
t-ou
tput
cont
rolla
bilit
yof
the
“pla
nt”
[Pu
Pr]
(or
Pr
ifth
ese
t
u1
isem
pty)
with
dist
urba
nce
mod
el
Pd
,to
bebe
tter
than
that
ofth
epl
ant[
G11
G12]
(or
G12
)w
ithdi
stur
banc
em
odel
Gd1
.
Rem
ark.
Mos
tof
the
argu
men
tsgi
ven
inSe
ctio
n10
.2,f
orth
ese
para
tion
into
anop
timiz
atio
nan
da
cont
roll
ayer
,and
inSe
ctio
n10
.3,f
orth
ese
lect
ion
ofco
ntro
lled
outp
uts,
appl
yto
casc
ade
cont
roli
fthe
term
“opt
imiz
atio
nla
yer”
isre
plac
edby
“pri
mar
yco
ntro
ller”
,and
“con
trol
laye
r”is
repl
aced
by“s
econ
dary
cont
rolle
r”.
Exe
rcis
e10
.6T
hebl
ock
diag
ram
inF
igur
e10
.5sh
ows
aca
scad
eco
ntro
lsy
stem
whe
reth
epr
imar
you
tput
y 1
depe
nds
dire
ctly
onth
eex
tra
mea
sure
men
t
y 2
,so
G12
=
G1G2
,
G22
=
G2
,
Gd1
=
[I
G1]
and
Gd2
=
[0
I].
Show
that
Pd
=
[I
0]
and
Pr
=
G1
and
disc
uss
the
resu
lt.
Not
eth
at
Pr
isth
e“
new
”pl
ant
asit
appe
ars
wit
hth
ein
ner
loop
clos
ed.
436
MULTIVARIABLEFEEDBACKCONTROL
10.7
.3“T
rue”
part
ialc
ontr
ol
We
here
cons
ider
the
case
whe
rew
eat
tem
ptto
leav
ea
set
ofpr
imar
you
tput
s
y1
unco
ntro
lled.
Thi
s“t
rue”
part
ialc
ontr
olm
aybe
poss
ible
inca
ses
whe
reth
eou
tput
sar
eco
rrel
ated
such
that
cont
rolli
ngth
eou
tput
s
y2
indi
rect
lygi
ves
acce
ptab
leco
ntro
lof
y1
.One
just
ifica
tion
forp
artia
lcon
trol
isth
atm
easu
rem
ents
,act
uato
rsan
dco
ntro
llin
ksco
stm
oney
,and
we
ther
efor
epr
efer
cont
rols
chem
esw
ithas
few
cont
roll
oops
aspo
ssib
le.
Toan
alyz
eth
efe
asib
ility
ofpa
rtia
lcon
trol
,con
side
rthe
effe
ctof
dist
urba
nces
onth
eun
cont
roll
edou
tput
(s)y
1
asgi
ven
by(1
0.28
).Su
ppos
eal
lvar
iabl
esha
vebe
ensc
aled
asdi
scus
sed
inSe
ctio
n1.
4.T
hen
we
have
that
:
�
Ase
tof
outp
uts
y1
may
bele
ftun
cont
roll
edon
lyif
the
effe
cts
ofal
ldi
stur
banc
eson
y1
,as
expr
esse
dby
the
elem
ents
inth
eco
rres
pond
ing
part
iald
istu
rban
cega
inm
atri
x
Pd
,are
less
than
1in
mag
nitu
deat
allf
requ
enci
es.
The
refo
re,t
oev
alua
teth
efe
asib
ility
ofpa
rtia
lco
ntro
lon
em
ust
for
each
choi
ceof
cont
rolle
dou
tput
s(y
2
)an
dco
rres
pond
ing
inpu
ts(u
2
),re
arra
nge
the
syst
emas
in(1
0.24
)and
(10.
25)a
ndco
mpu
te
Pd
usin
g(1
0.28
).
The
rem
ayal
sobe
chan
ges
in
r2
(of
mag
nitu
de
R2
)w
hich
may
bere
gard
edas
dist
urba
nces
onth
eun
cont
rolle
dou
tput
s
y1
.Fro
m(1
0.28
)the
n,w
eal
soha
veth
at:
�
Ase
tof
outp
uts
y1
may
bele
ftun
cont
roll
edon
lyif
the
effe
cts
ofal
lre
fere
nce
chan
ges
inth
eco
ntro
lled
outp
uts
(y2
)on
y1
,as
expr
esse
dby
the
elem
ents
inth
em
atri
x
G12G�122R2
,are
less
than
1
inm
agni
tude
atal
lfre
quen
cies
.
One
unco
ntro
lled
outp
utan
don
eun
used
inpu
t.“T
rue”
part
ial
cont
rol
isof
ten
cons
ider
edif
we
have
an
m�m
plan
t
G(s)
whe
reac
cept
able
cont
rol
ofal
l
m
outp
uts
isdi
fficu
lt,an
dw
eco
nsid
erle
avin
gon
ein
put
uj
unus
edan
don
eou
tput
y i
unco
ntro
lled.
Inth
isca
se,
asan
alte
rnat
ive
tore
arra
ngin
g
y
into
� y 1 y2
� and
u
into
� u 1 u2
� for
each
cand
idat
eco
ntro
lcon
figur
atio
nan
dco
mpu
ting
Pd
from
(10.
28),
we
may
dire
ctly
eval
uate
the
part
ial
dist
urba
nce
gain
base
don
the
over
all
mod
el
y=Gu+Gdd
.The
effe
ctof
adi
stur
banc
e
dk
onth
eun
cont
rolle
dou
tput
yi
is
Pdk
=� @yi
@dk
� uj=0;yl6=i=0
=[G�1Gd] jk
[G�1] ji
(10.
29)
whe
re“u
j=0;yl6=i=0
”m
eans
that
inpu
tuj
isco
nsta
nt(u
nuse
d)an
dth
ere
mai
ning
outp
uts
yl6=i
are
cons
tant
(per
fect
lyco
ntro
lled)
.
Pro
ofof
(10.
29):
The
proo
fis
from
Skog
esta
dan
dW
olff
(199
2).R
ewri
te
y=Gu+[Gd] kdk
as
u=G�1y�[G�1] kGdd
.Set
yl=0
for
alll
6=i.
The
nuj=[G�1] jiyi�[G�1Gd] jkdk
and
byse
tting
uj=0
we
find
yi=dk=[G�1Gd] jk=[G�1] ji
.
2
CONTROLSTRUCTUREDESIGN
437
We
wan
t
Pdk
smal
lso
from
(10.
29)
we
deri
vedi
rect
insi
ght
into
how
tose
lect
the
unco
ntro
lled
outp
utan
dun
used
inpu
t:
1.Se
lect
the
unus
edin
putu
j
such
that
the
j’th
row
in
G�1Gd
has
smal
lel
emen
ts.
Tha
tis,
keep
the
inpu
tcon
stan
t(un
used
)if
itsde
sire
dch
ange
issm
all.
2.Se
lect
the
unco
ntro
lled
outp
ut
yi
and
unus
edin
putu
j
such
that
the
ji
’th
elem
ent
in
G�1
isla
rge.
Tha
tis,
keep
anou
tput
unco
ntro
lled
ifit
isin
sens
itive
toch
ange
sin
the
unus
edin
putw
ithth
eot
her
outp
uts
cont
rolle
d.
Exa
mpl
e10
.6C
onsi
der
the
FC
Cpr
oces
sin
Exe
rcis
e6.
16on
page
250
wit
h
G(0)=
" 16:8
30:5
4:30
�16:7
31:0
�1:41
1:27
54:1
5:40
# ;
G�1(0)=
" 0:09
0:02
�0:06
0:03
0:03
�0:02
�0:34
�0:32
0:38
#
whe
rew
ew
antt
ole
ave
one
inpu
tunu
sed
and
one
outp
utun
cont
roll
ed.F
rom
the
seco
ndru
le,
sinc
eal
lel
emen
tsin
the
thir
dro
wof
G�1
are
larg
e,it
seem
sre
ason
able
tole
tin
put
u3
beun
used
,as
isdo
nein
Exe
rcis
e6.
16.(
The
outp
uts
are
mai
nly
sele
cted
toav
oid
the
pres
ence
ofR
HP
-zer
os,s
eeE
xerc
ise
6.16
).
(10.
29)
may
bege
nera
lized
toth
eca
sew
ithse
vera
lun
cont
rolle
dou
tput
s/
unus
edin
puts
(Zha
oan
dSk
oges
tad,
1997
).W
efir
stre
orde
r
G
such
that
the
uppe
rle
ft
11
-su
bsys
tem
cont
ains
the
unco
ntro
lled
and
unus
edva
riab
les.
If
G
(and
thus
G11
)is
squa
re,w
eth
enha
ve
Pd=
�� G�1� 1
1� �1�G�1Gd
� 1
(10.
30)
Thi
sre
sult
isde
rive
dfr
omth
ede
finiti
onof
Pd
in(1
0.28
)by
mak
ing
use
ofth
eSc
hur
com
plem
enti
n(A
.7).
We
next
cons
ider
a
2�2
dist
illat
ion
proc
ess
whe
reit
isdi
fficu
ltto
cont
rol
both
outp
uts
inde
pend
ently
due
tost
rong
inte
ract
ions
,an
dw
ele
ave
one
outp
ut(y
1
)un
cont
rolle
d.To
impr
ove
the
perf
orm
ance
of
y1
we
also
cons
ider
the
use
offe
edfo
rwar
dco
ntro
lw
here
u1
isad
just
edba
sed
onm
easu
ring
the
dist
urba
nce
(but
we
need
nom
easu
rem
ento
f
y1
).
Exa
mpl
e10
.7P
arti
alan
dfe
edfo
rwar
dco
ntro
lof
2�2
dist
illat
ion
proc
ess.
Con
side
ra
dist
illa
tion
proc
ess
wit
h
2
inpu
ts(r
eflux
L
and
boil
up
V
),
2
outp
uts
(pro
duct
com
posi
tion
s
yD
and
xB
)an
d
2
dist
urba
nces
(fee
dflo
wra
te
F
and
feed
com
posi
tion
z F
).W
eas
sum
eth
atch
ange
sin
the
refe
renc
e(r
1
and
r 2
)ar
ein
freq
uent
and
they
wil
lnot
beco
nsid
ered
.Ats
tead
y-st
ate
(s=0
)w
eha
ve
G=
� 87:8
�86:4
108:2
�109:6
� ;Gd=
� 7:88
8:81
11:72
11:19
� ;G�1Gd=
� �0:54
�0:005
�0:64
�0:107
�
(10.
31)
Sinc
eth
ero
wel
emen
tsin
G�1Gd
are
sim
ilar
inm
agni
tude
asar
eal
soth
eel
emen
tsof
G�1
(bet
wee
n
0:3
and
0:4
),th
eru
les
foll
owin
g(1
0.29
)do
notc
lear
lyfa
vour
any
part
icul
arpa
rtia
l
438
MULTIVARIABLEFEEDBACKCONTROL
cont
rol
sche
me.
Thi
sis
confi
rmed
byth
eva
lues
of
Pd
,whi
char
ese
ento
bequ
ite
sim
ilar
for
the
four
cand
idat
epa
rtia
lcon
trol
sche
mes
:
P2;2d1
=� �1:36
�0:011
� T ;P
2;1d1
=� �1:63
�0:27
� T ;P
1;2d2
=� 1:72
0:014
� T ;P
1;1d2
=� 2:00
0:33
� T
The
supe
rscr
ipts
deno
teth
eco
ntro
lled
outp
utan
dco
rres
pond
ing
inpu
t.Im
port
antl
y,in
all
four
case
s,th
em
agni
tude
sof
the
elem
ents
in
Pd
are
muc
hsm
alle
rth
anin
Gd
,so
cont
rol
ofon
eou
tput
sign
ifica
ntly
redu
ces
the
effe
ctof
the
dist
urba
nces
onth
eun
cont
roll
edou
tput
.In
part
icul
ar,
this
isth
eca
sefo
rdi
stur
banc
e
2
,for
whi
chth
ega
inis
redu
ced
from
abou
t
10
to
0:33
and
less
.
Let
usco
nsid
erin
mor
ede
tail
sche
me
1
whi
chha
sth
esm
alle
stdi
stur
banc
ese
nsit
ivit
yfo
rth
eun
cont
roll
edou
tput
(P2;2d1
).T
his
sche
me
corr
espo
nds
toco
ntro
llin
gou
tput
y 2
(the
bott
omco
mpo
siti
on)
usin
g
u2
(the
boil
up
V
)an
dw
ith
y 1
(the
top
com
posi
tion
)un
cont
roll
ed.W
eus
ea
dyna
mic
mod
elw
hich
incl
udes
liqu
idflo
wdy
nam
ics;
the
mod
elis
give
nin
Sect
ion
12.4
.F
requ
ency
-dep
ende
ntpl
ots
of
Gd
and
Pd
show
that
the
conc
lusi
onat
stea
dyst
ate
also
appl
ies
athi
gher
freq
uenc
ies.
Thi
sis
illu
stra
ted
inF
igur
e10
.9,
whe
rew
esh
owfo
rth
eun
cont
roll
edou
tput
y 1
and
the
wor
stdi
stur
banc
e
d 1
both
the
open
-loo
pdi
stur
banc
ega
in(G
d11
,Cur
ve
1
)an
dth
epa
rtia
ldi
stur
banc
ega
in(P
2;2d11
,Cur
ve
2
).Fo
rdi
stur
banc
e
d2
the
part
ial
dist
urba
nce
gain
(not
show
n)re
mai
nsbe
low
1
atal
lfre
quen
cies
.
10−
310
−2
10−
110
010
110
−2
10−
1
100
101
1.
Gd11
2.
P2;2d11
3.A
ddst
atic
Feed
-For
war
d
4.20
%er
ror
inFF
5.A
ddfil
ter
toFF
Freq
uenc
y[r
ad/m
in]
Magnitude
Fig
ure
10.9
:Eff
ecto
fdi
stur
banc
e1
onou
tput
1fo
rdi
still
atio
nco
lum
nex
ampl
e
The
part
iald
istu
rban
cega
info
rdi
stur
banc
e
d 1
(the
feed
flow
rate
F
)is
som
ewha
tabo
ve1
atlo
wfr
eque
ncie
s(P
d(0)=�1:36
),so
letu
sne
xtco
nsid
erho
ww
em
ayre
duce
its
effe
cton
y 1
.O
neap
proa
chis
tore
duce
the
dist
urba
nce
itse
lf,fo
rex
ampl
e,by
inst
alli
nga
buffe
rta
nk(a
sin
pH-e
xam
ple
inC
hapt
er5.
16.3
).H
owev
er,
abu
ffer
tank
has
noef
fect
atst
eady
-sta
te,
soit
does
noth
elp
inth
isca
se.
Ano
ther
appr
oach
isto
inst
alla
feed
forw
ard
cont
roll
erba
sed
onm
easu
ring
d 1
and
adju
stin
g
u1
(the
reflu
x
L
)w
hich
isso
far
unus
ed.
Inpr
acti
ce,
this
isea
sily
impl
emen
ted
asa
rati
oco
ntro
ller
whi
chke
eps
L=F
cons
tant
.T
his
elim
inat
esth
est
eady
-sta
teef
fect
of
d 1
on
y 1
(pro
vide
dth
eot
her
cont
rol
loop
iscl
osed
).In
term
sof
our
line
arm
odel
,th
em
athe
mat
ical
equi
vale
nce
ofth
isra
tio
cont
roll
eris
tous
e
u1
=
0:54d1
,w
here
0:54
isth
e
1;1
-ele
men
tin
�G�1Gd
.T
heef
fect
ofth
edi
stur
banc
eaf
ter
incl
udin
gth
isst
atic
feed
forw
ard
cont
roll
er
CONTROLSTRUCTUREDESIGN
439
issh
own
ascu
rve
3
inF
igur
e10
.9.
How
ever
,du
eto
mea
sure
men
ter
ror
we
cann
otac
hiev
epe
rfec
tfee
dfor
war
dco
ntro
l,so
letu
sas
sum
eth
eer
ror
is
20
%,a
ndus
e
u1=1:2�0:54d1
.The
stea
dy-s
tate
effe
ctof
the
dist
urba
nce
isth
en
Pd(0)(1�1:2)=1:36�0:2=0:27
,w
hich
isst
illa
ccep
tabl
e.B
ut,a
sse
enfr
omth
efr
eque
ncy-
depe
nden
tpl
ot(c
urve
4),
the
effe
ctis
abov
e
0:5
athi
gher
freq
uenc
ies,
whi
chm
ayno
tbe
desi
rabl
e.T
here
ason
for
this
unde
sira
ble
peak
isth
atth
efe
edfo
rwar
dco
ntro
ller
,whi
chis
pure
lyst
atic
,rea
cts
too
fast
,and
infa
ctm
akes
the
resp
onse
wor
seat
high
erfr
eque
ncie
s(a
sse
enw
hen
com
pari
ngcu
rves
3
and
4
wit
hcu
rve
2
).To
avoi
dth
isw
efil
ter
the
feed
forw
ard
acti
onw
ith
ati
me
cons
tant
of
3
min
resu
ltin
gin
the
foll
owin
gfe
edfo
rwar
dco
ntro
ller
:
u1=
0:54
3s+1
d1
(10.
32)
Tobe
real
isti
cw
eag
ain
assu
me
aner
ror
of20
%.
The
resu
ltin
gef
fect
ofth
edi
stur
banc
eon
the
unco
ntro
lled
outp
utis
show
nby
curv
e5
,and
we
see
that
the
effe
ctis
now
less
than
0:27
atal
lfre
quen
cies
,so
the
perf
orm
ance
isac
cept
able
.
Rem
ark.
Inth
eex
ampl
eth
ere
are
four
alte
rnat
ive
part
ial
cont
rols
chem
esw
ith
quit
esi
mil
ardi
stur
banc
ese
nsit
ivit
yfo
rth
eun
cont
roll
edou
tput
.To
deci
deon
the
best
sche
me,
we
shou
ldal
sope
rfor
ma
cont
roll
abil
ity
anal
ysis
ofth
efe
edba
ckpr
oper
ties
ofth
efo
ur
1�1
prob
lem
s.Pe
rfor
min
gsu
chan
anal
ysis
,w
efin
dth
atsc
hem
es
1
(the
one
chos
en)
and
4
are
pref
erab
le,
beca
use
the
inpu
tin
thes
etw
oca
ses
has
am
ore
dire
ctef
fect
onth
eou
tput
,and
wit
hle
ssph
ase
lag.
Inco
nclu
sion
,for
this
exam
ple
itis
diffi
cult
toco
ntro
lbot
hou
tput
ssi
mul
tane
ousl
yus
ing
feed
back
cont
rol
due
tost
rong
inte
ract
ions
.How
ever
,we
can
alm
ost
achi
eve
acce
ptab
leco
ntro
lof
both
outp
uts
byle
avin
g
y1
unco
ntro
lled.
The
effe
ctof
the
mos
tdi
fficu
ltdi
stur
banc
eon
y1
can
befu
rthe
rre
duce
dus
ing
asi
mpl
efe
edfo
rwar
dco
ntro
ller
(10.
32)f
rom
dist
urba
nce
d1
to
u1
.
10.7
.4M
easu
rem
ent
sele
ctio
nfo
rin
dire
ctco
ntro
l
Ass
ume
the
over
all
goal
isto
keep
som
eva
riab
le
y1
ata
give
nva
lue
(set
poin
t)
r1
,e.
g.ou
rob
ject
ive
isto
min
imiz
e
J=ky1�r 1k.
We
assu
me
we
cann
otm
easu
re
y1
,an
din
stea
dw
eat
tem
ptto
achi
eve
our
goal
byco
ntro
lling
y2
ata
cons
tant
valu
e
r 2
.Fo
rsm
allc
hang
esw
em
ayas
sum
elin
eari
tyan
dw
rite
y 1=G1u+Gd1d
(10.
33)
y 2=G2u+Gd2d
(10.
34)
With
feed
back
cont
rolo
f
y2
we
get
y2=r 2+e 2
whe
re
e 2
isth
eco
ntro
lerr
or.W
eno
wfo
llow
the
deri
vatio
nth
atle
dto
Pd
in(1
0.28
):So
lvin
gfo
r
u2
in(1
0.34
)an
dsu
bstit
utin
gin
to(1
0.33
)yie
lds
y 1=(Gd1�G1G�12
Gd2)d+G1G�12
(r2+e 2)
440
MULTIVARIABLEFEEDBACKCONTROL
With
e 2=0
and
d=0
this
give
s
y1=G1G�12
r 2
,so
r 2
mus
tbe
chos
ensu
chth
at
r 1=G1G�12
r 2
(10.
35)
heco
ntro
lerr
orin
the
prim
ary
outp
utis
then
y 1�r 1=(Gd1�G1G�12
Gd2)
|{z}
Pd
d+G1G�12
|{z}
Pr
e 2
(10.
36)
Tom
inim
ize
J=ky1�r 1kw
esh
ould
ther
efor
ese
lect
cont
rolle
dou
tput
ssu
chth
at
kPddka
nd
kPre 2ka
resm
all.
Not
eth
at
Pd
depe
nds
onth
esc
alin
gof
dist
urba
nces
d
and
“pri
mar
y”ou
tput
s
y1
(and
isin
depe
nden
tof
the
scal
ing
ofin
puts
u
and
sele
cted
outp
uts
y2
,atl
east
for
squa
repl
ants
).T
hem
agni
tude
ofth
eco
ntro
lerr
or
e2
depe
nds
onth
ech
oice
ofou
tput
s
y2
.B
ased
on(1
0.36
)a
proc
edur
efo
rse
lect
ing
cont
rolle
dou
tput
s
y2
may
besu
gges
ted:
Scal
eth
edi
stur
banc
es
d
tobe
ofm
agni
tude
1(a
sus
ual)
,an
dsc
ale
the
outp
uts
y2
soth
atth
eex
pect
edco
ntro
ler
ror
e2
(mea
sure
men
tno
ise)
isof
mag
nitu
de1
for
each
outp
ut(t
his
isdi
ffer
entf
rom
the
outp
utsc
alin
gus
edin
othe
rca
ses)
.The
nto
min
imiz
eth
eco
ntro
ler
ror
for
the
prim
ary
outp
uts,
J=ky1�r1k,
we
shou
ldse
lect
sets
ofco
ntro
lled
outp
uts
whi
ch:
Minimizesk[Pd
Pr]k
(10.
37)
Rem
ark
1T
hech
oice
ofno
rmin
(10.
37)
isus
ually
ofse
cond
ary
impo
rtan
ce.T
hem
axim
umsi
ngul
arva
lue
aris
esif
kdk 2�1
and
ke2k 2�1
,and
we
wan
tto
min
imiz
e
ky1�r 1k 2
.
Rem
ark
2Fo
rth
ech
oice
y 2=y1
we
have
that
r 1=r 2
isin
depe
nden
tof
d
and
the
mat
rix
Pd
in(1
0.36
)an
d(1
0.37
)is
zero
.How
ever
,Pr
isst
illno
n-ze
ro.
Rem
ark
3In
som
eca
ses
this
mea
sure
men
tse
lect
ion
prob
lem
invo
lves
atr
ade-
off
betw
een
wan
ting
kPdks
mal
l(w
antin
ga
stro
ngco
rrel
atio
nbe
twee
nm
easu
red
outp
uts
y 2
and
“pri
mar
y”ou
tput
s
y 1
)an
dw
antin
g
kPrks
mal
l(w
antin
gth
eef
fect
ofco
ntro
lerr
ors
(mea
sure
men
tnoi
se)
tobe
smal
l).F
orex
ampl
e,th
isis
the
case
ina
dist
illat
ion
colu
mn
whe
nw
eus
ete
mpe
ratu
res
insi
deth
eco
lum
n(y
2
)fo
rin
dire
ctco
ntro
lof
the
prod
uctc
ompo
sitio
ns(y
1
).Fo
rahi
gh-p
urity
sepa
ratio
n,w
eca
nnot
plac
eth
em
easu
rem
entt
oocl
ose
toth
eco
lum
nen
ddu
eto
sens
itivi
tyto
mea
sure
men
terr
or(kP
rkb
ecom
esla
rge)
,and
we
cann
otpl
ace
itto
ofa
rfr
omth
eco
lum
nen
ddu
eto
sens
itivi
tyto
dist
urba
nces
( kPdkb
ecom
esla
rge)
.
Rem
ark
4In
dire
ctco
ntro
lis
rela
ted
toth
eid
eaof
infe
rent
ial
cont
rol
whi
chis
com
mon
lyus
edin
the
proc
ess
indu
stry
.H
owev
er,
inin
fere
ntia
lco
ntro
lth
eid
eais
usua
llyto
use
the
mea
sure
men
tof
y 2
toes
timat
e(i
nfer
)
y1
and
then
toco
ntro
lth
ises
timat
era
ther
than
cont
rolli
ng
y 2
dire
ctly
,e.
g.se
eSt
epha
nopo
ulos
(198
4).
How
ever
,th
ere
isno
univ
ersa
lag
reem
ento
nth
ese
term
s,an
dM
arlin
(199
5)us
esth
ete
rmin
fere
ntia
lcon
trol
tom
ean
indi
rect
cont
rola
sdi
scus
sed
abov
e.
Rem
ark
5T
hepr
oble
mof
indi
rect
cont
rol
iscl
osel
yre
late
dto
that
ofca
scad
eco
ntro
ldi
scus
sed
inSe
ctio
n10
.7.2
.The
mai
ndi
ffer
ence
isth
atin
casc
ade
cont
rolw
eal
som
easu
rean
dco
ntro
ly1
inan
oute
rlo
op.I
nth
isca
sew
ew
antk
[Pd
Pr]k
smal
lonl
yat
high
freq
uenc
ies
beyo
ndth
eba
ndw
idth
ofth
eou
ter
loop
invo
lvin
g
y 1
.
CONTROLSTRUCTUREDESIGN
441
10.8
Dec
entr
aliz
edfe
edba
ckco
ntro
l
d d
q q
- -
-
-
- -
- -
? 6
r 1 r 2
++- -
k1
k2
u1
u2
G(s)
K(s)
y 1 y 2
Fig
ure
10.1
0:D
ecen
tral
ized
diag
onal
cont
rolo
fa
2�2
plan
t
Inth
isse
ctio
n,
G(s)
isa
squa
repl
ant
whi
chis
tobe
cont
rolle
dus
ing
adi
agon
alco
ntro
ller
(see
Figu
re10
.10)
K(s)=diagfki(s)g=
2 6 6 6 4k1(s)
k2(s)
. . .
km(s)
3 7 7 7 5
(10.
38)
Thi
sis
the
prob
lem
ofde
cent
raliz
eddi
agon
alfe
edba
ckco
ntro
l.T
hede
sign
ofde
cent
raliz
edco
ntro
lsys
tem
sin
volv
estw
ost
eps:
1.T
hech
oice
ofpa
irin
gs(c
ontr
olco
nfigu
ratio
nse
lect
ion)
2.T
hede
sign
(tun
ing)
ofea
chco
ntro
ller,
ki(s)
.
The
optim
also
lutio
nto
this
prob
lem
isve
rydi
fficu
ltm
athe
mat
ical
ly,
beca
use
the
optim
alco
ntro
ller
isin
gene
ral
ofin
finite
orde
ran
dm
aybe
non-
uniq
ue;
we
dono
tad
dres
sit
inth
isbo
ok.
The
read
eris
refe
rred
toth
elit
erat
ure
(e.g
.So
urla
san
dM
anou
siou
thak
is,
1995
)fo
rm
ore
deta
ils.
Rat
her
we
aim
atpr
ovid
ing
sim
ple
tool
sfo
rpa
irin
gse
lect
ions
(ste
p1)
and
for
anal
yzin
gth
eac
hiev
able
perf
orm
ance
(con
trol
labi
lity)
ofdi
agon
ally
cont
rolle
dpl
ants
(whi
chm
ayas
sist
inst
ep2)
.
Not
atio
nfo
rde
cent
raliz
eddi
agon
alco
ntro
l.
G(s)
deno
tes
asq
uare
m�m
plan
tw
ithel
emen
ts
g ij
.Gij(s)
deno
tes
the
rem
aini
ng
(m�1)�(m�1)
plan
tobt
aine
dby
rem
ovin
gro
w
i
and
colu
mn
j
in
G(s).
With
apa
rtic
ular
choi
ceof
pair
ing
we
can
rear
rang
eth
eco
lum
nsor
row
sof
G(s)
such
that
the
pair
edel
emen
tsar
eal
ong
the
diag
onal
of
G(s).
We
then
have
that
the
cont
rolle
rK(s)
isdi
agon
al(d
iagfkig)
,and
we
also
intr
oduc
e
e G,diagfgiig=
2 6 6 6 4g11
g22
. ..
gmm
3 7 7 7 5
(10.
39)
442
MULTIVARIABLEFEEDBACKCONTROL
asth
em
atri
xco
nsis
ting
ofth
edi
agon
alel
emen
tsof
G
.T
helo
optr
ansf
erfu
nctio
nin
loop
i
isde
note
d
Li=g iiki,
whi
chis
also
equa
lto
the
i’th
diag
onal
elem
ento
f
L=GK
.
10.8
.1R
GA
asin
tera
ctio
nm
easu
refo
rde
cent
raliz
edco
ntro
l
We
here
follo
wB
rist
ol(1
966)
,an
dsh
owth
atth
eR
GA
prov
ides
am
easu
reof
the
inte
ract
ions
caus
edby
dece
ntra
lized
diag
onal
cont
rol.
Let
uj
and
yi
deno
tea
part
icul
arin
puta
ndou
tput
fort
hem
ultiv
aria
ble
plan
tG(s),
and
assu
me
that
ourt
ask
isto
use
uj
toco
ntro
lyi.
Bri
stol
argu
edth
atth
ere
will
betw
oex
trem
eca
ses:
�
Oth
erlo
ops
open
:A
llot
her
inpu
tsar
eco
nsta
nt,i
.e.u
k=0;8k6=j.
�
Oth
erlo
ops
clos
ed:
All
othe
rou
tput
sar
eco
nsta
nt,i
.e.y
k=0;8k6=i.
Inth
ela
tter
case
,it
isas
sum
edth
atth
eot
her
loop
sar
ecl
osed
with
perf
ect
cont
rol.
Perf
ect
cont
rol
ison
lypo
ssib
leat
stea
dy-s
tate
,bu
tit
isa
good
appr
oxim
atio
nat
freq
uenc
ies
with
inth
eba
ndw
idth
ofea
chlo
op.W
eno
wev
alua
teth
eef
fect
@yi=@uj
of“o
ur”
give
nin
putu
j
on“o
ur”
give
nou
tput
yi
for
the
two
extr
eme
case
s.W
ege
t
Oth
erlo
ops
open
:
� @yi
@uj
� uk=0;k6=j
=g ij
(10.
40)
Oth
erlo
ops
clos
ed:
� @yi
@uj
� yk=0;k6=i
,bg ij
(10.
41)
Her
e
g ij=[G] ij
isth
e
ij
’th
elem
ent
of
G
,w
here
as
bg ijisth
ein
vers
eof
the
ji
’th
elem
ento
f
G�1
bg ij=1=[G�1] ji
(10.
42)
Tode
rive
(10.
42)n
ote
that
y=Gu
)
� @yi
@uj
� uk=0;k6=j
=[G] ij
(10.
43)
and
inte
rcha
nge
the
role
sof
G
and
G�1
,of
u
and
y
,and
of
i
and
j
toge
t
u=G�1y
)
� @uj
@y i
� yk=0;k6=i
=[G�1] ji
(10.
44)
and
(10.
42)
follo
ws.
Bri
stol
argu
edth
atth
era
tiobe
twee
nth
ega
ins
in(1
0.40
)an
d(1
0.41
),co
rres
pond
ing
toth
etw
oex
trem
eca
ses,
isa
usef
ulm
easu
reof
inte
ract
ions
,an
dhe
intr
oduc
edth
ete
rm,i
j’th
rela
tive
gain
defin
edas
�ij,
g ij bg ij=[G] ij[G�1] ji
(10.
45)
CONTROLSTRUCTUREDESIGN
443
The
Rel
ativ
eG
ain
Arr
ay(R
GA
)is
the
corr
espo
ndin
gm
atri
xof
rela
tive
gain
s.Fr
om(1
0.45
)w
ege
t
�(G)=G�(G�1)T
whe
re
�
deno
tes
elem
ent-
by-e
lem
ent
mul
tiplic
atio
n(t
heSc
hur
prod
uct)
.T
his
isid
entic
alto
our
defin
ition
ofth
eR
GA
-m
atri
xin
(3.6
9).
Intu
itive
ly,w
ew
ould
like
topa
irva
riab
les
uj
and
yi
soth
at�ij
iscl
ose
to1,
beca
use
this
mea
nsth
atth
ega
infr
om
uj
to
y i
isun
affe
cted
bycl
osin
gth
eot
her
loop
s.M
ore
prec
icel
y,w
ew
ould
like
topa
irsu
chth
atth
ere
arra
nged
syst
em,
with
the
pair
ings
alon
gth
edi
agon
al,h
asa
RG
Am
atri
xcl
ose
toid
entit
y(s
eePa
irin
gR
ule
1,pa
ge44
5).
10.8
.2F
acto
riza
tion
ofse
nsit
ivit
yfu
ncti
on
The
mag
nitu
deof
the
off-
diag
onal
elem
ents
in
G
(the
inte
ract
ions
)re
lativ
eto
itsdi
agon
alel
emen
tsar
egi
ven
byth
em
atri
xE,(G�e G)e G�1
(10.
46)
An
impo
rtan
tre
latio
nshi
pfo
rde
cent
raliz
edco
ntro
lis
give
nby
the
follo
win
gfa
ctor
izat
ion
ofth
ere
turn
diff
eren
ceop
erat
or:
(I+GK)
|{z}
overall
=(I+Ee T)|{z}
interactions
(I+e GK)|{z}
individualloops
(10.
47)
oreq
uiva
lent
lyin
term
sof
the
sens
itivi
tyfu
nctio
n
S=(I+GK)�1
,
S=e S(I+Ee T)�1
(10.
48)
Her
e
e S,(I+e GK)�1=diagf
1
1+g iiki
gand
e T=I�e S
(10.
49)
cont
ain
the
sens
itivi
tyan
dco
mpl
emen
tary
sens
itivi
tyfu
nctio
nsfo
rth
ein
divi
dual
loop
s.N
ote
that
e Sisno
teq
ual
toth
em
atri
xof
diag
onal
elem
ents
of
S
.(1
0.48
)fo
llow
sfr
om(A
.139
)w
ith
G
=
e Gand
G0=
G
.T
here
ader
isen
cour
aged
toco
nfirm
that
(10.
48)
isco
rrec
t,be
caus
em
ost
ofth
eim
port
ant
resu
ltsfo
rst
abili
tyan
dpe
rfor
man
ceus
ing
dece
ntra
lized
cont
rolm
aybe
deri
ved
from
this
expr
essi
on.
10.8
.3St
abili
tyof
dece
ntra
lized
cont
rols
yste
ms
Con
side
ra
squa
repl
antw
ithsi
ngle
-loo
pco
ntro
llers
.For
a
2�2
plan
tthe
rear
etw
oal
tern
ativ
epa
irin
gs,
a
3�3
plan
tof
fers
6,a
4�4
plan
t24
,an
dan
m�m
plan
tha
s
m!
alte
rnat
ives
.Thu
s,to
ols
are
need
edw
hich
are
capa
ble
ofqu
ickl
yev
alua
ting
444
MULTIVARIABLEFEEDBACKCONTROL
alte
rnat
ive
pair
ings
.In
this
sect
ion
we
first
deri
vesu
ffici
ent
cond
ition
sfo
rst
abili
tyw
hich
may
beus
edto
sele
ctpr
omis
ing
pair
ings
.The
segi
veru
les
inte
rms
ofdi
agon
aldo
min
ance
.We
then
deri
vene
cess
ary
cond
ition
sfo
rst
abili
tyw
hich
may
beus
edto
elim
inat
eun
desi
rabl
epa
irin
gs.
A.S
uffic
ient
cond
itio
nsfo
rst
abili
ty
For
dece
ntra
lized
diag
onal
cont
rol,
itis
desi
rabl
eth
atth
esy
stem
can
betu
ned
and
oper
ated
one
loop
ata
time.
Ass
ume
ther
efor
eth
at
G
isst
able
and
each
indi
vidu
allo
opis
stab
leby
itsel
f(e San
d
e Tarest
able
).T
hen
from
the
fact
oriz
atio
n
S=
e S(I+Ee T)�1
in(1
0.47
)an
dth
ege
nera
lized
Nyq
uist
theo
rem
inL
emm
aA
.5(p
age
540)
,it
follo
ws
that
the
over
all
syst
emis
stab
le(S
isst
able
)if
and
only
if
det(I+Ee T(s))
does
not
enci
rcle
the
orig
inas
s
trav
erse
sth
eN
yqui
st
D
-con
tour
.Fr
omth
esp
ectr
alra
dius
stab
ility
cond
ition
in(4
.107
)w
eth
enha
veth
atth
eov
eral
lsy
stem
isst
able
if
�(Ee T(j!))<1;8!
(10.
50)
Thi
ssu
ffici
ent
cond
ition
for
over
all
stab
ility
can,
asdi
scus
sed
byG
rosd
idie
ran
dM
orar
i(19
86),
beus
edto
obta
ina
num
ber
ofev
enw
eake
rst
abili
tyco
nditi
ons.
Suffi
cien
tco
ndit
ions
inte
rms
of
E
.T
hele
ast
cons
erva
tive
appr
oach
isto
split
up
�(Ee T)u
sing
the
stru
ctur
edsi
ngul
arva
lue.
From
(8.9
2)w
eha
ve
�(Ee T)�
�(E)��(T)
and
from
(10.
50)
we
get
the
follo
win
gth
eore
m(a
sfir
stde
rive
dby
Gro
sdid
ier
and
Mor
ari,
1986
):
The
orem
10.2
Ass
ume
G
isst
able
and
that
the
indi
vidu
allo
ops
are
stab
le(
e T isst
able
).T
hen
the
enti
resy
stem
iscl
osed
-loo
pst
able
(T
isst
able
)if
��(e T)=maxi
je t ij<1=�(E)
8!
(10.
51)
Her
e
�(E)
isca
lled
the
stru
ctur
edsi
ngul
arva
lue
inte
ract
ion
mea
sure
,an
dis
com
pute
dw
ithre
spec
tto
the
diag
onal
stru
ctur
eof
e T ,whe
rew
em
ayvi
ew
e T asth
e“d
esig
nun
cert
aint
y”.W
ew
ould
like
tous
ein
tegr
alac
tion
inth
elo
ops,
that
is,
we
wan
t
e T�I
atlo
wfr
eque
ncie
s,i.e
.
��(e T)�1
.Thu
s,in
orde
rto
satis
fy(1
0.51
)w
ene
ed
�(E)�1
atlo
wfr
eque
ncie
sw
here
we
have
tight
cont
rol.
Thi
sgi
ves
the
follo
win
gru
le:
Pre
fer
pair
ings
for
whi
chw
eha
ve
�(E)<1
(“ge
nera
lize
ddi
agon
aldo
min
ance
”)
atlo
wfr
eque
ncie
s.
Ano
ther
appr
oach
isto
use
Ger
shgo
rin’
sth
eore
m,
see
page
514.
From
(10.
50)
we
then
deri
veth
efo
llow
ing
suffi
cien
tco
nditi
onfo
rov
eral
lst
abili
tyin
term
sof
the
row
sof
G
:
je t ij<jg iij=
X j6=i
jg ijj8i;8!
(10.
52)
CONTROLSTRUCTUREDESIGN
445
oral
tern
ativ
ely,
inte
rms
ofth
eco
lum
ns,
je t ij<jg iij=
X j6=i
jg jij8i;8!
(10.
53)
Thi
sgi
ves
the
impo
rtan
tins
ight
that
we
pref
erto
pair
onla
rge
elem
ents
in
G
.
Rem
ark
1W
eca
nnot
say
that
(10.
51)
isal
way
sle
ssco
nser
vativ
eth
an(1
0.52
)an
d(1
0.53
).It
istr
ueth
atth
esm
alle
stof
the
i=1;:::m
uppe
rbo
unds
in(1
0.52
)or
(10.
53)
isal
way
ssm
alle
r(m
ore
rest
rict
ive)
than
1=�(E)
in(1
0.51
).H
owev
er,(
10.5
1)im
pose
sth
esa
me
boun
don
je t ijfor
each
loop
,whe
reas
(10.
52)
and
(10.
53)
give
indi
vidu
albo
unds
,som
eof
whi
chm
aybe
less
rest
rict
ive
than
1=�(E).
Rem
ark
2A
noth
erde
finiti
onof
gene
raliz
eddi
agon
aldo
min
ance
isth
at
�(jEj)<1
,whe
re
�(jEj)
isth
ePe
rron
root
;see
(A.1
27).
How
ever
,sin
ce
�(E)=�(DED�1),
see
(8.8
4)w
here
D
inth
isca
seis
diag
onal
,it
follo
ws
from
(A.1
27)
that
�(E)��(jEj),
and
itis
bette
r(l
ess
rest
rict
ive)
tous
e
�(E)
tode
fine
diag
onal
dom
inan
ce.
Rem
ark
3C
ondi
tion
(10.
51)
and
the
use
of
�(E)
for
(nom
inal
)st
abili
tyof
the
dece
ntra
lized
cont
rol
syst
emca
nbe
gene
raliz
edto
incl
ude
robu
stst
abili
tyan
dro
bust
perf
orm
ance
;se
eeq
uatio
ns(3
1a-b
)in
Skog
esta
dan
dM
orar
i(19
89).
Suffi
cien
tcon
diti
ons
for
stab
ility
inte
rms
ofR
GA
.We
now
wan
tto
show
that
for
clos
ed-l
oop
stab
ility
itis
desi
rabl
eto
sele
ctpa
irin
gssu
chth
atth
eR
GA
iscl
ose
toth
eid
entit
ym
atri
xin
the
cros
sove
rre
gion
.The
next
sim
ple
theo
rem
,whi
chap
plie
sto
atr
iang
ular
plan
t,w
illen
able
usto
doth
is:
The
orem
10.3
Supp
ose
the
plan
t
G(s)
isst
able
.If
the
RG
A-m
atri
x
�(G)=I8!
then
stab
ilit
yof
each
ofth
ein
divi
dual
loop
sim
plie
sst
abil
ity
ofth
een
tire
syst
em.
Pro
of:
From
the
defin
ition
ofth
eR
GA
itfo
llow
sth
at
�(G)=
I
can
only
aris
efr
oma
tria
ngul
ar
G(s)
orfr
om
G(s)-
mat
rice
sth
atca
nbe
mad
etr
iang
ular
byin
terc
hang
ing
row
san
dco
lum
nsin
such
aw
ayth
atth
edi
agon
alel
emen
tsre
mai
nth
esa
me
buti
na
diff
eren
tord
er(t
hepa
irin
gsre
mai
nth
esa
me)
.Apl
ant
with
a“t
rian
gula
rize
d”tr
ansf
erm
atri
x(a
sde
scri
bed
abov
e)co
ntro
lled
bya
diag
onal
cont
rolle
rha
son
lyon
e-w
ayco
upli
ngan
dw
illal
way
syi
eld
ast
able
syst
empr
ovid
edth
ein
divi
dual
loop
sar
est
able
.Mat
hem
atic
ally
,E=(G�e G)e G�1
can
bem
ade
tria
ngul
ar,
and
sinc
eth
edi
agon
alel
emen
tsof
E
are
zero
,it
follo
ws
that
all
eige
nval
ues
of
Ee Tare
zero
,so
�(Ee T)=0
and
(10.
50)
issa
tisfie
d.
2
RG
Aat
cros
sove
rfr
eque
ncie
s.In
mos
tcas
es,i
tis
suffi
cien
tfor
over
alls
tabi
lity
tore
quir
eth
at
G(j!)
iscl
ose
totr
iang
ular
(or
�(G)�I
)at
cros
sove
rfre
quen
cies
:
Pai
ring
Rul
e1.
Toac
hiev
est
abil
ity
wit
hde
cent
rali
zed
cont
rolp
refe
rpa
irin
gssu
chth
atat
freq
uenc
ies
!
arou
ndcr
osso
ver,
the
rear
rang
edm
atri
x
G(j!)
(wit
hth
epa
ired
elem
ents
alon
gth
edi
agin
al)
iscl
ose
totr
iang
ular
.Thi
sis
equi
vale
ntto
requ
irin
g
�(G(j!))�I
,i.e
.the
RG
A-n
umbe
r
k�(G(j!))�Ik sum
shou
ldbe
smal
l.
446
MULTIVARIABLEFEEDBACKCONTROL
Der
ivat
ion
ofPa
irin
gru
le1.
Ass
ume
that
e Sisst
able
,an
dth
at
e S�(s)=
e Se G(s)G(s)�1
isst
able
and
has
noR
HP-
zero
s(w
hich
isal
way
ssa
tisfie
dif
both
G
and
e G are
stab
lean
dha
veno
RH
P-ze
ros)
.T
hen
from
(10.
60)
the
over
all
syst
emis
stab
le(S
isst
able
)if
and
only
if
(I+e S(��I))�1
isst
able
.H
ere
e S(��I)
isst
able
,so
from
the
spec
tral
radi
usst
abili
tyco
nditi
onin
(4.1
07)
the
over
alls
yste
mis
stab
leif
�(e S(��I)(j!))<1;
8!
(10.
54)
At
low
freq
uenc
ies,
this
cond
ition
isus
ually
satis
fied
beca
use
e Sissm
all.
At
high
erfr
eque
ncie
s,w
here
the
elem
ents
in
e S=diagfes ig
appr
oach
and
poss
ibly
exce
ed1
inm
agni
tude
,(1
0.54
)m
aybe
satis
fied
if
G(j!)
iscl
ose
totr
iang
ular
.T
his
isbe
caus
e
��I
and
thus
e S(��I)
are
then
clos
eto
tria
ngul
ar,
with
diag
onal
elem
ents
clos
eto
zero
,so
the
eige
nval
ues
of
e S(��I)(j!)
are
clos
eto
zero
,(1
0.54
)is
satis
fied
and
we
have
stab
ility
of
S
.Thi
sco
nclu
sion
also
hold
sfo
rpl
ants
with
RH
P-ze
ros
prov
ided
they
are
loca
ted
beyo
ndth
ecr
osso
ver
freq
uenc
yra
nge.
2
Exa
mpl
e.C
onsi
der
apl
anta
ndit
sR
GA
-mat
rix
G=
� �51
6
2� ;
�(G)=
� 0:625
0:375
0:375
0:625
�
The
RG
Ain
dica
tes
that
G
isdi
agon
ally
dom
inan
tand
that
we
wou
ldpr
efer
tous
eth
edi
agon
alpa
irin
g.T
his
isco
nfirn
edby
com
puti
ngth
ere
lati
vein
tera
ctio
nsas
give
nby
the
mat
rix
E
:
e G=� �50
0
2� ;
E=(G�e G)e G�1=
� 0:375
0:3125
�0:75
0:375
�
.T
heSS
V-i
nter
acti
onm
easu
reis
�(E)=0:6124
,so
the
plan
tis
diag
onal
lydo
min
ant,
and
from
(10.
51)
stab
ilit
yof
the
indi
vidu
allo
ops
wil
lgua
rant
eest
abil
ity
ofth
eov
eral
lclo
sed-
loop
syst
em.N
ote
that
the
Perr
onro
ot
�(jEj)=0:8591
whi
chsh
ows
that
the
use
of
�(E)
isle
ssco
nser
vati
ve.
Itis
not
poss
ible
inth
isca
seto
conc
lude
from
the
Ger
shgo
rin
boun
dsin
(10.
52)
and
(10.
53)
that
the
plan
tis
diag
onal
lydo
min
ant,
beca
use
the
off-
diag
onal
elem
ento
f6is
larg
erth
anan
yof
the
diag
onal
elem
ents
.
B.N
eces
sary
stea
dy-s
tate
cond
itio
nsfo
rst
abili
ty
Ade
sira
ble
prop
erty
ofa
dece
ntra
lized
cont
rols
yste
mis
that
itha
sin
tegr
ity,
i.e.t
hecl
osed
-loo
psy
stem
shou
ldre
mai
nst
able
assu
bsys
tem
cont
rolle
rsar
ebr
ough
tin
and
out
ofse
rvic
e.M
athe
mat
ical
ly,
the
syst
empo
sses
ses
inte
grity
ifit
rem
ains
stab
lew
hen
the
cont
rolle
r
K
isre
plac
edby
EK
whe
re
E
=diagf�ig
and
� im
ayta
keon
the
valu
esof
� i=0
or
� i=1
.
An
even
stro
nger
requ
irem
ent
isth
atth
esy
stem
rem
ains
stab
leas
the
gain
inva
riou
slo
ops
are
redu
ced
(det
uned
)by
anar
bitr
ary
fact
or,
i.e.
0��i
�1
(“co
mpl
ete
detu
nabi
lity”
).D
ecen
tral
ized
inte
gral
cont
rolla
bilit
y(D
IC)
isco
ncer
ned
with
whe
ther
this
ispo
ssib
lein
tegr
alco
ntro
l:
CONTROLSTRUCTUREDESIGN
447
Defi
niti
on10
.1D
ecen
tral
ized
Inte
gral
Con
trol
labi
lity
(DIC
).T
hepl
ant
G(s)
(cor
resp
ondi
ngto
agi
ven
pair
ing
wit
hth
epa
ired
elem
ents
alon
git
sdi
agon
al)
isD
ICif
ther
eex
ists
ast
abil
izin
gde
cent
rali
zed
cont
roll
erw
ith
inte
gral
acti
onin
each
loop
such
that
each
indi
vidu
allo
opm
aybe
detu
ned
inde
pend
entl
yby
afa
ctor
�i
(0�� i�1
)w
itho
utin
trod
ucin
gin
stab
ilit
y.
Not
eth
atD
ICco
nsid
ers
the
exis
tenc
eof
aco
ntro
ller,
soit
depe
nds
only
onth
epl
ant
G
and
the
chos
enpa
irin
gs.T
hest
eady
-sta
teR
GA
prov
ides
ave
ryus
eful
tool
tote
stfo
rD
IC,a
sis
clea
rfr
omth
efo
llow
ing
resu
ltw
hich
was
first
prov
edby
Gro
sdid
ier
etal
.(19
85):
The
orem
10.4
Stea
dy-s
tate
RG
Aan
dD
IC.C
onsi
der
ast
able
squa
repl
antG
and
adi
agon
alco
ntro
ller
K
wit
hin
tegr
alac
tion
inal
lel
emen
ts,
and
assu
me
that
the
loop
tran
sfer
func
tion
GK
isst
rict
lypr
oper
.Ifa
pair
ing
ofou
tput
san
dm
anip
ulat
edin
puts
corr
espo
nds
toa
nega
tive
stea
dy-s
tate
rela
tive
gain
,th
enth
ecl
osed
-loo
psy
stem
has
atle
asto
neof
the
foll
owin
gpr
oper
ties
:(a
)T
heov
eral
lclo
sed-
loop
syst
emis
unst
able
.(b
)T
helo
opw
ith
the
nega
tive
rela
tive
gain
isun
stab
leby
itse
lf.(c
)T
hecl
osed
-loo
psy
stem
isun
stab
leif
the
loop
wit
hth
ene
gati
vere
lati
vega
inis
open
ed(b
roke
n).
Thi
sca
nbe
sum
mar
ized
asfo
llow
s:
Ast
able
(reo
rder
ed)p
lant
G(s)
isD
ICon
lyif
�ii(0)�0
for
alli
.(1
0.55
)
Pro
of:
The
theo
rem
may
bepr
oved
byse
tting
e T=I
in(1
0.47
)an
dap
plyi
ngth
ege
nera
lized
Nyq
uist
stab
ility
cond
ition
.A
ltern
ativ
ely,
we
can
use
The
orem
6.5
onpa
ge24
5an
dse
lect
G0=diagfgii;Giig.
Sinc
e
detG0=g iidetGii
and
from
(A.7
7)
�ii=giidetGii
detG
we
have
detG0 =detG=�ii
and
The
orem
10.4
follo
ws.
2
Eac
hof
the
thre
epo
ssib
lein
stab
ilitie
sin
The
orem
10.4
resu
lting
from
pair
ing
ona
nega
tive
valu
eof
�ij(0)
isun
desi
rabl
e.T
hew
orst
case
is(a
)whe
nth
eov
eral
lsys
tem
isun
stab
le,b
utsi
tuat
ion
(c)
isal
sohi
ghly
unde
sira
ble
asit
will
impl
yin
stab
ility
ifth
elo
opw
ithth
ene
gativ
ere
lativ
ega
inso
meh
owbe
com
esin
activ
e,fo
rexa
mpl
e,du
eto
inpu
tsat
urat
ion.
Situ
atio
n(b
)is
unac
cept
able
ifth
elo
opin
ques
tion
isin
tend
edto
beop
erat
edby
itsel
f,or
ifal
lthe
othe
rlo
ops
may
beco
me
inac
tive,
e.g.
due
toin
put
satu
ratio
n.
Rem
arks
onD
ICan
dR
GA
.
1.D
ICw
asin
trod
uced
bySk
oges
tad
and
Mor
ari(
1988
b).A
deta
iled
surv
eyof
cond
ition
sfo
rD
ICan
dot
her
rela
ted
prop
ertie
sis
give
nby
Cam
poan
dM
orar
i(19
94).
2.U
nsta
ble
plan
tsar
eno
tD
IC.
The
reas
onis
that
with
all
� i=
0
we
are
left
with
the
unco
ntro
lled
plan
t G
,and
the
syst
emw
illbe
(int
erna
lly)
unst
able
if
G(s)
isun
stab
le.
448
MULTIVARIABLEFEEDBACKCONTROL
3.Fo
r
� i=0
we
assu
me
that
the
inte
grat
orof
the
corr
espo
ndin
gSI
SOco
ntro
ller
has
been
rem
oved
,oth
erw
ise
the
inte
grat
orw
ould
yiel
din
tern
alin
stab
ility
.4.
For
2�2
and
3�3
plan
tsw
eha
veev
entig
hter
cond
ition
sfo
rD
ICth
an(1
0.55
).Fo
r
2�2
plan
ts(S
koge
stad
and
Mor
ari,
1988
b)
DIC
,
�11(0)>0
(10.
56)
For
3�3
plan
tsw
ithpo
sitiv
edi
agon
alR
GA
-ele
men
tsof
G(0)
and
of
Gii(0);i=1;2;3
(its
thre
epr
inci
pals
ubm
atri
ces)
we
have
(Yu
and
Fan,
1990
)
DIC
,
p � 11(0)+
p � 22(0)+
p � 33(0)�1
(10.
57)
(Str
ictly
spea
king
,as
poin
ted
outb
yC
ampo
and
Mor
ari(
1994
),w
edo
noth
ave
equi
vale
nce
for
the
case
whe
n
p � 11(0)+
p � 22(0)+
p � 33(0)
isid
entic
alto
1,bu
tth
isha
slit
tlepr
actic
alsi
gnifi
canc
e).
5.O
neca
nnot
expe
cttig
htco
nditi
ons
forD
ICin
term
sof
the
RG
Afo
r4�4
syst
ems
orhi
gher
.T
here
ason
isth
atth
eR
GA
esse
ntia
llyon
lyco
nsid
ers
“cor
ner
valu
es”,
� i=0
or
� i=1
(int
egri
ty),
for
the
detu
ning
fact
orin
each
loop
inth
ede
finiti
onof
DIC
.Thi
sis
clea
rfr
om
the
fact
that
�ii=
giidetGii
detG
,whe
re
G
corr
espo
nds
to
� i=1
for
alli
,gii
corr
espo
nds
to
� i=1
with
the
othe
r
� k=0
,and
Gii
corr
espo
nds
to
� i=0
with
the
othe
r
� k=1
.6.
Det
erm
inan
tcon
diti
ons
for
inte
grit
y(D
IC).
The
follo
win
gco
nditi
onis
conc
erne
dw
ithw
heth
erit
ispo
ssib
leto
desi
gna
dece
ntra
lized
cont
rolle
rfo
rth
epl
ant
such
that
the
syst
empo
sses
ses
inte
grity
,w
hich
isa
prer
equi
site
for
havi
ngD
IC:
Ass
ume
wit
hout
loss
ofge
nera
lity
that
the
sign
sof
the
row
sor
colu
mns
of
G
have
been
adju
sted
such
that
all
diag
onal
elem
ents
of
G
are
posi
tive
,i.e
.
g ii(0)�0
.T
hen
one
may
com
pute
the
dete
rmin
ant
of
G(0)
and
all
its
prin
cipa
lsu
bmat
rice
s(o
btai
ned
byde
leti
ngro
ws
and
corr
espo
ndin
gco
lum
nsin
G(0))
,whi
chsh
ould
allh
ave
the
sam
esi
gnfo
rD
IC.
Thi
sde
term
inan
tcon
ditio
nfo
llow
sby
appl
ying
The
orem
6.5
toal
lpos
sibl
eco
mbi
natio
nsof
� i=0
or1
asill
ustr
ated
inth
epr
oof
ofT
heor
em10
.4,a
ndis
equi
vale
ntto
requ
irin
gth
atth
eso
-cal
led
Nie
derl
insk
iind
ices
,
NI=detG(0)=�ig ii(0)
(10.
58)
of
G(0)
and
itspr
inci
pals
ubm
atri
ces
are
allp
ositi
ve.A
ctua
lly,t
his
yiel
dsm
ore
info
rmat
ion
than
the
RG
A,
beca
use
inth
eR
GA
the
term
sar
eco
mbi
ned
into
�ii
=
giidetGii
detG
sow
em
ayha
veca
ses
whe
retw
one
gativ
ede
term
inan
tsre
sult
ina
posi
tive
RG
A-e
lem
ent.
Nev
erth
eles
s,th
eR
GA
isus
ually
the
pref
erre
dto
olbe
caus
eit
does
not
have
tobe
reco
mpu
ted
for
each
pair
ing.
7.D
ICis
also
clos
ely
rela
ted
to
D
-sta
bilit
y,se
epa
pers
byY
uan
dFa
n(1
990)
and
Cam
poan
dM
orar
i(19
94).
The
theo
ryof
D-s
tabi
lity
prov
ides
nece
ssar
yan
dsu
ffici
entc
ondi
tions
exce
ptin
afe
wsp
ecia
lca
ses,
such
asw
hen
the
dete
rmin
ant
ofon
eor
mor
eof
the
subm
atri
ces
isze
ro.
8.If
we
assu
me
that
the
cont
rolle
rsha
vein
tegr
alac
tion,
then
T(0)=
I,
and
we
can
deri
vefr
om(1
0.51
)th
ata
suffi
cien
tco
nditi
onfo
rD
ICis
that
G
isge
nera
lized
diag
onal
lydo
min
anta
tste
ady-
stat
e,th
atis
,
�(E(0))<1
Thi
sis
prov
edby
Bra
atz
(199
3,p.
154)
.H
owev
er,
the
requ
irem
ent
ison
lysu
ffici
ent
for
DIC
and
ther
efor
eca
nnot
beus
edto
elim
inat
ede
sign
s.Sp
ecifi
cally
,fo
ra
2�2
syst
em
CONTROLSTRUCTUREDESIGN
449
itis
easy
tosh
ow(G
rosd
idie
ran
dM
orar
i,19
86)
that
�(E(0))<
1
iseq
uiva
lent
to
�11(0)>
0:5
,w
hich
isco
nser
vativ
ew
hen
com
pare
dw
ithth
ene
cess
ary
and
suffi
cien
tco
nditi
on
�11(0)>0
in(1
0.56
).9.
Ifth
epl
ant
has
j!
-axi
spo
les,
e.g.
inte
grat
ors,
itis
reco
mm
ende
dth
at,p
rior
toth
eR
GA
-an
alys
is,
thes
ear
em
oved
slig
htly
into
the
LH
P(e
.g.
byus
ing
very
low
-gai
nfe
edba
ck).
Thi
sw
illha
veno
prac
tical
sign
ifica
nce
for
the
subs
eque
ntan
alys
is.
10.
Sinc
eT
heor
em6.
5ap
plie
sto
unst
able
plan
ts,
we
may
also
easi
lyex
tend
The
orem
10.4
toun
stab
lepl
ants
(and
inth
isca
seon
em
ayac
tual
lyde
sire
topa
iron
ane
gativ
eR
GA
-el
emen
t).
Thi
sis
show
nin
Hov
dan
dSk
oges
tad
(199
4a).
Alte
rnat
ivel
y,on
em
ayfir
stim
plem
ent
ast
abili
zing
cont
rolle
ran
dth
enan
alyz
eth
epa
rtia
llyco
ntro
lled
syst
emas
ifit
wer
eth
epl
antG
(s).
11.
The
abov
ere
sults
only
addr
ess
stab
ility
.Per
form
ance
isan
alyz
edin
Sect
ion
10.8
.5.
10.8
.4T
heR
GA
and
righ
t-ha
lfpl
ane
zero
s:F
urth
erre
ason
sfo
rno
tpa
irin
gon
nega
tive
RG
Ael
emen
ts
Bri
stol
(196
6)cl
aim
edth
atne
gativ
eva
lues
of
�ii(0)
impl
ied
the
pres
ence
ofR
HP-
zero
s.T
his
isin
deed
true
asill
ustr
ated
byth
efo
llow
ing
two
theo
rem
s:
The
orem
10.5
(Hov
dan
dSk
oges
tad,
1992
)C
onsi
der
atr
ansf
erfu
ncti
onm
atri
x
G(s)
wit
hno
zero
sor
pole
sat
s=0
.Ass
ume
lims!1�ij(s)
isfin
ite
and
diffe
rent
from
zero
.If
�ij(j1)
and
�ij(0)
have
diffe
rent
sign
sth
enat
leas
ton
eof
the
foll
owin
gm
ustb
etr
ue:
a)T
heel
emen
tgij(s)
has
aR
HP
-zer
o.b)
The
over
allp
lant
G(s)
has
aR
HP
-zer
o.c)
The
subs
yste
mw
ith
inpu
tj
and
outp
ut
i
rem
oved
,Gij(s),
has
aR
HP
-zer
o.
The
orem
10.6
(Gro
sdid
ier
etal
.,19
85)
Con
side
ra
stab
letr
ansf
erfu
ncti
onm
atri
x
G(s)
wit
hel
emen
ts
g ij(s).
Let
bg ij(s)
deno
teth
ecl
osed
-loo
ptr
ansf
erfu
ncti
onbe
twee
nin
put
uj
and
outp
ut
yi
wit
hal
lth
eot
her
outp
uts
unde
rin
tegr
alco
ntro
l.A
ssum
eth
at:
(i)
gij(s)
has
noR
HP
-zer
os,
(ii)
the
loop
tran
sfer
func
tion
GK
isst
rict
lypr
oper
,(ii
i)al
loth
erel
emen
tsof
G(s)
have
equa
lor
high
erpo
leex
cess
than
g ij(s).
We
then
have
:
If
�ij(0)<0
then
bg ij(s)
has
anod
dnu
mbe
rof
RH
P-p
oles
and
RH
P-z
eros
.
Neg
ativ
eR
GA
-ele
men
tsan
dde
cent
raliz
edpe
rfor
man
ceW
ithde
cent
raliz
edco
ntro
lw
eus
ually
desi
gnan
dim
plem
ent
the
cont
rolle
rby
tuni
ngan
dcl
osin
gon
elo
opat
atim
ein
ase
quen
tialm
anne
r.A
ssum
eth
atw
epa
iron
ane
gati
vest
eady
-sta
teR
GA
-ele
men
t,
�ij(0)<0
,ass
ume
that
�ij(1)
ispo
sitiv
e(i
tis
usua
llycl
ose
to1,
see
Pair
ing
rule
1),
and
assu
me
that
the
elem
ent
gij
has
noR
HP-
zero
.T
hen
take
nto
geth
erth
eab
ove
two
theo
rem
sth
enha
veth
efo
llow
ing
impl
icat
ions
:
450
MULTIVARIABLEFEEDBACKCONTROL
(a)I
fwe
star
tby
clos
ing
this
loop
(inv
olvi
ngin
putu
i
and
outp
ut
yj
),th
enw
ew
illge
ta
RH
P-ze
ro(i
n
Gij(s))
whi
chw
illlim
itth
epe
rfor
man
cein
the
othe
rout
puts
(fol
low
sfr
omT
heor
em10
.5by
assu
min
gth
at
G
has
noR
HP-
zero
and
that
�ij(1)>0
).
(b)
Ifw
een
dby
clos
ing
this
loop
,the
nw
ew
illge
ta
RH
P-ze
roi(
in
bg ij(s))
whi
chw
illlim
itth
epe
rfor
man
cein
outp
ut
yi
(fol
low
sfr
omT
heor
em10
.6).
Inco
nclu
sion
,pa
irin
gon
ane
gativ
eR
GA
-ele
men
tw
ill,
inad
ditio
nto
resu
lting
inpo
tent
ial
inst
abili
tyas
give
nin
The
orem
10.4
,al
solim
itth
ecl
osed
-loo
pde
cent
raliz
edpe
rfor
man
ce.
We
have
then
firm
lyes
tabl
ishe
dth
efo
llow
ing
rule
:
Pai
ring
Rul
e2.
For
ast
able
plan
tav
oid
pair
ings
that
corr
espo
ndto
nega
tive
stea
dy-s
tate
RG
A-e
lem
ents
,�ij(0)<0
.
The
RG
Ais
ave
ryef
ficie
ntto
olbe
caus
eit
does
not
have
tobe
reco
mpu
ted
for
each
poss
ible
choi
ceof
pair
ing.
Thi
sfo
llow
ssi
nce
any
perm
utat
ion
ofth
ero
ws
and
colu
mns
of
G
resu
ltsin
the
sam
epe
rmut
atio
nin
the
RG
Aof
G
.To
achi
eve
DIC
one
has
topa
iron
apo
sitiv
eR
GA
(0)-
elem
enti
nea
chro
wan
dco
lum
n,an
dth
eref
ore
one
can
ofte
nel
imin
ate
man
yal
tern
ativ
epa
irin
gsby
asi
mpl
egl
ance
atth
eR
GA
-mat
rix.
Thi
sis
illus
trat
edby
the
follo
win
gex
ampl
e.
Exa
mpl
e10
.8C
onsi
der
a
3�3
plan
twit
h
G(0)=
" 10:2
5:6
1:4
15:5
�8:4
�0:7
18:1
0:4
1:8
# ;
�(0)=
" 0:96
1:45
�1:41
0:94
�0:37
0:43
�0:90
�0:07
1:98
#
(10.
59)
For
a
3�3
plan
tth
ere
are
6al
tern
ativ
epa
irin
gs,b
utfr
omth
est
eady
-sta
teR
GA
we
see
that
ther
eis
only
one
posi
tive
elem
enti
nco
lum
n
2
(�12=1:45
),an
don
lyon
epo
siti
veel
emen
tin
row
3
(�33
=1:98
),an
dth
eref
ore
ther
eis
only
one
poss
ible
pair
ing
wit
hal
lR
GA
-ele
men
tspo
siti
ve(u
1$y2
,u2$y1
,u3$y3
).T
hus,
ifw
ere
quir
eD
ICw
eca
nfr
oma
quic
kgl
ance
atth
est
eady
-sta
teR
GA
elim
inat
efiv
eof
the
six
alte
rnat
ive
pair
ings
.
Exa
mpl
e10
.9C
onsi
der
the
plan
tand
RG
A
G(s)=
(�s+1)
(5s+1)2
" 14:19
�25:96
6:19
1
�25:96
1
1
1
# ;
�(G)=
" 15
�5
�51
5
5
�51
#N
ote
that
the
RG
Ais
cons
tant
,ind
epen
dent
offr
eque
ncy.
Onl
ytw
oof
the
six
poss
ible
pair
ings
give
posi
tive
stea
dy-s
tate
RG
A-e
lem
ents
(Pai
ring
Rul
e2)
:(a
)T
he(d
iago
nal)
pair
ing
onal
l
�ii=1
.(b)
The
pair
ing
onal
l�ii=5
.Int
uiti
vely
,one
may
expe
ctpa
irin
g(a
)to
beth
ebe
stsi
nce
itco
rres
pond
sto
pair
ing
onR
GA
-ele
men
tseq
ual
to1.
How
ever
,the
RG
A-m
atri
xis
far
from
iden
tity
,and
the
RG
A-n
umbe
r,
k��Ik sum
,is
30fo
rbo
thal
tern
ativ
es.T
hus,
none
ofth
etw
oal
tern
ativ
essa
tisf
yPa
irin
gR
ule
1,an
dw
ear
ele
dto
conc
lude
that
dece
ntra
lize
dco
ntro
lsh
ould
notb
eus
edfo
rth
ispl
ant.
Rem
ark.
(Hov
dan
dSk
oges
tad,
1992
)co
nfirm
this
conc
lusi
onby
desi
gnin
gP
Ico
ntro
ller
sfo
rth
etw
oca
ses.
Surp
risi
ngly
,the
yfo
und
pair
ing
(a)c
orre
spon
ding
to
� ii=1
tobe
sign
ifica
ntly
CONTROLSTRUCTUREDESIGN
451
wor
seth
an(b
)wit
h
�ii=5
.The
yfo
und
the
achi
evea
ble
clos
ed-l
oop
tim
eco
nsta
nts
tobe
1160
and
220,
resp
ecti
vely
,whi
chin
both
case
sis
very
slow
com
pare
dto
the
RH
P-z
ero
whi
chha
sa
tim
eco
nsta
ntof
1.
Exe
rcis
e10
.7(a
)A
ssum
eth
atth
e
4�4
mat
rix
in(A
.82)
repr
esen
tsth
est
eady
-sta
tem
odel
ofa
plan
t.Sh
owth
at
20
ofth
e
24
poss
ible
pair
ings
can
beel
imin
ated
byre
quir
ing
DIC
.(b)
Con
side
rth
e
3�3
FC
Cpr
oces
sin
Exe
rcis
e6.
16on
page
250.
Show
that
5of
the
6po
ssib
lepa
irin
gsca
nbe
elim
inat
edby
requ
irin
gD
IC.
10.8
.5P
erfo
rman
ceof
dece
ntra
lized
cont
rols
yste
ms
Abo
vew
eus
edth
efa
ctor
izat
ion
S=
e S(I+Ee T)�1
in(1
0.48
)to
stud
yst
abili
ty.
Her
ew
ew
ant
toco
nsid
erpe
rfom
ance
.Are
late
dfa
ctor
izat
ion
whi
chfo
llow
sfr
om(A
.140
)is
S=(I+e S(��I))�1e S�
(10.
60)
whe
re
�
isth
ePe
rfor
man
ceR
elat
ive
Gai
nA
rray
(PR
GA
),
�(s),
e G(s)G�1(s)
(10.
61)
whi
chis
asc
aled
inve
rse
ofth
epl
ant.
Not
eth
at
E
=
��1�I
.A
tfr
eque
ncie
sw
here
feed
back
isef
fect
ive
(e S�0
),(1
0.60
)yi
elds
S�e S�w
hich
show
sth
at
�
isim
port
antw
hen
eval
uatin
gpe
rfor
man
cew
ithde
cent
raliz
edco
ntro
l.T
hedi
agon
alel
emen
tsof
the
PRG
A-m
atri
xar
eeq
ual
toth
edi
agon
alel
emen
tsof
the
RG
A,
ii=�ii
,and
this
isth
ere
ason
for
itsna
me.
Not
eth
atth
eof
f-di
agon
alel
emen
tsof
the
PRG
Ade
pend
onth
ere
lativ
esc
alin
gon
the
outp
uts,
whe
reas
the
RG
Ais
scal
ing
inde
pend
ent.
On
the
othe
rha
nd,
the
PRG
Am
easu
res
also
one-
way
inte
ract
ion,
whe
reas
the
RG
Aon
lym
easu
res
two-
way
inte
ract
ion.
We
will
also
mak
eus
eof
the
rela
ted
Clo
sed-
Loo
pD
istu
rban
ceG
ain
(CL
DG
)mat
rix,
defin
edas
e G d(s),�(s)Gd(s)=
e G(s)G�1(s)Gd(s)
(10.
62)
The
CL
DG
depe
nds
onbo
thou
tput
and
dist
urba
nce
scal
ing.
Inth
efo
llow
ing,
we
cons
ider
perf
orm
ance
inte
rms
ofth
eco
ntro
lerr
or
e=y�r=Gu+Gdd�r
(10.
63)
Supp
ose
the
syst
emha
sbe
ensc
aled
asou
tline
din
Sect
ion
1.4,
such
that
atea
chfr
eque
ncy:
1.E
ach
dist
urba
nce
isle
ssth
an1
inm
agni
tude
,jdkj<1
.2.
Eac
hre
fere
nce
chan
geis
less
than
the
corr
espo
ndin
gdi
agon
alel
emen
tin
R
,
jr jj<Rj
.
452
MULTIVARIABLEFEEDBACKCONTROL
3.Fo
rea
chou
tput
the
acce
ptab
leco
ntro
lerr
oris
less
than
1,
jeij<1
.
For
SISO
syst
ems,
we
foun
din
Sect
ion
5.10
that
inte
rms
ofsc
aled
vari
able
sw
em
usta
tall
freq
uenc
ies
requ
ire
j1+Lj>jGdjand
j1+Lj>jRj
(10.
64)
for
acce
ptab
ledi
stur
banc
ere
ject
ion
and
com
man
dtr
acki
ng,r
espe
ctiv
ely.
Not
eth
at
L;Gd
and
R
are
alls
cala
rsin
this
case
.For
dece
ntra
lized
cont
rolt
hese
requ
irem
ents
may
bedi
rect
lyge
nera
lized
byin
trod
ucin
gth
ePR
GA
-mat
rix,
�=
e GG�1
,and
the
CL
DG
-mat
rix,
e G d=�Gd
.T
hese
gene
raliz
atio
nsw
illbe
pres
ente
dan
ddi
scus
sed
next
,and
then
subs
eque
ntly
prov
ed.
Sing
ledi
stur
banc
e.C
onsi
der
asi
ngle
dist
urba
nce,
inw
hich
case
Gd
isa
vect
or,
and
let
g di
deno
teth
e
i’th
elem
ent
of
Gd
.L
et
Li=g iiki
deno
teth
elo
optr
ansf
erfu
nctio
nin
loop
i.C
onsi
der
freq
uenc
ies
whe
refe
edba
ckis
effe
ctiv
eso
e S�issm
all
(and
(10.
67)
isva
lid).
The
nfo
rac
cept
able
dist
urba
nce
reje
ctio
n(je
ij<1
)w
em
ust
with
dece
ntra
lized
cont
rolr
equi
refo
rea
chlo
op
i,
j1+Lij>jeg dij8i
(10.
65)
whi
chis
the
sam
eas
the
SISO
-con
ditio
n(5
.52)
exce
ptth
at
Gd
isre
plac
edby
the
CL
DG
,eg di.In
wor
ds,eg di
give
sth
e“a
ppar
ent”
dist
urba
nce
gain
asse
enfr
omlo
op
i
whe
nth
esy
stem
isco
ntro
lled
usin
gde
cent
raliz
edco
ntro
l.
Sing
lere
fere
nce
chan
ge.
Sim
ilarl
y,co
nsid
era
chan
gein
refe
renc
efo
rou
tput
j
ofm
agni
tude
Rj
.Con
side
rfr
eque
ncie
sw
here
feed
back
isef
fect
ive
(and
(10.
67)
isva
lid).
The
nfo
rac
cept
able
refe
renc
etr
acki
ng(je
ij<1
)w
em
ust
requ
ire
for
each
loop
i
j1+Lij>j ijj�jRjj8i
(10.
66)
whi
chis
the
sam
eas
the
SISO
-con
ditio
nex
cept
for
the
PRG
A-f
acto
r,
j ijj.I
not
her
wor
ds,w
hen
the
othe
rloo
psar
ecl
osed
the
resp
onse
inlo
op
i
gets
slow
erby
afa
ctor
j iij.
Con
sequ
ently
,for
perf
orm
ance
itis
desi
rabl
eto
have
smal
lel
emen
tsin
�
,at
leas
tat
freq
uenc
ies
whe
refe
edba
ckis
effe
ctiv
e.H
owev
er,
atfr
eque
ncie
scl
ose
tocr
osso
ver,
stab
ility
isth
em
ain
issu
e,an
dsi
nce
the
diag
onal
elem
ents
ofth
ePR
GA
and
RG
Aar
eeq
ual,
we
usua
llypr
efer
toha
ve
ii
clos
eto
1(r
ecal
lPai
ring
Rul
e1
onpa
ge44
5).
Pro
ofs
of(1
0.65
)an
d(1
0.66
):A
tfre
quen
cies
whe
refe
edba
ckis
effe
ctiv
e,
e Sissm
all,
so
I+e S(��I)�I
(10.
67)
and
from
(10.
60)
we
have
S�e S�
(10.
68)
The
clos
ed-l
oop
resp
onse
then
beco
mes
e=SGdd�Sr�e Se G dd�e S�r
(10.
69)
CONTROLSTRUCTUREDESIGN
453
and
the
resp
onse
inou
tput
i
toa
sing
ledi
stur
banc
e
d k
and
asi
ngle
refe
renc
ech
ange
r j
is
e i�es ieg dikdk�es i ikr k
(10.
70)
whe
re
es i=1=(1+g iiki)
isth
ese
nsiti
vity
func
tion
for
loop
i
byits
elf.
Thu
s,to
achi
eve
je ij<1
for
jd kj=1
we
mus
treq
uire
jes ieg dikj<1
and
(10.
65)
follo
ws.
Sim
ilarl
y,to
achi
eve
je ij<1
for
jr jj=jRjjw
em
ust
requ
ire
js i ikRjj<1
and
(10.
66)
follo
ws.
Als
ono
teth
at
js i ikj<1
will
impl
yth
atas
sum
ptio
n(1
0.67
)is
valid
.Sin
ceR
usua
llyha
sal
lof
itsel
emen
tsla
rger
than
1,in
mos
tcas
es(1
0.67
)w
illbe
auto
mat
ical
lysa
tisfie
dif
(10.
66)
issa
tisfie
d,so
we
norm
ally
need
notc
heck
assu
mpt
ion
(10.
67).
2
Rem
ark
1(1
0.68
)m
ayal
sobe
deri
ved
from
(10.
48)
byas
sum
ing
e T�I
whi
chyi
elds
(I+Ee T)�1�(I+E)�1=�
.
Rem
ark
2C
onsi
der
apa
rtic
ular
dist
urba
nce
with
mod
el
g d
.It
sef
fect
onou
tput
i
with
noco
ntro
lis
g di
,and
the
ratio
betw
een
eg di(the
CL
DG
)an
d
g di
isth
ere
lati
vedi
stur
banc
ega
in(R
DG
)(�
i
)of
Stan
ley
etal
.(19
85)
(see
also
Skog
esta
dan
dM
orar
i(19
87b)
):
�i,eg di=gdi=[e GG�1g d] i=[gd] i
(10.
71)
Thu
s
�i
,whi
chis
scal
ing
inde
pend
ent,
give
sth
ech
ange
inth
eef
fect
ofth
edi
stur
banc
eca
used
byde
cent
raliz
edco
ntro
l.It
isde
sira
ble
toha
ve
�i
smal
l,as
this
mea
nsth
atth
ein
tera
ctio
nsar
esu
chth
atth
eyre
duce
the
appa
rent
effe
ctof
the
dist
urba
nce,
such
that
one
does
not
need
high
gain
s
jLijin
the
indi
vidu
allo
ops.
10.8
.6Su
mm
ary:
Con
trol
labi
lity
anal
ysis
for
dece
ntra
lized
cont
rol
Whe
nco
nsid
erin
gde
cent
raliz
eddi
agon
alco
ntro
lof
apl
ant,
one
shou
ldfir
stch
eck
that
the
plan
tis
cont
rolla
ble
with
any
cont
rolle
r.If
the
plan
tis
unst
able
,th
enas
usua
lth
eun
stab
lem
odes
mus
tbe
cont
rolla
ble
and
obse
rvab
le.
Inad
ditio
n,th
eun
stab
lem
odes
mus
tno
tbe
dece
ntra
lize
dfix
edm
odes
,ot
herw
ise
the
plan
tca
nnot
best
abili
zed
with
adi
agon
alco
ntro
ller
(Lun
ze,1
992)
.For
exam
ple,
this
isth
eca
sefo
ra
tria
ngul
arpl
anti
fth
eun
stab
lem
ode
appe
ars
only
inth
eof
f-di
agon
alel
emen
ts.
The
next
step
isto
com
pute
the
RG
A-m
atri
xas
afu
nctio
nof
freq
uenc
y,an
dto
dete
rmin
eif
one
can
find
ago
odse
tof
inpu
t-ou
tput
pair
sbe
arin
gin
min
dth
efo
llow
ing:
1.Pr
efer
pair
ings
whi
chha
veth
eR
GA
-mat
rix
clos
eto
iden
tity
atfr
eque
ncie
saro
und
cros
sove
r,i.e
.th
eR
GA
-num
ber
k�(j!)�Ik
shou
ldbe
smal
l.T
his
rule
isto
ensu
reth
atin
tera
ctio
nsfr
omot
her
loop
sdo
not
caus
ein
stab
ility
asdi
scus
sed
follo
win
g(1
0.54
).2.
Avo
ida
pair
ing
ij
with
nega
tive
stea
dy-s
tate
RG
Ael
emen
ts,�
ij(G(0))
.3.
Pref
era
pair
ing
ij
whe
re
gij
puts
min
imal
rest
rict
ions
onth
eac
hiev
able
band
wid
th.S
peci
fical
ly,t
hefr
eque
ncy
!uij
whe
re
\g ij(j!uij)=�180Æ
shou
ldbe
asla
rge
aspo
ssib
le.
454
MULTIVARIABLEFEEDBACKCONTROL
Thi
sru
lefa
vour
spa
irin
gon
vari
able
s“c
lose
toea
chot
her”
,whi
chm
akes
itea
sier
tosa
tisfy
(10.
65)a
nd(1
0.66
)phy
sica
llyw
hile
atth
esa
me
time
achi
evin
gst
abili
ty.
Itis
also
cons
iste
ntw
ithth
ede
sire
that
�(j!)
iscl
ose
to
I
.
Whe
na
reas
onab
lech
oice
ofpa
irin
gsha
sbe
enm
ade,
one
shou
ldre
arra
nge
G
toha
veth
epa
ired
elem
ents
alon
gth
edi
agon
alan
dpe
rfor
ma
cont
rolla
bilit
yan
alys
is.
4.C
ompu
teth
eC
LD
Gan
dPR
GA
,and
plot
thes
eas
afu
nctio
nof
freq
uenc
y.5.
For
syst
ems
with
man
ylo
ops,
itis
best
tope
rfor
mth
ean
alys
ison
elo
opat
the
time,
that
is,
for
each
loop
i,pl
ot
jeg dikjf
orea
chdi
stur
banc
e
k
and
plot
j ijj
for
each
refe
renc
e
j
(ass
umin
ghe
refo
rsi
mpl
icity
that
each
refe
renc
eis
ofun
itm
agni
tude
).Fo
rpe
rfor
man
ce,w
ene
ed
j1+Lijt
obe
larg
erth
anea
chof
thes
e
Performance:
j1+Lij>maxk;jfjeg dikj;j ijjg
(10.
72)
Toac
hiev
est
abili
tyof
the
indi
vidu
allo
ops
one
mus
tan
alyz
e
gii(s)
toen
sure
that
the
band
wid
thre
quir
edby
(10.
72)
isac
hiev
able
.Not
eth
atR
HP-
zero
sin
the
diag
onal
elem
ents
may
limit
achi
evab
lede
cent
raliz
edco
ntro
l,w
here
asth
eym
ayno
tpo
sean
ypr
oble
ms
for
am
ultiv
aria
ble
cont
rolle
r.Si
nce
with
dece
ntra
lized
cont
rol
we
usua
llyw
ant
tous
esi
mpl
eco
ntro
llers
,th
eac
hiev
able
band
wid
thin
each
loop
will
belim
ited
byth
efr
eque
ncy
whe
re
\gii
is
�180Æ
(rec
all
Sect
ion
5.12
).6.
As
alre
ady
men
tione
don
em
aych
eck
for
cons
trai
nts
byco
nsid
erin
gth
eel
emen
tsof
G�1Gd
and
mak
ing
sure
that
they
dono
texc
eed
one
inm
agni
tude
with
inth
efr
eque
ncy
rang
ew
here
cont
rol
isne
eded
.Equ
ival
ently
,one
may
for
each
loop
i
plot
jg iij,
and
the
requ
irem
enti
sth
enth
at
Toavoidinputconstraints:
jg iij>jeg dikj;8k
(10.
73)
atfr
eque
ncie
sw
here
jeg dikjis
larg
erth
an1
(thi
sfo
llow
ssi
nce
e G d=e GG�1Gd
).T
his
prov
ides
adi
rect
gene
raliz
atio
nof
the
requ
irem
ent
jGj>jGdjf
orSI
SOsy
stem
s.T
head
vant
age
of(1
0.73
)com
pare
dto
usin
g
G�1Gd
isth
atw
eca
nlim
itou
rsel
ves
tofr
eque
ncie
sw
here
cont
roli
sne
eded
tore
ject
the
dist
urba
nce
(whe
re
jeg dikj>1
).
Ifth
epl
anti
sno
tcon
trol
labl
e,th
enon
em
ayco
nsid
eran
othe
rcho
ice
ofpa
irin
gsan
dgo
back
toSt
ep4.
Ifon
est
illca
nnot
find
any
pair
ings
whi
char
eco
ntro
llabl
e,th
enon
esh
ould
cons
ider
mul
tivar
iabl
eco
ntro
l.
7.If
the
chos
enpa
irin
gis
cont
rolla
ble
then
the
anal
ysis
base
don
(10.
72)
tells
usdi
rect
lyho
wla
rge
jLij=jg iikijm
ustb
e,an
dca
nbe
used
asa
basi
sfo
rdes
igni
ngth
eco
ntro
ller
ki(s)
for
loop
i.
Rem
ark.
Inso
me
case
s,pa
irin
gsw
hich
viol
ate
the
abov
eru
les
may
bech
osen
.For
exam
ple,
one
may
even
choo
seto
pair
onel
emen
tsw
ith
g ii=0
whi
chyi
eld
�ii=0
.One
then
relie
son
CONTROLSTRUCTUREDESIGN
455
the
inte
ract
ions
toac
hiev
eth
ede
sire
dpe
rfor
man
ceas
loop
i
byits
elfh
asno
effe
ct.A
nex
ampl
eof
this
isin
dist
illat
ion
cont
rolw
hen
the
LV
-con
figur
atio
nis
notu
sed,
see
Exa
mpl
e10
.5.
Exa
mpl
e10
.10
App
licat
ion
todi
still
atio
npr
oces
s.In
orde
rto
dem
onst
rate
the
use
ofth
efr
eque
ncy
depe
nden
tR
GA
and
CL
DG
for
eval
uati
onof
expe
cted
diag
onal
cont
rol
perf
orm
ance
,w
eco
nsid
erag
ain
the
dist
illa
tion
proc
ess
used
inE
xam
ple
10.7
.T
he
LV
confi
gura
tion
isus
ed,
that
is,
the
man
ipul
ated
inpu
tsar
ere
flux
L
(u1
)an
dbo
ilup
V
(u2
).O
utpu
tsar
eth
epr
oduc
tco
mpo
siti
ons
y D
(y1
)an
d
xB
(y2
).D
istu
rban
ces
infe
edflo
wra
te
F
(d1
)an
dfe
edco
mpo
siti
on
z F
(d2
),ar
ein
clud
edin
the
mod
el.T
hedi
stur
banc
esan
dou
tput
sha
vebe
ensc
aled
such
that
am
agni
tude
of
1
corr
espo
nds
toa
chan
gein
F
of
20
%,a
chan
gein
z F
of
20
%,
and
ach
ange
in
xB
and
y D
of
0:01
mol
efr
acti
onun
its.
The
5
stat
edy
nam
icm
odel
isgi
ven
inSe
ctio
n12
.4.
Init
ial
cont
rolla
bilit
yan
alys
is.G
(s)
isst
able
and
has
noR
HP
-zer
os.
The
plan
tan
dR
GA
-m
atri
xat
stea
dy-s
tate
are
G(0)=
� 87:8
�86:4
108:2
�109:6
� ;
�(0)=
� 35:1
�34:1
�34:1
35:1
�
(10.
74)
The
RG
A-e
lem
ents
are
muc
hla
rger
than
1
and
indi
cate
apl
ant
that
isfu
ndam
enta
lly
diffi
cult
toco
ntro
l.Fo
rtun
atel
y,th
eflo
wdy
nam
ics
part
iall
yde
coup
leth
ere
spon
seat
high
erfr
eque
ncie
s,an
dw
efin
dth
at
�(j!)�I
atfr
eque
ncie
sab
ove
abou
t0:5
rad/
min
.The
refo
reif
we
can
achi
eve
suffi
cien
tly
fast
cont
rol,
the
larg
est
eady
-sta
teR
GA
-ele
men
tsm
aybe
less
ofa
prob
lem
.
10−
310
−2
10−
110
010
110
−1
100
101
gd11
�gd12
gd22
gd21
Freq
uenc
y[r
ad/m
in]
Magnitude
Fig
ure
10.1
1:D
istu
rban
cega
ins
jg dikj,f
oref
fect
ofdi
stur
banc
e
k
onou
tput
i
The
stea
dy-s
tate
effe
ctof
the
two
dist
urba
nces
isgi
ven
by
Gd(0)=
� 7:88
8:81
11:72
11:19
�
(10.
75)
and
the
mag
nitu
des
ofth
eel
emen
tsin
Gd(j!)
are
plot
ted
asa
func
tion
offr
eque
ncy
inF
igur
e10
.11.
Fro
mth
ispl
otth
etw
odi
stur
banc
esse
emto
beeq
uall
ydi
fficu
ltto
reje
ctw
ith
mag
nitu
des
larg
erth
an
1
upto
afr
eque
ncy
ofab
out0
:1
rad/
min
.We
conc
lude
that
cont
roli
sne
eded
upto
0.1
rad/
min
.The
mag
nitu
deof
the
elem
ents
in
G�1Gd(j!)
(not
show
n)ar
eal
lle
ssth
an
1
atal
lfr
eque
ncie
s(a
tlea
stup
to
10
rad/
min
),an
dso
itw
illb
eas
sum
edth
atin
put
cons
trai
nts
pose
nopr
oble
m.
456
MULTIVARIABLEFEEDBACKCONTROL
10−
310
−2
10−
110
010
110
−1
100
101
102
eg d11 eg d12eg d22eg d21 Fr
eque
ncy
[rad
/min
]
Magnitude
Fig
ure
10.1
2:C
lose
d-lo
opdi
stur
banc
ega
ins,
jeg dikj,f
oref
fect
ofdi
stur
banc
e
k
onou
tput
i
Cho
ice
ofpa
irin
gs.
The
sele
ctio
nof
u1
toco
ntro
l
y 1
and
u2
toco
ntro
l
y 2
,co
rres
pond
sto
pair
ing
onpo
siti
veel
emen
tsof
�(0)
and
�(j!)�I
athi
ghfr
eque
ncie
s.T
his
seem
sse
nsib
le,
and
isus
edin
the
foll
owin
g.
Ana
lysi
sof
dece
ntra
lized
cont
rol.
The
elem
ents
inth
eC
LD
Gan
dP
RG
Am
atri
ces
are
show
nas
func
tion
sof
freq
uenc
yin
Fig
ures
10.1
2an
d10
.13.
Ats
tead
y-st
ate
we
have
�(0)=
� 35:1
�27:6
�43:2
35:1
� ;
e G d(0)=�(0)Gd(0)=
� �47:7
�0:40
70:5
11:7
�
(10.
76)
Inth
ispa
rtic
ular
case
the
off-
diag
onal
elem
ents
ofR
GA
(�
)an
dP
RG
A(�
)ar
equ
ite
sim
ilar
.W
eno
teth
at
e G d(0)
isve
rydi
ffere
ntfr
om
Gd(0),
and
this
also
hold
sat
high
erfr
eque
ncie
s.Fo
rdi
stur
banc
e
1
(firs
tcol
umn
in
e G d)w
efin
dth
atth
ein
tera
ctio
nsin
crea
seth
eap
pare
ntef
fect
ofth
edi
stur
banc
e,w
here
asth
eyre
duce
the
effe
ctof
dist
urba
nce
2
,atl
east
onou
tput
1
.
10−
310
−2
10−
110
010
110
−1
100
101
102
11= 22
12
21
Freq
uenc
y[r
ad/m
in]
Magnitude
Fig
ure
10.1
3:PR
GA
-ele
men
ts
j ijjfo
ref
fect
ofre
fere
nce
j
onou
tput
i
We
now
cons
ider
one
loop
ata
tim
eto
find
the
requ
ired
band
wid
th.F
orlo
op1
(out
put
1
)w
eco
nsid
er
11
and
12
for
refe
renc
es,a
nd
eg d11
and
eg d12
for
dist
urba
nces
.Dis
turb
ance
1
isth
em
ost
diffi
cult
,an
dw
ene
ed
j1+L1j>jbg d11ja
tfr
eque
ncie
sw
here
jbg d11ji
sla
rger
than
1
,w
hich
isup
toab
out0
:2
rad/
min
.The
mag
nitu
deof
the
PR
GA
-ele
men
tsar
eso
mew
hats
mal
ler
than
jeg d11j(a
tlea
stat
low
freq
uenc
ies)
,so
refe
renc
etr
acki
ngw
illb
eac
hiev
edif
we
can
reje
ctdi
stur
banc
e
1
.F
rom
eg d12
we
see
that
dist
urba
nce
2
has
alm
ost
noef
fect
onou
tput
1
unde
rfe
edba
ckco
ntro
l.
CONTROLSTRUCTUREDESIGN
457
010
2030
4050
6070
8090
100
−0.
4
−0.
20
0.2
0.4
y2
y1
Tim
e[m
in]
Fig
ure
10.1
4:D
ecen
tral
ized
PI-c
ontr
ol.R
espo
nses
toa
unit
step
in
d 1
at
t=0
and
aun
itst
epin
d2
at
t=50
min
Als
o,fo
rlo
op
2
we
find
that
dist
urba
nce
1is
the
mos
tdi
fficu
lt,
and
from
eg d12
we
requ
ire
alo
opga
inla
rger
than
1
upto
abou
t0:3
rad/
min
.Aba
ndw
idth
ofab
out
0:2
to
0:3
rad/
min
inea
chlo
op,i
sre
quir
edfo
rre
ject
ing
dist
urba
nce
1
,and
shou
ldbe
achi
evab
lein
prac
tice
.
Obs
erve
dco
ntro
lpe
rfor
man
ce.
Toch
eck
the
vali
dity
ofth
eab
ove
resu
lts
we
desi
gned
two
sing
le-l
oop
PI
cont
roll
ers:
k1(s)=0:261
1+3:76s
3:76s
;
k2(s)=�0:375
1+3:31s
3:31s
(10.
77)
The
loop
gain
s,Li=g iiki
,wit
hth
ese
cont
roll
ers
are
larg
erth
anth
ecl
osed
-loo
pdi
stur
banc
ega
ins,
jÆ ikj,a
tfre
quen
cies
upto
cros
sove
r.C
lose
d-lo
opsi
mul
atio
nsw
ith
thes
eco
ntro
ller
sar
esh
own
inF
igur
e10
.14.
The
sim
ulat
ions
confi
rmth
atdi
stur
banc
e
2
ism
ore
easi
lyre
ject
edth
andi
stur
banc
e
1
.
Insu
mm
ary,
ther
eis
anex
celle
ntag
reem
entb
etw
een
the
cont
rolla
bilit
yan
alys
isan
dth
esi
mul
atio
ns,a
sha
sal
sobe
enco
nfirm
edby
anu
mbe
rof
othe
rex
ampl
es.
10.8
.7Se
quen
tial
desi
gnof
dece
ntra
lized
cont
rolle
rs
The
resu
ltspr
esen
ted
inth
isse
ctio
non
dece
ntra
lized
cont
rol
are
mos
tus
eful
for
the
case
whe
nth
elo
cal
cont
rolle
rs
ki(s)
are
desi
gned
inde
pend
entl
y,th
atis
,ea
chco
ntro
ller
isde
sign
edlo
cally
and
then
all
the
loop
sar
ecl
osed
.As
disc
usse
dab
ove,
one
prob
lem
with
this
isth
atth
ein
tera
ctio
nsm
ayca
use
the
over
all
syst
em(T
)to
beun
stab
le,e
ven
thou
ghth
elo
cal
loop
s(
e T)are
stab
le.T
his
will
not
happ
enif
the
plan
tis
diag
onal
lydo
min
ant,
such
that
we
satis
fy,f
orex
ampl
e,
��(e T)<1=�(E)
in(1
0.51
).
The
stab
ility
prob
lem
isav
oide
dif
the
cont
rolle
rsar
ede
sign
edse
quen
tial
lyas
isco
mm
only
done
inpr
actic
ew
hen,
for
exam
ple,
the
band
wid
ths
ofth
elo
ops
are
quite
diff
eren
t.In
this
case
the
oute
rlo
ops
are
tune
dw
ithth
ein
ner
(fas
t)lo
ops
inpl
ace,
and
each
step
may
beco
nsid
ered
asa
SISO
cont
rol
prob
lem
.In
part
icul
ar,
over
all
stab
ility
isde
term
ined
by
m
SISO
stab
ility
cond
ition
s.H
owev
er,
the
issu
e
458
MULTIVARIABLEFEEDBACKCONTROL
ofpe
rfor
man
ceis
mor
eco
mpl
icat
edbe
caus
eth
ecl
osin
gof
alo
opm
ayca
use
“dis
turb
ance
s”(i
nter
actio
ns)i
nto
apr
evio
usly
desi
gned
loop
.The
engi
neer
mus
tthe
ngo
back
and
rede
sign
alo
opth
atha
sbe
ende
sign
edea
rlie
r.T
hus
sequ
entia
lde
sign
may
invo
lve
man
yite
ratio
ns;
see
Hov
dan
dSk
oges
tad
(199
4b).
The
perf
orm
ance
boun
dsin
(10.
72)
are
usef
ulfo
rde
term
inin
gth
ere
quir
edba
ndw
idth
inea
chlo
opan
dm
ayth
ussu
gges
tasu
itabl
ese
quen
cein
whi
chto
desi
gnth
eco
ntro
llers
.
Alth
ough
the
anal
ysis
and
deri
vatio
nsgi
ven
inth
isse
ctio
nap
ply
whe
nw
ede
sign
the
cont
rolle
rsse
quen
tially
,it
isof
ten
usef
ul,
afte
rha
ving
desi
gned
alo
wer
-lay
erco
ntro
ller
(the
inne
rfa
stlo
ops)
,to
redo
the
anal
ysis
base
don
the
mod
elof
the
part
ially
cont
rolle
dsy
stem
usin
g(1
0.26
)or
(10.
28).
For
exam
ple,
this
isus
ually
done
ford
istil
latio
nco
lum
ns,w
here
we
base
the
anal
ysis
ofth
eco
mpo
sitio
nco
ntro
lpr
oble
mon
a
2�2
mod
elof
the
part
ially
cont
rolle
d
5�5
plan
t,se
eE
xam
ples
10.5
and
10.1
0.
10.8
.8C
oncl
usio
nson
dece
ntra
lized
cont
rol
Inth
isse
ctio
n,w
eha
vede
rive
da
num
ber
ofco
nditi
ons
for
the
stab
ility
,e.g
.(10
.51)
and
(10.
55),
and
perf
orm
ance
,e.
g.(1
0.65
)an
d(1
0.66
),of
dece
ntra
lized
cont
rol
syst
ems.
The
cond
ition
sm
aybe
usef
ulin
dete
rmin
ing
appr
opri
ate
pair
ings
ofin
puts
and
outp
uts
and
the
sequ
ence
inw
hich
the
dece
ntra
lized
cont
rolle
rssh
ould
bede
sign
ed.R
ecal
l,ho
wev
er,t
hati
nm
any
prac
tical
case
sde
cent
raliz
edco
ntro
llers
are
tune
dba
sed
onlo
calm
odel
sor
even
on-l
ine.
The
cond
ition
s/bo
unds
are
also
usef
ulin
anin
put-
outp
utco
ntro
llabi
lity
anal
ysis
for
dete
rmin
ing
the
viab
ility
ofde
cent
raliz
edco
ntro
l.
Som
eex
erci
ses
whi
chin
clud
ea
cont
rolla
bilit
yan
alys
isof
dece
ntra
lized
cont
rola
regi
ven
atth
een
dof
Cha
pter
6.
10.9
Con
clus
ion
The
issu
eof
cont
rol
stru
ctur
ede
sign
isve
ryim
port
ant
inap
plic
atio
ns,
but
itha
sre
ceiv
edre
lativ
ely
little
atte
ntio
nin
the
cont
rolc
omm
unity
duri
ngth
ela
st
40
year
s.In
this
chap
ter,
we
have
disc
usse
dth
eis
sues
invo
lved
,and
we
have
prov
ided
som
eid
eas
and
tool
s.T
here
iscl
earl
ya
need
for
bette
rto
ols
and
theo
ryin
this
area
.
11
M
ODEL
REDUCTION
Thi
sch
apte
rde
scri
bes
met
hods
forr
educ
ing
the
orde
rof
apl
anto
rco
ntro
ller
mod
el.W
epl
ace
cons
ider
able
emph
asis
onre
duce
dor
derm
odel
sob
tain
edby
resi
dual
izin
gth
ele
ssco
ntro
llabl
ean
dob
serv
able
stat
esof
aba
lanc
edre
aliz
atio
n.W
eal
sopr
esen
tthe
mor
efa
mili
arm
etho
dsof
bala
nced
trun
catio
nan
dop
timal
Han
keln
orm
appr
oxim
atio
n.
11.1
Intr
oduc
tion
Mod
ern
cont
rolle
rde
sign
met
hods
such
as
H1
and
LQ
G,
prod
uce
cont
rolle
rsof
orde
rat
leas
teq
ualt
oth
atof
the
plan
t,an
dus
ually
high
erbe
caus
eof
the
incl
usio
nof
wei
ghts
.T
hese
cont
rol
law
sm
aybe
too
com
plex
with
rega
rds
topr
actic
alim
plem
enta
tion
and
sim
pler
desi
gns
are
then
soug
ht.F
orth
ispu
rpos
e,on
eca
nei
ther
redu
ceth
eor
dero
fthe
plan
tmod
elpr
iort
oco
ntro
llerd
esig
n,or
redu
ceth
eco
ntro
ller
inth
efin
alst
age,
orbo
th.
The
cent
ral
prob
lem
we
addr
ess
is:
give
na
high
-ord
erlin
ear
time-
inva
rian
tst
able
mod
el
G
,find
alo
w-o
rder
appr
oxim
atio
n
Ga
such
that
the
infin
ity(H
1
or
L 1
)nor
mof
the
diff
eren
ce,k
G�Gak 1
,is
smal
l.B
ym
odel
orde
r,w
em
ean
the
dim
ensi
onof
the
stat
eve
ctor
ina
min
imal
real
izat
ion.
Thi
sis
som
etim
esca
lled
the
McM
illan
degr
ee.
Sofa
rin
this
book
we
have
only
been
inte
rest
edin
the
infin
ity(H
1
)no
rmof
stab
lesy
stem
s.B
utth
eer
ror G
�Ga
may
beun
stab
lean
dth
ede
finiti
onof
the
infin
ityno
rmne
eds
tobe
exte
nded
toun
stab
lesy
stem
s.
L 1
defin
esth
ese
tof
ratio
nal
func
tions
whi
chha
veno
pole
son
the
imag
inar
yax
is,i
tinc
lude
s
H1
,and
itsno
rm(l
ike
H1
)is
give
nby
kGk 1=supw��(G(jw))
.
We
will
desc
ribe
thre
em
ain
met
hods
for
tack
ling
this
prob
lem
:bal
ance
dtr
unca
tion,
bala
nced
resi
dual
izat
ion
and
optim
alH
anke
lno
rmap
prox
imat
ion.
Eac
hm
etho
dgi
ves
ast
able
appr
oxim
atio
nan
da
guar
ante
edbo
und
onth
eer
ror
inth
eap
prox
imat
ion.
We
will
furt
her
show
how
the
met
hods
can
beem
ploy
edto
redu
ceth
eor
der
ofan
unst
able
mod
el
G
.All
thes
em
etho
dsst
artf
rom
asp
ecia
lsta
te-s
pace
