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Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order...
77
1.1 Mario Garcia-Sanz Mario Garcia-Sanz Automatic Control & Computer Science Department Public University of Navarra 31006 Pamplona, Spain (Email: [email protected]) NATO. RTO-LS-SCI-195, May-June 2008 1 st Session Beyond the Classical Performance Limitations Controlling Uncertain MIMO Systems: UAV Applications
Transcript of Beyond the Classical Performance Limitations Controlling … · 2011. 5. 14. · ÎMinimum Order...
1.1
Mar
io G
arci
a-Sa
nz
Mar
io G
arci
a-Sa
nzA
utom
atic
Con
trol &
Com
pute
r Sci
ence
Dep
artm
ent
Publ
ic U
nive
rsity
of N
avar
ra31
006
Pam
plon
a, S
pain
(Em
ail:
mgs
anz@
unav
arra
.es)
NA
TO
. R
TO
-LS-
SCI-
195,
May
-Jun
e 20
08
1stSe
ssio
n
Bey
ond
the
Cla
ssic
al P
erfo
rman
ce
Lim
itatio
ns C
ontr
ollin
g U
ncer
tain
MIM
O
Syst
ems:
UA
V A
pplic
atio
ns
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4. TITLE AND SUBTITLE Beyond the Classical Performance Limitations Controlling UncertainMIMO Systems: UAV Applications
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13. SUPPLEMENTARY NOTES See also ADM002223. Presented at the NATO/RTO Systems Concepts and Integration Panel LectureSeries SCI-195 on Advanced Autonomous Formation Control and Trajectory Management Techniques forMultiple Micro UAV Applications held in Glasgow, United Kingdom on 19-21 May 2008.
14. ABSTRACT
15. SUBJECT TERMS
16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT Same as
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Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18
1.2
Mar
io G
arci
a-Sa
nz
Out
line
1.-
QFT
Con
trolle
r Des
ign
Tech
niqu
e Fu
ndam
enta
ls
2.-
Rea
l-wor
ld Q
FT c
ontro
l app
licat
ions
and
exa
mpl
es
3.-
Non
-dia
gona
l MIM
O Q
FT c
ontro
ller d
esig
n m
etho
dolo
gies
4.-A
pplic
atio
n: R
obus
t QFT
con
trol f
or a
MIM
O S
pace
craf
t with
flex
ible
suns
hiel
d
5.-S
witc
hing
robu
st c
ontro
l: B
eyon
d th
e lin
ear l
imita
tions
.
6.-E
xam
ple:
Sw
itchi
ng c
ontro
l for
Unm
anne
d V
ehic
les
1.3
Mar
io G
arci
a-Sa
nz
1.-Q
FT C
ontr
olle
r D
esig
n T
echn
ique
Fun
dam
enta
ls
1.1.
-Int
rodu
ctio
n
Ach
ieve
s rea
sona
bly
low
loop
gai
ns, i
.e.,
avoi
ds o
r min
imiz
es: S
enso
r noi
se
ampl
ifica
tion,
Sat
urat
ion,
Hig
h Fr
eque
ncy
unce
rtain
ties.
Look
s for
a d
esig
n th
at c
ombi
nes:
Mod
el +
Par
amet
er U
ncer
tain
ty. (
Rob
ustn
ess)
.Pe
rfor
man
ce S
peci
ficat
ions
.M
inim
um O
rder
Con
trolle
r. Tr
ansp
aren
cy o
f the
Tec
hniq
ue.
Freq
uenc
y D
omai
n Te
chni
que.
Use
s the
Nic
hols
Cha
rt (N
C).
The
QFT
desi
gn o
bjec
tive
is to
des
ign
and
impl
emen
t a ro
bust
con
trol f
or a
sy
stem
with
Unc
erta
inty
that
satis
fies t
he d
esire
d Pe
rfor
man
ce S
peci
ficat
ions
.
Qua
ntita
tive
Feed
back
The
ory
(Q.F
.T.).
Intro
duce
d by
Pro
f. Is
aac
Hor
owitz
.19
59. F
irst i
deas
.19
72. T
he n
ame.
1973
. Hor
owitz
at t
he A
ir Fo
rce
Off
ice
of S
cien
tific
Res
earc
h (f
irst g
rant
).19
92. P
rof.
Hou
piso
rgan
izes
the
Firs
t Int
erna
tiona
l Sym
posi
um o
n Q
FT.
Unt
il th
en, I
nt. S
ympo
sia
ever
y tw
o ye
ars:
199
5, 1
997,
199
9, 2
001,
200
3, 2
005,
200
7.
a re
liabl
e co
ntro
l des
ign
met
hodo
logy
1.4
Mar
io G
arci
a-Sa
nz
Perf
orm
ance
Spec
ifica
tions
(P.S
.)Pl
antM
odel
+ U
ncer
tain
ty QFT
Con
trol
ler
Des
ign
Per
form
ance
Rob
ustn
ess
F ,
GF
, G
Min
imum
Ord
erCo
ntro
ller
Tran
spar
ency
ofth
ete
chni
que
Min
imum
Cost
ofFe
edba
ck
Bou
nds
Loop
Shap
ing
L 0
PD
1D
2
GF
RY
_+
++
++
QFT
. A su
cces
sful
robu
st c
ontro
l th
eory
for r
eal-w
orld
ap
plic
atio
ns:
Stab
le a
nd U
nsta
ble
Syst
ems,
SISO
and
MIM
O P
roce
sses
, A
nalo
g an
d D
iscr
ete
Syst
ems,
Line
ar a
nd N
on-li
near
Pla
nts,
Proc
esse
s with
con
stan
t and
va
riabl
e pa
ram
eter
s, M
inim
um a
nd N
on-m
inim
um
Phas
e Sy
stem
s, C
asca
de C
ontro
l Sys
tem
s, et
c.
1.5
Mar
io G
arci
a-Sa
nz
Step
1: C
ontro
l Spe
cific
atio
ns: S
tabi
lity
Step
2: C
ontro
l Spe
cific
atio
ns: P
erfo
rman
ceSt
ep 3
: Spe
cify
Pla
nt m
odel
s + U
ncer
tain
tySt
ep 4
: Obt
ain
tem
plat
es a
t spe
cifie
d ω i
(des
crib
es u
ncer
tain
ty)
Step
5: S
elec
t nom
inal
pla
nt P
o(s)
Step
6: D
eter
min
e st
abili
ty c
onto
ur (U
-con
tour
) on
N.C
.St
eps 7
-9: D
eter
min
e tra
ckin
g, d
istur
banc
e, &
opt
imal
bou
nds
Step
10:
Syn
thes
ize
nom
inal
Lo(
s) =
G(s
)Po(s
)--
Satis
fies a
ll bo
unds
& st
abili
ty c
onto
ur--
Obt
ain
G(s
) = L
o(s)/P
o(s)
Step
11:
Syn
thes
ize
pref
ilter
F(s)
.St
ep 1
2: S
imul
ate
linea
r sys
tem
(J ti
me
resp
onse
s)St
ep 1
3: S
imul
ate
with
nonl
inea
ritie
s
QFT
Des
ign
Proc
edur
eQ
FT D
esig
n Pr
oced
ure
Con
trol
Sp
ecifi
catio
ns
Mod
el +
un
cert
aint
y
Bou
nds
Loo
psha
ping
Pref
ilter
1.2.
-MIS
O a
nalo
g co
ntro
l sys
tem
des
ign
1.6
Mar
io G
arci
a-Sa
nz
Step
1, 2
: Con
trol
Spe
cific
atio
ns:
Step
1, 2
: Con
trol
Spe
cific
atio
ns:
Stab
ility
and
Per
form
ance
Stab
ility
and
Per
form
ance
RR
efer
ence
WC
ontro
llerI
nput
Dis
turb
ance
VPl
antI
nput
Dis
turb
ance
DPl
antO
utpu
t Dis
turb
ance
NN
oise
FU
EY
N
DV
WR
Pref
ilter
Sens
or
Plan
tC
ontro
ller
-G
P
H
)VD
N(1
)RW(
1
N1
R1
1W
1V
1D
1
N1
)RW(
1V
1D
11
PH
GPHG
FH
GPG
U
HG
PHF
HG
PH
GP
HG
PH
GPHP
HG
PHE
HG
PH
GP
FH
GPGP
HG
PPH
GP
Y
++
+−
++
=
+−
++
++
++
+−
=
+−
++
++
++
=
Spec
ifica
tions
in te
rms o
f T.F
.Fo
r exa
mpl
e:
)(
11
1ω
sW
HG
P≤
+
(1)
(2)
(3)
1.7
Mar
io G
arci
a-Sa
nz
Gai
n M
argi
n:G
M ≥
1 +
1/μ
(mag
nitu
de)
Phas
e M
argi
n:PM
≥18
0º-θ
(deg
)
whe
re:
μis
the
circ
le M
spec
ifica
tion
in m
agni
tude
: M
m=
20 lo
g 10(
μ)θ
= 2
cos-1
(0.5
/μ) ∈
[0, 1
80º]
YP(
s)G
(s)
R
Plan
tC
ontro
ller
-
1po
int
at18
018
0po
int
at1
1
1
=+
=−
==
==
=≤
+=
=
)j
(Lº
PM
ºM/
GM
eM
)j
(G
)j
(P
)j
(L
W)
j(
T
)j
(G
)j
(P
)j
(G
)j
(P
)j
(T
)j
(R
)j
(Y
j
s
ωψ
ψωω
ω
μω
ωω
ωω
ωωω
ψ
The
Stab
ility
(Gai
n an
d Ph
ase
Mar
gins
) is
rel
ated
with
the
Max
imum
clo
sed-
loop
R
eson
ance
Mm
spec
ifica
tion.
PGH
PGH
Ws
11
+≤
Ws1
GM
PMO
vers
hoot
1.1
(0.8
dB
)1.
99 (5
.9 d
B)
55º
∼11
%1.
2 (1
.58
dB)
1.83
(5.2
dB
)49
º∼
18 %
1.3
(2.2
8 dB
)1.
77 (5
.0 d
B)
45º
∼22
%
1.4
(2.9
dB
)1.
71 (4
.7 d
B)
41.8
º∼
27 %
Stab
ility
-350
-300
-250
-200
-150
-100
-50
0-4
0
-30
-20
-10010203040
6 db3 db1
db0.5
db0.25
db
0 db
-1
db -3 d
b
-6 d
b
-12
db
-20
db
-40
db
Phas
e. O
pen
loop
(deg
)
Magnitude. Openloop(dB)-1
80
-5 º
-10
º
-2 º
-330
º-3
0 º
-300
º-6
0 º
-90
º-2
70 º
Circ
leM
m=
2 db
2 db
L =
G P
-350
-300
-250
-200
-150
-100
-50
0-4
0
-30
-20
-10010203040
6 db3 db1
db0.5
db0.25
db
0 db
-1
db -3 d
b
-6 d
b
-12
db
-20
db
-40
db
Phas
e. O
pen
loop
(deg
)
Magnitude. Openloop(dB)-1
80
-5 º
-10
º
-2 º
-330
º-3
0 º
-300
º-6
0 º
-90
º-2
70 º
Circ
leM
m=
2 db
2 db
L =
G P
1.8
Mar
io G
arci
a-Sa
nz
Tim
e-D
omai
n Sp
ecifi
catio
ns:
Des
ire s
yste
m o
utpu
t y(t)
to li
e be
twee
n sp
ecifi
ed u
pper
and
low
er b
ound
s, y(
t) Uan
d y(
t) L, r
espe
ctiv
ely.
Figu
res o
f mer
it(F
OM
), ba
sed
upon
a st
ep in
put s
igna
l r(t)
= R
0u -
1(t),
Mp
peak
ove
rsho
ot; t
rris
e tim
e; t p
peak
tim
e; a
nd t s
settl
ing
time.
Dep
ends
on
requ
irem
ents
that
the
desi
gner
wan
ts fo
r the
spec
ific
Pla
nt:
Airp
lane
, Hea
ting
syst
em, M
achi
nery
, Win
d Tu
rbin
e, e
tc...
Des
ired
syst
em p
erfo
rman
ce
spec
ifica
tions
: tim
e do
mai
n re
spon
se sp
ecifi
catio
ns;
t r L
t r U
MP
t Pt
y(t) L
y(t) U
r(t)
1.0
0.9
0.1
Acc
epta
ble
Perf
orm
ance
Are
a
t s
Settl
ing
Tim
eTo
lera
nce
Trac
king
Perf
orm
ance
WPG PG
HW
sa
sb
77
1≤
+≤
1.9
Mar
io G
arci
a-Sa
nz
Tran
slate
d in
toth
e fr
eque
ncy
dom
ain
are,
BU
and
B L, t
he u
pper
and
low
er
boun
ds re
spec
tivel
y: P
eak
over
shoo
t Lm
Mm
& fr
eque
ncy
band
wid
th ω
h.
(Not
e: in
crea
sing
δ δδδR(j
ω ωωωi)
abov
e 0
dB c
ross
ing)
Des
ired
syst
em p
erfo
rman
ce
spec
ifica
tions
: fr
eque
ncy
dom
ain
resp
onse
sp
ecifi
catio
ns.
+ 0 - -12
dB
LmT R
ωω
hω
i
LmM
m
BU
= Lm
T RU
= Lm
BR
U
δ hf
BL
= Lm
T R L =
LmB
R L
δ δδδ R(jω ωωω
)
Ban
dwid
th
Trac
king
WPG PG
HW
sa
sb
77
1≤
+≤
1.10
Mar
io G
arci
a-Sa
nz
22
2
72
)()
/(
nn
nU
Rb
ss
s
as
aT
Wω
ωξω
++
+=
=
)s()
s()s(
TW
LR
as
32
17
kσ
σσ
++
+=
=
Upp
erfu
nctio
n
Low
erfu
nctio
n
01
23
45
67
89
1000.2
0.4
0.6
0.811.2
1.4
Tim
e (s
)
t r(L)
t r(U
)
Mp
y(t) U
y(t) L
ω(r
ad/s
)
BL(
jω)
BU(jω
)
10-1
100
101
-40
-35
-30
-25
-20
-15
-10-505
δ R(jω
)
Mm
2 db
ω(ra
d/s)
10-1
100
101
05101520dB
δ R(jω
)
δ δδδ R(j
ω ωωω) =
BU
(jω ωωω
) -B
L(jω ωωω
)cl
osed
loop
Trac
king
WPG PG
HW
sa
sb
77
1≤
+≤
1.11
Mar
io G
arci
a-Sa
nz
Freq
uenc
y-D
omai
n Sp
ecifi
catio
ns:
Des
ire T
F sy
stem
Y(ω
)/U(ω
) to
lie
un
der a
spec
ific
boun
d, B
(ω)
Figu
res o
f mer
it(F
OM
), R
eson
ance
; Ban
dwid
th; R
oll-o
ff; L
ow F
requ
ency
, etc
.
Dep
ends
on
requ
irem
ents
that
the
desi
gner
wan
ts fo
r the
spec
ific
Pla
nt:
Stru
ctur
e re
sona
nce,
Noi
se m
easu
rem
ent,
Dis
turb
ance
s, St
eady
Sta
te E
rror
s, et
c...
+ 0 - -12
dB
Lm(Y
(ω)/U
(ω))
ωω
hω
r
Res
onan
ceB
(ω)
Ban
dwid
th
Rol
l-Off
Dist
urba
nce
reje
ctio
n1
12
+≤
PG
HW
s
1.12
Mar
io G
arci
a-Sa
nz
22
2
22
21
1
nn
nd
ss
s
ss
HG
PT
Wω
ωξωξ
++
+≤
+=
=
-30
-20
-10010
Magnitude (dB)
10-1
100
101
102
04590Phase (deg)
Bode
Dia
gram
Freq
uenc
y (r
ad/s
ec)
Dis
turb
ance
reje
ctio
n
Dist
urba
nce
reje
ctio
n1
12
+≤
PG
HW
s
1.13
Mar
io G
arci
a-Sa
nz
Tran
sfer
func
tions
and
spec
ifica
tions
Eq
.No.
{} 1
11
1),
()
()
(1
)(
)(
)(
)(
)(
)(
)(
)(
)(
)(
ωω
ωδ
ωω
ωω
ωωωω
ωω
ωω
∈≤
⋅+
⋅=
==
⋅=
jj
Gj
Pj
Gj
Pj
Nj
Yj
Dj
Uj
Fj
Rj
Yj
T(T
1)
{}
22
22
),(
)(
)(
11
)(
)(
)(
ωω
ωδ
ωω
ωωω
∈≤
⋅+
==
jG
jP
jD
jY
jT
(T2)
{}
33
13
),(
)(
)(
1)
()
()
()
(ω
ωω
δω
ωω
ωωω
∈≤
⋅+
==
jG
jP
jP
jD
jY
jT
(T3)
{}
44
24
),(
)(
)(
1)
()
()
()
()
()
()
()
()
(ω
ωω
δω
ωω
ωω
ωωω
ωωω
∈≤
⋅+
=⋅
==
=j
Gj
Pj
Gj
Fj
Rj
Uj
Nj
Uj
Dj
Uj
T(T
4)
{}
5su
p5
5in
f5
),(
)(
)(
1)
()
()
()
()
()
()
(ω
ωω
δω
ωω
ωω
ωωω
ωδ
∈≤
⋅+
⋅=
=≤
jG
jP
jG
jP
jF
jR
jY
jT
(T5)
Clo
sed
loop
spec
ifica
tions
are
usu
ally
des
crib
ed in
term
s of f
requ
ency
func
tions
δk(
ω)
that
are
impo
sed
on th
e m
agni
tude
of t
he sy
stem
tran
sfer
func
tions
|Tk|
, k =
1, .
.. 5
(1) r
obus
t sta
bilit
y, c
ontro
l eff
ort l
imit
in th
e in
put d
istu
rban
ce re
ject
ion,
sen
sor n
oise
atte
nuat
ion
(2) o
utpu
t sys
tem
dis
turb
ance
reje
ctio
n
(3) i
nput
syst
em d
istu
rban
ce re
ject
ion
(4) c
ontro
l eff
ort l
imit
in th
e ou
tput
dis
turb
ance
reje
ctio
n, n
oise
atte
nuat
ion,
and
trac
king
(5) s
igna
l tra
ckin
g
1.14
Mar
io G
arci
a-Sa
nz
Why
Unc
erta
inty
?W
hy U
ncer
tain
ty?
Step
3: P
lant
Mod
el +
Unc
erta
inty
Step
3: P
lant
Mod
el +
Unc
erta
inty
Tes
t-be
d fo
r L
arge
Mul
tipol
eG
ener
ator
s
M.T
orre
s
Up
to fo
ur
500
kW
Mot
ors
driv
es a
30
00 k
W
Gen
erat
or
1.15
Mar
io G
arci
a-Sa
nz
Shad
ed r
egio
nre
pres
ents
th
e re
gion
of p
lant
un
certa
inty
.
Mot
or r
epre
sent
edby
6
LTI t
rans
fer f
unct
ions
Pι
(ι=
1,2,
…,J)
.
23
4
16
5
a
K 0
Reg
ion
of p
lant
para
met
er u
ncer
tain
tyK
max
Reg
ion
of p
lant
par
amet
er u
ncer
tain
ty.
Km
in
a max
a min
)(
)(
)(
)(
as
sK
as
Vs
sP
fm+
=Θ
=ι
A Si
mpl
e M
athe
mat
ical
Des
crip
tion
Mot
ortra
nsfe
r fun
ctio
n is
:
Para
met
ers K
and
ava
ry:
K∈
(Km
in, K
max
) an
d a
∈(a
min
, am
ax)
Vf
Ra
Rf
i fω
Lf
40 20 0
-20
-40
-60
-80
-100
Magnitude (dB)
110
100
Plan
t 1Pl
ant 2
Plan
t 3Pl
ant 4
Plan
t 5Pl
ant 6
Freq
uenc
y (ω
)
Freq
uenc
y (ω
)1
1010
0-1
80
-170
-160
-150
-140
-130
-120
-110
-100-9
0
Plan
t 1,2
Plan
t 3,6
Plan
t 4,5
Phase (degrees)
1.16
Mar
io G
arci
a-Sa
nz
Step
4: T
empl
ates
Step
4: T
empl
ates
Plan
t Tem
plat
eob
tain
ed b
y m
appi
ngPo
ints
of u
ncer
tain
ty re
gion
into
poi
nts o
n to
the
N.C
.C
urve
dra
wn
thro
ugh
poin
ts –
Shad
ed a
rea
labe
led
ℑP(
j1)
ℑP (
j1)
CB A
D
K=1
0
a=10
a=1
K=1
20 17 dB 0 -3-1
80°
-90°
-135
°-0
.04
dB
Phas
e
dB
N.C
. cha
ract
eriz
ing
P(s)
ove
r the
regi
on
of u
ncer
tain
ty.
a)s(
sKa
(s)
P+
=ι
ω =
1
rad/
s
23
4
16
5
a
K 0
Reg
ion
of
plan
t par
amet
erun
certa
inty
Km
ax=1
0
Km
in=1
a max
=10
a min
=1
1.17
Mar
io G
arci
a-Sa
nz
Tem
plat
es fo
r oth
er v
alue
s of ω
iar
e ob
tain
ed
Cha
ract
eris
tic o
f tem
plat
es:
Star
ting
from
low
val
ues o
f ωi,
(nar
row
wid
th),t
he a
ngul
ar w
idth
be
com
es la
rger
(med
ium
freq
.)
For i
ncre
asin
g va
lues
of ω
i
tem
plat
es b
ecom
e na
rrow
er a
gain
.
Even
tual
ly a
ppro
ach
stra
ight
lin
e: h
eigh
t V d
BPh
ase
Magnitude
ω 1 ω 2 ω 3
ω 4
ω 5
ω 6
ω 6> ω
5 >
…>
ω 1
dBV
Klo
g20
Klo
g20
]P
log
20P
log
[20
limm
inm
axm
inm
ax10
1010
10=
−=
−=
Δ∞
→ω
1.18
Mar
io G
arci
a-Sa
nz
-140
-120
-100
-80
-60
-40
-30
-28
-26
-24
-22
-20
-18
-16
-14
5
Pha
se(d
egre
es)
Magnitude(dB)
-140
-120
-100
-80
-60
-40
-30
-28
-26
-24
-22
-20
-18
-16
-14
5
Pha
se(d
egre
es)
Magnitude(dB)
128
punt
os
-140
-120
-100
-80
-60
-40
-30
-28
-26
-24
-22
-20
-18
-16
-14
5
Pha
se(d
egre
es)
Magnitude(dB)
Who
leTe
mpl
ate
Edge
sTem
plat
e
Con
tour
Tem
plat
e
[]
[]
[]
[]
∈τ∈
∈∈
τ−+
=
=2.0,1.0
,2,1
,12,
10,
5,1
)ex
p()
( 1
1
kb
a
sb
sa
ks
P
PExam
ple:
4-di
men
sion
al p
aram
eter
spac
e
forω
= 5
rad/
sec
Ω4
Ω1
1.19
Mar
io G
arci
a-Sa
nz
Cho
se a
ny p
lant
Kee
p th
e sa
me
plan
t(s
et o
f par
amet
ers)
as th
e no
min
al fo
r all
freq
uenc
ies
ℑP (
j1)
C
B A
D
K=1
0
a=10
a=1
K=1
20 17 dB 0 -3-1
80°
-90°
-135
°-0
.04
dB
Phas
e
dB
Step
5: N
omin
al P
lant
Step
5: N
omin
al P
lant
1.20
Mar
io G
arci
a-Sa
nz
The
Forb
idde
n R
egio
n is
ex
tend
ed b
y V
dB
.
-350
-300
-250
-200
-150
-100
-50
0-4
0
-30
-20
-10010203040
6 db3
db1 db0.
5 db0.
25 d
b0
db
-1 d
b
-3 d
b
-6 d
b
-12
db
-20
db
-40
db
Pha
se. O
pen
loop
(deg
)
Magnitude. Openloop(dB)
-180
-5 º
-10
º
-2 º
-330
º-3
0 º
-300
º
-60
º
-90
º-2
70 º
2 db
V
L o(jω
) -no
min
alL(
jω) -
max
imum
due
to u
ncer
tain
ty
Tem
plat
eV
Ps
Ks
z
sp
Ks
iim
iin
nm
()
()
()
=+
∏
+∏
⎯→
⎯⎯⎯
= =
→∞
−1 1
ωU
nive
rsal
Hig
hFr
eque
ncy
Bou
nd(U
HFB
)
Step
6: U
Step
6: U
-- Con
tour
(Sta
bilit
y bo
unds
)C
onto
ur (S
tabi
lity
boun
ds)
1.21
Mar
io G
arci
a-Sa
nz
22
2 2)
()/
(
nn
nU
Rs
s
as
aT
ωωξ
ω+
++
=
)()
()(
k3
21
σσ
σ+
++
=s
ss
TL
R
Upp
erfu
nctio
n
Step
7: T
rack
ing
Bou
nds o
n L
Step
7: T
rack
ing
Bou
nds o
n L
oo
Low
erfu
nctio
n
01
23
45
67
89
1000.2
0.4
0.6
0.811.2
1.4
Tim
e (s
)
t r(L)
t r(U
)
Mp
y(t) U
y(t) L
y(t) U
. ( ω
n=
1; a
= 1
; ξ=
0.6
)
y(t) L.
( σ1
= 0.
5; σ
2=
1; σ
3=
2; k
= 1
)
dB
ω(r
ad/s
)
BL(
jω)
BU(jω
)
10-1
100
101
-40
-35
-30
-25
-20
-15
-10-505
δ R(jω
)
Mm
2 db
ω(ra
d/s)
10-1
100
101
05101520dB
δ R(jω
)
δ R(jω
) = B
U(jω
) -B
L(jω
)cl
osed
loop
YF
PG
UE
D
R
Pre-
filte
rPl
ant
Con
trolle
r-
1.22
Mar
io G
arci
a-Sa
nz
Solu
tion
forB
R(jω
i) re
quire
s:
be sa
tisfie
d fo
r all
L ι(jω
i).
And
ar
e be
twee
n B
Uan
d B
L
dB)
(jL
B-)(j
UB
)(j
)(j
Ti
ii
Ri
Rω
ωω
δω
=≤
Δ
+ 0 -
dBLm
ω
BU
δ L(jω
1)δ L(
jω2)
δ L(jω
3)δ L(
jω4)
BL
δ R(jω
4)
δ R(jω
3)
δ R(jω
2)
δ R(jω
1)
TL ι
Clo
sed-
loop
resp
onse
s:
LTI p
lant
s with
onl
y G
(s)
Clo
sed-
loop
resp
onse
s:
LTI p
lant
s with
G(s
) and
F(s
)
+ 0 -
dBLm
ω
BU
BL
T Rι
)(j
Ti
Rω
Δ
YF
PG
UE
D
R
Pre-
filte
rPl
ant
Con
trolle
r-
1.23
Mar
io G
arci
a-Sa
nz
Bas
ed o
nth
eno
min
al p
lant
At s
peci
fied
ωi
By
use
of te
mpl
ates
Alo
ng e
very
NC
ph
ase
grid
line
-350
-300
-250
-200
-150
-100
-50
0-4
0
-30
-20
-10010203040
6db3
db1db0.
5db0.
25db
0db
-1db
-3db
-6db
-12
db
-20d
b
-40
db
Pha
se. O
pen
loop
(deg
)
Magnitude. Open loop(dB)
-180
-5 º
-10
º
-2 º
-330
º-3
0 º
-300
º-6
0 º
-90
º-2
70 º
Rep
eat
proc
edur
eon
su
ffic
ient
ph
ase
grid
lines
to p
rovi
de
enou
gh p
oint
s to
dra
w B
R(ω
i)
BR(jω
i)fo
r suf
ficie
nt
phas
e gr
id li
nes
The
cla
ssic
gra
phic
al
proc
edur
e
1.24
Mar
io G
arci
a-Sa
nz
Cas
e 1
Dis
turb
ance
at P
lant
Out
put [
d 2(t)
= D
ou-1
,(t),
d 1(t)
= 0
]th
e di
stur
banc
e co
ntro
l rat
io fo
r inp
ut d
2(t) i
s,
Subs
titut
ing
L =
G P
= 1
/lyi
elds
whi
ch h
as th
e m
athe
mat
ical
form
at re
quire
d to
use
the
N.C
.
L1
1(s
)DY
(s)
(s)
D2
T2
+=
=
+=
=1
(s)
DY(s
)(s
)T
2D
2
PD
1D
2
GF
RY
_++
++
+
A 2
DO
F fe
edba
ck st
ruct
ure.
Step
8: D
istu
rban
ce R
ejec
tion
Bou
nds
Step
8: D
istu
rban
ce R
ejec
tion
Bou
nds
The
cla
ssic
gra
phic
al
proc
edur
e
1.25
Mar
io G
arci
a-Sa
nz
0° °°°-2
0° °°°-4
0° °°°-6
0° °°°-8
0° °°°-1
00° °°°
-120
° °°°-1
40° °°°
-160
° °°°-1
80° °°°
-200
° °°°-2
20° °°°
-240
° °°°-2
60° °°°
-280
° °°°-3
00° °°°
12 8 6 5 4 3 2 1dB
0.5
0.35
-24 -18
-12
-9 -6 -5 -4 -3 -2dB -1
0dB
-0.2
5
-0.5
AB
C D
β CD
ω2
ωb
ω1ωa
Lm l
280° or -80°
240° or
-120
°
(Not
e: U
se th
e ne
gativ
e an
gle
for
L si
nce
n >
w)
L =
1/l
24 d
B
18 d
B
12 d
B
For -
Lm
TD o
r BD(jω
i) ev
alua
tion
24 1620 12 8 4 0 -4 -8 -12
-16
-20
-24
-28
-24
-16
-20
-12 -8 -4 0 4 8 12 16 20 24 28
Lm L
Lm l
Attenuation, dB
Phas
e an
gle,
φ, d
eg
Rot
ated
Nic
hols
cha
rt.
∠L
∠l
Tem
plat
e of
[1/P
(jωi)]
BD
2(jω
i)
A N
.C.
is ro
tate
d 18
0o
Cha
nge
of si
gn
of th
e ve
rtica
l ax
isin
dB
, and
ho
rizon
tal a
xis
in d
eg.
Sim
ilar t
hat
used
for t
he
track
ing
boun
ds
Now
look
ing
for:
()
()
dB)(j
L/11
L/1lo
g20
L1
1lo
g20
)(j
)(j
log
20
)(j
Tlo
g20
iD
210
1010
iD
210
ωδ
ωωω
≤+
=
=+
==
=
2DY
1.26
Mar
io G
arci
a-Sa
nz
Ineq
ualit
ies B
ound
s Exp
ress
ions
(S
teps
6 to
8)
G(s
)F(
s)P(
s)
D1(
s)D
2(s)
R(s
)
N(s
)
Y(s
)
U(s
)
E(s)
+ -
++
++
+
+
Pre-
filte
rC
ontro
ller
Plan
t with
unce
rtain
ty
H(s
)
Let’s
con
side
r the
two-
degr
ees-
of-f
reed
om fe
edba
ck sy
stem
.
In a
gen
eral
real
-wor
ld p
robl
em P
(s)w
ill p
rese
nt u
ncer
tain
ty{P
}.
The
com
pens
ator
G(s
)and
a th
e pr
e-fil
ter F
(s)w
ill b
e de
sign
ed to
mee
t rob
ust s
tabi
lity
and
robu
st p
erfo
rman
ce sp
ecifi
catio
ns,
and
to d
eal w
ithre
fere
nces
R(s
), di
stur
banc
es D
1,2(
s), s
igna
l noi
se N
(s)a
nd sa
tura
ble
cont
rol e
ffor
t U(s
),
min
imiz
ing
the
‘cos
t of t
he fe
edba
ck’(
exce
ssiv
e ba
ndw
idth
)
The
mod
ern
proc
edur
e
1.27
Mar
io G
arci
a-Sa
nz
Tran
sfer
func
tions
and
spec
ifica
tions
Eq.N
o.
{} 1
11
1),
()
()
(1
)(
)(
)(
)(
)(
)(
)(
)(
)(
)(
ωω
ωδ
ωω
ωω
ωωωω
ωω
ωω
∈≤
⋅+
⋅=
==
⋅=
jj
Gj
Pj
Gj
Pj
Nj
Yj
Dj
Uj
Fj
Rj
Yj
T(1
)
{}
22
22
),(
)(
)(
11
)(
)(
)(
ωω
ωδ
ωω
ωωω
∈≤
⋅+
==
jG
jP
jD
jY
jT
(2)
{}
33
13
),(
)(
)(
1)
()
()
()
(ω
ωω
δω
ωω
ωωω
∈≤
⋅+
==
jG
jP
jP
jD
jY
jT
(3)
{}
44
24
),(
)(
)(
1)
()
()
()
()
()
()
()
()
(ω
ωω
δω
ωω
ωω
ωωω
ωωω
∈≤
⋅+
=⋅
==
=j
Gj
Pj
Gj
Fj
Rj
Uj
Nj
Uj
Dj
Uj
T(4
)
{}
5su
p5
5in
f5
),(
)(
)(
1)
()
()
()
()
()
()
(ω
ωω
δω
ωω
ωω
ωωω
ωδ
∈≤
⋅+
⋅=
=≤
jG
jP
jG
jP
jF
jR
jY
jT
(5)
Clo
sed
loop
spec
ifica
tions
are
usu
ally
des
crib
ed in
term
s of f
requ
ency
func
tions
δk(
ω)
that
are
impo
sed
on th
e m
agni
tude
of t
he sy
stem
tran
sfer
func
tions
|Tk|
, k =
1, .
.. 5
(1) r
obus
t sta
bilit
y, c
ontro
l eff
ort l
imit
in th
e in
put d
istu
rban
ce re
ject
ion,
sen
sor n
oise
atte
nuat
ion
(2) o
utpu
t sys
tem
dis
turb
ance
reje
ctio
n
(3) i
nput
syst
em d
istu
rban
ce re
ject
ion
(4) c
ontro
l eff
ort l
imit
in th
e ou
tput
dis
turb
ance
reje
ctio
n, n
oise
atte
nuat
ion,
and
trac
king
(5) s
igna
l tra
ckin
g
Tabl
e 1
1.28
Mar
io G
arci
a-Sa
nz
k-pr
oble
mB
ound
Qua
drat
ic In
equa
lity
10
1)
cos(
21
12
2 1
2≥
+⋅
+⋅
⋅+
⋅−
⋅g
pg
pθ
φδ
20
11
)co
s(2
2 2
22
≥−
+⋅
+⋅
⋅+
⋅δ
θφ
gp
gp
30
1)
cos(
22 32
22
≥−
+⋅
+⋅
⋅+
⋅δ
θφ
pg
pg
p
40
1)
cos(
21
22 4
2≥
+⋅
+⋅
⋅+
⋅−
gp
gp
θφ
δ
50
)co
s()
cos(
21
12 52
22 5
22 5
22
≥−
+⋅
+−
++
⋅−
δθ
φδ
θφ
δd
ee
dd
ed
ed
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pp
gp
pp
pg
pp δ 5=δ
5sup
/δ5i
nfφ∠
=ω
g)
j(G
i{} 1
,...,
0,
)(
)(
−=
∠=
=m
rp
jP
jP
ir
iθ
ωω
Plan
t
Con
trolle
r
Each
pla
nt in
the
ω i-tem
plat
e an
d th
e co
ntro
ller c
an b
e ex
pres
sed
in it
s pol
ar fo
rm:
Then
, sub
stitu
ting
and
rear
rang
ing
the
ineq
ualit
ies -
Eq. (
1) to
(5) i
n Ta
ble
1-, t
hey
can
be re
duce
d to
the
quad
ratic
ineq
ualit
ies –
k-pr
oble
m(1
) to
(5) i
n Ta
ble
2-.
Solv
ing
equa
litie
s suc
h as
ag2 +
bg+c
= 0
the
set o
f ω ωωωi-b
ound
sfor
{δ k=
1,..,
5} is
com
pute
d.
Tabl
e 2
1.29
Mar
io G
arci
a-Sa
nz
Alg
orith
mto
com
pute
th
ebo
unds
1.D
iscr
etiz
eth
edo
mai
n{
}k
ωin
toa
finite
set
{} k
ik
ni
,...,
1,
==
ωΩ
.2.
Esta
blis
hth
eun
certa
inLT
I pla
ntm
odel
s{
} )(
ω jP
=℘
and
map
itsbo
unda
ryfo
reac
hfr
eque
ncy
ki
Ωω
∈on
the
Nic
hols
char
t. A
seto
fn
tem
plat
es {
} )(
ij
Pω
, n
i,..
.,1
=is
obta
ined
. Eac
hte
mpl
ate
{} 1
,...,
0,
)(
)(
−=
∠=
=m
rp
jP
jP
ir
iθ
ωω
cont
ains
mpo
ints
orpl
ants
. Sel
ecto
neof
them
as th
eno
min
al p
lant
00
0)
(θ
ω∠
=p
jP
i.
3.N
ow, t
heco
nditi
onst
om
eetb
y th
eco
ntro
ller
φω
∠=
gj
Gi)
(ha
veto
be c
ompu
ted.
4.D
efin
e a
rang
e, Φ
, for
the
com
pens
ator
’sph
aseφ
, and
disc
retiz
eit;
fore
xam
ple
[] º0:º5:º
360
−=
∈Φ
φ.
5.C
hoos
ea
sing
le fr
eque
ncy
ki
Ωω
∈.
6.C
hoos
ea
sing
le c
ontro
ller’
spha
seΦ
φ∈
.7.
Cho
ose
a si
ngle
pla
ntin
theω
i-tem
plat
e:
θω
∠=
pj
Pi
r)
(.
8.A
tthi
sste
p, th
ek
feed
back
pro
blem
isre
duce
dto
solv
ea
kqu
adra
ticin
equa
lity
with
outu
ncer
tain
ty. T
hefe
edba
ck
prob
lem
sin
equa
tions
(1) t
o(5
) in
Tabl
eI a
re re
duce
dto
ineq
ualit
iesi
n Ta
ble
II.9.
Com
pute
the
max
imum
)(
rm
axm
axP
gg
=an
dth
em
inim
um)
(r
min
min
Pg
g=
ofth
etw
oro
ots
g 1an
dg 2
that
solv
eth
ek
quad
ratic
ineq
ualit
y,.
10.R
epea
tSte
ps6
and
7 fo
rthe
mpl
ants
1,..
.0
),(
−=
mr
jP
ir
ωin
the
iω
tem
plat
e)
(i
jP
ω.
11.C
hoos
eth
em
ostr
estri
ctiv
eof
the
m)
(r
max
Pg
and
the
m)
(r
min
Pg
. Thu
s, )
(Pg m
axan
d)
(Pg m
inar
e ob
tain
ed. T
hey
are
the
max
imum
and
min
imum
boun
dpo
ints
fort
heco
ntro
llerm
agni
tude
g at
a ph
ase
φ .12
.Rep
eatS
tep
5 ov
erth
era
nge
Φ. T
heun
ion
of)
(Pg m
axan
d)
(Pg m
info
reac
hΦ
φ∈
give
sφ
∠m
axg
and
φ∠
min
g,
resp
ectiv
ely.
13.N
owth
ebo
unds
fort
heop
enlo
optra
nsm
issio
n0
00
)(
ψω
∠=
lj
Li
are
com
pute
d. S
etφ
ψ∠
⋅=
∠m
axm
axg
pl
00
0an
dφ
ψ∠
⋅=
∠m
inm
ing
pl
00
0, b
eing
[] 0:º5:º
360
, 00
−=
+=
φθ
φψ
. The
sebo
unds
will
be la
belle
d)
(i
kj
Bω
.14
.Rep
eatS
tep
4 ov
erth
era
nge
kΩ
. The
seto
fbou
ndsf
orth
ek
con
trolp
robl
em{
}k
ii
kj
BΩ
ωω
∈),
(ha
s jus
tbee
nco
mpu
ted.
Y. C
hait,
and
O. Y
aniv
, “M
ulti-
inpu
t/sin
gle-
outp
ut
com
pute
r-ai
ded
cont
rol d
esig
n us
ing
the
Qua
ntita
tive
Feed
back
The
ory,
”In
t. J.
Rob
ust &
N
on-l
inea
r C
ontr
ol, v
ol.3
, pp.
47-
54, 1
993.
1.30
Mar
io G
arci
a-Sa
nz
-350
-300
-250
-200
-150
-100
-50
0-5
0
-40
-30
-20
-1001020
1
1 11
2 2 23 3 3
Pha
se (d
egre
es)
Magnitude (dB)
All B
ound
s
The
set o
f ω ωωωi-b
ound
sPe
rfor
man
ceSp
ecifi
catio
ns(P
.S.)
Plan
tMod
el+
Unc
erta
inty
+
1.31
Mar
io G
arci
a-Sa
nz
Step
9: O
ptim
al B
ound
s St
ep 9
: Opt
imal
Bou
nds
BBoo(
j(jω ωωωω ωωω
ii))
-350
-300
-250
-200
-150
-100
-50
0
-30
-20
-1001020
Pha
se (d
egre
es)
Magnitude (dB)
Inte
rsec
tion
of B
ound
s
-350
-300
-250
-200
-150
-100
-50
0-5
0
-40
-30
-20
-1001020
1
1 11
2 2 23 3 3
Pha
se (d
egre
es)
Magnitude (dB)
All B
ound
s
Inte
rsec
tion
of ω ωωω
i-bou
nds
One
line
for
each
fr
eque
ncy
ω ωωω iTh
e m
ost d
eman
ding
one
1.32
Mar
io G
arci
a-Sa
nz
Step
10:
Syn
thes
izin
g G
(s)
Step
10:
Syn
thes
izin
g G
(s)
or L
oop
Shap
ing
Lor
Loo
p Sh
apin
g L
oo(s)(s)
Shap
ing
ofL o
(jω) =
Po(
jω) G
(jω)
Onl
y on
e L
= L o
to b
e sh
ape!
!
Lo(
jωi)
mus
t be
at e
very
ωi:
-out
side
the
U-c
onto
ur-a
bove
the
cont
inuo
us b
ound
s B0(
jω)
-bel
ow th
e di
scon
t. B
ound
s B0(
jω)
Synt
hesi
ze ra
tiona
l fun
ctio
nL o
(s)
Bui
ld u
p G
(jω) t
erm
-by-
term
add
ing
som
e el
emen
ts li
ke: g
ain,
real
pol
es a
nd z
eros
, co
mpl
ex p
oles
and
zer
os, i
nteg
rato
rs,
diff
eren
tiato
rs, l
ead/
lag
netw
orks
, not
ch
filte
rs, s
econ
d or
der T
F, e
tc.
Com
pens
ator
: G(s
) = L
o(s)
/Po(
s)
Prob
ably
one
of t
he m
ost d
iffic
ult s
teps
of
the
met
hodo
logy
for t
he b
egin
ner.
-180
°
8db
16db
24db 0db
-8db
-16d
b
-24d
b
-140
°-1
00°
-60°
Bo(j
1)
Bo(j
2) Bo(j
3)
Bo(j
4)
Bo(j
6)
Bo(j
10)
Bo(j
20)
ω=1
ω=2 ω=3
ω=5 ω
=10
ω=2
0
ω=3
0
ω=4
0ω
=60
ω=1
00
U-c
onto
ur(S
tabi
lity
Bou
nd)
LmL
o(jω
)
LmL
o(jω
)
A
ω=5
0
[B0(j
ω)]
ω≥ω
h=40
1.33
Mar
io G
arci
a-Sa
nz
1.
Gai
n
2.
Rea
l Pol
e
3.
Rea
l Zer
o
4.
Com
plex
Pol
e
5.
Com
plex
Zer
o)1
(1
s2
s
)1(
1s
2s
1
1zs
1ps1
k
n
2 n2
n
2 n2
<ζ
+ωζ
+ω
<ζ
+ωζ
+ω
++
Proc
edur
e:B
uild
up
ter
m-b
y-te
rm a
ddin
g so
me
elem
ents
like
:
1s
2s
1s
2s
1ps
1zsss1
1s
bs
b1
sa
sa
n22
n2n1
2
n2
nn
22
1
22
1
+ωζ
+ω
+ωζ
+ω
++
++
++
6.
2ºor
der /
2ºo
rder
7.
Inte
grat
or
8.
Diff
eren
ciat
or
9.
Lead
/Lag
N
etw
ork
10. N
otch
Filt
eret
c...
∏ ==
n 0i
)](j i
[GK
)(j
P)
(jL
oo
ωω
ω
1.34
Mar
io G
arci
a-Sa
nz
The
Inte
ract
ive
Des
ign
Env
iron
men
t(ID
E):
(Ter
asof
t, ve
rsio
n2)
Inte
ract
ive
tool
tode
sign
the
cont
rolle
rG(s
)Fu
nctio
n:
lpsh
ape(
...)
Onl
y on
e L
= L o
to b
e sh
ape!
!
& the
cont
rolle
ris
for t
hew
hole
set o
fun
certa
inpl
ants
1.35
Mar
io G
arci
a-Sa
nz
Step
11:
St
ep 1
1: P
refil
ter
Pref
ilter
Des
ign
Des
ign
F(s)
F(s)
+ 0 -
dBLm
ω
BU
δ L(jω
1)δ L(
jω2)
δ L(jω
3)δ L(
jω4)
BL
δ R(jω
4)
δ R(jω
3)
δ R(jω
2)
δ R(jω
1)
TL
ι
Fig.
9 C
lose
d-lo
op re
spon
ses:
LT
I pla
nts w
ith o
nly
G(s
)
Fig.
10
Clo
sed-
loop
resp
onse
s:
LTI p
lant
s with
G(s
) and
F(s
)
+ 0 -
dBLm
ω
BU
BL
TR
ι
to li
ebe
twee
n B U
&B L
for
all J
pla
nts
A P
refil
terF
(s) i
s nee
ded
YF
PG
UE
D
R
Pre-
filte
rPl
ant
Con
trolle
r-
1.36
Mar
io G
arci
a-Sa
nz
ω(r
ad/s
)
LmT R
L
LmT R
U
db
101
0
δ R(jω
i)
LmT m
ax
LmT m
in
ωi
100
10-1
δ R(jω
i)
LmT
LmT
BLm
TLm
F
LmT
LmT
BLm
TLm
F
RU
max
Um
axm
ax
RL
min
Lm
inm
in
−=
−=
−=
−=
Lm F
max
Lm F
min
+ 0 -
dBLm
ω
LmT R
L-L
mT m
in
0db/
dec
-20d
b/de
c
-40d
b/de
c
LmT R
U-L
mT m
ax
Freq
uenc
y bo
unds
on
the
pref
ilter
F(s)
.
F(s)
is sy
nthe
size
d,
in d
ashe
s, th
at li
es w
ithin
th
e up
per &
low
er
plot
s
Lm F
jj
jm
axm
in(
) <
Lm
F(
) <
Lm
F(
)ω
ωω
limF
ss→
=0
1(
)
We
have
tom
ove
dow
nm
ore
than
F max
but
less
than
F min
.
1.37
Mar
io G
arci
a-Sa
nz
(Ter
asof
t, ve
rsio
n2)
1.38
Mar
io G
arci
a-Sa
nz
Step
12:
Sim
ulat
e lin
ear
syst
em (J
tim
e re
spon
ses)
Step
12:
Sim
ulat
e lin
ear
syst
em (J
tim
e re
spon
ses)
Step
13:
Sim
ulat
e w
ithSt
ep 1
3: S
imul
ate
with
nonl
inea
ritie
sno
nlin
eari
ties
10-2
100
102
104
-70
-60
-50
-40
-30
-20
-100
Wei
ght:
--
Magnitude (dB)
Freq
uenc
y(ra
d/se
c)
10-2
10-1
100
101
-40
-35
-30
-25
-20
-15
-10-505
Wei
ght:
--
Magnitude (dB)
Freq
uenc
y(ra
d/se
c)
Tim
e (s
ec.)
Amplitude
Ste
p R
espo
nse
00.
20.
40.
60.
81
1.2
1.4
1.6
1.8
20
0.2
0.4
0.6
0.81
Freq
uenc
y D
omai
n
Tim
e D
omai
n
1.39
Mar
io G
arci
a-Sa
nz
Des
ign
proc
edur
e:
Pref
ilter
desi
gnF(
s)
Satis
fact
ory
sim
ulat
ion
?
Rea
djus
tmen
t
Indu
stri
al
Indu
stri
al
Impl
emen
tatio
nIm
plem
enta
tion
Exp
erim
enta
l Ver
ifica
tion
?
Ver
ifica
tion
In th
efr
eq. &
tim
edo
mai
n?
Ver
ifica
tion
In th
efr
eque
ncy
dom
ain?
Boun
dsCo
mpu
tatio
n
Yes
Spec
ifica
tions
disc
ussio
nM
athe
mat
ical
mod
elre
visio
n
Non
linea
rm
odel
s
Spec
ifica
tions
Tem
plat
es
Line
ariza
tion
Mat
hem
atic
alm
odel
Doe
saSo
lutio
nex
ist?
Loop
-sha
ping
G(s
)
Rea
djus
tmen
t
Rea
djus
tmen
t
No
No
No
Star
t
End
1.40
Mar
io G
arci
a-Sa
nz
•B
ooks
:
–H
OR
OW
ITZ,
I. M
., 19
93, Q
uant
itativ
e F
eedb
ack
Des
ign
Theo
ry (Q
FT)
.Q
FT P
ub.,
660
Sout
h M
onac
o Pa
rkw
ay, D
enve
r, C
olor
ado
8022
4-12
29.
–Y
AN
IV, O
., 19
99, Q
uant
itativ
e F
eedb
ack
Des
ign
of L
inea
r an
d N
on-l
inea
r C
ontr
ol S
yste
ms.
Klu
verA
cade
mic
Pub
., IS
BN
: 0-7
923-
8529
-2.
–SI
DI,
M..,
200
2, D
esig
n of
Rob
ust C
ontr
ol S
yste
ms:
Fro
m c
lass
ical
to
mod
ern
prac
tical
app
roac
hes.
Krie
gerP
ublis
hing
.
–H
OU
PIS,
CH
., R
ASM
USS
EN, S
J., G
AR
CIA
-SA
NZ,
M.,
2006
, Q
uant
itativ
e F
eedb
ack
Theo
ry. F
unda
men
tals
and
App
licat
ions
. 2nd
Editi
on. A
CR
C b
ook,
Tay
lor a
nd F
ranc
is.
1.3.
-Ref
eren
ces
1.41
Mar
io G
arci
a-Sa
nz
C.H
. Hou
pis,
S.J.
Ras
mus
sen
and
M. G
arcí
a-Sa
nz"Q
uant
itativ
e Fe
edba
ck T
heor
y.
Fund
amen
tals
and
appl
icat
ions
”.
2nded
ition
, 624
pag
es, a
CR
C P
ress
boo
k,
Tayl
or &
Fra
ncis,
Boc
a R
atón
, Flo
rida,
U
SA, I
SBN
: 084
9333
709,
Janu
ary
2006
. C.H
. Hou
pis,
S.J.
Ras
mus
sen
and
M. G
arcí
a-Sa
nz"S
olut
ions
Man
ual t
o Q
uant
itativ
e Fe
edba
ck
Theo
ry. F
unda
men
tals
and
appl
icat
ions
”. 2
nd
editi
on, 9
0 pa
ges,
Tayl
or &
Fra
ncis,
Boc
a R
atón
, Flo
rida,
USA
, Jan
uary
2006
.
Air
Forc
eIn
stitu
teof
Tech
nolo
gyW
right
-Pat
ters
onA
FB,
Day
ton,
Ohi
o, U
SA, 2
003
1.42
Mar
io G
arci
a-Sa
nz
•In
tern
atio
nal S
ympo
sia
on Q
uant
itativ
e Fe
edba
ck T
heor
y an
d R
obus
t Fr
eque
ncy
Dom
ain
Met
hods
–U
p to
now
ther
e ha
ve b
een
eigh
t Int
. Sym
p. o
n Q
FT:
1.-H
oupi
s, C
.H.,
Cha
nder
, P. (
Edito
rs).
Writ
ghtP
atte
rson
Airf
orce
Bas
e, D
ayto
n,
Ohi
o, U
SA, A
ugus
t 199
2.
2.-N
wok
ah, O
.D.I.
, Cha
nder
, P.(
Edito
rs).
Purd
ue U
nive
rsity
, Wes
t Laf
ayet
te, I
ndia
na,
USA
, Aug
ust 1
995.
3.-P
etro
poul
akis
, L.,
Lei
thea
d, W
.E.(E
dito
rs).
Uni
vers
ity o
f Stra
thcl
yde,
Gla
sgow
, Sc
otla
nd, U
K, A
ugus
t 199
7.
4.-B
oje,
E.,
and
Eite
lber
g, E
.(Ed
itors
). U
nive
rsity
of N
atal
, Dur
ban,
Sou
th A
fric
a,
Aug
ust 1
999.
5.-G
arcí
a-Sa
nz, M
.(Ed
itor)
. Pub
lic U
nive
rsity
of N
avar
ra, P
ampl
ona,
Spa
in, A
ugus
t20
01.
6.-B
oje,
E.,
and
Eite
lber
g, E
.(Ed
itors
). U
nive
rsity
of C
ape
Tow
n, C
ape
Tow
n, S
outh
A
fric
a, D
ecem
ber 2
003.
7.-C
olgr
en, R
.(Ed
itor)
. Uni
vers
ity o
f Kan
sas,
Law
renc
e, K
ansa
s, U
SA, A
ugus
t 200
5.
8.-G
utm
an, P
-O.(
Edito
r). T
echn
ion,
Hai
fa, I
srae
l, Ju
ly 2
007.
1.43
Mar
io G
arci
a-Sa
nz
•Sp
ecia
lIss
ues:
1.-N
wok
ah, O
.D.I.
(Gue
st E
dito
r). H
orow
itz a
nd Q
FT D
esig
n M
etho
ds. S
peci
al Is
sue.
In
tern
atio
nal J
ourn
al o
f Rob
ust a
nd N
onlin
ear
Con
trol
. Vol
. 4, N
um 1
, Jan
uary
-Fe
brua
ry 1
994.
Wile
y.
2.-H
oupi
s, C
.H. (
Gue
st E
dito
r). Q
uant
itativ
e Fe
edba
ck T
heor
y. S
peci
al Is
sue.
In
tern
atio
nal J
ourn
al o
f Rob
ust a
nd N
onlin
ear
Con
trol
. Vol
. 7, N
um 6
, Jun
e 19
97.
Wile
y.
3.-E
itelb
erg,
Edu
ard
(Gue
st E
dito
r). I
saac
Hor
owitz
. Spe
cial
Issu
e. In
tern
atio
nal J
ourn
al
of R
obus
t and
Non
linea
r C
ontr
ol. P
art 1
, Vol
. 11,
Num
10,
Aug
ust 2
001
and
Part
2,
Vol
. 12,
Num
4, A
pril
2002
. Wile
y.
4.-G
arci
a-Sa
nz, M
ario
(Gue
st E
dito
r). R
obus
t Fre
quen
cy D
omai
n. S
peci
al Is
sue.
In
tern
atio
nal J
ourn
al o
f Rob
ust a
nd N
onlin
ear
Con
trol
. Vol
. 13,
Num
7, J
une
2003
. W
iley.
5.-G
arci
a-Sa
nz, M
ario
and
Hou
pis,
Con
stan
tine
(Gue
st E
dito
rs).
Qua
ntita
tive
Feed
back
T
heor
y. In
Mem
oria
m o
f Isa
ac H
orow
itz. S
peci
al Is
sue.
Inte
rnat
iona
l Jou
rnal
of
Rob
ust a
nd N
onlin
ear
Con
trol
. Vol
. 17,
Num
2-3
, Jan
uary
200
7. W
iley.
1.44
Mar
io G
arci
a-Sa
nz
•BO
RG
HES
AN
I, C
., C
HA
IT, Y
., Y
AN
IV, O
.,20
02,
Qua
ntita
tive
Fee
dbac
k Th
eory
Tool
box
-For
use
with
MA
TLA
B, 2
nd E
d.U
SA
ht
tp://
ww
w.te
raso
ft.co
m/p
rodu
cts/
qft/
•G
UTM
AN
, P.O
., Q
syn.
H
aifa
, Isr
ael.
•H
OU
PIS,
C.H
., R
ASM
USS
EN
, S.,
GA
RC
IA-S
AN
Z, M
., 20
06Q
FT
CA
D T
oolf
orM
ISO
and
MIM
O s
yste
ms.
With
the
book
, Qua
ntita
tive
Fee
dbac
k Th
eory
. Fun
dam
enta
ls a
nd a
pplic
atio
nsTa
ylor
and
Fran
cis,
2nd
editi
onU
SA
1.4.
-QFT
Sof
twar
e to
ols
9.50
1.45
Mar
io G
arci
a-Sa
nz
Out
line
1.-
QFT
Con
trolle
r Des
ign
Tech
niqu
e Fu
ndam
enta
ls
2.-
Rea
l-wor
ld Q
FT c
ontro
l app
licat
ions
and
exa
mpl
es
3.-
Non
-dia
gona
l MIM
O Q
FT c
ontro
ller d
esig
n m
etho
dolo
gies
4.-A
pplic
atio
n: R
obus
t QFT
con
trol f
or a
MIM
O S
pace
craf
t with
flex
ible
suns
hiel
d
5.-S
witc
hing
robu
st c
ontro
l: B
eyon
d th
e lin
ear l
imita
tions
.
6.-E
xam
ple:
Sw
itchi
ng c
ontro
l for
Unm
anne
d V
ehic
les
1.46
Mar
io G
arci
a-Sa
nz
Mul
tipol
e,
Var
iabl
e Sp
eed,
Dire
ct D
rive
1650
kW
Win
d Tu
rbin
e.TW
T165
0. M
.Tor
res(
Spai
n).
•M
ore
than
20
cont
rol l
oops
•V
ery
Non
-line
ar m
odel
s•
MIM
O p
lant
•Pa
ram
eter
unc
erta
inty
•H
igh
relia
bilit
y ne
eded
•O
ptim
um e
ffic
ienc
y
E. T
orre
s, M
. Gar
cía-
Sanz
"Exp
erim
enta
l Res
ults
of t
he V
aria
ble
Spee
d, D
irec
t Dri
ve
Mul
tipol
eSy
nchr
onou
s W
ind
Turb
ine:
TW
T165
0". W
ind
Ene
rgy,
200
4.
2.-R
eal-w
orld
QFT
con
trol
app
licat
ions
: C
ontr
ol o
fa L
arge
Win
dTu
rbin
e
1.47
Mar
io G
arci
a-Sa
nz
TWT1
650.
Mec
hani
cal D
iagr
am
Pate
nts b
y M
.Tor
res
Win
d
Tran
sf
Grid
Gea
rbox
Indu
ctio
nG
ener
atorC
lass
ical
Sys
tem
Win
d
Dire
ctD
rive
Sync
hron
ous
gene
rato
r
Mul
tipol
eSy
stem
1.48
Mar
io G
arci
a-Sa
nz
TWT1
650.
Ele
ctri
cal B
lock
Dia
gram
Vca
pM
ultip
ole
Sync
hron
ous
Gen
erat
or
AC
-DC
con
v
I exc
I gen
Con
verte
rs
Win
d
Pitc
hM
otor
s
Tran
sf
DC
-AC
con
v
Win
d
Line
Yaw
Mot
ors
Con
trol S
yste
m
Safe
ty R
edun
dant
Sys
tem
Exci
tatio
n
IGB
TsIG
BTs
Var
iabl
e fr
eque
ncy
from
2.5
Hz
to 1
2.5
Hz
Con
stan
t fr
eque
ncy
50 H
z
DC
link
Var
iabl
e Sp
eed
M.T
orre
s
1.49
Mar
io G
arci
a-Sa
nz
TWT-
1650
1.50
Mar
io G
arci
a-Sa
nz
TWT1
650.
Fir
st P
roto
type
bui
lt in
May
200
1
Firs
t Pro
toty
pe a
t Cab
anill
asW
ind
Farm
(Spa
in)
M.T
orre
s
1.51
Mar
io G
arci
a-Sa
nz
TWT1
650.
Fir
st P
roto
type
bui
lt in
May
200
1
Tow
er: 7
0 m
; B
lade
: 40
mR
otor
: 82
m ;
Iner
tia ro
tor:
5,00
0,00
0 K
g m
2
M.T
orre
s
1.52
Mar
io G
arci
a-Sa
nz
Con
trol L
oop
1:In
put:
Win
d Tu
rbin
e R
otor
Spe
ed
Ref
. = 2
0 rp
mO
utpu
t:Pi
tch
Ang
le R
efer
ence
.
Con
trol L
oop
2:In
put:
Pitc
h A
ngle
Ref
eren
ce.
Out
put:
Pitc
h Sp
eed
Ref
eren
ce.
Con
trol L
oop
3:In
put
Pitc
h Sp
eed
Ref
eren
ce.
Out
put:
Mot
or C
urre
nt R
efer
ence
.
P 3(s
)P 2
(s)
C2(
s)
Pitc
hA
ngle
Ref
I(s)
--
Pitc
han
gle
cont
rolle
rPitc
hSp
eed
Ref
C3(
s)
Pitc
hsp
eed
cont
rolle
r
P 1(s
)
Pitc
hSp
eed
Pitc
hA
ngle
Rot
or
Spee
dC
1(s)
Rot
or
Spee
dR
ef
-R
otor
spee
dco
ntro
ller
Con
trol S
yste
m
TWT1
650.
Act
ual R
esul
ts.
Exam
ple
of th
ree
of m
ore
than
20
loop
s.M
.Tor
res
Targ
et: T
o co
ntro
l the
Win
d Tu
rbin
e R
otor
Spe
ed .
Act
uato
rs: P
itch
angl
e m
ovem
ent.
3 In
depe
nden
t dr
iven
bla
des.
1.53
Mar
io G
arci
a-Sa
nz
06.A
pril.
2003
. Cab
anill
asW
ind
Farm
(Spa
in).
Ver
y H
igh
Win
d:A
ver.
24
m/s
TWT1
650.
Act
ual R
esul
ts.
Exam
ple
of th
ree
of m
ore
than
20
loop
s.
Targ
et: c
ontro
l Rot
or S
peed
(r.p
.m.).
Set-p
oint
: 20
rpm
Act
uato
rs: P
itch
angl
e m
ovem
ent.
3 In
depe
nden
t driv
en b
lade
s.
010
020
030
040
050
060
00510152025
Tim
e (s
ec)
Rot
or S
peed
-rpm
-
010
020
030
040
050
060
01415161718192021222324
Tim
e (s
ec)
Con
trol P
itch
Ang
le: r
ef(k
),1(r)
,2(b
),3(g
) (de
g bl
ade)
010
020
030
040
050
060
01618202224262830
Tim
e (s
ec)
Win
d S
peed
-m/s
-
M.T
orre
s
1.7
MW
800
kNm
4.5
MW
2140
kN
m
Win
dG
ust
24
m/s
Test
at v
ery
high
Win
d Sp
eed.
Ver
y N
on-li
near
mod
el.
()
3p
2ai
rV
CR
5.0Po
wer
πρ
=
Cp =
f(V
,Ω,β
)
1.54
Mar
io G
arci
a-Sa
nz
TWT1
650.
Act
ual R
esul
ts.
Exam
ple
of th
ree
of m
ore
than
20
loop
s.
Targ
et: c
ontro
l Rot
or S
peed
(r.p
.m.).
Set-p
oint
: 20
rpm
Act
uato
rs: P
itch
angl
e m
ovem
ent.
3 In
depe
nden
t driv
en b
lade
s.
M.T
orre
s15
m
/s
410
420
430
440
450
460
470
480
490
500
510
1213141516171819
Tim
e (s
ec)
Win
d S
peed
-m/s
-
410
420
430
440
450
460
470
480
490
500
510
024681012141618202224
Tim
e (s
ec)
Rot
or S
peed
-rpm
-
410
420
430
440
450
460
470
480
490
500
510
456789101112
Tim
e (s
ec)
Con
trol P
itch
Ang
le: r
ef(k
),1(r)
,2(b
),3(g
) (de
g bl
ade)
Pitc
h A
ngle
Ref
eren
ce
Pitc
h A
ngle
04.F
ebru
ary.
2003
. Cab
anill
asW
ind
Farm
(Spa
in).
Med
ium
Win
d:A
vera
ge 1
5 m
/s
Test
at v
ery
high
Win
d Sp
eed.
Ver
y N
on-li
near
mod
el.
()
3p
2ai
rV
CR
5.0Po
wer
πρ
=
Cp =
f(V
,Ω,β
)
1.55
Mar
io G
arci
a-Sa
nz
M.T
orre
s
Vid
eo:
TWT1
650
1.56
Mar
io G
arci
a-Sa
nz
Out
line
1.-
QFT
Con
trolle
r Des
ign
Tech
niqu
e Fu
ndam
enta
ls
2.-
Rea
l-wor
ld Q
FT c
ontro
l app
licat
ions
and
exa
mpl
es
3.-
Non
-dia
gona
l MIM
O Q
FT c
ontro
ller d
esig
n m
etho
dolo
gies
4.-A
pplic
atio
n: R
obus
t QFT
con
trol f
or a
MIM
O S
pace
craf
t with
flex
ible
suns
hiel
d
5.-S
witc
hing
robu
st c
ontro
l: B
eyon
d th
e lin
ear l
imita
tions
.
6.-E
xam
ple:
Sw
itchi
ng c
ontro
l for
Unm
anne
d V
ehic
les
1.57
Mar
io G
arci
a-Sa
nz
This
sect
ion
disc
usse
s how
the
QFT
tech
niqu
e ca
n be
ap
plie
d to
the
desi
gn o
f MIM
O c
ontro
l sys
tem
s.
2x2
Exam
ple
of a
MIM
O s
yste
m
T 1N
1
T 2Q
p 11 p 12 p 21 p 22
g 11 g 22
+ +-
-e 1
e 2
r 1 r 2
2x2
Plan
t
N1
N2
T 1 T 2
QH
OT
WA
TER
TAN
K
HEA
TEX
CH
AN
GER
PRO
DU
CT
HO
TW
ATE
R
New
Pro
blem
s: In
tera
ctio
nbe
twee
n co
ntro
l lo
ops,
Inpu
t and
out
put D
irect
ions
, Inp
ut-
outp
ut P
airin
g, T
rans
mis
sion
Zer
os(R
HP)
.
New
Too
ls: R
GA
, SV
D,
Smith
-McM
illan
…
3.N
on-d
iago
nal M
IMO
QFT
C
ontr
olle
r D
esig
n M
etho
dolo
gies
g 21 g 12
(non
-dia
gona
l con
trol
ler)
1.58
Mar
io G
arci
a-Sa
nz
A fu
lly p
opul
ated
(non
-dia
gona
l) m
atrix
com
pens
ator
allo
ws t
he d
esig
ner m
uch
mor
e de
sign
flex
ibili
tyto
gov
ern
MIM
O sy
stem
s tha
n th
e cl
assi
cal d
iago
nal c
ontro
ller.
This
sess
ion
exte
ndst
he c
lass
ical
dia
gona
l QFT
com
pens
ator
desi
gn to
a fu
lly
popu
late
d m
atrix
com
pens
ator
desi
gn.
In th
is se
ssio
n th
ree
case
sare
stud
ied:
-t
he re
fere
nce
track
ing,
-t
he e
xter
nal d
istu
rban
ce re
ject
ion
at th
e pl
ant i
nput
and
-t
he e
xter
nal d
istu
rban
ce re
ject
ion
at p
lant
out
put.
The
defin
ition
of t
hree
cou
plin
g m
atri
ces(
C1,
C2,
C3)
of t
he n
on-d
iago
nal e
lem
ents
ar
e us
ed to
qua
ntify
the
amou
nt o
f loo
p in
tera
ctio
n an
d to
des
ign
the
non-
diag
onal
co
mpe
nsat
ors r
espe
ctiv
ely.
This
yie
lds a
crit
erio
n to
pro
pose
a se
quen
tial d
esig
n m
etho
dolo
gyof
the
fully
po
pula
ted
mat
rix c
ompe
nsat
orin
the
QFT
robu
st c
ontro
l fra
me.
3.1.
-Int
rodu
ctio
n
Gar
cía-
Sanz
M.,
Egañ
aI.
(200
2). Q
uant
itativ
e N
on-d
iago
nal
Con
trol
ler
Des
ign
for
Mul
tivar
iabl
e Sy
stem
s with
Unc
erta
inty
. In
t. J.
Rob
ust N
onlin
ear
Con
trol
, Vol
. 12,
No.
4, p
p. 3
21-3
33.
Gar
cía-
Sanz
M.,
Egañ
aI.,
Bar
rera
s M. (
2005
). D
esig
n of
qu
antit
ativ
e fe
edba
ck th
eory
non
-dia
gona
l con
trol
lers
for
use
in
unce
rtai
n m
ultip
le-in
put m
ultip
le-o
utpu
t sys
tem
s. IE
EC
ontr
ol
Theo
ry a
nd A
pplic
atio
ns. V
ol. 1
52, N
. 02,
pp.
177
-187
.
1.59
Mar
io G
arci
a-Sa
nz
Con
side
r an
nxn
linea
r mul
tivar
iabl
e sy
stem
(see
Fig
ure)
, com
pose
d of
a
plan
t P, a
fully
pop
ulat
ed m
atrix
com
pens
ator
G, a
nd a
pre
filte
rF:
MIM
O S
yste
m
=
==
nnn
n
nn
nnn
n
nn
nnn
n
nn
ff
f
ff
f
ff
f
gg
g
gg
g
gg
g
pp
p
pp
p
pp
p
.........
;
.........
;
.........
21
222
21
112
11
21
222
21
112
11
21
222
21
112
11
F
GP
F(s)
G(s
)P(
s)u
r’r
y
T(s)
-
P do(
s) d od o’
P di(s
) d id i’
whe
re P
∈,
and
is th
e se
t of
poss
ible
pla
nts d
ue to
unc
erta
inty
.
1.60
Mar
io G
arci
a-Sa
nz
The
plan
t inv
erse
, den
oted
by
P*
, is p
rese
nted
in th
e fo
llow
ing
form
at:
[]
+=
+=
==
0...
p...
0...
p...
0
p0
00
00
0p
** 1n
* n1
* nn
* 11* ij
1-B
PP
p
and
whe
re th
e co
mpe
nsat
or m
atrix
is b
roke
n up
into
two
parts
asf
ollo
ws:
+=
+=
0...
g...
0...
g...
0
g0
00
...0
00
g
n1
1n
nn
11
bd
GG
G
The
follo
win
g in
trodu
ces a
mea
sure
men
t in
dex
to q
uant
ify th
e lo
op in
tera
ctio
n in
the
thre
e cl
assi
cal c
ases
: ref
eren
ce tr
acki
ng,
exte
rnal
dis
turb
ance
s at t
he p
lant
inpu
t, an
d th
e ex
tern
al d
istu
rban
ces a
t the
pla
nt o
utpu
t.
F(s)
G(s
)P(
s)u
r’r
y
T(s)
-
P do(
s) d od o’
P di(s
) d id i’
1.61
Mar
io G
arci
a-Sa
nz
Ref
eren
ce T
rack
ing.
The
trans
fer f
unct
ion
mat
rix o
f the
con
trol s
yste
m fo
r the
refe
renc
e tra
ckin
g pr
oble
m, w
ithou
t any
ext
erna
l dis
turb
ance
, is w
ritte
n as
follo
ws:
()
'/
/1
rr
rr
yr
yF
TT
GP
GP
Iy
==
+=
−
()
()
()
[] r
rr
r-
-y/
ry/
rT
GG
GI
GG
IT
bb
1-1
d1
d1-
1d
1+
−+
++
=−
−B
and
appl
ying
the
defin
ition
s of :
P* =
Λ ΛΛΛ+
Ban
d
G=
Gd
+ G
b
A d
iago
nal t
erm
A n
on-d
iago
nal t
erm
F(s)
G(s
)P(
s)u
r’r
y
T(s)
-
P do(
s) d od o’
P di(s
) d id i’
1.62
Mar
io G
arci
a-Sa
nz
A d
iago
nal t
erm
:
A d
iago
nal
term(
)d
1-1
d1
y/r_
dG
GI
T−
+=
-
g ii
r iy i
t i(s)
-
u i*1 iip
A n
on-d
iago
nal t
erm
: (
)(
)[
](
)1
1-1
d1-
bb
1-1
d1
y/r_
bC
GI
TG
GG
IT
y/r
B−
−=
+=
+−
+-
A n
on-d
iago
nal
term
g ii
u i0
y i
t i(s)
-
=
=n 1
kj
ij1j
rc
d
*1 iip
()
=+
−=
mm
mm
mm
bb
cc
c
cc
c
cc
c
12
11
1
21221
211
11121
111
1y/
rT
GB
GC
()
()
()
−+
−−
==n 1
kik
kjik
* ikij
ijij1
1g
p1
gc
t
C1
repr
esen
ts th
e co
uplin
g m
atri
x C
of t
he e
quiv
alen
t sys
tem
for r
efer
ence
trac
king
pro
blem
s
≠⇔
==
⇔=
=i
k0
ik
1kiki
ki
1.63
Mar
io G
arci
a-Sa
nz
The
trans
fer f
unct
ion
mat
rix o
f the
con
trol s
yste
m fo
r the
ext
erna
l dis
turb
ance
re
ject
ion
at p
lant
inpu
tpro
blem
, with
out a
ny e
xter
nal d
istu
rban
ce, i
s writ
ten
as,
and
appl
ying
the
defin
ition
s of
P*
= Λ ΛΛΛ
+ B
and
G
= G
d+
Gb
A d
iago
nal
term
A n
on-d
iago
nal
term
Ext
erna
l dist
urba
nce
reje
ctio
n at
pla
nt in
put.
()
'/
/1
idi
diy
idi
yi
dP
dd
yT
TP
GP
I=
=+
=−
()
()
()
[] i
-i
-i
y/di
dd
dy/
dib
1-1
d1
1-1
d1
TG
GI
GI
T+
+−
+=
−−
B
F(s)
G(s
)P(
s)u
r’r
y
T(s)
-
P do(
s) d od o’
P di(s
) d id i’
1.64
Mar
io G
arci
a-Sa
nz
A d
iago
nal t
erm
:
A d
iago
nal
term(
)1-
1d
1y/
di_d
−+
=G
IT
-
g ii
0y i
t i(s)
-
u i
dii
*1 iip
A n
on-d
iago
nal t
erm
:
A n
on-d
iago
nal
term
()
()
()
21-
1d
1-b
1-1
d1
y/di
_bB
CG
IT
GG
IT
y/di
−−
+=
++
=-
g ii
u i0
y i
t i(s)
-
=
=n 1
kj
ij2j
dic
d
*1 iip
C2
repr
esen
ts th
e co
uplin
g m
atri
x of
the
equi
vale
nt sy
stem
for e
xter
nal d
istu
rban
ce re
ject
ion
at th
e pl
ant i
nput
pro
blem
s
≠⇔
==
⇔=
=i
k0
ik
1kiki
ki
()
y/di
TG
Cb
+=
B2
)1(
t)g
p(c
ikkj
ikn 1
k* ik
ij2δ
−+
==
1.65
Mar
io G
arci
a-Sa
nz
The
trans
fer f
unct
ion
mat
rix o
f the
con
trol s
yste
m fo
r the
ext
erna
l dis
turb
ance
re
ject
ion
at p
lant
out
putp
robl
em, w
ithou
t any
ext
erna
l dis
turb
ance
, is w
ritte
n as
,
and
appl
ying
the
defin
ition
s of
P*
= Λ ΛΛΛ
+ B
and
G
= G
d+
Gb
A d
iago
nal
term
A n
on-d
iago
nal
term
Ext
erna
l dist
urba
nce
reje
ctio
n at
pla
nt o
utpu
t.
()
'/
/1
odo
doy
odo
yo
dd
dy
PT
TG
PI
==
+=
−
()
()
()
[] o
-o
-o
dd
dy/
doy/
doT
GG
IG
IT
b1-
1d
11
d1
+−
++
+=
−−
BB
F(s)
G(s
)P(
s)u
r’r
y
T(s)
-
P do(
s) d od o’
P di(s
) d id i’
1.66
Mar
io G
arci
a-Sa
nz
≠⇔
==
⇔=
=i
k0
ik
1kiki
ki
()
y/do
TG
BB
Cb
+−
=3
)1(
)(
)1(
ikkj
ik1
* ikij
* ij3i
jδ
δ−
+−
−=
=
tg
pp
cn k
C3
repr
esen
ts th
e co
uplin
g m
atri
x of
the
equi
vale
nt sy
stem
for e
xter
nal d
istu
rban
ce re
ject
ion
at th
e pl
ant o
utpu
t pro
blem
s
A d
iago
nal t
erm
:
A d
iago
nal
term(
)1d
1y/
do_d
−+
=G
IT
-
g ii
0y i
t i(s)
-
u i
doi
*1 iip
A n
on-d
iago
nal t
erm
:
A n
on-d
iago
nal
term
()
()
[]
()
31-
1d
1-b
1-1
d1
y/do
_bC
GI
TG
GI
Ty/
doB
B−
−+
=+
−+
=-
g ii
u i0
y i
t i(s)
-
=
=n 1
kj
ij3j
doc
d
*1 iip
1.67
Mar
io G
arci
a-Sa
nz
The
Cou
plin
g el
emen
ts
To d
esig
n a
MIM
O c
ompe
nsat
or w
ith a
low
cou
plin
g le
vel,
it is
nec
essa
ry to
stud
y th
e in
fluen
ce o
f eve
ry n
on-d
iago
nal e
lem
ent g
ijon
the
coup
ling
elem
ents
c1i
j, c 2i
jan
d c 3i
j.
Hyp
othe
sis
Thus
,
()
()
jjkj
ik* ik
ij* ij
tof
band
wid
th
in th
ean
dj,
kfo
r,
tg
pjjt
gp
≠+
>>+
jjkj
jj t
ofba
ndw
idth
in
the
and
j,k
for
,≠
>>t
t
1.68
Mar
io G
arci
a-Sa
nz
Due
to h
ypot
hesi
s, th
e co
uplin
g ef
fect
s c1i
j,c 2
ij,c 3
ijar
e co
mpu
ted
as,
track
ing
dist
urba
nce
reje
ctio
n at
the
plan
t inp
ut
dist
urba
nce
reje
ctio
n at
pla
nt o
utpu
t
()
()
ji
;g
p
gp
gg
cjj
* jj
ij* ij
jjij
ij1≠
+
+−
=
()
()
ji
;g
p
gp
cjj
* jj
ij* ij
ij2≠
++=
()
()
ji
;g
p
gp
pp
cjj
* jj
ij* ij
* jj* ij
ij3≠
+
+−
=
1.69
Mar
io G
arci
a-Sa
nz
The
Opt
imum
non
-dia
gona
l con
trol
ler
The
optim
um n
on-d
iago
nal c
ompe
nsat
ors f
or th
e th
ree
case
s (tra
ckin
g an
d di
stur
banc
e re
ject
ion
at p
lant
inpu
t and
out
put)
are
obta
ined
mak
ing
last
thre
e Eq
s. eq
ual t
o ze
ro.
whe
re th
e fu
nctio
n F
pd(A
) mea
ns in
eve
ry c
ase
a st
able
pro
per f
unct
ion
mad
e fr
om
the
dom
inan
t pol
es a
nd z
eros
of t
he e
xpre
ssio
n A
.
ji
for
,pp
gg
* jj* ijjj
opt
ij≠
=NN
()
ji
for
,p
gN
* ijop
tij
≠−
=pd
F
ji
for
,pp
gg
N* jjN* ij
jjop
tij
≠=
track
ing
dist
urba
nce
reje
ctio
n at
the
plan
t inp
ut
dist
urba
nce
reje
ctio
n at
pla
nt o
utpu
t
1.70
Mar
io G
arci
a-Sa
nz
Ever
y un
certa
in p
lant
can
be
any
plan
t rep
rese
nted
by
the
fam
ily:
whe
re
i
s the
nom
inal
pla
nt (≠
P o), a
nd
is th
e m
axim
um o
f the
non
-pa
ram
etric
unc
erta
inty
radi
i
{}
()
n1,
...,
ji,
for
,p
0,
1p
p* ij
ijij
N* ij
* ij=
≤≤
+=
N* ijp
* ijpΔij
The
nom
inal
pla
nts
min
imis
e th
e m
axim
um o
f the
non
-par
amet
ric
unce
rtain
ty ra
dii
and
th
at
com
pris
e th
e pl
ant t
empl
ates
.
N* ijp * ijpΔ
* jjpΔ
* ijpΔ
N * ijp* ijpΔ
Imag
ReN * ijp
ij
ij
1.71
Mar
io G
arci
a-Sa
nz
Des
ign
Met
hodo
logy
Step
A.C
ontr
olle
r St
ruct
ure,
C
oupl
ing
Ana
lysi
s, In
put/o
utpu
t P
airi
ng a
nd lo
op o
rder
ing.
Firs
t, th
e m
etho
dolo
gy b
egin
s pai
ring
the
plan
t inp
uts a
nd o
utpu
ts a
nd se
lect
ing
the
cont
rolle
rst
ruct
ure
with
the
Rel
ativ
e G
ain
Ana
lysi
s (R
GA
-Bris
tol)
tech
niqu
e.
This
is fo
llow
ed b
y ar
rang
ing
the
mat
rix
P* so
that
ha
s the
smal
lest
pha
se
mar
gin
freq
uenc
y,
the
next
smal
lest
ph
ase
mar
gin
freq
uenc
y, a
nd so
on.
The
sequ
entia
ltec
hniq
ue, c
ompo
sed
of n
stag
es (n
loop
s), r
epea
ts st
eps (
B a
nd C
) fo
r eve
ry c
olum
n k
= 1
to n
.
The
com
pens
ator
des
ign
met
hod
is a
sequ
entia
l pr
oced
ure
by c
losin
g lo
ops.
()1
* 11p−
()1
* 22p−
nnnk
2n1n
knkk
2k1k
n2k2
2221
n1k1
1211
g...
g...
gg
......
...
g...
g...
gg
......
...
g...
g...
gg
g...
g...
gg
Step
n
...
=
2n1n
2k1k
2221
1211
1n1k2111
...
0...
0...
gg
......
...
0...
0...
gg
......
...
0...
0...
gg
0...
0...
gg
0...
0...
0g
......
...
0...
0...
0g
......
...
0...
0...
0g
0...
0...
0g
G
Step
1St
ep 2
1.72
Mar
io G
arci
a-Sa
nz
Step
B.D
esig
n of
the
diag
onal
co
mpe
nsat
or e
lem
ents
gkk
.
This
des
ign
of th
e el
emen
t gkk
is
calc
ulat
ed u
sing
the
stan
dard
QFT
lo
op-s
hapi
ng te
chni
que
for t
he
inve
rse
of th
e eq
uiva
lent
pla
nt
in
ord
er to
ach
ieve
robu
st st
abili
ty
and
robu
st p
erfo
rman
ce
spec
ifica
tions
.
nnnk
2n1n
knkk
2k1k
n2k2
2221
n1k1
1211
g...
g...
gg
......
...
g...
g...
gg
......
...
g...
g...
gg
g...
g...
gg
Step
n
...
=
2n1n
2k1k
2221
1211
1n1k2111
...
0...
0...
gg
......
...
0...
0...
gg
......
...
0...
0...
gg
0...
0...
gg
0...
0...
0g
......
...
0...
0...
0g
......
...
0...
0...
0g
0...
0...
0g
G
Step
1St
ep 2
()1
p*e− k
kk
[]
[]
[]
() [
](
)[
](
)[
]1
1*
1,1
1*
1,1
,11
*,1
1,
1*
1,
1*
*
−−
−−
−−
−−
−−
−−
−
=
+
++
−=
Pp
gp
gp
gp
pp
e ik
ii
ie
ii
ki
ie
ki
ii
ie i
i
ie ik
ie ik
k
e kkk
kp
pp
pp
p
pp
p
pp
p
*1
* 21
* 1
1* 33
1* 32
1* 31
1* 23
1* 22
1* 21
1* 13
1* 12
1* 11 fu
nctio
nof
1.73
Mar
io G
arci
a-Sa
nz
Step
C.D
esig
n of
the
non-
diag
onal
co
mpe
nsat
or e
lem
ents
gij
The
(n-1
) non
-dia
gona
l ele
men
ts g
ik
(i ≠
k, i
= 1,
2,...
n) o
f the
k-th
com
pens
ator
col
umn
are
desi
gned
to,
min
imis
e th
e cr
oss-
coup
ling
term
s cik
.nn
nk2n
1n
knkk
2k1k
n2k2
2221
n1k1
1211
g...
g...
gg
......
...
g...
g...
gg
......
...
g...
g...
gg
g...
g...
gg
Step
n
...
=
2n1n
2k1k
2221
1211
1n1k2111
...
0...
0...
gg
......
...
0...
0...
gg
......
...
0...
0...
gg
0...
0...
gg
0...
0...
0g
......
...
0...
0...
0g
......
...
0...
0...
0g
0...
0...
0g
G
Step
1St
ep 2
Step
D.T
he d
esig
n of
the
pref
ilter
Fdo
es n
ot p
rese
nt a
ny d
iffic
ulty
be
caus
e th
e fin
al T
y/rfu
nctio
n sh
ows
less
loop
inte
ract
ion.
The
refo
re, t
he
pref
ilter
Fca
n be
dia
gona
l.
()
()
0g
p
gp
gg
cjj
* jj
ij* ij
jjij
ij1=
+
+−
=
()
()
0g
p
gp
cjj
* jj
ij* ij
ij2=
++=
()
()
0g
p
gp
pp
cjj
* jj
ij* ij
* jj* ij
ij3=
+
+−
=
1.74
Mar
io G
arci
a-Sa
nz
The
sequ
entia
l non
-dia
gona
l MIM
O Q
FT te
chni
que
intro
duce
d he
re a
rriv
es a
t a
robu
st st
able
clos
ed-lo
op sy
stem
if ,
for e
ach
P∈
TP
a)ea
ch L
i(s)
= g ii(
s) (
p ii*e)−1
, i=
1, .
.., n
, sa
tisfie
s th
e N
yqui
sten
circ
lem
ent
cond
ition
,
b)no
RH
P po
le-z
ero
canc
ella
tions
occ
ur b
etw
een
g ii(s)
and
(pii*e
)−1, i
=1, .
.., n
,
c)no
Sm
ith-M
cMill
an p
ole-
zero
can
cella
tions
occ
ur b
etw
een
P(s)
and
G(s
), an
d
d)no
Sm
ith-M
cMill
an p
ole-
zero
can
cella
tions
occ
ur in
⏐P*
(s) +
G(s
)⏐
Rob
ust S
tabi
lity
of th
e M
IMO
syst
em
Che
cked
at
eac
h lo
op
Che
cked
at
the
end
1.75
Mar
io G
arci
a-Sa
nz
Des
ign
of th
e co
ntro
ller
mat
rix
G(s
)
Non
-dia
gona
l MIM
O Q
FT m
etho
d
Cal
cula
tion
of tr
ansm
issi
on z
eros
Smith
-McM
illan
form
Mod
ifica
tion
of R
HP
tran
smis
sion
zer
os
Non
-dia
gona
l ele
men
ts g
ij(i
≠j) p
lace
d in
the
last
col
umn
of th
e m
atrix
con
trolle
r G(s
)
Are
ther
e R
HP
trans
mis
sion
zer
os?
Yes
No E
ND
nnnk
2n1n
knkk
2k1k
n2k2
2221
n1k1
1211
g...
g...
gg
......
...
g...
g...
gg
......
...
g...
g...
gg
g...
g...
gg
To re
mov
e th
e R
HP
tra
nsm
issi
on z
eros
th
at c
ould
be
intro
duce
d in
the
prev
ious
con
trolle
r de
sign
RH
P tr
ansm
issio
n ze
ros o
f the
M
IMO
syst
em
1.76
Mar
io G
arci
a-Sa
nz
Any
ques
tions
?
Than
ks
End
first
part
