TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 13, No. 2, Februa
ry 20
15, pp. 282 ~ 286
DOI: 10.115
9
1
/telkomni
ka.
v
13i2.688
6
282
Re
cei
v
ed O
c
t
ober 2
0
, 201
4; Revi
se
d Decem
b
e
r
18, 2014; Accept
ed Ja
nua
ry 3,
2015
Solving Method of H-Infinity Model Mat
c
hing Based on
the Theory of the Model Reduction
Li Minzhi, Ca
o Xinjun
T
he School of Electron
ics an
d Information E
ngi
neer
in
g,
La
nzho
u Jia
o
tong
Universit
y
, La
nzho
u, Chi
n
a
*Corres
p
o
ndi
n
g
author, em
ail
:
sqlmz@sina.c
o
m
A
b
st
r
a
ct
Peop
le
used t
o
solv
e hi
gh-
o
r
der
H
mo
del
matchin
g
b
a
sed
on
H
control t
heory, it is t
o
o
difficult. In this pap
er, w
e
use mo
de
l reducti
o
n
theory to sol
v
e hig
h
-ord
er
H
mo
de
l matchin
g
prob
le
m, A
new
method t
o
solv
e
H
mod
e
l
match
i
ng
p
r
obl
em
bas
ed
on the th
eor
y of the
mod
e
l re
ductio
n
i
s
prop
osed
．
T
h
e si
mulati
on
re
sults sh
ow
that
the
metho
d
h
a
s b
e
tter a
ppl
i
c
abil
i
ty a
n
d
ca
n g
e
t the
ex
pe
cted
perfor
m
a
n
ce
．
Ke
y
w
ords
:
hi
g
h
-ord
er mode
l, reducti
on the
o
r
y
,
H
mo
de
l matchin
g
Copy
right
©
2015 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1.
Introduc
tion
H-infinity (
H
) o
p
timal
control
theo
ry of lin
ear sy
st
ems i
s
a
ne
w
kin
d
of de
sign
met
hod
develop
ed in
the end
of 19
80, and
is th
e
very acti
ve frontier
su
bje
c
t in current
co
ntrol the
o
ry. In
many co
ntrol
system
s, in o
r
de
r to imp
r
o
v
e t
he stea
dy and dyn
a
mic perfo
rman
ce
of system, th
e
approp
riate
corre
c
tion d
e
vice ne
ed
s to be
add
ed in the
system, ma
king th
e ou
tput
cha
r
a
c
t
e
ri
st
ic
s of
t
h
e
sy
st
em m
eet
all
of the de
ma
nd for pe
rfor
mance
spe
c
if
ics. T
h
is is t
he
model m
a
tch
i
ng p
r
oble
m
. In solvin
g the mo
del m
a
tchin
g
p
r
obl
em, it is m
o
stly solved
by
conve
r
ting to
H
standa
rd con
t
rol pro
b
lem [1-2]. Che
n
Yongjin p
r
op
osed a kind of u
pper b
oun
d
method of sea
r
ching fo
r multi-blo
cks of model
matching [
3
]. Zhuge
Hai propo
se
d an
approximate
method
of impre
c
ise mo
de
l matchi
ng
[4]
.
These meth
ods
are ea
sy
to be a
c
hi
eved
for gen
eral
systems, but t
hese metho
d
s
are mo
re complicated fo
r high o
r
d
e
r
system m
ode
l.
Moore propo
sed th
e bala
n
c
e o
r
de
r redu
ction p
r
obl
em
of system in
1981 [5], the
n
the metho
d
is
improve
d
co
n
s
tantly [6], and some n
e
w
redu
ction alg
o
r
ithms
were p
u
t forwa
r
d [7-9].
Due to th
e hi
gh o
r
de
r p
r
ob
lem of sy
ste
m
model i
n
H
model m
a
tchi
ng, co
mbinin
g with
the
model order redu
ction
theory,
H
mod
e
l matchi
ng resolvin
g met
hod i
s
propo
sed ba
sed
on
model
red
u
cti
on the
o
ry. Th
e an
alysis an
d sim
u
lation
sho
w
th
at the
method
ha
s
good
matchin
g
cha
r
a
c
t
e
ri
st
ic
s.
2.
H
Model Matching Probl
em
P
s
K
s
z
y
u
w
Figure 1. Prin
ciple figu
re of
H
standa
rd problem
I
n
co
nt
rol
sy
st
em,
ma
ny
H
optimization
pro
b
lem
s
of different
re
q
u
irem
ents ca
n be
conve
r
ted int
o
H
stan
da
rd p
r
oble
m
. As
shown in Fig
u
re 1,
w
is the
external i
nput,
z
is control
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TELKOM
NIKA
ISSN:
2302-4
046
Solving M
e
th
od of
H
Model Matchin
g
Based on the Th
eory of the Model… (LI Minzhi
)
283
output, an
d
u
is
the
control
input,
y
is th
e outp
u
t of
measurement
.
Ps
is the
gen
erali
z
ed
controlled o
b
j
e
ct,
Ks
is desi
g
n
ed co
ntrolle
r.
State equatio
n of the gene
ralized obj
ect
Ps
is described
as:
12
x
Ax
B
w
B
u
(
1)
11
1
1
2
zC
x
D
w
D
u
(
2)
22
1
2
2
y
Cx
D
w
D
u
(
3)
Tran
sfe
r
function is:
12
11
12
11
1
1
2
21
22
22
1
2
2
A
BB
PP
Ps
C
D
D
PP
CD
D
(
4)
Usi
ng the lin
ear fra
c
tional
transfo
rmati
on (LFT
), tra
n
sfer fun
c
tion
from
w
to
z
can be
descri
bed a
s
:
1
11
12
22
21
,
l
GF
P
K
P
P
K
I
P
K
P
(
5)
The
H
standa
rd cont
rol pro
b
lem is fo
r a regul
ar
cont
roller
K
, making
the clo
s
ed
-lo
op
of system sta
b
le, and
,
l
FP
K
less than a given
,
0
.
w
z
1
T
G
2
T
K
Figure 2. Matchin
g
prin
cipl
e figure of
H
standa
rd control model
H
standa
rd co
ntrol model
matchin
g
is shown
as fig.2. Using th
re
e transfe
r function
matrix series
1
T
,
K
,
2
T
to approa
ch tra
n
sfe
r
functio
n
G
, the approxim
ation deg
ree
will b
e
measured by
12
GT
K
T
. The gene
rali
zed
controlle
d obje
c
t:
1
2
0
GT
Ps
T
(
6)
The co
ntroll
er is:
K
K
(
7)
A measu
r
e of
model match
i
ng deg
ree
ca
n be expre
ssed as:
12
GT
K
T
. When
1
T
and
2
T
are
reversibl
e
, then
the
e
x
pressio
n
of
model
matchi
ng m
e
a
s
ure
m
ent i
s
:
11
12
TG
T
K
. So
11
12
ˆ
GT
G
T
,
r
GK
, then, solving proble
m
of
H
model m
a
tchin
g
can
be tran
sfo
r
m
ed into
solving the m
odel re
du
ctio
n probl
em
s, makin
g
ˆ
r
GG
withi
n
a requi
re
d range.
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ISSN: 23
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TELKOM
NI
KA
Vol. 13, No. 2, Februa
ry 2015 : 282 – 286
284
To make
ˆ
A
B
Gs
CD
a balan
ce a
c
hiev
ement.
Definition 1.
Controllability
and ob
servability Gram
m
a
trix of
system
A
BC
D
，，
，
are
defined
sep
a
rately as follo
ws:
0
At
T
A
t
Pe
B
B
e
d
t
(
8)
0
At
T
A
t
Qe
C
C
e
d
t
(
9)
A
denote
s
the
transpo
se o
f
matrix
A
. It
can b
e
se
en
that the two matrices a
r
e
symmetri
c
po
sitive semi
-de
f
inite matrixes
, whi
c
h satisfy the Lyapunov equation b
e
low:
0
AP
P
A
BB
(10)
0
QA
A
Q
C
C
(
11
)
Diag
onali
z
ati
on of the matrix
,
PQ
, then:
11
12
1
(,
,
,
,
)
kk
n
TP
T
T
Q
T
d
i
ag
(12)
Whe
r
e
12
1
0
kk
n
.
The sy
stem
A
BC
D
，，
，
and
can be
separated into
blocks:
11
12
1
12
21
22
2
,,
AA
B
A
BC
C
C
AA
B
(13)
12
(14)
Whe
r
e
()
()
12
,
kk
n
k
n
k
RR
.
Theo
rem
1
[6]. Given a
s
ymptotically
stabl
e mini
mum
system
ˆ
G
ha
s Lya
p
unov
equilibrium form as follows:
11
12
1
21
22
2
12
ˆ
A
AB
AB
G
s
AAB
CD
CC
D
(
15
)
And there a
r
e
:
12
()
PQ
d
i
a
g
，
(
1
6
)
Whe
r
e
11
(,
)
k
di
a
g
,
21
(,
)
kn
dia
g
.
Red
u
ced ord
e
r model
11
1
1
r
A
B
Gs
CD
which is trun
cate
d is asymptot
ically stable a
nd
minimum sy
stem, and meet
:
1
ˆ
2
rk
n
Gs
G
s
(17
)
The re
du
ced
orde
r mod
e
l
r
Gs
is
the
K
in the matchin
g
mo
del we a
r
e a
s
king for.
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TELKOM
NIKA
ISSN:
2302-4
046
Solving M
e
th
od of
H
Model Matchin
g
Based on the Th
eory of the Model… (LI Minzhi
)
285
3. Simulation Examples
The mathe
m
atical expression
s for
state equat
io
n model of DC motor d
r
ive system is
[10]:
4
00
0
0
0
0
0
0
1
.
4
01
0
0
0
0
0
0
0
0
0
13
0
0
1
0
0
0
0
0
0
0
0
0
1
00
0.44
0
0
0
0
0
0
0
2
00
0.
88
1
1
.
7
6
10
0
0
0
0
0
00
0
0
0
1
0
0
0
0
1
.
4
0
0
0
0
100
10
0
0
0
0
0
0
0
29
4.1
2
9
.
4
1
19
.6
1
1
4
9
.3
0
27.56
0
0
0
0
0
0
1.045
1
0
6.66
7
A
0
1
0
0000
00
T
B
1
3
000
0000
00
C
0
D
As
1
TI
and
2
TI
, output image for
H
model match
i
ng of system
is sho
w
n a
s
Figure 3(a
)
,
the model ma
tching
solutio
n
is:
2
22
15
2.92
47
4
.
9
6
2
5
5
.
7
2
8
050
19.4
7
14
1.7
3
6.75
65
9.7
ss
s
K
ss
s
s
(a)
(b)
Figure 3. Output image of
H
model mat
c
hi
ng
As
1
1
100
T
s
and
2
1
5
T
s
,
st
e
p
re
spo
n
s
e
f
o
r
H
model ma
tching
of the
system i
s
sho
w
n a
s
fig.3 (b), the mo
del matchi
ng
solutio
n
is:
2
2
126
11
.7
07
3
3
369
2
5
2
8
7.70
5
1
63
9
1
58.1
4
1.78
7.2
0
6
2
7.46
3
3
4
.
3
ss
s
s
K
ss
s
s
s
From th
e
ste
p
re
sp
on
se i
m
age
of
H
model mat
c
hin
g
, it can
be
see
n
that the
ma
tching
model
got
b
y
ord
e
r
re
du
ction
metho
d
and
the
st
ep
re
spon
se
of the
ori
g
i
nal
system
are
c
o
mpletely c
o
ns
is
tent.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
50
100
150
200
250
300
350
S
t
e
p
R
e
sp
o
n
se
Ti
m
e
(
s
e
c
)
A
m
pl
i
t
ude
原系
统
模型匹
配系
统
0
0.1
0.2
0.
3
0.
4
0.5
0.6
0.
7
0
50
100
150
200
250
300
350
S
t
ep R
e
s
pons
e
Ti
m
e
(
s
e
c
)
A
m
pl
i
t
ude
原系
统
模型匹配系
统
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ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 13, No. 2, Februa
ry 2015 : 282 – 286
286
4. Conclusio
n
Usi
ng the pri
n
cipl
e of model ord
e
r red
u
ction to sol
v
e
H
model matchin
g
, from
the
step
response curve, it
can be
seen that the system
has
goo
d tracking
ability. The controller
got
by
this
d
e
sig
n
ing
m
e
thod has a certai
n pra
c
tica
l
appli
c
ation val
ue,
and m
odel m
a
tchin
g
p
r
obl
em
of high order
system
will be solved
well.
Referen
ces
[1]
Yuan SZ
. Des
i
gn
of pro
puls
i
on o
n
l
y
emerg
enc
y fli
ght co
n
t
rol s
y
st
em usi
ng
H
model
ma
tching
.
Flight Dyn
a
m
ic
s
. 2001; 19(
1): 85-8
8
.
[2]
Shao KY, Jin
g
YW
, Li YS, H
uan
g W
D
. Rob
u
st
control s
y
s
t
em base
d
on
mode
l matchi
n
g
.
Journ
a
l of
Daqi
ng Petr
ole
u
m Institute.
1
999; 23(
3): 35-
37.
[3]
Chen YJ, Zuo ZQ,
Wen SH, Ci CL. A solution of
H
control-
mode
l-matchi
n
g
prob
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Jo
urna
l of
Yansh
an U
n
ive
r
sity
. 2001; 25(
z): 37-40.
[4] Z
huge
H
．
A
n
ine
x
act model
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g
ap
pr
oach a
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atio
ns.
T
he Journ
a
l of S
ystems a
n
d
Softw
are (S0164-1
212)
. 2
003
; 67(3): 201-2
1
2
.
[5]
Moore B
C
. Pri
n
cip
a
l com
p
o
n
ent an
al
ys
is in
lin
ear
s
y
stem
s: Control
l
ab
ilit
y, o
b
serva
lil
it
y and m
o
d
e
l
reducti
on.
IEEE Trans Automatic Control
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81; ACO26(1):
17-3
1
.
[6]
K Z
hou, JC Do
yl
e, K Glover. Rob
u
st and o
p
ti
mal co
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e
w
J
e
rse
y
: Pre
n
tice-H
all. 19
9
6
.
[7]
W
ang G, Sree
ram V, Liu
W
Q. Balanc
ed
performa
nce
p
r
eservin
g
co
nt
roller
red
u
ctio
n. S
y
stem
&
control l
e
tters. 200
2; 46: 99-1
1
0
．
[8] Serkan
G
uger
cin
，
Atha
nasi
o
s C.Antou
l
as
．
A Surve
y
of Mode
l R
e
d
u
c
t
ion b
y
Ba
lanc
ed T
r
uncatio
n
and Som
e
Ne
w
R
e
sults. Inte
rnatio
nal Jo
urn
a
l of Contro
l. 2004: 77(
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[9]
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ang G, Sreer
am V, Liu W
Q. Performanc
e
Preservi
ng C
o
ntroll
er Re
ducti
on vi
a Ad
ditive
Perturbati
o
n
of the Clos
ed-
Loo
p T
r
ansfer F
unction.
IEEE Transactions on Autom
a
tic Control
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0
0
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; 46(5): 77
1-
775
．
[10]
Xu
e DY. Desi
g
n
and a
n
a
l
ysis
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ontro
l s
y
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ng: T
s
ingh
ua Un
iversit
y
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r
ess, 2000.
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