TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 11, Novembe
r
2014, pp. 77
1
6
~ 772
0
DOI: 10.115
9
1
/telkomni
ka.
v
12i11.66
03
7716
Re
cei
v
ed
Jun
e
26, 2014; Revi
sed Aug
u
st
28, 2014; Accepted Sept
em
ber 15, 20
14
Solving Method of H-infinity Model Mat
c
hing Based on
the Theory of the Model Reduction
Li Minzhi*, Cao 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, e-ma
i
l
: sqlmz@sin
a
.com
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
©
2014 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
optimal
control theo
ry of linear sy
stems
is a
ne
w ki
n
d
of de
sign
method
devel
oped
in the end of
1980, an
d is the very a
c
tive frontie
r subj
ect in
cu
rre
nt co
ntrol
theory. In many
control
syste
m
s, in
o
r
de
r to imp
r
ove
the st
e
ady a
nd dyn
a
mic
perfo
rman
ce
of
system,
the
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
Ps
K
s
z
y
u
w
Figure 1. Prin
ciple Fig
u
re o
f
H
Standard Proble
m
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
sta
nda
rd
probl
em. As
sho
w
n i
n
fig.1,
w
is the
external in
put,
z
is cont
ro
l
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Solving M
e
th
od of H-infinit
y
Model M
a
tching Based o
n
the Theo
ry
of the Model
… (Li Min
z
hi)
7717
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 fractio
nal
transfo
rm
ati
on (L
FT), tra
n
sfer fu
nctio
n
from
w
to
z
ca
n be
descri
bed a
s
:
1
11
12
22
21
,
l
GF
P
K
P
P
K
I
P
K
P
(5)
The
H
stand
ard co
ntrol
pro
b
lem i
s
for
a
regul
ar
co
ntroller
K
, ma
k
i
ng
th
e
c
l
os
ed
-
l
oo
p
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
Standard Co
ntrol Model
H
standa
rd
co
ntrol mod
e
l matchin
g
is
sho
w
n a
s
Fi
gure 2.
Usi
n
g three tra
n
s
fer
function
matri
x
seri
es
1
T
,
K
,
2
T
to approa
ch tra
n
sfer fun
c
tion
G
, the ap
proximation d
egre
e
will
be mea
s
u
r
ed
by
12
GT
K
T
. The gene
ralized contro
lled obje
c
t:
1
2
0
GT
Ps
T
(6)
The co
ntroll
er is:
K
K
(7)
A measure
of model m
a
tch
i
ng de
gre
e
ca
n be exp
r
e
s
sed a
s
:
12
GT
K
T
. When
1
T
an
d
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.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 11, Novem
ber 20
14: 77
16 – 772
0
7718
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
den
otes th
e
tran
spo
s
e
of
matrix
A
. It can b
e
seen
that the t
w
o
matrices are
symmetri
c
positive semi-definite matri
x
es, whi
c
h sa
tisfy the Lyapunov equ
atio
n belo
w
:
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
,
k
k
nk
nk
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
，
(16)
Whe
r
e
11
(,
)
k
di
ag
,
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
:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Solving M
e
th
od of H-infinit
y
Model M
a
tching Based o
n
the Theo
ry
of the Model
… (Li Min
z
hi)
7719
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.
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
10
0
T
s
and
2
1
5
T
s
, step respon
se
for
H
mod
e
l m
a
tching
of the
system
is
sho
w
n a
s
Fig
u
re 3 (b), the
model mat
c
hi
ng sol
u
tion is:
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
p
lit
u
d
e
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
10
0
15
0
20
0
25
0
30
0
35
0
S
t
ep R
e
s
pons
e
Ti
m
e
(
s
e
c
)
A
m
p
lit
u
d
e
原系
统
模
型匹配
系
统
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ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 11, Novem
ber 20
14: 77
16 – 772
0
7720
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.
4. Conclusio
n
Usi
ng the pri
n
cipl
e of mod
e
l orde
r re
du
ction to solve
H
model match
i
ng, from the
step re
sp
on
se curve, it ca
n be se
en tha
t
the sy
stem has g
ood tra
c
king a
b
ility. T
he co
ntrolle
r
got
by this desi
g
n
i
ng method h
a
s a certain p
r
acti
cal ap
plication value, a
nd model m
a
tchin
g
pro
b
le
m
of high order
system
will be solved
well.
Referen
ces
[1]
Yuan SZ
. Des
i
gn of pro
puls
i
on on
l
y
emerg
enc
y
fl
ig
ht con
t
rol s
y
stem usi
ng
H
mode
l 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
contro
l-
mode
l-matchi
n
g
pro
b
l
e
m.
Jo
urna
l of
Yansh
an U
n
ive
r
sity.
2001; 25(
z): 37-40.
[4]
Z
huge H. An i
n
e
x
act mod
e
l
matchin
g
ap
pr
oach a
nd its a
pplic
atio
ns. T
h
e
Journ
a
l of S
ystems a
n
d
Softw
are (S0164-1
212)
. 2
003
; 67(3): 201-2
1
2
．
[5] Moore
BC
．
Pr
incip
a
l com
pon
ent ana
l
y
sis in
linear s
y
stem
s
：
Contro
lla
bil
i
t
y
，
o
b
serva
lil
ity an
d mode
l
reducti
on.
IEEE Trans Automatic Control.
19
81; ACO26(1):
17-3
1
．
[6] K
Zhou
，
J C D
o
yle a
nd K Glo
v
er
．
Ro
bust a
nd optim
al co
ntrol. Ne
w
Jers
e
y
; Prentic
e-Ha
ll
. 1996
．
[7]
W
ang G, Sreeram V, Liu
W
Q
．
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d
perform
ance
preservi
ng c
o
ntroll
er red
u
cti
on. S
ystem
&
control letters
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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
．
Int
e
rnati
ona
l Jour
nal of Co
ntrol
. 200
4; 77(8): 74
8-76
6.
[9] W
ang
G
，
Sreeram V, Liu WQ
．
Perfo
rmance Pres
e
r
ving C
ontrol
l
e
r Red
u
ction
via Additiv
e
Perturbati
on of
the Close
d-L
o
op T
r
ansfer F
unction
．
IEEE
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a
tic Control.
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01;
46(5): 77
1-7
7
5
．
[10]
Xu
e DY. Desi
g
n
and a
n
a
l
ysis
for feedback c
ontro
l s
y
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ng: T
s
ingh
ua Un
iversit
y
P
r
ess. 2000.
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