Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
6, N
o
. 1
,
Febr
u
a
r
y
201
6,
pp
. 17
7
~
18
7
I
S
SN
: 208
8-8
7
0
8
,
D
O
I
:
10.115
91
/ij
ece.v6
i
1.8
838
1
77
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
New Stability Conditions for Nonl
inear Systems Described by
Multiple Model Approach
Ameur
Sassi, Afe
f
Abdelkri
m
Labora
t
oire
de
r
echer
che
en
auto
m
a
tique (
L
A.R.
A), Na
tion
a
l Sch
ool of
Engin
eers
of Tun
i
s (ENIT)
, Tun
i
sia
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Aug 17, 2015
Rev
i
sed
O
c
t 19
, 20
15
Accepte
d Nov 1, 2015
This paper studies the global as
y
m
p
t
ot
i
c
s
t
abi
lit
y and
the tr
acki
ng contro
l
problem of an u
n
certain non stationar
y
continuo
us sy
stem described b
y
th
e
m
u
ltiple m
odel
approach
. It
is b
a
sed on th
e cons
truction
of a
bas
i
s of m
odels
conta
i
ning four extrem
e m
odels and possibilit
y of addition of an avera
g
e
model. Once
the basis of models is gene
rated
,
an
operation of fusion of thes
e
differen
t
m
odels is
m
a
de to the leve
l of the ele
m
entar
y
control
law and the
partial outpu
t using the g
e
ometric met
hod. New
sufficien
t conditions for th
e
stability
are d
e
rived via Ly
apuno
v techni
qu
e.
The matrices of feed
back gains
and tracking gains are determined while
solving
s
y
stems of LMI constrain
t
s
(Linear Matrix I
n
equalities). Th
e cas
e of an unstable continuou
s nonlinear
model of electrical
circu
it oper
a
ting in
pseudo-p
e
riodic s
y
stem is considered
to illustrate
the p
r
oposed appro
a
ch.
Keyword:
Electrical circuit
Lin
ear m
a
trix
in
equ
a
lities
Mu
ltip
le m
o
d
e
l
Nonlinea
r syst
e
m
Q
u
adr
a
tic Lyap
uno
v fun
c
tio
n
Copyright ©
201
6 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Am
eur Sassi,
Laboratoi
r
e de Recherc
h
e
e
n
Aut
o
m
a
tique (LA.R.A),
Ecol
e Nat
i
onal
e
d
’
I
ngé
ni
eu
rs de
T
u
ni
s,
B
P
3
7
, 1
0
0
2
Tu
ni
s B
e
l
v
édère
,
T
u
ni
si
a
Tel
:
+21
6
71
8
7
4
70
0,
Fax
:
+2
16
71
8
2
729
Em
ai
l: A
m
eu
r.Sassi@en
it.rnu
.
tn
1.
INTRODUCTION
Th
e m
u
ltip
le
m
o
d
e
l ap
p
r
o
a
ch
is p
r
ov
ing
v
e
ry in
tere
stin
g
wh
en
ev
er
we are con
f
ron
t
ed
with
co
m
p
lex
sy
st
em
s and/
o
r
n
onl
i
n
ear.
It
i
s
t
o
re
prese
n
t
t
h
e sy
st
em
st
udi
ed
by
a
fam
i
ly
o
f
si
m
p
l
e
r an
d e
a
si
er t
o
m
a
ni
pul
at
e
math
e
m
atica
l
m
o
d
e
ls [1
], [2
], [3
]. Th
is ap
pro
ach
h
a
s b
een
recen
tly d
e
v
e
lop
e
d
in sev
e
ral science an
d
en
g
i
n
eeri
n
g
do
m
a
in
s, with
typ
i
cal ap
p
licatio
n
s
in
th
e electrical and mechanical
engi
neeri
ng area
s, with
ap
p
lication
to
m
o
d
e
llin
g
,
co
n
t
ro
l and
/
or fau
lt d
e
tectio
n. It was in
trod
u
c
ed
as
an e
fficient and powe
rful
m
e
thod t
o
c
o
pe with m
odelling and c
o
ntrol difficulties
when com
p
lex
non linear an
d/or unce
rtain
processe
s
are con
c
ern
e
d
.
Th
e m
u
ltip
le m
o
d
e
l ap
proach
assu
m
e
s t
h
at it is po
ssi
b
l
e to
rep
l
ace a un
iqu
e
n
on lin
ear
represe
n
tation
by a c
o
m
b
ination of sim
p
ler m
odels t
hus
building a
so-called m
odel-base. Us
ually, each
m
odel
of t
h
i
s
base
de
scri
bes t
h
e co
nsi
d
e
r
ed
pr
oce
ss at
a speci
fi
c operat
i
ng p
o
i
n
t
[4]
.
The i
n
t
e
ract
i
on
bet
w
e
e
n t
h
e
di
ffe
re
nt
m
odel
s
of t
h
e
base t
h
r
o
ug
h
no
rm
ali
zed act
i
v
atio
n fun
c
tion
s
allows th
e m
o
d
e
llin
g
of th
e
g
l
ob
al n
o
n
-
lin
ear and
com
p
lex
syste
m
.
Th
e stab
ility
o
f
th
ese m
o
d
e
l
s
is,
m
o
st o
f
th
e ti
m
e
, stu
d
i
ed
u
s
i
n
g
th
e
q
u
ad
ratic
Lyap
uno
v
appro
a
ch
[5
], [6
], [7
], [8
], [9
], [10
]
. Th
e ob
tained
con
d
ition
s
are g
i
v
e
n
in
term
s o
f
Lin
ear Matrix
In
equ
a
lities (LMI) and
can
b
e
efficien
tly so
lv
ed b
y
con
v
e
x
p
r
og
rammin
g
t
ech
n
i
q
u
e
s [11
]
, [12
]
, [13
]
, [14
]
.
Th
e stab
ility c
o
nd
itio
ns b
a
sed
on
t
h
e u
s
e of th
e
q
u
a
dratic Lyap
uno
v
fu
ncti
o
n
are con
s
erv
a
tiv
e as a
sin
g
l
e co
mm
o
n
symmetric p
o
sitiv
e d
e
fi
n
ite
m
a
trix
v
e
rifyin
g
all Lyap
uno
v
in
equ
a
lities is req
u
i
red
[13
]
. It is
also re
jected
by certain system
s su
ch as the
saturated system
s, the
piecewise linear system
s, etc. So
m
e
works
show t
h
e c
o
ntribution
of the
polyqua
dratic and the
piecewi
s
e qua
d
ratic L
y
apunov functi
ons
,
[15], [16],
[17].
The st
u
d
y
p
r
o
pos
ed i
n
t
h
i
s
p
a
per
foc
u
ses
o
n
a cl
ass of
un
cert
a
i
n
sy
st
em
s and c
o
m
p
l
e
x cont
i
n
ui
n
g
bo
u
nde
d
para
m
e
t
e
rs [1
8]
, [
1
9]
. T
h
e gl
obal
m
odel
can be
obt
ai
ne
d ei
t
h
e
r
by
usi
n
g
t
h
e s
w
i
t
c
hi
n
g
ope
ra
t
i
on
o
r
fusi
on
.
In
t
h
i
s
st
udy
,
we
are i
n
t
e
rest
ed
i
n
t
h
e
f
u
si
o
n
usi
n
g t
h
e
geom
et
ri
c m
e
t
hod
[
19]
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
6, No
. 1, Feb
r
uar
y
20
1
6
:
17
7 – 18
7
17
8
Thi
s
m
e
t
hod i
s
seen at
bot
h
t
h
e el
em
ent
a
ry
cont
r
o
l
s
at
t
h
e l
e
vel
of
pa
rt
i
a
l
out
p
u
t
s
a
nd
d
r
i
v
e
n
by
v
a
lid
ity in
d
i
ces.
Our
n
e
w stab
il
ity co
n
d
ition
s
are pro
p
o
s
ed
i
n
th
e case
o
f
m
u
l
tip
le
m
o
d
e
l stab
ilizin
g
st
ate feedb
a
ck
co
n
t
ro
l
with
referen
ce,
p
e
rm
i
t
s
to
l
o
cally find
a
symmetr
ic
p
o
s
itiv
e d
e
fin
i
t
e
m
a
trix
to
each
m
o
d
e
l of t
h
e
b
a
sis.
At th
e en
tran
ce to
each
b
a
se
m
o
d
e
l in
clo
s
ed
loop
con
t
ro
l
is
app
lied
o
v
erall
in
ferred weig
h
t
s o
f
elem
e
n
tary
l
i
n
ear fee
dbac
k
co
nt
r
o
l
wi
t
h
refe
rence st
at
e
and t
o
c
o
nve
r
g
e t
h
e
gl
o
b
al
out
put
of t
h
e c
ont
rol
l
e
d sy
st
e
m
t
o
a
d
e
sired
traj
ect
o
r
y. Matrices g
a
in
s b
y
state feedb
a
ck
and
track
i
n
g
are
d
e
term
in
ed
b
y
so
lv
ing
syste
m
s s
t
ab
ility
co
nd
itio
ns
g
i
v
e
n
in term
s o
f
LMI [2
0
]
, [21
]
, [2
2
]
, [2
3
]
.
Th
is
p
a
p
e
r is org
a
n
i
zed as
fo
llo
ws. In Section
2
,
we in
trod
uce briefly so
me b
a
sic
no
tion
s
of m
u
ltip
le
m
o
d
e
l ap
p
r
ao
ch
and
th
e th
eoretical to
o
l
u
s
ed
in
th
is work
, n
a
m
e
ly
th
e alg
e
braic m
e
th
od
. In
sectio
n
3, th
e
strateg
i
es
o
f
mu
ltip
le m
o
d
e
l co
n
t
ro
l
b
a
sed
on
g
e
o
m
etric meth
od
is
proposed
. Stab
ility co
nd
itio
ns
u
s
ing
n
e
w
LM
I ap
p
r
oac
h
base
d o
n
st
at
e
feed
bac
k
wi
t
h
refe
rence
are
p
r
o
v
i
d
e
d
i
n
sect
i
on
4.
I
n
t
h
e
se
ct
i
on
5 a
n
e
x
a
m
ple
of an
un
st
abl
e
seco
nd o
r
der c
ont
i
n
u
o
u
s
m
odel
of el
ect
ri
cal
ci
rcui
t
operat
i
ng i
n
pseu
d
o
-
p
eri
o
di
c sy
st
em
i
s
is
stu
d
i
ed
to illu
strate th
e efficien
cy of t
h
is apprach. Con
c
lusio
n
is
d
r
awn in
sectio
n
6
.
No
ta
tio
n
s
The sy
m
bol
(
*
)
d
e
no
tes th
e
tran
sp
o
s
e elem
e
n
ts in th
e symmetric p
o
s
itions, fo
r ex
am
p
l
e,
00
T
XX
X
and
00
T
AB
AB
C
BC
LMI: Lin
e
ar M
a
trix
In
eq
u
a
lities
2.
PROBLEM STATEMENT
The ev
ol
ut
i
on
of a c
ont
i
n
u
o
u
s
no
n-
st
at
i
ona
r
y
uncert
a
i
n
pa
r
a
m
e
t
e
rs and
b
o
u
n
d
ed c
o
m
p
l
e
x sy
st
em
is
descri
bed
by
t
h
e f
o
l
l
o
wi
ng
di
f
f
ere
n
t
i
a
l
eq
uat
i
on:
11
01
1
.
.
...
.
nn
n
ay
t
a
y
t
a
y
t
y
t
u
t
(1
)
with the symbol
.
represents the set of vari
ables,
unc
er
tainties, noise
or
distu
r
ba
nc
es acting
on t
h
e
coefficients
of
this system
as, for:
0,
1
,
,
1
in
,
.
ii
i
aa
a
with
mi
n
.
ii
i
aa
and
ma
x
.
ii
i
aa
.
Model
(1) m
a
y be
given i
n
t
h
e followi
ng controllab
ility com
p
anion
form
[15]:
x
tA
x
t
B
u
t
yt
C
x
t
(2
)
whe
r
e
A
,
B
and
C
are re
spectively
the state m
a
tr
ices, c
ontr
o
l a
n
d
out
put
with:
01
2
1
01
0
0
01
0
,0
0
1
a
n
d
1
0
0
.
0
00
0
1
..
.
.
T
n
AB
C
aa
a
a
T
h
e
state
m
a
trices
characte
r
izing
the
four
m
odels
i
M
of t
h
e base
are gi
ven
by
10
1
2
3
,,
,
,
Aa
a
a
a
,
20
12
3
,,
,
,
Aa
a
a
a
,
301
2
3
,,
,
,
Aa
a
a
a
and
40
12
3
,,
,
,
Aa
a
a
a
.
In addition to these m
odels, it is worth adding the
average m
odel as a
fifth in the database, it somehow
rep
r
ese
n
ts the
bary
ce
ntre
of t
h
e e
x
trem
e
m
odels a
n
d
para
m
e
ters are d
e
f
i
ned
by
the
ari
t
hm
etic
m
ean of t
h
e
four param
e
ters extrem
e
m
o
dels with
5
2
ii
i
aa
a
fo
r
0,
1
,
,
1
in
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
New S
t
ab
ility Co
nd
itio
n
s
fo
r
No
n
lin
ea
r
S
y
st
ems
Descri
b
e
d b
y
Mu
ltip
le Mo
d
e
l
App
r
o
a
c
h
(Am
e
u
r
Sa
ssi)
17
9
A com
p
rehe
nsi
v
e desc
ri
pt
i
o
n
of t
h
e st
udi
e
d
sy
st
em
can be defi
ned
fr
om
an i
n
t
e
r
pol
at
i
o
n
m
odel
i
M
base
d on
t
h
e fo
l
l
o
wi
n
g
st
at
e r
e
prese
n
t
a
t
i
o
n
[
18]
:
1
1
r
ii
i
i
r
ii
i
x
tt
A
x
t
B
u
t
yt
t
C
x
t
(3
)
whe
r
e
i
A
,
i
B
and
i
C
are re
spectively
the state m
a
tric
es,
c
ont
r
o
l
a
n
d
out
put
o
f
eac
h of
t
h
e
i
M
basi
c m
odel
.
In
th
is
p
a
p
e
r, th
e m
a
in
o
b
j
ect
iv
e is to
con
v
erg
e
t
h
e ou
tpu
t
yt
of t
h
e gl
obal
s
y
st
em
(3) t
o
a
desi
re
d
trajectory
d
yt
,
i.e.
,
0
d
yt
y
t
as
t
with
:
1
r
di
i
d
i
yt
t
C
x
t
(4
)
d
x
t
is
the desire
d state
vector.
3.
MULTIPLE
MODEL CONTROL
STRATEG
Y USI
N
G
GEO
M
ETRIC AP
PROACH
The ge
om
et
ri
c
m
e
t
hod i
s
use
d
t
o
cal
cul
a
t
e
t
h
e di
st
ance
i
dt
b
e
t
w
een
the p
a
rtial o
u
t
pu
ts
i
yt
and
the de
sired tra
j
ectory
d
yt
,
s
u
ch
as
[1
0
]
,[
11
]:
id
i
dt
y
t
y
t
(5
)
The norm
alize
d
distance
i
n
i
s
gi
ven
by
:
1
1
r
ii
j
j
nt
d
t
d
t
(6
)
Th
e g
e
o
m
etric
v
a
lid
ities
i
are
g
i
ven
by
:
1
1
r
ii
j
j
tt
t
(7
)
with
:
2
1
11
e
x
p
r
j
ii
j
ji
nt
tn
t
(8
)
re
prese
n
ts a
variable re
gulating param
e
ter with
00
.
9
.
The validities
i
satisfy the foll
owi
n
g c
ond
itio
n
s
o
f
conv
ex
ity:
01
i
and
1
1
r
i
i
.
The st
rategy
of
m
u
ltiple m
odel cont
rol
desc
ribe
d
by
ge
om
etric ap
pr
oac
h
is
give
n in
Fig
u
r
e
1.
The
bloc
k “
F
u
s
ion
o
f
elem
entary
c
ontr
o
ls
”
aim
s
to calculate the gl
obal
control
ut
of t
h
e f
u
sio
n
of
n
ele
m
entary controls
i
ut
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
IJEC
E V
o
l. 6, No
. 1, Feb
r
uar
y
20
1
6
: 17
7 – 18
7
18
0
The
global c
o
ntrol is
dri
v
en
by the indices
of validity
i
t
.
In
our case, inputs
e
re
fers to
the state vect
or
12
,,
,
n
x
tx
t
x
t
x
t
and the
desi
red state vect
or
12
,,
,
dd
d
n
d
x
tx
t
x
t
x
t
of system
in the closed
loop.
T
h
ese i
n
puts are
rele
vant to t
h
e
determ
ination of
validities.
Figure 1.
Block di
agram
of the proposed m
u
ltiple
m
odel control strategy
The
global control resulting
of th
e fusion is
written as follows:
1
r
ii
i
ut
t
u
t
(9
)
The
glo
b
al
out
put
o
f
sy
stem
is gi
ven
by
:
1
r
ii
i
yt
t
x
t
(1
0)
4.
NEW ST
ABI
L
ITY C
O
N
D
I
T
IONS
BASE
D O
N
STATE
FEEDBA
CK
WITH
REFE
REN
C
E
The obj
ective
in this section
is to find
from such
a
fo
rm
ulation LM
I m
a
tri
ces state feedback
gains
i
K
and those tracking
i
N
for
whic
h the
state vector
12
n
x
tx
t
x
t
x
t
of
com
p
le
x
contin
u
o
u
s
sy
s
t
em
describe
d by
(3
)
c
o
nve
r
ge to a desire
d state vector
12
dd
d
n
d
x
tx
t
x
t
x
t
,
n
is the
or
de
r
of
the co
ntr
o
lled
sy
stem
[5]
,
[
9
]
.
For
each base
m
odel is applied a
basic c
o
ntrol
i
ut
state fee
d
back
with
refe
re
nce to the
form
:
ii
i
d
ut
K
x
t
N
x
t
(1
1)
The
global c
o
ntrol low
ut
is gi
v
e
n
by
:
1
r
ii
i
d
i
ut
t
K
x
t
N
x
t
(1
2)
d
x
M
ode
l
1
M
ode
l
r
y
System
Fusion of
elem
ent
a
r
y
controls
ii
u
i
M
x
i
u
u
M
ode
l
1
M
ode
l
r
e
Librar
y
of contro
ls
Librar
y
of models
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN:
208
8-8
7
0
8
New Stability Conditions for
N
onlinear Syst
ems Descri
bed
by Multiple
M
odel Approach
(Ameu
r
Sa
ssi)
18
1
The
relations (3) and
(10) allow to
have
the representation of
m
u
ltiple
m
o
del and can
be
writing as
follows:
11
1
rr
i
j
ij
ij
d
ij
r
ii
i
xt
xt
t
t
G
H
x
t
yt
t
C
x
t
(1
3)
with
ij
i
i
j
GA
B
K
and
ij
i
j
H
BN
.
Whet
her
et
is the tracking e
r
ror bet
w
een t
h
e
state vector
x
t
of co
n
tinuou
s sy
ste
m
(
1
3)
and
th
e d
e
sired
steady state ve
ctor
d
x
t
with:
d
et
x
t
x
t
(1
4)
The deri
vative of
the
trac
ki
ng
err
o
r
de
t
et
dt
is written as follows:
11
rr
i
j
ij
ij
d
ij
d
xt
et
t
t
G
H
xt
xt
(1
5)
For
a c
o
nstant
desire
d stea
dy
state vecto
r
d
x
t
, the expression (1
5) can be written:
11
rr
i
j
ij
ij
d
ij
et
et
t
t
G
x
t
(1
6)
with
ij
ij
ij
Z
GH
In case the cont
rol m
a
trice
s
i
B
are square and invertibl
e
, th
ere are new stability c
o
nditions,
fo
rm
ulated in t
e
rm
s of LM
I
,
a
r
e
devel
ope
d.
If we use
d
t
h
e follo
win
g
c
o
ns
traints:
0
,
1
,
2
,
..
.,
const
a
nt
ij
d
Z
ij
r
xt
(1
7)
The deri
vative
et
is written as
follows:
11
rr
ij
i
j
ij
et
t
t
G
e
t
(1
8)
Theorem
:
Th
e eq
u
ilibr
i
u
m
o
f
th
e con
tin
uous syste
m
(
1
0
)
i
s
g
l
ob
ally asym
p
t
o
tical
ly sta
b
le if
ex
ists a
matr
ix
1
QP
suc
h
t
h
at
0
T
QQ
and
square m
a
trices
i
R
wit
h
size
n
, such as:
*0
ii
i
AQ
B
R
fo
r
1
,
2
,
..
.,
ir
(1
9)
11
1
*0
22
2
ij
i
j
j
i
AA
Q
B
R
B
R
(2
0)
fo
r
1
ij
r
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
IJEC
E V
o
l. 6, No
. 1, Feb
r
uar
y
20
1
6
: 17
7 – 18
7
18
2
The gains
by stabilizing feedback
i
K
state are c
a
lculated
by:
ii
K
RP
fo
r
1
,
2
,
..
.,
ir
(2
1)
The trac
ki
ng gains
i
N
are give
n by
:
1
ij
j
j
i
NB
A
B
K
for
,1
,
2
,
.
.
.
,
ij
r
(2
2)
Proof:
Based
on the
Lyap
unov stability conditio
ns can
be written:
0
T
Ve
t
e
t
P
e
t
(2
3)
0
TT
Ve
t
e
t
P
e
t
e
t
P
e
t
(2
4)
The de
velo
pm
ent
o
f
Ve
t
gi
ves:
11
*0
rr
T
ij
i
j
ij
Ve
t
e
t
t
t
P
G
e
t
(2
5)
From
the c
onst
r
aint
0
ij
Z
fo
r
,1
,
2
,
,
ij
r
, it follows t
h
at:
0
ii
j
i
j
AB
K
B
N
f
or
,1
,
2
,
,
ij
r
(2
6)
whe
n
ce
t
h
e relation (2
2)
.
5.
CASE OF ST
UDY
:
ELECTRICAL CIRCUIT
Consi
d
er a
n
electric circuit, Figu
re 2, m
a
y
models a di
ffe
re
nt dy
nam
i
c process (therm
al, m
echanical,
hydraulic,…
)
,
whic
h i
n
cludes two resistors
12
,
RR
and two ca
paci
tors
12
,
CC
sup
p
lied
by
tw
o
v
o
ltage
s
12
,
ut
u
t
.
Figu
re
2.
Electrical C
i
rcuit
5.
1. Sys
t
em
M
o
del
i
n
g
The m
a
in obje
c
tive is to converge
1
C
vt
and
2
C
vt
to co
nstant
val
u
es
5v
and
24
v
respect.
Following c
h
aracteristics:
12
10
0
CC
F
,
1
1
RK
and
2
10
RK
.
The
dif
f
ere
n
tial equatio
n
s
des
c
ri
bin
g
t
h
e circ
uit are
give
n
b
y
:
1
R
1
u
1
i
1
C
1
C
v
2
R
2
u
2
C
v
2
C
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN:
208
8-8
70
8
N
e
w
Sta
b
ility C
o
n
d
itions
for
N
online
a
r
Syst
ems
Descri
bed
by
M
u
ltiple M
odel
Ap
pr
oac
h
(Ameu
r
Sa
ssi)
18
3
11
1
2
1
C
ut
R
i
t
i
t
v
t
(2
7.a
)
12
2
2
2
CC
vt
R
i
tv
t
u
t
(2
7.
b)
1
11
1
1
C
C
dv
t
vt
i
t
dt
C
(2
7.c
)
2
22
2
1
C
C
dv
t
vt
i
t
dt
C
(2
7.
d)
By choosi
ng the state vect
or
12
T
CC
x
tv
t
v
t
and the
control
vect
or
12
T
ut
u
t
u
t
, we
ca
n
write the
electric circuit in t
h
e
followi
ng m
a
trix
form
:
x
tA
x
t
B
u
t
yt
C
x
t
(2
8)
with:
12
12
1
2
1
22
22
1
11
RR
RR
C
R
C
A
RC
RC
,
11
2
1
22
11
1
0
RC
R
C
B
RC
a
n
d
10
01
C
.
For t
h
e co
nstr
u
c
tion o
f
m
odel’s libra
ry, we a
ssum
e
that the resistors
1
R
and
2
R
have a
n
unce
r
t
a
inty
suc
h
as
111
R
RR
and
22
2
R
RR
with
:
112
2
0.
8
,
1.2
,
8.5
,
11.5
.
R
KR
K
R
K
R
K
.
The
fo
u
r
m
ode
ls in the
data
ba
se are
de
fine
d f
o
r
com
b
ination
s
12
,
RR
,
12
,
RR
,
12
,
RR
and
12
,
RR
.
The following m
a
trices
i
A
and
i
B
are calculated:
12
3
4
1
3
.67
6
5
1
.1
76
5
1
3
.
3
6
9
6
0
.
86
96
9.50
98
1.17
65
9.20
29
0.8
6
9
6
,,
a
n
d
.
1
.
17
65
1
.
17
65
0.8
6
9
6
0
.
86
96
1.1
7
6
5
1
.
17
65
0.8
6
9
6
0.8
6
9
6
AA
A
A
12
3
4
12
.
5
1.176
5
1
2.5
0
.86
9
6
8
.333
3
1
.
1
765
8
.
3
333
0.869
6
,,
a
n
d
.
0
1
.
1
765
0
0
.869
6
0
1.17
65
0
0
.869
6
BB
B
B
Solvi
n
g
the
pr
op
ose
d
LM
I sy
stem
stability
con
d
ition
s
prese
nted i
n
T
h
e
ore
m
,
the m
a
trices
of feedb
ack
gains
i
K
define
d i
n
(2
1)
are
gi
ve
n
by
:
12
3
4
0.2
666
0.
654
5
0
.44
6
7
0
.47
4
9
0
.2
534
0
.
632
0
0
.433
5
0
.4
52
5
,,
a
n
d
.
8
.
37
51
0.
577
0
8
.383
9
0
.43
5
4
5
.873
6
0
.583
9
5
.87
6
9
0
.443
1
KK
K
K
The m
a
tr
ic
es
of
t
r
acking
gai
n
s
i
N
d
e
fine
d in
(
2
2
)
a
r
e
give
n
by
:
12
3
4
0.73
34
0
.
65
45
0
.
55
33
0
.
474
9
0
.746
6
0
.6
320
0.5
665
0
.
452
5
,,
,
.
7
.
375
1
0
.423
0
7
.38
3
9
0
.564
6
4
.8
736
0.4
161
4
.
87
69
0
.
556
9
NN
N
N
5.
2. Si
mul
a
ti
o
n
Res
u
l
t
s
In this sectio
n,
the pr
o
p
o
s
ed c
ont
rol is ap
plied to
re
g
u
late uncertain
no
n st
ationary
n
o
n
lin
ear sy
stem
s
of electrical ci
rcuit de
scribe
d
by
m
u
ltiple
m
odel appr
oac
h
.
In t
h
e sim
u
lations,
we e
m
ploy
ed
Sim
u
link
of
®
MA
T
L
A
B
an
d the
S
o
lve
r
o
p
tion
is “
ode
45”
.
The objective is
to
st
abilize the voltages
1
C
vt
an
d
2
C
vt
across t
h
e
capacitors
1
C
and
2
C
of
electrical circuit to cons
tant d
e
sired v
o
ltages
1
5v
d
x
and
2
24
v
d
x
.
The ev
ol
utions
of state va
riab
les
1
i
x
t
and
2
i
x
t
for each base m
odel
are gi
ven in
Figures
3 and
4
respectively.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
IJEC
E V
o
l. 6, No
. 1, Feb
r
u
ar
y
20
1
6
: 17
7 – 18
7
18
4
The pe
rf
orm
an
ce of tra
j
ector
y
tracking
of t
h
e state variab
les
1
x
t
and
2
x
t
starting system
in
closed
lo
op
, a
n
d
desire
d state
varia
b
les co
nst
a
nt,
1
d
x
and
2
d
x
are il
lustrated i
n
Fi
g
ure
5
.
Figu
re 6 s
h
o
w
s the evolutio
n
s
of the v
o
ltag
e
1
ut
to the input
of electrical circuit and f
o
u
r
basic
voltage
s
1
i
ut
.
Figu
re
7 s
h
ows
the e
vol
utio
ns
of
the
v
o
ltage
2
ut
and
f
o
ur
basic
voltage
s
1
i
ut
.
The state varia
b
les
1
x
t
and
2
x
t
of t
h
e starting syste
m
in closed
lo
op
conv
erg
e
resp
ectiv
ely to
desire
d v
o
ltag
e consta
nts values
1
5v
d
x
and
2
24
v
d
x
with
sm
all oscillations
(Fig
ur
e 5)
.
This is a so-
called pseu
d
o
-
p
eri
odic oscill
ations w
h
ose am
plitude, ex
pr
essed in v
o
lts,
is not consta
n
t
and dec
r
ease
s
(The
oscillations a
r
e
dam
p
ed)
.
Fig
ure
3
.
E
v
olutio
ns
of
the
state varia
b
le
1
x
t
for each m
odel of the
base
Fi
gu
re
4.
Ev
olutio
ns
o
f
the
state va
riable
2
x
t
fo
r eac
h m
odel
o
f
the
bas
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN:
208
8-8
70
8
N
e
w
Sta
b
ility C
o
n
d
itions
for
N
online
a
r
Syst
ems
Descri
bed
by
M
u
ltiple M
odel
Ap
pr
oac
h
(Ameu
r
Sa
ssi)
18
5
Figu
re
5.
The
per
f
o
r
m
a
nce of
traject
ory
trac
kin
g
of
1
C
vt
and
2
C
vt
w
ith
the desire
d con
s
tant voltag
e
s
The m
u
ltiple
m
odel control
in whic
h we
w
i
ll be in
terested co
nsists in th
e fusio
n
of
par
t
ial controls.
For t
h
at we
ha
ve to c
o
m
pute the validities of each
pa
rtial m
odel and associate the sub
cont
rols
weighted by
the co
rre
sp
o
n
d
e
nt coe
fficie
n
ts.
The
obtained result will cont
ro
l
th
e g
l
ob
al p
r
o
cess (2
9)
.
The glo
b
al volt
a
ges
1
ut
and
2
ut
are give
n by
the f
o
l
lowin
g
relations:
11
1
22
1
r
ii
i
r
ii
i
ut
t
u
t
ut
t
u
t
(2
9)
Figu
re
6.
Sim
u
lation res
u
lts
o
f
new
LM
I a
p
p
roac
h
,
(a
)
E
vol
ution
s
of f
o
u
r
basic voltage
s
1
i
ut
,
(
b) Ev
olutio
n of
the v
o
ltage
1
ut
with
th
e pro
posed
state
feed
b
ack with refe
re
nce
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
IJEC
E V
o
l. 6, No
. 1, Feb
r
u
ar
y
20
1
6
: 17
7 – 18
7
18
6
Figu
re
7.
Sim
u
lation res
u
lts
o
f
new
LM
I a
p
p
roac
h
,
(a
)
E
vol
ution
s
of f
o
u
r
basic voltage
s
2
i
ut
,
(
b) Ev
olutio
n of
the v
o
ltage
2
ut
with
th
e pro
posed
state
feed
b
ack with refe
re
nce
Whe
n
the ne
w
conditions of stab
ility are used, the fee
d
back
i
K
and the tracki
ng
i
N
gains guara
n
tee
the stability of the continuous syste
m
of ele
c
trical circuit,
initially unsta
b
l
e, desc
ribe
d
b
y
the m
u
ltiple
m
odel
approach. Thus, all the traj
ectories of the voltages
1
C
vt
and
2
C
vt
across the capa
c
itors
1
C
and
2
C
can
track the de
sired constant vol
t
ages
1
d
x
and
2
d
x
ver
y
well and the glo
b
al
m
odel r
e
sp
onse is bett
er in the ne
w
app
r
o
ach
(state
fee
dbac
k
with
refe
rence
)
case
than
in t
h
e
or
d
i
nary
state fee
d
back
co
ntr
o
l.
We ca
n c
oncl
ude
that eac
h
capacitor
disc
har
g
es a
n
d rec
h
ar
ges with s
m
all
oscillations
a
r
o
u
n
d
the
value
o
f
the
de
sired c
o
nstant
voltage
s.
6.
CO
NCL
USI
O
N
In this pa
pe
r, a contri
butio
n
in trackin
g co
ntr
o
l
fo
r a class of co
ntin
uo
u
s
no
n stationa
r
y
uncertain
sy
stem
with lim
ited param
e
ters is c
o
nside
r
ed. B
a
se
d
o
n
t
h
e
geom
etric app
r
o
ach
, a lib
r
a
ry
o
f
fo
ur
m
odels
was
built. Ne
w sta
b
ility
conditio
n
s ha
ve
been
d
e
velo
ped i
n
the case of m
u
ltiple m
odel cont
rol in the case
of state
feedbac
k
with
refe
rence
.
T
h
e
s
e conditions a
llow to
r
eachi
ng t
h
e state va
riables of the
closed-loop m
u
ltiple
m
odel descri
bing
the
com
p
lex sy
stem
starting
to
the
desi
red
state va
ria
b
les.
The
M
a
trices
gains
by
state
feed
bac
k
an
d the m
a
trices of trackin
g gai
n
s
are give
n
by
solvin
g a sy
stem
of LM
I co
n
d
itions
. C
a
se study
o
f
an
electr
i
cal cir
c
u
i
t
o
f
second or
d
e
r
op
er
ative in
m
o
d
e
p
s
eud
o-p
er
iod
i
c is
co
ndu
cted to
co
m
p
ar
e th
e pr
op
o
s
ed
app
r
o
ach
with
the or
dina
ry
state feedbac
k
.
Finally
, th
e result of com
put
er sim
u
lation has dem
onstrat
ed the
ef
f
ectiv
en
ess
of
th
e pr
opo
sed
ap
pro
ach.
REFERE
NC
ES
[1]
H. Su, Y. Qu, S. Gao, H.
Song and K. W
a
ng, “A m
odel of feedback c
ontro
l s
y
s
t
em
on network and its stability
anal
ysis”
,
Comm
un Nonlin
ear Sci Numer Simul
, v
o
l. 18
, pp
. 1822-
31, 2013
.
[2]
M
.
K
s
ouri-Lahm
ari, P
.
Borne and M
.
Benreje
b
, “
M
ultiple m
odel: the cons
tru
c
tion of m
odel bas
e
s
”
,
Studies in
Informatics and
Control,
vo
l. 3,
no. 3
,
pp
. 199-2
10, 2004
.
[3]
N
.
Elf
e
ll
y,
J.Y
.
D
i
eulot
,
M
.
Ben
r
ejeb
and P
.
Bo
rne,
“Multim
odel contro
l design
usi
ng unsupervised classifiers
”
,
Studies
in Infor
m
atics and Con
t
rol
, ISSN 1220-1766, vol. 21
(1)
,
pp
. 101-108
, 2
012.
[4]
T.A
.
J
ohans
en,
B.A
.
F
o
s
s
,
E. “M
ultiple m
odel
approach
es
to m
odelling and c
ontrol”
,
International Journal of
Control
, Vol. 72
, pp
. 575
, 1999
.
[5]
H.O. W
a
ng, K. Tanak
a
a
nd M.F. Griffin, “An approach to fuzzy
control
of nonlin
ear s
y
stem
s: stab
ility
and design
”,
i
ssues
IEEE Trans. on
Fuzzy Sys
t
, vo
l. 4, no. 4, p
p
. 14-23
, Febru
a
r
y
1996.
Evaluation Warning : The document was created with Spire.PDF for Python.