Inter
national
J
our
nal
of
P
o
wer
Electr
onics
and
Dri
v
e
Systems
(IJPEDS)
V
ol.
8,
No.
3,
September
2017,
pp.
990
–
1001
ISSN:
2088-8694
990
Stability
and
Rob
ust
Stabilization
of
2-D
Continuous
Systems
in
Roesser
Model
Based
on
GKYP
Lemma
Ismail
Er
Rachid
and
Abdelaziz
Hmamed
LESSI,
Department
of
Ph
ysics
F
aculty
of
Sciences
Dhar
El
Mehraz,
B.P
.
1796
Fes-Atlas
Morocco
Article
Inf
o
Article
history:
Recei
v
ed
Jun
16,
2016
Re
vised
Feb
17,
2017
Accepted
Feb
28,
2017
K
eyw
ord:
Stability
Rob
ust
Stabilization
2-D
Continuous
Systems
GKYP
Lemma
LMI
ABSTRA
CT
This
paper
is
concerned
with
the
stability
and
Rob
ust
stabilization
prob-
lem
for
2-D
continuous
systems
in
Roesser
model,
based
on
Generalized
Kalman
Y
akubo
vich
Popo
v
lemma
in
combination
with
frequenc
y-partitioning
ap-
proach.
Suf
ficient
conditions
of
stability
of
the
systems
are
formulated
via
linear
ma-
trix
inequali
ty
technique.
Finally
,
numerical
e
xamples
are
gi
v
en
to
illustrate
the
ef
fec-
ti
v
eness
of
the
proposed
method.
Copyright
c
2017
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Ismail
Er
Rachid
LESSI,
Department
of
Ph
ysics
F
aculty
of
Sciences
Dhar
El
Mehraz
B.P
.
1796
Fes-Atlas
Morocco
ismail.errachid@gmail.com
1.
INTR
ODUCTION
Stability
of
2-D
continuous
systems
is
the
major
aim
in
all
researches,
in
order
to
guarantee
the
nor
-
mal
operation
of
systems.
In
relation
with
these
researches,
there
are
v
arious
results
in
the
past
decades.
F
or
e
xample,
the
stability
of
2-D
continuous
systems
has
been
solv
ed
lately
in
[1],
the
stability
mar
gin
of
2-D
con-
tinuous
systems
ha
v
e
been
computed
wit
h
a
ne
w
method
in
[2],
LMI
based
stability
analysis
for
2-D
continuous
systems
w
as
considered
in
[3],
Rob
ust
stability
analysis
for
2-D
continuous-time
systems
w
as
obtained
in
[4],
the
stability
analysis
based
on
the
quadratic
L
yapuno
v
function
w
as
obtained
in
[5],
H
1
filtering
of
uncertain
2-D
continuous
systems
with
time-v
arying
delays
w
as
considered
in
[6].
In
addition,
the
Rob
ust
stabilization
and
control
design
ha
v
e
been
studied
in
some
papers
as
well,
to
list
some
of
these,
authors
in
[7]
proposed
the
rob
ust
sta
te
feedback
H
1
control
for
uncertain
2-D
continuous
state
delayed
Roesser
systems.
H
1
control
of
2-D
continuous
switched
systems
ha
v
e
been
in
v
estig
ated
in
[8],
multiobjecti
v
e
H
2
=H
1
control
design
w
as
considered
in
[9],
LMI
based
rob
ust
PID
control
ha
s
been
solv
ed
in
[10],
and
s
tabilization
of
tw
o-dimensional
continuous
systems
ha
v
e
been
in
v
estig
ated
in
[11].
Recently
,
attention
has
been
de
v
oted
to
w
ards
the
Kalman
Y
akubo
vich
Popo
v
(KYP)
lemma
in
[12],
this
lemma
mak
es
equi
v
alence
between
frequenc
y
domain
inequality
(FDI)
characterizing
a
class
of
properties
of
a
transfer
function,
and
a
linear
matrix
inequality
(LMI)
in
[13],
for
its
state
space
realization.
Therefore,
authors
in
[14]
has
proposed
an
e
xtension
of
the
KYP
lemma,
which
is
kno
wn
as
Generalized
KYP
(GKYP)
lemma
for
the
case
of
finite
frequenc
y
domain.
The
2-D
GKYP
lemma
is
obtained
for
Roesser
model
of
2-D
continuous
systems
in
[15],
and
for
2-D
discrete
systems,
for
both
cases:
F
ornasini-Marchesini
(FM)
and
for
Roesser
models,
in
[16]
and
[17],
respect
i
v
ely
.
The
GKYP
combined
with
the
frequenc
y-partitioning
approach
to
stability
analysis,
were
obtained
in
[18]
for
2-D
discrete
system,
and
for
h
ybrid
systems
in
[19,
20].
Moti
v
ated
by
the
Pre
vious
research,
in
this
paper
,
we
suggest
a
suf
ficient
conditions
of
stability
of
2-D
continuous
Roesser
systems,
via
GKYP
lemma
and
frequenc
y-partitioning
approach,
in
order
to
reduce
the
conserv
ati
v
eness
of
the
e
xisting
simple
2-D
continuous
L
yapuno
v
inequality
.
Generally
,
in
order
to
realize
J
ournal
Homepage:
http://iaesjournal.com/online/inde
x.php/IJPEDS
,
DOI:
10.11591/ijpeds.v8i3.pp990-1001
Evaluation Warning : The document was created with Spire.PDF for Python.
IJPEDS
ISSN:
2088-8694
991
a
series
of
no
v
el
stability
conditions
for
our
system,
the
GKYP
lemma
is
applied
on
each
one
of
the
N
interv
als
of
the
entire
frequenc
y
domain.
Moreo
v
er
,
rob
ust
stabilization
is
also
considered
based
on
the
proposed
stabil-
ity
conditions.
Finally
,
numerical
e
xamples
are
gi
v
en
to
de
monstrate
the
ef
fecti
v
eness
of
the
proposed
method.
Notation:
we
use
the
follo
wing
notati
on
throughout
this
paper
.
The
superscript
T
,
*,
-1
stand
for
m
atrix
transpose,
matrix
comple
x
conjug
ate
transpose,
and
matrix
in
v
erse,
respecti
v
ely
.
I
denotes
an
identity
matrix
with
appropriate
dimension.
The
notation
P
>
0
(
P
<
0
)
means
that
matrix
P
is
positi
v
e
(ne
g
ati
v
e)
definite.
diag
f
:
g
stands
for
the
block
diagonal
matrix,
R
e
(
:
)
is
the
real
of
eigen
v
alue
of
a
square
m
atrix.
Matrices,
if
their
dimensions
are
not
e
xplicitly
stated,
are
assumed
to
be
compatible
for
algebraic
operations.
2.
PR
OBLEM
FORMULA
TION
AND
PRELIMIN
ARIES
Consider
the
follo
wing
Roesser
model
for
2-D
continuous
systems
:
"
@
x
h
(
t
1
;t
2
)
@
t
1
@
x
v
(
t
1
;t
2
)
@
t
2
#
=
A
1
A
2
A
3
A
4
x
h
(
t
1
;
t
2
)
x
v
(
t
1
;
t
2
)
+
B
1
B
2
u
(
t
1
;
t
2
)
(1)
where
x
h
(
t
1
;
t
2
)
2
R
nh
,
x
v
(
t
1
;
t
2
)
2
R
nv
and
u
(
t
1
;
t
2
)
2
R
m
are
the
horizontal
state,
v
ertical
state
and
input
of
system,
respecti
v
ely
,
and
A
1
,
A
2
,
A
3
,
A
4
,
B
1
and
B
2
,
are
real
matrices
with
appropriate
dimensions.
F
or
real
numbers
t
1
and
t
2
,
we
introduce
notations
X
h
=
sup
t
2
k
x
h
(0
;
t
2
)
k
;
X
v
=
sup
t
1
k
x
v
(
t
1
;
0)
k
:
Assumption
1
lim
t
1
!1
k
x
(
t
1
;
0)
k
=
0
and
lim
t
2
!1
k
x
(0
;
t
2
)
k
=
0
:
The
y
ar
e
inferr
ed
to
the
initial
condition
for
the
system
(1).
In
the
stability
analysis
of
2-D
continuous
system
(1),
it
is
required
to
consider
the
zeros
of
the
2-D
characteristic
polynomial
gi
v
en
by
C
(
s
1
;
s
2
)
=
det
s
1
I
nh
A
1
A
2
A
3
s
2
I
nv
A
4
(2)
It
is
kno
wn
in
the
literature
that
the
2-D
continuous
syst
em
is
asymptotically
stable
if
and
only
if
C
(
s
1
;
s
2
)
6
=
0
8
(
s
1
;
s
2
)
:
R
e
(
s
1
)
0
and
R
e
(
s
2
)
0
.
In
general,
this
condition
is
dif
ficult
to
use
in
practice
to
v
erify
the
stability
,
therefore,
another
method
will
be
used
via
LMI.
Lemma
1
Simple
necessary
conditions
for
asymptotic
stability
of
the
2-D
continuous
system
(1)
ar
e
as
follows:
i)
A
1
is
Hurwitz
(i.e
.
all
its
eig
en
values
have
ne
gative
r
eal
parts,
R
e
i
(
A
1
)
<
0
;
i
=
1
;
::;
n
h
).
ii)
A
4
is
Hurwitz.
Proof
From
(1)
for
A
2
=
A
3
=
A
4
=
0
,
we
obtain
the
state
equation
of
the
continuous
system
(
for
the
fix
ed
0
t
2
2
R
)
@
x
h
(
t
1
;
t
2
)
@
t
1
=
A
1
x
h
(
t
1
;
t
2
)
(3)
The
system
(3)
is
asymptotically
stable
if
the
matrix
A
1
is
Hurwitz.
Similarly
,
we
can
proof
ii).
Therefore,
we
assume
the
follo
wing
throughout
the
paper
.
Assumption
2
The
matrices
A
1
and
A
4
ar
e
Hurwitz.
Lemma
2
Let
the
assumption
2
be
satisfied,
the
2-D
continuous
system
(1)
is
asymptotically
stable
if
and
only
if
S
(
s
)
=
A
3
(
sI
A
1
)
1
A
2
+
A
4
;
s
=
j
!
(4)
is
Hurwitz
matrix
for
!
2
R
.
Stability
and
Rob
ust
Stabilization
of
2-D
Roesser
Continuous
Systems
...
(Ismail
Er
Rac
hid)
Evaluation Warning : The document was created with Spire.PDF for Python.
992
ISSN:
2088-8694
Proof
Let
u
(
t
1
;
t
2
)
=
0
,
and
taking
the
Laplace
transformation
of
system
(1)
for
t
1
only
,
and
under
zero
initial
condition,
we
get
sX
h
(
s;
t
2
)
@
X
v
(
s;t
2
)
@
t
2
=
A
1
A
2
A
3
A
4
X
h
(
s;
t
2
)
X
v
(
s;
t
2
)
(5)
Solving
(5),
we
obtain
@
X
v
(
s;
t
2
)
@
t
2
=
[
A
3
(
sI
A
1
)
1
A
2
+
A
4
]
X
v
(
s;
t
2
)
(6)
System
(6)
can
be
re
g
arded
as
a
1-D
continuous
system
with
comple
x
v
ariable
s
,
and
we
notice
that
the
v
ariable
t
2
of
the
system
doesn’
t
depend
on
the
v
ariable
s
.
So
the
1-D
continuous
system
(6)
is
asymptot
ically
stable
if
and
only
if
[
A
3
(
sI
A
1
)
1
A
2
+
A
4
]
is
Hurwitz
matrix
for
R
e
(
s
)
=
0
.
Hence,
the
2-D
continuous
system
(1)
is
asymptotically
stable
if
and
only
if
[
A
3
(
j
!
I
A
1
)
1
A
2
+
A
4
]
is
Hurwitz
matrix
8
w
2
R
.
Remark
1
Notice
that
when
inter
c
hanging
A
1
with
A
4
,
and
A
2
with
A
3
,
the
2-D
continuous
system
(1)
is
asymptotically
stable
if
A
2
(
j
!
I
A
4
)
1
A
3
+
A
1
is
Hurwitz
matrix
8
!
2
R
.
W
e
will
use
the
follo
wing
lemmas,
kno
wn
as
the
KYP
lemma,
the
GKYP
,
and
the
Projection
Lemma,
respec-
ti
v
ely
.
Lemma
3
[12]
Let
matrices
A,
B,
and
=
T
be
given,
if
det
(
j
w
I
A
)
6
=
0
8
!
2
R
.
Then
the
following
two
statements
ar
e
equivalent.
(i)
F
or
any
!
2
R
[
1
,
(
j
w
I
A
)
1
B
I
(
j
w
I
A
)
1
B
I
<
0
(7)
(ii)
Ther
e
e
xists
a
symmetric
matrix
P
suc
h
that
A
B
I
0
0
P
P
0
A
B
I
0
+
<
0
(8)
Lemma
4
[14]
Let
the
matrices
,
F
,
and
be
given,
and
denote
N
!
is
the
null
space
of
T
!
F
,
wher
e
T
!
=
I
j
!
I
.
The
inequality
N
!
N
!
<
0
;
w
ith
!
2
[
!
1
;
!
2
]
;
(9)
holds
if
and
only
if
ther
e
e
xist
Q
>
0
and
a
symmetric
matrix
P
,
suc
h
that
F
(
P
+
Q
)
F
+
<
0
(10)
wher
e
=
0
1
1
0
,
=
1
j
!
c
j
!
c
!
1
!
2
,
w
c
=
(
!
1
+
!
2
)
2
.
Lemma
5
[13]
Given
a
symmetric
matrix
2
R
p
p
and
two
matrices
X
,
Z
of
column
dimensi
on
p
,
ther
e
e
xists
a
matrix
Y
suc
h
that
the
LMI
+
sy
mX
T
Y
Z
<
0
(11)
holds
if
and
only
if
the
following
two
pr
ojection
inequalities
with
r
espect
to
Y
ar
e
satisfied:
X
?
T
X
?
<
0
;
Z
?
T
Z
?
<
0
:
(12)
wher
e
X
?
and
Z
?
ar
e
arbitr
ary
matrices
whose
columns
form
a
basis
of
the
null
spaces
of
X
and
Z
,
r
espec-
tively
.
IJPEDS
V
ol.
8,
No.
3,
September
2017:
990
–
1001
Evaluation Warning : The document was created with Spire.PDF for Python.
IJPEDS
ISSN:
2088-8694
993
3.
ST
ABILITY
AN
AL
YSIS
we
are
no
w
in
a
position
to
present
a
ne
w
condition
for
checking
the
stability
of
2-D
continuous
systems
of
Roesser
model.
Lemma
6
The
2-D
continuous
system
(1)
is
asymptotically
stable
if
ther
e
e
xist
P
1
>
0
and
P
2
>
0
suc
h
that
the
LMI
A
T
P
+
P
A
<
0
(13)
is
feasible
.
Wher
e
P
=
diag
f
P
1
;
P
2
g
and
A
=
A
1
A
2
A
3
A
4
.
Proof
LMI
(13)
can
be
re
written
as
P
1
A
1
+
A
T
1
P
1
A
T
3
P
2
+
P
1
A
2
P
2
A
4
+
A
T
4
P
2
<
0
(14)
and
this
latter
LMI
can
be
re
written
as
A
1
A
2
I
0
T
0
P
1
P
1
0
A
1
A
2
I
0
+
<
0
(15)
where
=
A
3
A
4
0
I
T
0
P
2
P
2
0
A
3
A
4
0
I
(16)
by
Lemma
3,
(15)
is
equi
v
alent
to
(
j
w
I
A
1
)
1
A
2
I
(
j
w
I
A
1
)
1
A
2
I
<
0
;
or
S
(
j
!
)
I
0
P
2
P
2
0
S
(
j
!
)
I
<
0
(17)
where
S
(
j
w
)
is
the
frequenc
y
response
matrix
obtained
from
S
(
s
)
of
Lemma
4,
moreo
v
er
,
(17)
can
be
written
as
S
(
j
w
)
P
2
+
P
2
S
(
j
w
)
<
0
:
(18)
and
the
e
xistence
of
a
P
2
>
0
satisfying
this
last
condition
immediately
implies
that
all
eigen
v
alues
of
S
(
j
!
)
must
ha
v
e
strictly
ne
g
ati
v
e
real
parts,
8
!
2
R
[
1
,
that
is,
feasibility
of
(13)
guarantees
that
condition
of
Lemma
2
holds.
Moreo
v
er
,
feasibility
of
(13)
implies
that
P
1
A
1
+
A
T
1
P
1
<
0
;
P
2
A
4
+
A
T
4
P
2
<
0
;
and,
since
P
1
>
0
and
P
2
>
0
,
all
eigen
v
alues
of
the
matrices
A
1
and
A
4
must
ha
v
e
strictly
ne
g
ati
v
e
real
parts,
and
feasibility
of
(13)
guarantees
that
Lemma
1
and
Lemma
2
are
satisfied.
Lemma
6
proposes
an
LMI
conditi
on
for
the
asymptotical
stability
of
the
system
in
(1),
there
e
xists
some
conserv
ati
v
eness
due
to
the
requirement
of
a
constant
matrix
P
2
for
all
!
2
R
[
1
,
though.
F
ollo
wing
the
similar
line
of
[18,
19,
20],
the
e
xistence
of
P
2
(
j
!
)
>
0
such
that
S
(
j
w
)
P
2
(
j
w
)
+
P
2
(
j
w
)
S
(
j
w
)
<
0
;
8
!
2
R
[
1
;
is
a
suf
ficient
condition
for
asymptotical
stability
of
the
system
(1).
Based
on
this
result,
and
in
order
to
reduce
the
conserv
ati
v
eness
of
Lemma
6,
we
attempt
to
obtain
a
piece
wise
constant
matrices
P
2
(
j
!
)
via
a
frequenc
y-
partitioning
appoach,
o
v
er
the
entire
frequenc
y
field.
Denote
=
R
[
1
,
and
due
to
S
(
j
!
)
=
S
(
j
!
)
,
the
follo
wing
identities
hold:
sup
!
2
R
e
(
S
(
j
w
))
=
sup
!
2
+
R
e
(
S
(
j
w
))
=
sup
!
2
R
e
(
S
(
j
w
))
(19)
Stability
and
Rob
ust
Stabilization
of
2-D
Roesser
Continuous
Systems
...
(Ismail
Er
Rac
hid)
Evaluation Warning : The document was created with Spire.PDF for Python.
994
ISSN:
2088-8694
where
+
=
[0
;
1
]
,
=
[
1
;
0]
.
Therefore,
it
suf
fices
to
consider
the
half
frequenc
y
field
+
.
No
w
,
for
a
gi
v
en
positi
v
e
inte
ger
N
,
di
viding
the
frequenc
y
domain
+
into
N
interv
als,
such
that
+
=
N
[
i
=1
i
;
i
=
[
w
i
1
;
w
i
]
;
!
0
=
0
;
!
N
=
1
;
(20)
then
applying
the
result
of
Lemma
4
on
each
interv
al,
we
obtain
the
follo
wing
theorem.
Theorem
1
F
or
a
given
positive
inte
g
er
N
,
define
fr
equency
intervals
+
as
in
(20).
System
(1)
is
asymp-
totically
stable
if
ther
e
e
xist
P
sj
>
0
,
j=1,2.
P
1
i
,
P
2
i
,
Q
i
>
0
,
i
=
1
;
2
;
:::;
N
A
T
P
i
+
P
i
A
+
F
T
(
i
Q
i
)
F
<
0
(21)
A
T
j
j
P
sj
+
P
sj
A
j
j
<
0
;
j
=
1
;
2
:
(22)
wher
e
A
=
A
1
A
2
A
3
A
4
,
F
=
A
1
A
2
I
0
,
and
P
i
=
diag
f
P
1
i
;
P
2
i
g
.
F
or
i
=
2
;
3
;
:::;
N
1
,
i
=
1
j
w
ci
j
w
ci
w
i
1
w
i
;
w
ith
w
ci
=
(
w
i
1
+
w
i
)
2
:
(23)
F
or
i
=
1
and
i
=
N
,
1
=
1
0
0
w
2
1
and
N
=
1
0
0
w
2
N
1
(24)
r
espectively
.
Proof
By
Assumption
2,
we
ha
v
e
R
e
(
A
1
)
<
0
and
R
e
(
A
4
)
<
0
if
and
only
if
there
e
xist
P
s
1
>
0
and
P
s
1
>
0
such
that
A
T
1
P
s
1
+
P
s
1
A
1
<
0
,
A
T
4
P
s
2
+
P
s
2
A
4
<
0
,
LMIs
in
(22)
are
satisfied.
F
or
i
=
2
;
:::;
N
1
,
and
f
i
=
1
,
i
=
N
g
,
according
to
[14]
the
matrix
i
should
be
taking
as
(23)
and
(24),
respecti
v
ely
.
The
condition
in
(21),
can
be
written
as
F
T
(
P
1
i
+
i
Q
i
)
F
+
i
<
0
(25)
where
=
0
1
1
0
(26)
and
i
=
A
3
A
4
0
I
T
(
P
2
i
)
A
3
A
4
0
I
(27)
Denote
G
(
j
!
)
=
(
j
!
I
A
1
)
1
A
2
,
and
S
(
j
!
)
=
A
4
+
A
3
(
j
!
I
A
1
)
1
A
2
=
A
3
G
(
j
!
)
+
A
4
,
by
lemma
4,
the
follo
wing
inequality
follo
ws:
G
(
j
!
)
I
i
G
(
j
!
)
I
<
0
;
8
!
2
+
(28)
or
S
(
j
!
)
I
(
P
(
l
)
2
)
S
(
j
!
)
I
<
0
;
8
!
2
+
(29)
or
in
a
more
compact
form
S
(
j
!
)
P
2
i
+
P
2
i
S
(
j
!
)
<
0
;
P
2
i
>
0
;
8
!
2
+
(30)
So
R
e
(
S
(
j
!
))
<
0
is
finally
guaranteed
for
all
!
2
+
.
Combining
R
e
(
A
1
)
<
0
,
R
e
(
A
4
)
<
0
and
R
e
(
S
(
j
!
))
<
0
,
we
conclude
that
system
(1)
is
asymptotically
stable
based
on
Lemma
2.
The
proof
is
completed.
IJPEDS
V
ol.
8,
No.
3,
September
2017:
990
–
1001
Evaluation Warning : The document was created with Spire.PDF for Python.
IJPEDS
ISSN:
2088-8694
995
Remark
2
When
N
=
1
,
and
if
letting
Q
i
=
0
,
P
1
i
>
0
and
P
2
i
>
0
be
r
eal,
then
(21)
r
educes
to
(13),
that
is,
Lemma
6
is
a
special
case
of
Theor
em
1.
Theorem
2
F
or
a
given
positive
inte
g
er
N
,
define
fr
equency
intervals
+
as
in
(20).
System
(1)
is
asymp-
totically
stable
if
ther
e
e
xist
P
sj
>
0
,
j=1,2.
P
1
i
,
P
2
i
,
Q
i
>
0
,
W
1
i
,
W
2
i
,
i
=
2
;
3
;
:::;
N
1
,
suc
h
that
i
=
2
6
6
4
11
i
12
i
0
14
i
22
i
23
i
24
i
33
i
34
i
44
i
3
7
7
5
<
0
(31)
sj
i
=
sj
1
i
sj
2
i
sj
3
i
<
0
;
j
=
1
;
2
:
(32)
11
i
=
Q
i
W
1
i
W
T
1
i
,
12
i
=
P
1
i
+
j
w
ci
Q
i
W
T
1
i
+
W
1
i
A
1
,
14
i
=
W
1
i
A
2
,
22
i
=
w
i
1
w
i
Q
i
+
A
T
1
W
T
1
i
+
W
1
i
A
1
,
23
i
=
A
T
3
W
T
2
i
,
24
i
=
A
T
3
W
T
2
i
+
W
1
i
A
2
,
33
i
=
W
2
i
W
T
2
i
,
34
i
=
P
2
i
W
T
2
i
+
W
2
i
A
4
,
44
i
=
W
2
i
A
4
+
A
T
4
W
T
2
i
.
sj
1
i
=
W
j
i
W
T
j
i
,
sj
2
i
=
P
sj
W
T
1
i
+
W
j
i
A
j
j
,
sj
3
i
=
A
T
j
j
W
T
j
i
+
W
j
i
A
j
j
,
j=1,2.
F
or
i=1,
we
r
eplace
11
i
,
and
12
i
in
(31)
by
121
=
P
11
W
T
11
+
W
11
A
1
,
221
=
w
2
1
Q
1
+
A
T
1
W
T
11
+
W
11
A
1
,
r
espectively
.
F
or
i=N,
we
r
eplace
11
i
,
12
i
and
22
i
in
(31)
by
11
N
=
Q
N
W
1
N
W
T
1
N
,
12
N
=
P
1
N
W
T
1
N
+
W
1
N
A
1
,
22
N
=
w
2
N
1
Q
N
+
A
T
1
W
T
1
N
+
W
1
N
A
1
,
r
espectively
.
Proof
From
Theorem
1,
let
=
P
1
i
+
i
Q
i
0
P
2
i
;
(33)
According
to
[14],
for
i
=
2
;
:::;
N
1
,
i
=
1
and
i
=
N
,
i
as
in
(23),
(24),
and
as
in
(26).
Let
Y
=
2
6
6
4
W
1
i
0
W
1
i
0
0
W
2
i
0
W
2
i
3
7
7
5
,
Z
=
I
A
1
0
A
2
0
A
3
I
A
4
,
X
=
I
,
(31)
is
equi
v
alent
to
sy
m
(
X
T
Y
Z
)
+
<
0
(34)
since
one
can
choose
X
?
=
0
,
the
first
inequality
in
(12)
v
anishes,
and
then
by
lemma
5,
(34)
hold
for
some
Y
if
and
only
if
Z
?
T
Z
?
<
0
.
Note
that
Z
?
can
be
selected
as
Z
?
=
2
6
6
4
A
1
A
2
I
0
A
3
A
4
0
I
3
7
7
5
,
and
then
by
calculation,
we
can
obtain
the
equi
v
alence
between
Z
?
T
Z
?
<
0
and
(21).
Consequently
(21)
is
equi
v
alent
to
(31).
In
addition
from
(22),
we
get
A
j
j
I
T
0
P
sj
P
sj
0
A
j
j
I
<
0
;
j
=
1
;
2
:
(35)
Stability
and
Rob
ust
Stabilization
of
2-D
Roesser
Continuous
Systems
...
(Ismail
Er
Rac
hid)
Evaluation Warning : The document was created with Spire.PDF for Python.
996
ISSN:
2088-8694
Figure
1:
R
e
max
(
S
(
j
!
))
and
i
The
equi
v
alence
between
(35)
and
(32)
can
be
similarly
found
by
re-introducing
=
0
P
sj
P
sj
0
,
Y
=
W
j
i
W
j
i
,
Z
=
I
A
j
j
,
X
=
I
,
j=1,2.
Thus,
Theorem
2
is
equi
v
alent
to
Theorem
1.
Example
1
In
this
part,
we
pr
o
vide
an
e
xample
to
illustr
ate
the
application
of
the
pr
oposed
method.
consider
system
in
(1),
wher
e
the
matrices
in
the
system
ar
e
obtained
by
a
suitable
tr
ansformation
fr
om
the
original
system
matrices
[21]:
A
c
=
A
1
A
2
A
3
A
4
=
(
A
d
1)(
A
d
+
1)
1
The
matrices
in
the
original
pr
oblem
ar
e
as
the
following
form
[18]:
A
d
=
A
d
1
A
d
2
A
d
3
A
d
4
,
A
d
1
=
0
:
5
0
:
5
0
:
1
0
:
1
,
A
d
2
=
0
:
4
1
:
1
0
:
6
0
:
1
,
A
d
3
=
0
:
1
0
:
1
0
:
2
0
:
6
,
A
d
4
=
0
:
5
0
:
5
0
:
1
0
:
7
.
we
obtain
A
1
=
5
:
2979
16
:
0426
4
:
2128
12
:
7447
,
A
2
=
20
:
5957
23
:
9149
16
:
4255
16
:
5106
,
A
3
=
5
:
7872
16
:
2553
7
:
4894
22
:
2128
,
A
4
=
22
:
5745
23
:
4894
26
:
9787
30
:
5745
.
Denote
i
as
the
minimum
value
of
i
that
satisfies
sup
!
2
i
R
e
max
(
S
(
j
!
))
<
i
<
0
i
could
be
computed
fr
om
(31)
by
r
eplacing
P
2
i
in
(33)
by
0
P
2
i
P
2
i
i
and
minimizing
i
.
F
igur
e
1
shows
R
e
max
(
S
(
j
!
))
and
the
e
xecuted
i
by
Theor
em
2
with
N
=
1
;
2
;
4
;
8
.
The
stabilit
y
of
the
abo
ve
system
is
verified,
since
R
e
(
max
(
S
(
j
!
)))
<
0
is
e
vident.
W
ith
N
gr
owing
,
It
is
further
shown
that
i
tends
to
the
value
of
R
e
max
(
S
(
j
!
))
o
ver
+
.
Theor
em
2
with
N
=
1
fails
to
decide
the
stability
of
the
abo
ve
system.
By
incr
easing
N
,
it
is
found
that
Theor
em
2
with
N
=
2
;
4
;
8
succeeds,
note
that,
the
abo
ve
system
is
asymptotically
stable
only
for
i
=
2
;
:::;
N
.
But
for
i
=
1
,
whate
ver
N
,
and
whate
ver
the
way
of
partitioning
the
entir
e
interval,
always
system
is
not
stable
.
This
is
due
to
the
r
apid
IJPEDS
V
ol.
8,
No.
3,
September
2017:
990
–
1001
Evaluation Warning : The document was created with Spire.PDF for Python.
IJPEDS
ISSN:
2088-8694
997
variation
of
the
curve
of
R
e
max
(
S
(
j
!
))
in
the
vicinity
of
!
0
=
0
.
Remark
3
In
the
following
domain
[16,
+
1
],
R
e
max
(
S
(
j
!
))
r
emains
r
elatively
stationary
to
the
value
R
e
max
(
S
(
j
1
))
=
R
e
max
(
A
4
)
=
1
:
0850
,
then
i
also
tends
to
this
value
thr
oughout
the
domain.
Even
if
it
decomposed,
we
find
very
similar
values
to
1
:
0850
.
That’
s
why
we
work
ed
on
just
the
domain
[0
;
16]
(F
igur
e
1).
4.
CONTR
OL
LA
W
DESIGN
In
this
section,
Theorem
2
is
further
de
v
eloped
for
state-feedback
control
of
t
he
uncertain
2-D
continuous
sys-
tems.
Consider
a
2-D
continuous
system
of
Roesser
model
with
norm-bounded
uncertainty:
"
@
x
h
(
t
1
;t
2
)
@
t
1
@
x
v
(
t
1
;t
2
)
@
t
2
#
=
A
1
+
A
1
A
2
+
A
2
A
3
+
A
3
A
4
+
A
4
x
h
(
t
1
;
t
2
)
x
v
(
t
1
;
t
2
)
+
B
1
+
B
1
B
2
+
B
2
u
(
t
1
;
t
2
)
(36)
where
the
uncertain
matrices
A
q
,
q
=
1
;
2
;
3
;
4
and
B
p
,
p
=
1
;
2
formed
as
A
1
A
2
B
1
=
H
1
E
1
E
2
L
1
A
3
A
4
B
2
=
H
2
E
3
E
4
L
2
(37)
where
H
1
,
H
2
,
E
1
,
E
2
,
E
3
,
E
4
,
L
1
and
L
2
are
kno
wn
constant
matric
es,
is
norm-bounded
parameter
uncertainty
satis-
fying
T
I
.
Suppose
the
system
(36)
is
controlled
by
a
state-feedback
controller:
u
(
t
1
;
t
2
)
=
K
1
K
2
x
h
(
t
1
;
t
2
)
x
v
(
t
1
;
t
2
)
(38)
where
K
1
and
K
2
are
the
controller
g
ains
to
be
found,
then
the
closed-loop
system
is
gi
v
en
by:
"
@
x
h
(
t
1
;t
2
)
@
t
1
@
x
v
(
t
1
;t
2
)
@
t
2
#
=
A
c
1
+
A
c
1
A
c
2
+
A
c
2
A
c
3
+
A
c
3
A
c
4
+
A
c
4
x
h
(
t
1
;
t
2
)
x
v
(
t
1
;
t
2
)
(39)
where
A
c
1
=
A
1
+
B
1
K
1
,
A
c
2
=
A
2
+
B
1
K
2
,
A
c
3
=
A
3
+
B
2
K
1
,
A
c
4
=
A
4
+
B
2
K
2
,
A
c
1
=
A
1
+
B
1
K
1
,
A
c
2
=
A
2
+
B
1
K
2
,
A
c
3
=
A
3
+
B
2
K
1
,
A
c
4
=
A
4
+
B
2
K
2
.
Our
objecti
v
e
is
to
find
a
state-feedback
controller
in
(38)
for
the
system
(36)
such
that
the
closed-loop
system
(39)
is
asymptotically
stable
for
all
possible
uncertainties.
Before
we
proceed,
the
follo
wing
lemma
which
is
usually
used
in
the
rob
ust
control
of
systems
will
be
gi
v
en
first.
Lemma
7
[22]
Let
1
,
2
and
be
r
eal
matrices
with
appr
opriate
dimensions
suc
h
that
T
I
.
Then,
for
any
scalar
"
>
0
the
following
inequality
holds:
1
2
+
T
2
T
T
1
"
1
1
T
1
+
"
T
2
2
(40)
No
w
,
based
on
Theorem
2,
we
ha
v
e
the
follo
wing
analysis
result
on
rob
ust
stabilization
of
the
2-D
continuous
system
(39).
Proposition
1
F
or
a
given
positive
inte
g
er
N
,
define
fr
equency
intervals
+
as
in
(20).
System
(39)
is
asymptoti-
cally
stable
for
all
satisfying
T
I
,
if
ther
e
e
xist
matrices
P
sj
>
0
,
j
=
1
;
2
:
P
s
2
>
0
,
P
1
i
,
P
2
i
,
Q
i
>
0
,
W
1
,
W
2
,
and
scalar
s
"
i
>
0
,
i
=
1
;
2
;
:::;
N
,
suc
h
that
=
1
2
3
<
0
(41)
sj
=
sj
1
sj
2
sj
3
<
0
;
j
=
1
;
2
:
(42)
1
=
2
6
6
4
11
12
0
14
22
23
24
33
34
44
3
7
7
5
,
2
=
1
"
i
2
,
3
=
diag
f
"
i
I
;
"
i
I
;
"
i
I
;
"
i
I
g
.
11
=
Q
i
W
1
W
T
1
,
12
=
P
1
i
+
j
!
ci
Q
i
W
T
1
+
W
1
A
c
1
,
14
=
W
1
A
c
2
,
22
=
!
i
1
!
i
Q
i
+
A
T
c
1
W
T
1
+
W
1
A
c
1
,
Stability
and
Rob
ust
Stabilization
of
2-D
Roesser
Continuous
Systems
...
(Ismail
Er
Rac
hid)
Evaluation Warning : The document was created with Spire.PDF for Python.
998
ISSN:
2088-8694
23
=
A
T
c
3
W
T
2
,
24
=
A
T
c
3
W
T
2
+
W
1
A
c
2
,
33
=
W
2
W
T
2
,
34
=
P
2
i
W
T
2
+
W
2
A
c
4
,
44
=
W
2
A
c
4
+
A
T
c
4
W
T
2
.
1
=
2
6
6
4
W
1
H
1
0
W
1
H
1
0
0
W
2
H
2
0
W
2
H
2
3
7
7
5
,
2
=
2
6
6
4
0
0
(
E
1
+
L
1
K
1
)
T
(
E
3
+
L
2
K
1
)
T
0
0
(
E
2
+
L
1
K
2
)
T
(
E
4
+
L
2
K
2
)
T
3
7
7
5
,
sj
1
=
sj
1
sj
2
sj
3
,
sj
2
=
sj
1
"
i
sj
2
,
sj
3
=
diag
f
"
i
I
;
"
i
I
g
.
sj
1
=
W
j
W
T
j
,
sj
2
=
P
sj
W
T
j
+
W
j
A
cj
j
,
sj
3
=
A
T
cj
j
W
T
j
+
W
j
A
cj
j
,
sj
1
=
W
j
H
j
W
j
H
j
,
sj
2
=
0
(
E
j
j
+
L
j
K
j
)
T
,
j
=
1
;
2
.
Proof
From
Theorem
2,
by
replacing
W
1
i
and
W
2
i
with
W
1
and
W
2
respecti
v
ely
.
System
(39)
is
asymptotically
stable
if
there
e
xist
P
sj
>
0
,
j
=
1
;
2
:
P
1
i
,
P
2
i
,
Q
i
>
0
such
that
inequalities
in
(31)
and
(32)
satisfied,
in
which
A
q
should
be
A
cq
+
A
cq
for
q
=
1
;
2
;
3
;
4
.
The
abo
v
e
LMIs
can
be
re-written
into
the
follo
wing
form:
1
+
1
T
2
+
2
T
T
1
<
0
:
(43)
sj
1
+
sj
1
T
sj
2
+
sj
2
T
T
sj
1
<
0
:
j
=
1
;
2
:
(44)
According
to
Lemma
7,
the
abo
v
e
inequalities
holds
for
all
if
and
only
if
there
e
xist
some
scalars
"
i
>
0
such
that
1
+
"
1
i
1
T
1
+
"
i
2
T
2
<
0
(45)
sj
1
+
"
1
i
sj
1
T
sj
1
+
"
i
sj
2
T
sj
2
<
0
(46)
which,
by
the
Schur
complement
in
[23],
(45)
and
(46)
gi
v
e
rise
to
(41)
and
(42).
Remark
4
It
is
inter
esting
t
o
note
that,
as
the
LMIs
including
their
pr
oofs
for
i
=
1
and
i
=
N
ar
e
similar
to
those
for
cases
of
i
=
2
;
:::;
N
1
,
we
give
these
Pr
oposition
1
in
one
unified
form
for
all
possible
value
of
i
for
r
eason
of
space
.
The
same
e
xpr
ession
applies
to
the
following
Theor
em
3.
No
w
,
based
on
Proposition
1,
we
are
in
a
position
to
gi
v
e
a
ne
w
method
of
state-feedback
stabilization
controller
design
for
the
Reosser
model.
Theorem
3
F
or
a
given
positive
inte
g
er
N
,
define
fr
equency
intervals
+
as
in
(20).
System
(39)
is
asymptotically
stable
for
all
satisfying
T
I
,
by
a
state
feedbac
k
contr
oller
in
(38),
if
ther
e
e
xist
matrices
e
P
sj
>
0
,
j
=
1
;
2
:
e
P
1
i
,
e
P
2
i
,
e
Q
i
>
0
,
f
W
1
,
f
W
2
,
N
1
,
N
2
and
scalar
s
i
>
0
,
i
=
1
;
2
;
:::;
N
,
suc
h
that
e
=
"
e
1
e
2
e
3
#
<
0
(47)
e
sj
=
"
e
sj
1
e
sj
2
e
sj
3
#
<
0
(48)
e
1
=
2
6
6
6
4
e
11
e
12
0
e
14
e
22
e
23
e
24
e
33
e
34
e
44
3
7
7
7
5
,
e
2
=
h
i
e
1
e
2
i
,
e
3
=
diag
f
i
I
;
i
I
;
i
I
;
i
I
g
e
11
=
e
Q
i
f
W
1
f
W
T
1
,
e
12
=
e
P
1
i
+
j
!
c
e
Q
i
f
W
1
+
A
1
f
W
T
1
+
B
1
N
1
,
e
14
=
A
2
f
W
T
2
+
B
1
N
2
,
e
22
=
!
i
1
!
i
e
Q
i
+
f
W
1
A
T
1
+
A
1
f
W
T
1
+
B
1
N
1
+
N
T
1
B
T
1
,
e
23
=
f
W
1
A
T
3
+
N
T
1
B
T
2
,
e
24
=
A
2
f
W
T
2
+
f
W
1
A
T
3
+
N
T
1
B
T
2
+
B
1
N
2
,
e
33
=
f
W
2
f
W
T
2
,
e
34
=
e
P
2
i
f
W
2
+
A
4
f
W
T
2
+
B
2
N
2
,
IJPEDS
V
ol.
8,
No.
3,
September
2017:
990
–
1001
Evaluation Warning : The document was created with Spire.PDF for Python.
IJPEDS
ISSN:
2088-8694
999
Figure
2:
Closed-loop
responses
of
x
h
1
(
t
1
;
t
2
)
and
x
v
1
(
t
1
;
t
2
)
.
e
44
=
A
4
f
W
T
2
+
B
2
N
2
+
f
W
2
A
T
4
+
N
T
2
B
2
.
e
1
=
2
6
6
4
H
1
0
H
1
0
0
H
2
0
H
2
3
7
7
5
,
e
2
=
2
6
6
4
0
0
f
W
1
E
T
1
+
N
T
1
L
T
1
f
W
1
E
T
3
+
N
T
1
L
T
1
0
0
f
W
2
E
T
2
+
N
T
2
L
T
1
f
W
2
E
T
4
+
N
T
2
L
T
2
g
3
7
7
5
.
e
sj
1
=
"
e
sj
1
e
sj
2
e
sj
3
#
,
e
sj
2
=
h
i
e
sj
1
e
sj
2
i
,
e
sj
3
=
diag
f
i
I
;
i
I
g
,
e
sj
1
=
f
W
j
f
W
T
j
,
e
sj
2
=
e
P
sj
f
W
j
+
A
j
j
f
W
T
j
+
B
j
N
j
,
e
sj
3
=
f
W
j
A
T
j
j
+
A
j
j
f
W
T
j
+
B
j
N
j
+
N
T
j
B
T
j
,
e
sj
1
=
H
j
H
j
,
e
sj
2
=
0
f
W
j
E
T
j
j
+
N
T
j
L
T
j
,
j
=
1
;
2
:
If
the
abo
ve
conditions
ar
e
satisfied,
a
stabilizing
contr
ol
law
K
1
K
2
is
given
by
K
1
=
N
1
f
W
T
1
,
K
2
=
N
2
f
W
T
2
.
Proof
If
(42)
holds,
W
1
and
W
2
are
nonsingular
.
Pre-and
post-multiplying
(41)
by
nonsingular
matrices:
diag
f
W
1
1
;
W
1
1
;
W
1
2
;
W
1
2
;
"
1
i
I
;
"
1
i
I
;
"
1
i
I
;
"
1
i
I
g
and
diag
f
W
T
1
;
W
T
1
;
W
T
2
;
W
T
2
;
"
T
i
I
;
"
T
i
I
;
"
T
i
I
;
"
T
i
I
g
,
and
Pre-and
post-multiplying
(42)
by
nonsingular
matrices:
diag
f
W
1
j
;
W
1
j
;
"
1
i
I
;
"
1
i
I
g
and
diag
f
W
T
j
;
W
T
j
;
"
T
i
I
;
"
T
i
I
g
,
j
=
1
;
2
.
Making
change
of
v
ariables
as
follo
ws:
f
W
1
=
W
1
1
,
f
W
2
=
W
1
2
,
i
=
"
1
i
,
e
Q
i
=
f
W
1
Q
i
f
W
T
1
,
e
P
1
i
=
f
W
1
P
1
i
f
W
T
1
,
e
P
2
i
=
f
W
2
P
2
i
f
W
T
2
,
e
P
s
1
=
f
W
1
P
s
1
f
W
T
1
,
e
P
s
2
=
f
W
2
P
s
2
f
W
T
2
.
W
e
can
obtain
the
equi
v
alence
between
Theorem
3
and
Proposition
1,
where
N
1
=
K
1
f
W
T
1
,
N
2
=
K
2
f
W
T
2
.
In
the
follo
wing,
we
pro
vide
an
e
xample
to
demonstrate
the
ef
fecti
v
eness
of
the
proposed
method
in
this
section.
Example
2
Consider
the
uncertain
2-D
continuous
system
in
(39)
with
par
ameter
s
given
by:
[11]
A
1
=
2
4
1
:
2
0
:
3
0
:
7
1
0
:
5
0
:
6
0
0
:
2
1
:
8
3
5
,
A
2
=
2
4
0
:
7
0
:
2
0
:
5
1
0
:
5
0
3
5
,
A
3
=
0
:
9
0
1
:
5
0
0
:
2
0
:
1
,
A
4
=
0
:
8
0
:
2
0
:
1
0
:
6
,
B
1
=
2
4
0
:
3
0
:
1
0
:
5
1
0
:
5
0
1
0
:
2
0
:
6
3
5
,
B
2
=
1
0
:
3
0
:
2
1
0
:
6
0
:
5
,
H
1
=
2
4
0
:
1
0
0
:
2
3
5
,
H
2
=
0
:
1
0
,
E
1
=
0
:
1
0
0
:
2
,
E
2
=
0
:
1
0
:
2
,
L
1
=
0
:
1
0
:
2
0
,
E
3
=
E
1
,
E
4
=
E
2
,
L
2
=
L
1
.
Because
the
eig
en
values
of
matrices
A
1
and
A
4
contain
positive
eig
en
values
given
by
0
:
4072
and
0
:
6141
,
r
e-
spectively
.
Ther
efor
e
,
the
nominal
2D
continuous
system
under
consider
ation
is
not
asymptotically
stable
.
The
aim
of
this
e
xample
is
to
design
a
fr
equency-partitioning
state
feed-bac
k
contr
oller
suc
h
that
the
r
esulting
closed-loop
system
is
Stability
and
Rob
ust
Stabilization
of
2-D
Roesser
Continuous
Systems
...
(Ismail
Er
Rac
hid)
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