Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
4, N
o
. 4
,
A
ugu
st
2014
, pp
. 60
3
~
61
3
I
S
SN
: 208
8-8
7
0
8
6
03
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
Robust-Neural Observer Design for
Dis
c
ret
e
-Time Uncert
ain Non-Affi
ne Nonlinear Sys
t
em
Som
a
yeh Rahimi*,
Saeed
Mo
ha
mma
d
-Hoseini*
*
* Department of
Control
Engineering,
S
c
ien
ce
an
d Res
ear
ch br
an
ch, Is
l
a
m
i
c A
zad
Univers
i
t
y
,
Bor
oujed,
Iran
** Departmen
t
o
f
Electr
i
cal
Engineering
,
Malek-
as
htar Univ
ersity
of Techno
log
y
, Isfah
a
n, Iran
.
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Apr 28, 2014
Rev
i
sed
Jun
16,
201
4
Accepte
d J
u
l
5, 2014
This paper proposed a new non
linear di
screte-time robust-neural observer
(DTRNO) which capab
le to g
i
v
e
estim
at
ion for
the sta
t
es of Di
screte-
T
im
e
Uncerta
in Non-affine Non-l
i
n
ear S
y
s
t
em
s
in pres
ence of
extern
a
l
disturbances. Th
e Neural network is
a k
i
nd of d
i
scret
e
-tim
e Mul
ti L
a
yer
e
d
Perceptron
(ML
P
) which Tr
ain
e
d with
an
Ext
e
nded Ka
lm
an-
F
ilter
(EKF)
based algorithm, which th
is neur
al observ
e
r is
ro
bus
t in pres
ence
of ex
tern
a
l
and int
e
rnal
unc
erta
inti
es,
using
a parallel conf
ig
ura
tion
.
This wor
k
includ
es
the stab
ility
pro
o
f of the
estimation
error on
the basis of th
e Ly
apunov
approach
,
and f
o
r demonstrate
observer p
e
rfor
m
ance
an Uncertain
Non-
affine Nonlinear
Sy
stems have been si
mulated to formulations
valid
ate th
e
theore
tic
al
. Sim
u
lation
resul
t
s c
onfirm
the pro
f
i
c
ien
c
y of
the
DT
RNO even a
t
the d
i
ffer
e
nt op
erating
conditions
and pr
esence of
parameters
uncertainties.
Keyword:
Discrete-Tim
eNon
lin
ear
Ex
tend
ed
Kalman
-Filter
Mu
lti
Layered
Percep
tron
Neural State E
s
tim
a
tion
Obse
r
v
er
Ro
bu
st-Neu
ral
Copyright ©
201
4 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
:
Som
a
y
e
h R
a
hi
m
i
Depa
rt
m
e
nt
of C
ont
r
o
l
E
n
gi
ne
eri
n
g, Sci
e
nce and
R
e
searc
h
b
r
anc
h
,
Islamic Azad
Uni
v
er
sity, B
o
roujed, I
r
a
n
Pho
n
e
: +98
-
913
-95
832
21
E-m
a
i
l: S.rah
i
l
6
5@Gm
ai
l.Co
m
1.
INTRODUCTION
During t
h
e
rec
e
nt decades
, st
ate
estim
a
tion of dy
nam
i
c syste
m
s an
d the
state observation problem
has
been an ac
tive topic
of re
search
in
d
i
fferen
t areas su
ch
as au
to
m
a
tic co
n
t
ro
l app
licati
o
n
s
,
fau
lt
d
e
tectio
n,
m
oni
t
o
ri
ng
, m
odel
i
n
g
[
7
]
,
e
t
c. D
u
e t
o
c
o
st
, a
n
d t
e
c
h
nol
ogi
cal
c
o
n
s
t
r
ai
nt
s
us
ual
l
y
assum
e
co
m
p
l
e
t
e
accessibility for the system
st
ate, which is
not always possible [6].
It is
noted t
h
at m
o
st
practical system
s are
no
nl
i
n
ea
r an
d i
t
i
s
di
ffi
cul
t
t
o
desi
g
n
a per
f
o
r
m
ance cont
r
o
l
l
e
r or
obse
r
ver
.
So fa
r, t
h
e l
i
n
e
a
ri
zat
i
on t
ech
n
i
ques
can be
applied to
overc
o
me these probl
e
m
s. Howe
v
e
r, th
is lin
earizatio
n
can
limit en
o
r
m
o
u
s
ly th
e
per
f
o
r
m
a
nces of s
u
ch a
p
pr
oa
ches o
f
co
nt
r
o
l
and
obs
er
vat
i
o
n
.
I
n
t
h
i
s
cas
e, t
h
e use
of
n
e
ural
net
w
o
r
ks
(N
Ns)
p
e
rm
its to
app
r
o
x
i
m
a
te su
itab
l
y th
e non
lin
ear fu
n
c
tion
s
and
th
en
t
o
b
y
p
a
ss
th
e lin
earization
p
r
ob
lem
[1
], [2
].
Th
e state ob
serv
ation
p
r
o
b
l
e
m
h
a
s b
een
wid
e
ly d
e
v
e
lop
e
d
i
n
th
e literatu
re, and
u
s
ed
in
n
u
m
erous
applications.
Howe
ve
r in m
o
st cases, the state variable
s a
r
e rarely available for
direct online m
easurements.
Fu
rt
h
e
rm
o
r
e, t
h
ere is a su
b
s
t
a
n
tial requ
iremen
t fo
r
relia
bl
e rec
onst
r
uct
i
on
of t
h
e st
at
e vari
a
b
l
e
s, es
peci
al
l
y
whe
n
t
h
ey
are req
u
i
r
e
d
i
n
t
h
e
sy
nt
hesi
s of c
ont
rol
an
d o
b
s
e
rvat
i
o
n l
a
ws
or f
o
r
pr
ocess
m
oni
t
o
ri
ng
pu
r
pos
es
[4],
[26], [16]. Howe
ver, in
m
o
st realistic cases m
e
rely i
n
pu
t an
d
ou
tpu
t
of
the syst
e
m
are m
easu
r
able.
There
f
ore,
est
i
m
at
i
ng t
h
e st
a
t
e vari
a
b
l
e
s b
y
obse
r
vers
pl
ays a crucial role in t
h
e c
o
ntrol
of
process
e
s t
o
achi
e
ve b
e
t
t
e
r per
f
o
r
m
a
nces [20]
, [
1
3]
. Ob
s
e
rve
r
s desi
gn
pr
ocess i
s
t
oo
com
p
l
e
x have
a go
od
per
f
o
r
m
a
nce
ev
en in
pr
esence of
m
o
d
e
l an
d d
i
st
u
r
b
a
n
c
e un
cer
tain
ties
ar
e called
r
obust [
5
],
[7
], [11], [
2
7
]
.
N
e
w
l
y, o
t
h
e
r
ki
n
d
of
obse
r
v
e
rs has em
erged:
neu
r
al
ob
ser
v
ers
,
fo
r u
n
k
n
o
w
n
pl
a
n
t
dy
n
a
m
i
cs [9]
,
[15]
, [1
7]
, [1
8]
, [
2
2]
but
all th
e app
r
o
a
ch
es m
e
n
tio
n
e
d
ab
ov
e
n
e
ed
the pr
ev
i
o
u
s
know
ledg
e
o
f
th
e
p
l
an
t m
o
d
e
l, at
least p
a
r
tially.
Th
ere ex
ist d
i
fferen
t train
i
ng
alg
o
rith
m
s
fo
r
n
e
ur
al n
e
t
w
or
ks, wh
ich
,
ho
wev
e
r, no
r
m
ally
en
coun
ter
so
m
e
tech
n
i
cal p
r
ob
lem
s
su
ch
as lo
cal m
i
n
i
m
a
, slo
w
learn
i
n
g
, and
h
i
g
h
sen
s
itiv
ity to
in
itial co
n
d
itio
n
s
,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 4
,
N
o
. 4
,
Au
gu
st 2
014
:
60
3
–
61
3
60
4
am
ong ot
hers [10]. As
a viabl
e
alternative,
new traini
ng
algorithm
,
e.g.
, those
based on
Kalm
an filtering have
been
pr
op
ose
d
[1
4]
, [2
1]
, [
23]
, [2
4]
, [2
5]
. D
u
e t
o
t
h
e fact
t
h
at
t
r
ai
ni
ng a ne
ural
net
w
o
r
ks
t
y
pi
cal
l
y
resul
t
s
i
n
a
no
nl
i
n
ea
r
pr
ob
l
e
m
,
an ext
e
nd
ed
Kal
m
an fi
l
t
er (
E
KF
) i
s
re
qui
red
t
o
be
us
ed [
3
]
,
[1
4]
. E
K
F t
r
ai
ni
n
g
fo
r
N
N
s
al
l
o
ws t
h
e
re
d
u
ct
i
o
n
o
f
t
h
e e
poc
h si
ze a
n
d
t
h
e n
u
m
b
er o
f
r
e
qui
red
ne
ur
o
n
s
[1
4]
. C
o
nsi
d
eri
n
g t
h
ese t
w
o fact
s,
we propo
se the u
s
e of th
e EKF train
i
n
g
for DTNO i
n
o
r
d
e
r to
m
o
d
e
l co
m
p
lex
Discrete Ti
m
e
Un
certain
Non
lin
ear Syst
e
m
s (DTUNS).Param
e
ter estimatio
n
, an
d st
ate estim
a
tion
are relate
d in t
h
e se
nse
of how the
measurem
ent from
sensors c
a
n be use
d
to
obtain a
n
accu
rate
m
ode of t
h
e plant to be
controlled. So, the
learn
i
ng
algo
ri
th
m
fo
r th
e DTUNS is i
m
p
l
emen
ted
u
s
in
g
an
EKF. Th
e resp
ectiv
e stab
ility an
alysis, b
a
sed
on
th
e Lyapun
ov
ap
pro
ach, is inclu
d
e
d
for th
e
p
r
op
o
s
ed
sc
h
e
me. Th
e ap
p
licab
ility o
f
th
is
sch
e
m
e
is illu
s
t
rated
b
y
d
i
screte-time state esti
m
a
ti
on for a
nonlinear system
s.
2.
DISCRETE T
I
ME
NONLINEAR SYST
EM
In th
is sectio
n, im
p
o
r
tan
t
m
a
t
h
em
at
ical p
r
elimin
ar
ies req
u
i
red in fu
ture sectio
n
s
are
presen
ted and
then t
h
e state
of a
discrete-time nonlinea
r sys
t
e
m
,
whi
c
h
i
s
a
ssum
e
d t
o
be
o
b
ser
v
a
b
l
e
, i
s
pr
ovi
ded
.
2
.
1
.
Ma
thematica
l
Prelimina
r
ies
Thr
o
ug
h t
h
i
s
bri
e
f
,
we use
k as t
h
e sam
p
l
i
ng st
ep,
0
k
, as the absol
u
te value and, as the
Eucl
i
d
i
a
n
no
r
m
for vect
ors
and as any
a
d
eq
uat
e
n
o
rm
for m
a
trices wh
ich
clo
s
e fo
llo
ws [8
]. Con
s
id
er a
m
u
l
tip
le in
pu
t–m
u
l
tip
le o
u
t
p
u
t
(MIM
O)
no
n
l
i
n
ear system;
(k))
u
,
(k)
(x
F
1)
(k
x
(1
)
(k)
x
h
(k)
y
(2
)
Whe
r
e
n
x
,
m
u
and
n
m
n
F
is n
on-lin
ear
fu
n
c
tio
n
.
Definiti
on 1
:
S
y
st
em
(1) i
s
said t
o
be f
o
rce
d
,
or t
o
ha
ve i
n
pu
t
s
. In co
nt
rast
,
a sy
st
em
descri
bed
by
an eq
ua
t
i
o
n
with
ou
t ex
p
lici
t
p
r
esen
ce
of an
inpu
t
u
, th
at
is;
(k)
x
F
1)
(k
x
(3
)
is said
to b
e
unforced.
It can be ob
tain
ed after selectin
g th
e in
pu
t
u
as
a fee
dbac
k
function of the
state
(k)
x
u(k)
(4
)
Su
ch
su
bstitu
tio
n eli
m
in
ates
u
an
d y
i
el
ds a
n
un
f
o
rce
d
sy
st
e
m
[19]
;
Definiti
on 2
:
Th
e so
lu
tion
of (1
)–(3) is semi g
l
o
b
a
lly unifo
rm
l
y
ul
t
i
m
a
tel
y
bo
un
de
d (
S
GU
UB
)
,
i
f
fo
r
any
,
whi
c
h i
s
a
c
o
m
p
act
subset
o
f
n
and all
)
(
0
k
x
, th
ere ex
ists an
0
,
and a
num
b
er
))
(k
x
,
(
N
0
such
th
at
)
(
k
x
for all
N
k
k
0
, [2
9]
.I
n othe
r w
o
rds
,
the sol
u
tio
n o
f
(1
) is said
to be SG
UUB
if, fo
r any
a
p
r
i
o
ry g
i
v
e
n
(arb
itrarily larg
e) bo
und
ed
set
and
an
y a prio
ry g
i
v
e
n
(arb
itrarily s
m
al
l) set
0
, whic
h
cont
ai
n
s
)
0
,
0
(
as an
in
ter
i
o
r
po
in
t,
th
er
e ex
ists a co
n
t
r
o
l (3)
such
th
at ev
er
y tr
aj
ector
y of
th
e clo
s
ed
-
l
oop
syste
m
starting from
en
ters the set
}
<
(k)
x
|
(k)
{x
=
0
, i
n
a
fin
i
t
e
ti
m
e
an
d remain
s in
it th
ereafter, as is
d
i
sp
layed in
Fi
gu
re 1.
Figure
1
.
SGUUB, sch
e
m
a
tic
represen
tatio
n
0
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Rob
u
s
t
-
Neu
r
a
l
Ob
server
Desi
g
n
fo
r Discrete-Time Un
certa
i
n
N
o
n-Affin
e N
o
n
lin
ea
r S
y
stem (So
m
a
y
eh
R)
60
5
Theorem 1
:
Le
t
(k)
x
V
b
e
a
Lyapun
ov
fun
c
tion
fo
r a d
i
screte-time syste
m
(1
), wh
ich
satisfies th
e
fo
llowing
p
r
operties:
)
k
(
x
)
k
(
x
V
)
k
(
x
2
1
)
(
)
k
(
x
)
k
(
x
V
)
k
(
x
V
)
1
k
(
x
V
3
3
(6
)
Whe
r
e
ζ
is a p
o
s
itiv
e co
nstan
t
,
1
and
2
are strictly increasing functions
, and
3
is a co
n
tinuous
no
n
-
dec
r
easi
n
g
f
unct
i
o
n.
Th
us
i
f
0
)
(
x
V
For
)
(
k
x
(7
)
The
n
)
(
k
x
is u
n
i
formly u
lti
matel
y
b
oun
d
e
d
,
i.e., th
ere is a tim
e i
n
stan
t
T
k
s
u
ch
th
at
T
k
k
k
x
,
)
(
[8]
.
Definiti
on 3
:
A
s
u
b
s
et
n
S
is b
oun
d
e
d
i
f
th
ere ex
ists
0
r
such that
r
x
fo
r all
S
x
[1
9]
.
2.2.
Discrete
-
Time Nonlinear Sys
t
em
To est
i
m
at
e t
h
e st
at
e o
f
a
di
s
c
ret
e
-t
i
m
e nonl
i
n
ear sy
st
em
, whi
c
h i
s
a
ssu
m
e
d t
o
be
o
b
s
e
rva
b
l
e
,
gi
ve
n
by
;
d(k)
(k))
u
,
(k)
(x
F
1)
(k
x
(k)
x
h
(k)
y
(8
)
whe
r
e
n
x
is the state vector of the syste
m
,
m
k
u
)
(
i
s
t
h
e i
nput
vect
or
,
p
k
y
)
(
is th
e o
u
t
pu
t vecto
r
,
n
p
C
is a kn
own
ou
t
p
u
t
m
a
tr
ix
,
n
k
d
)
(
i
s
a di
st
ur
ba
nce ve
ct
or,
G
and
F
are sm
ooth vectors
field,
i
G
and
i
F
th
eirs en
t
r
ies. Hen
c
e, (8)
can
b
e
rewritten
as;
(k)
Cx
(k)
y
n
,
.
.
.
1,
i
,
(k)
d
(k)
u
,
(k)
x
F
1)
(k
x
(k)
d
.
.
.
(k)
d
.
.
.
(k)
d
(k)
d
,
(k)
x
.
.
.
(k)
x
.
.
.
(k)
x
(k)
x
i
i
i
T
n
i
1
T
n
i
1
(9
)
3.
NEURAL
ST
ATE ESTIMATION
A Mu
lti Layer Percep
tron
(M
LP) is a feed-fo
r
ward
artifici
a
l n
e
u
r
al n
e
t
w
o
r
k
m
o
d
e
l th
at
m
a
p
s
sets o
f
in
pu
t d
a
ta
on
to a set of ap
prop
riate
o
u
t
p
u
t
s.
A MLP
con
s
is
ts o
f
m
u
ltip
le l
a
yers of
n
o
d
e
s in
a
d
i
rected graph
,
with eac
h layer fully connect
ed to the
ne
xt
one
. E
x
ce
pt
for
th
e i
n
pu
t
n
o
d
e
s, each
no
de is a n
e
ur
on
with
a
no
nl
i
n
ea
r act
i
v
at
i
on f
unct
i
o
n.
M
L
P i
s
a
m
o
di
fi
cat
i
on
of t
h
e st
an
dar
d
l
i
n
ear per
cept
r
o
n
and ca
n di
st
i
n
g
u
i
s
h
dat
a
t
h
at
are n
o
t
l
i
n
earl
y
sep
a
rabl
e. T
h
e st
r
u
ct
u
r
e o
f
ne
ur
al
net
w
o
r
k
use
d
t
h
e p
r
op
ose
d
obse
r
ver M
P
L
neu
r
al
n
e
two
r
k
w
ith
fo
ur
i
n
pu
ts,
f
i
v
e
N
e
u
r
on
s in th
e h
i
dd
en
layer
an
d an ou
tpu
t
is sho
w
n
i
n
Fi
gur
e
2
.
Fi
gu
re 2.
The
s
t
ruct
u
r
e of
t
h
e neu
r
al
net
w
or
k
1
Iinput
2
Iinput
3
Iinput
4
Iinput
Layer
Iinput
Layer
Hidden
Layer
Output
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 4
,
N
o
. 4
,
Au
gu
st 2
014
:
60
3
–
61
3
60
6
B
y
addi
ng
an
d
subt
ract
i
n
g
P
h
r
a
se
k
Ax
of
eq
uat
i
o
n
(8
)
(k)
Cx
(k)
y
u(k)
x(k),
g
k
Ax
1)
(k
x
(1
0)
A
is Op
tion
a
l Horowitz
m
a
tr
ix
,
C)
(A,
are obse
rvable, a
n
d
u(k)
x(k),
g
include
s unce
rtain term
s
and
di
st
ur
banc
e sy
st
em
,
whe
r
e,
Ax(k)
-
d(k)
(k))
u
,
(k)
(x
F
u(k)
x(k),
g
(1
1)
The
key
t
o
des
i
gni
n
g
a
neu
r
o-
obs
er
ver i
s
t
o
e
m
pl
oy
a n
e
u
r
al
n
e
two
r
k
to
i
d
en
tify th
e non
lin
earity and
a co
nv
en
tio
n
a
l
o
b
s
erver to
esti
m
a
te th
e stat
es. B
y
i
nvo
ki
n
g
a Lue
n
ber
g
e
r
o
b
ser
v
e
r
[
28]
,
th
e ob
serv
er m
o
d
e
l
of
t
h
e sy
st
em
(10
)
ca
n
be
defi
ned
as
fol
l
o
ws;
(k))
x
C
-
(y(k)
u(k)
(k),
x
g
k
x
A
1)
(k
x
(k)
x
C
(k)
y
(12)
Whe
r
e
x
denotes
the state of the obse
rve
r
, a
nd t
h
e o
b
ser
v
er gai
n
m
n
is selected suc
h
that
C)
-
(A
b
eco
m
e
s a Hu
rwitz
m
a
trix
. It sh
ou
l
d
b
e
n
o
t
ed
th
at th
e g
a
i
n
is guara
nteed
to exist; since A can
be
selected suc
h
t
h
at
A)
(C,
i
s
obse
r
vab
l
e. The
st
r
u
ct
u
r
e o
f
a
ne
ur
o-
o
b
s
erve
r i
s
sh
o
w
n i
n
Fi
g
u
r
e
3.
Fi
gu
re 3.
The
s
t
ruct
u
r
e of
t
h
e pr
o
pose
d
ne
ura
l
net
w
or
k o
b
se
rve
r
To a
p
pr
oxi
m
a
te t
h
e
n
onl
i
n
ea
r
f
unct
i
o
n
u(k)
x(k),
g
a m
u
ltilayer NN is co
n
s
i
d
ered
. So
,
a m
u
lti
layer
NN
with
sufficien
tly larg
e n
u
m
b
er o
f
h
i
dd
en
layer n
e
urons can
esti
m
a
te
th
e u
nkn
own
fu
n
c
tion
u(k)
x(k),
g
as fo
llo
ws:
x
V
W
g
T
T
(1
3)
Whe
r
e
W
and
V
are
the
wei
ght
m
a
tr
ices of the
out
p
u
t
an
d
h
i
dd
en layer
s
, r
e
sp
ect
iv
ely,
u]
[x
=
x
,
is
t
h
e b
o
u
n
d
ed
ne
ural
n
e
t
w
or
k a
p
p
r
oxi
m
a
t
i
on err
o
r
,
an
d
i
s
t
h
e t
r
ansfe
r
f
u
nct
i
on
of t
h
e hi
dde
n ne
ur
o
n
s t
h
at
i
s
usu
a
l
l
y
consi
d
ered
as a t
a
ng
e
n
t
hy
per
b
ol
i
c
f
unct
i
o
n
p
r
esent
i
ng
bel
o
w:
1
exp
1
2
x
V
2
i
x
V
i
(1
4)
To
obtain a linear in-pa
r
am
eter ne
ur
al
net
w
or
k fi
xi
n
g
t
h
e
wei
g
ht
s i
s
re
q
u
i
red,
so t
h
e fi
rs
t
l
a
y
e
r as V
= I. The
n
, the
m
odel can be e
x
presse
d as
x
W
g
T
(1
5)
c
c
1
Z
)
(
),
(
k
u
k
x
F
network
neural
A
)
(
k
x
)
(
ˆ
k
x
g
ˆ
)
(
k
y
)
(
k
u
)
(
k
d
)
(
ˆ
k
y
1
Z
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Rob
u
s
t
-
Neu
r
a
l
Ob
server
Desi
g
n
fo
r Discrete-Time Un
certa
i
n
N
o
n-Affin
e N
o
n
lin
ea
r S
y
stem (So
m
a
y
eh
R)
60
7
The pr
o
p
o
s
ed
obs
er
ver
i
s
t
h
e
n
gi
ve
n by
;
u(k)
(k),
x
ˆ
tanh
W
u(k)
(k),
x
ˆ
g
ˆ
*T
(1
6)
On t
h
e ot
her
h
a
nd
, by
defi
ni
n
g
t
h
e st
at
e est
i
m
at
i
on err
o
r
(k)
x
ˆ
-
x(k)
(k)
x
~
and usi
n
g (1
2)
, and (1
6
)
,
the error
dyna
mics can be
express
e
d as;
u(k)
(k),
x
ˆ
tanh
W
ˆ
u(k)
(k),
x
ˆ
g
ˆ
T
(1
7)
,
(k)
x
C
-
(y(k)
u(k)
(k),
x
ˆ
tanh
W
ˆ
k
Ax
1)
(k
x
(k)
x
....
(k)
x
...
(k)
x
(k),
x
(k),
x
)
(k
x
T
n
i
3
2
1
(1
8)
Once the struct
ure of the ne
ural netw
or
k
is kn
own
,
a p
r
op
er lear
n
i
ng
r
u
le sh
ou
ld
b
e
d
e
f
i
ned
to
tr
ain
the network.
This weight-updating m
echanism
is usually define
d in such
a
way that the stability
of the
obs
er
ver
is g
u
a
r
antee
d
. T
h
e
r
ef
ore
,
the
wei
ght
estim
a
tion err
o
r
is de
fine
d as
;
*
i
i
i
W
)
k
(
w
)
k
(
W
~
(1
9)
And
)
k
(
x
ˆ
)
k
(
x
)
k
(
x
~
i
i
i
(2
0)
Since
*
i
W
is constant
Z
0
k
),
k
(
W
)
1
k
(
W
)
k
(
W
~
)
1
k
(
W
~
i
i
i
i
(2
1)
3.
1.
Extended Ka
lma
n
Filter
Kalm
an filter, whic
h is the
set of m
a
the
m
atical
equations, is considere
d
as
one
of the i
m
porta
nt
discoveries in
the control the
o
ry pr
inci
ples. E. Kalm
an’s article was pub
lishe
d in the
year 1960. Its
m
o
st
i
m
m
e
diate applications were
in control of com
p
lex
dynam
i
c system
s, such a
s
m
a
nufact
urin
g
pr
o
cesses,
aircrafts, ships or space
sh
i
p
s (it was part of the
Apollo onboa
rd
guidance system
).
It was and still
is
fre
que
ntly use
d
not only in autom
a
ti
on, but also in the gra
phical and eco
nom
ical applications
, etc.
Howeve
r,
the Extende
d
Kalm
an Filter
started to
appear in t
h
e ne
ural net
w
ork trai
ning applic
ations only relatively
recently, whic
h was ca
use
d
by the progress
of c
o
m
puter
syste
m
s developm
ent.
W
h
e
n
the
m
odel is nonlinear,
whic
h is t
h
e
case of
neural
networks,
we
ha
ve to
exte
nd Kalm
an filter usi
n
g line
a
rization
proc
edure.
Resulting filter is the
n
calle
d extended Kalm
an filter
(E
KF)
[12].T
he
wei
g
ht vectors a
r
e
updated
online
with
a dec
o
u
p
led
E
K
F,
desc
ribe
d
by
;
)
k
(
e
)
k
(
K
)
k
(
W
)
1
k
(
W
i
i
i
i
n
,...,
1
i
),
k
(
M
)
k
(
H
)
k
(
P
)
k
(
K
i
i
i
i
)
k
(
Q
)
k
(
P
)
k
(
H
)
k
(
K
)
k
(
P
)
1
k
(
P
i
i
T
i
i
i
i
(2
2)
Wi
t
h
1
i
i
T
i
i
i
)]
k
(
H
)
k
(
P
)
k
(
H
)
k
(
R
[
)
k
(
M
(2
3)
whe
r
e
)
k
(
W
i
is a vec
t
or
of all wei
g
hts,
i
is a fu
nction
retu
rni
n
g a
vecto
r
o
f
actua
l outp
u
ts,
K
is the
so called Kalm
an gain m
a
trix,
P
is the error c
o
varia
n
ce m
a
trix of the
state and
H is the
m
e
asurem
ent
m
a
t
r
ix
(Jacobian).
i
H
is t
h
e pa
rtial derivative of the MLP out
put wit
h
respect to the M
L
P netw
or
k param
e
ters at the
kth iteration
of the
Kalm
an recursi
o
n.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
I
J
ECE Vo
l.
4
,
No.
4
,
Au
gu
st 2
014
:
60
3
–
61
3
60
8
4.
PRO
O
F O
F
S
T
ABILITY:
L
YAP
U
N
O
V
’
S
DIRE
CT ME
THOD
Theore
m2
:
F
o
r uncertai
n
Discrete Tim
e
nonlinear
dynam
i
c syste
m
(8)
the Neur
al-
R
o
bust Ob
serv
er
give
n by
eq
u
a
tion (
1
2
)
w
h
ere
u(k)
(k),
x
ˆ
tanh
W
ˆ
u(k)
(k),
x
ˆ
g
ˆ
T
and
T
W
ˆ
trained with the E
K
F-base
d
algorithm
,
ensures that the
esti
m
a
ti
o
n
err
o
r
an
d th
e
ou
tpu
t
er
ro
r
ar
e
un
if
ormly u
lti
ma-
t
el
y b
oun
d
e
d
,
m
o
r
e
ov
er
netw
or
k
weig
h
t
s rem
a
in bo
un
ded
.
T
h
e
o
u
tp
u
t
err
o
r
)
k
(
y
ˆ
)
k
(
y
)
k
(
e
(2
4)
and the estim
ation err
o
r desc
ribe
d by
;
)
k
(
x
ˆ
)
k
(
x
)
k
(
x
~
i
i
i
then t
h
e dynam
i
cs of
)
1
k
(
x
i
can be
expresse
d as
)
k
(
x
ˆ
)
k
(
x
)
k
(
x
~
i
i
i
1
1
1
(2
5)
T
h
e
r
efo
r
e
(k))
x
C
-
(y(k)
(k)
x
ˆ
tanh
W
ˆ
k
x
ˆ
A
(k)
x
ˆ
tanh
W
k
Ax
)
1
k
(
x
~
)
1
k
(
x
ˆ
)
1
k
(
x
)
1
k
(
x
~
T
T
*
i
i
i
i
(k)
x
~
C
(k)
x
~
tanh
W
~
k
x
~
A
)
1
k
(
x
~
T
i
(2
6)
These dy
nam
i
cs can be co
n
s
idere
d
as a linear
sy
stem
,
whe
r
e A state
m
a
trix, I input
m
a
trix and
)
k
(
x
ˆ
C
(k)
x
ˆ
tanh
W
~
g
~
i
T
is input. If input a stable linear system
is bounde
d, the
n
output will be bounde
d,
therefore if
g
~
Re
m
a
in bounde
d
then t
h
e estim
a
tion e
r
ror is bounde
d
.
g
~
, a
n
d
are e
x
presse
d as follows:
)
k
(
.
)
k
(
,
)
k
(
g
ˆ
.
)
k
(
g
ˆ
)
k
(
g
ˆ
n
n
1
1
(2
7)
W
h
er
e
n
i
k
x
C
(k)
x
W
W
g
i
i
T
i
*T
i
i
,...
2
,
1
,
)
(
ˆ
ˆ
tanh
)
ˆ
(
~
(2
8)
*
W
is constant Mat
r
ix
but unknown;
Z
k
k
W
k
W
k
W
i
i
i
),
(
ˆ
)
(
)
(
~
(2
9)
According to the E
K
F algorithm
),
(
)
(
)
(
ˆ
)
1
(
ˆ
k
e
k
K
k
W
k
W
i
i
i
i
(3
0)
,
)
k
(
y
ˆ
)
k
(
y
)
k
(
e
(3
1)
)
k
(
e
)
k
(
K
)
k
(
W
~
)
1
k
(
W
~
i
i
i
i
(3
2)
It is e
v
ide
n
t that if
g
~
was b
ound
ed
th
en
g
ˆ
and e
r
ror will be
bounde
d.
In
orde
r to proof, consider t
h
e
candi
date Ly
ap
un
o
v
fu
nctio
n;
(k)
g
~
(k)
(k)P
g
~
(k)
W
~
(k)
P
W
~
(k)
V
i
T
i
i
T
i
i
(3
3)
Whose
first i
n
c
r
em
ent is defined as
(k)
V
-
1)
(k
V
(k)
V
i
i
i
(k)
g
~
(k)
(k)P
g
~
(k)
W
~
(k)
P
W
~
-
1)
(k
g
~
1)
(k
1)P
(k
g
~
1)
(k
W
~
1)
(k
P
W
~
(k)
V
i
T
i
i
T
i
i
T
i
i
T
i
i
(3
4)
Usin
g
(2
3)
a
n
d
(
2
0
)
in
(
4
3),
th
en
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
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I
S
SN:
208
8-8
7
0
8
Rob
u
s
t
-
Neu
r
a
l
Ob
server
Desi
g
n
fo
r Discrete-Time Un
certa
i
n
N
o
n-Affin
e N
o
n
lin
ea
r S
y
stem (So
m
a
y
eh
R)
60
9
(k)
g
~
(k)
(k)P
g
~
(k)
W
~
(k)
P
W
~
(k)
x
~
C
-
(k)
)
(
(k)
x
~
C
-
(k)
(k)e(k)
K
-
(k)
W
~
)
(
(k)e(k)
K
-
(k)
W
~
(k)
V
i
T
i
i
T
i
i
i
i
i
i
i
i
i
i
k
k
i
T
i
T
(3
5)
Wi
t
h
i
i
i
i
Q
k
-
(k)
P
k
)
(
)
(
(3
6)
i
i
(k)
x
tanh
(k)
W
(k)
ˆ
~
(3
7)
i
i
T
i
i
i
Q
k
P
k
H
k
K
k
)
(
)
(
)
(
)
(
(3
8)
Hence
,
(4
4)
ca
n be
e
x
p
r
esse
d as
(k)
g
~
(k)
(k)P
g
~
-
(k)
W
~
(k)P(k)
W
~
-
(k)
x
~
)
(
(k)
x
~
2
)
(
)
(
)
(
2
(k)
x
~
(k)C
K
)
(
)
(
x
~
2
(k)
W
~
)
(
(k)
W
~
2
-
(k)
W
~
(k)
(k)P
W
~
2
(k)
V
i
T
i
T
i
i
T
2
i
i
i
i
T
i
i
C
k
C
k
k
k
k
K
C
k
k
i
i
T
i
T
i
T
i
T
T
i
(3
9)
Using t
h
e ine
q
ualities
,
Y
X
2
Y
Y
X
X
T
T
T
(4
0)
,
Y
X
2
Y
Y
X
X
T
T
T
(4
1)
0
,
,
,
,
)
(
)
(
T
n
n
n
T
T
T
P
P
P
Y
X
X
X
P
PX
X
X
X
P
(4
2)
The
n
(48) can
be
rewritten as
)
(
2
)
(
4
)
(
.
(k)
W
~
)
(
(k)
x
~
(k)
V
)
(
(k)
x
~
)
(
(k)
x
~
2
)
(
(k)
x
~
tanh
(k)
W
~
4
)
(
4
)
(
C
K
(k)
x
~
2
(k))
(P
-
(k))
(P
(k)
W
~
(k)
V
2
2
i
2
i
2
2
2
2
2
i
2
2
i
i
2
i
i
2
i
i
k
k
k
k
k
P
k
C
g
k
k
k
i
i
i
i
i
i
i
i
i
i
i
i
(4
3)
Whe
r
e;
)
(
.
.
~
tanh
.
)
max(
)
(
,
)
(
~
tanh
4
)
(
,
)
(
)
(
2
)
(
2
)
(
*
2
2
2
k
x
W
W
k
k
(k)
x
(k))
(P
-
(k))
(P
k
k
P
k
C
g
k
C
K
k
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
(4
4)
As a
result,
0
(k)
V
i
when
1
i
i
max
2
zi
k
)
k
(
E
)
k
(
A
4
(k)
x
~
(4
5)
Or
2
i
i
max
2
zi
i
k
)
k
(
F
)
k
(
A
4
(k)
W
~
(4
6)
There
f
ore, t
h
e
solution of (12)
an
d (
3
2
)
is stable; henc
e the estim
a
tion e
r
r
o
r a
nd t
h
e DTR
N
O
weig
hts are
D
T
UN
NS
.C
o
n
si
deri
ng
(
9
) a
n
d
(2
4)
, it is easy
to see that t
h
e
out
put e
r
r
o
r ha
s an al
geb
r
aic r
e
lation
with
)
k
(
x
, for t
h
at reason,
)
k
(
x
ˆ
is bo
und
ed,
)
k
(
e
is bounde
d too. Fi
gure
4 i
llustrates the
ra
nge
1
k
, a
n
d
2
k
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
I
J
ECE Vo
l.
4
,
No.
4
,
Au
gu
st 2
014
:
60
3
–
61
3
61
0
Figu
re 4.
the ra
nge
1
k
and
2
k
5.
SIMULATION RESULTS
The
per
f
o
r
m
a
nce of t
h
e p
r
op
o
s
ed
obse
r
ver is
dem
onstrated
through sim
u
la
tion results. T
h
e exam
ple
is a Non-a
ffine Nonlinear
Syste
m
. The sim
u
la
tion is
pe
rf
orm
e
d in M
A
TL
AB
so
ftw
a
re. I
n
this sec
tion has
been
N
N
O
b
se
rve
r
by
E
K
F le
arni
ng
alg
o
rith
m
for a sec
o
nd
-o
rde
r
plant;
)
(
)
(
)
(
)
(
)
(
)
(
)
(
1
1
/
)
(
)
(
2
)
(
1
.
0
)
1
(
)
(
)
(
)
(
1
/
)
(
)
(
2
)
(
1
.
0
)
1
(
2
1
2
2
2
2
1
2
2
2
1
2
2
2
1
1
k
N
k
x
k
x
k
y
k
d
k
x
k
x
k
u
k
u
k
x
k
x
k
d
k
x
k
u
k
x
k
u
k
x
k
x
(4
7)
Whe
r
e
)
(
1
k
x
and
)
(
2
k
x
are state
varia
b
les,
u
is input system
,
)
(
1
k
d
and
)
(
2
k
d
are distu
r
bance
an
d
)
(
k
N
is m
easurem
e
n
t noise.T
h
e
num
e
rical values of
the
N
o
n-a
f
fine
N
o
n
-
linear
Sy
stem
param
e
ters an
d
obs
er
ver
are
de
scribe
d in
Ta
ble 1.
Table
1. T
h
e
Num
e
rical Values
Para
m
e
ter Values Para
m
e
ter
Values
Para
m
e
ter
Values
C
0
0
1
0
0
0
0
1
A
20
0
0
0
1
20
0
0
0
0
20
0
0
0
1
20
001
.
0
T
0
.
1
0
.
2
0
.
2
5
.
0
50
.
0
5
.
0
5
.
0
0
.
3
001
.
0
T
0.
001
The state an
d err
o
r estim
ation o
b
taine
d
by
ou
r p
r
op
ose
d
n
e
ural net
w
o
r
k
by
EKF lear
ni
ng al
go
rithm
fo
r
discrete-ti
m
e No
n-a
ffi
ne
No
nlinea
r sy
st
em
are sh
o
w
n
in Fi
gu
re
5.
Figu
re
5.
The
s
t
ate and
er
ro
r
r
e
sp
onses
to
sin
(
k
)
refe
re
nce
The state an
d err
o
r estim
ation o
b
taine
d
by
ou
r p
r
op
ose
d
n
e
ural net
w
o
r
k
by
EKF lear
ni
ng al
go
rithm
fo
r discrete-ti
m
e
No
n-a
ffi
ne
Nonlinear
syste
m
with input;
)
100
sin(
02
.
0
)
50
sin(
01
.
0
)
sin(
)
(
k
k
k
k
u
, the
results a
r
e
give
n in
Fig
u
r
e
6.
0
5
10
15
20
-1
-0
.
5
0
0.
5
1
a
Ti
m
e
(
s
e
c
)
0
5
10
15
20
-0
.
5
0
0.
5
b
Ti
m
e
(
s
e
c
)
X
2
(t
)
X
2
(t
)
E
rro
r
X
1
(t
)
X
1
(t
)
Er
r
o
r
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN:
208
8-8
7
0
8
Rob
u
s
t
-
Neu
r
a
l
Ob
server
Desi
g
n
fo
r Discrete-Time Un
certa
i
n
N
o
n-Affin
e N
o
n
lin
ea
r S
y
stem (So
m
a
y
eh
R)
61
1
Figu
re
6.
The
s
t
ate and
er
ro
r
r
e
sp
onses
to
)
100
sin(
02
.
0
)
50
sin(
01
.
0
)
sin(
)
(
k
k
k
k
u
ref
e
rence
The state estim
a
tion and er
r
o
r estim
ation obtaine
d
by
p
r
op
ose
d
ne
ural
netw
or
k by
E
K
F learni
ng
algorithm
for
syste
m
(56)
with input
)
sin(
)
(
k
k
u
and
In the
pre
s
enc
e
of m
easure
m
ent noise
is
shown i
n
Figu
re 7.
Figu
re
7.
The
s
t
ate and
er
ro
r
r
e
sp
onses
to
)
sin(
)
(
k
k
u
ref
e
rence
The state estim
a
tion and er
r
o
r estim
ation obtaine
d
by
p
r
op
ose
d
ne
ural
netw
or
k by
E
K
F learni
ng
algorithm
for syste
m
(47) with input
)
100
sin(
02
.
0
)
50
sin(
01
.
0
)
sin(
)
(
k
k
k
k
u
and
In t
h
e presence of
m
easurem
ent n
o
ise is s
h
ow
n i
n
Fi
gu
re
8.
Figu
re
8.
The
s
t
ate and
er
ro
r
r
e
sp
onses
to
)
100
sin(
02
.
0
)
50
sin(
01
.
0
)
sin(
)
(
k
k
k
k
u
ref
e
rence
The state estim
a
tion and er
r
o
r estim
ation obtaine
d
by
p
r
op
ose
d
ne
ural
netw
or
k by
E
K
F learni
ng
algo
rithm
for s
y
stem
(47
)
with in
p
u
t
)
sin(
)
(
k
k
u
an
d
I
n
t
h
e
prese
n
ce
o
f
distur
ba
nce a
r
e
sh
ow
n i
n
Fi
g
u
r
e
9.
Figu
re
9.
The
s
t
ate and
er
ro
r
r
e
sp
onses
to
)
sin(
)
(
k
k
u
refere
nce
0
2
4
6
8
10
12
14
16
18
20
-1
.
5
-1
-0
.
5
0
0.
5
1
1.
5
a
Ti
m
e
(
s
e
c
)
0
2
4
6
8
10
12
14
16
18
20
-0.
8
-0.
6
-0.
4
-0.
2
0
0.
2
0.
4
0.
6
b
Ti
m
e
(
se
c
)
X
1
(t
)
X
1
(t
)
E
rro
r
X
2
(t
)
X
2
(t
)
E
rro
r
0
2
4
6
8
10
12
14
16
18
20
-1
-0.
8
-0.
6
-0.
4
-0.
2
0
0.
2
0.
4
0.
6
0.
8
1
a
Ti
m
e
(
s
e
c
)
0
2
4
6
8
10
12
14
16
18
20
-0
.
5
-0
.
4
-0
.
3
-0
.
2
-0
.
1
0
0.
1
0.
2
0.
3
0.
4
b
Ti
m
e
(
s
e
c
)
X
1
(t
)
X
1
(t
)
Er
r
o
r
X
2
(t
)
X
2
(t
)
E
rro
r
0
2
4
6
8
10
12
14
16
18
20
-1.
5
-1
-0.
5
0
0.
5
1
1.
5
a
Ti
m
e
(
s
e
c
)
0
2
4
6
8
10
12
14
16
18
20
-0
.
5
-0
.
4
-0
.
3
-0
.
2
-0
.
1
0
0.
1
0.
2
0.
3
0.
4
0.
5
b
Ti
m
e
(
s
e
c
)
X
1
(t
)
X
1
(t
)
Er
r
o
r
X
2
(t
)
X
2
(t
)
E
rro
r
0
10
20
30
40
50
60
-1
-0.
8
-0.
6
-0.
4
-0.
2
0
0.
2
0.
4
0.
6
0.
8
1
a
Ti
m
e
(
s
e
c
)
0
10
20
30
40
50
60
-0.
4
-0.
3
-0.
2
-0.
1
0
0.
1
0.
2
0.
3
0.
4
b
Ti
m
e
(
s
e
c
)
X
1
(t
)
X
1
(t
)
Er
r
o
r
X
2
(t
)
X
2
(t
)
Er
r
o
r
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
I
J
ECE Vo
l.
4
,
No.
4
,
Au
gu
st 2
014
:
60
3
–
61
3
61
2
These results de
m
onstrate tha
t
the NN esti
m
a
tion er
r
o
r lear
ned
by
EKF al
go
rithm
is ver
y
low. The
stab
ility o
f
th
e ov
er
all system
was sho
w
n b
y
Lyapu
nov’
s
d
i
r
ect m
e
th
o
d
.
I
t
is
wor
t
h no
ting
th
at no
SPR
assum
p
tion
or
any other c
o
nstraints that rest
rict the ap
plicability of the a
p
proac
h
was im
posed
on the syste
m
.
The p
r
o
p
o
se
d
obse
r
ve
r can
be applied
b
o
th as an
onl
i
n
e and an off-line esti
m
a
tor. Si
m
u
lation results
per
f
o
r
m
e
d o
n
No
n-
af
fine
N
o
nlinear
Sy
stem
co
nfirm
the re
liable per
f
o
rm
ance
of
the
pr
o
p
o
se
d
obse
r
ver.
6.
CO
NCL
USI
O
N
A MLPNN structure was use
d
to design a neural
obse
rve
r
, nam
e
d DTRNO, for a class of Disc
rete
Tim
e
Uncertai
n
N
o
n-a
ffi
ne
No
nlinea
r Sy
stem
s (DT
U
N
N
S
); the
pr
o
pos
ed o
b
se
rve
r
w
a
s traine
d with
an E
K
F
base
d alg
o
rith
m
,
which was
im
plem
ented o
n
line in a
para
llel config
u
r
ation
.
T
h
e b
o
u
n
d
e
dne
ss o
f
the
out
put
,
state, and estim
ation errors
was establishe
d on t
h
e ba
sis of t
h
e Lyapunov a
p
proach. Discrete-tim
e results
show the e
ffec
tiveness of the
propose
d
obse
rver, as applied to a Non-a
ffi
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