I
nte
rna
t
io
na
l J
o
urna
l o
f
Ro
bo
t
ics a
nd
Aut
o
m
a
t
io
n (
I
J
R
A)
Vo
l.
7
,
No
.
4
,
Dec
em
b
er
201
8
,
p
p
.
2
3
2
~
2
5
0
I
SS
N:
2089
-
4
8
5
6
,
DOI
: 1
0
.
1
1
5
9
1
/
i
j
r
a
.
v
7
i
4
.
p
p
2
3
2
-
250
232
J
o
ur
na
l ho
m
ep
a
g
e
:
h
ttp
:
//ia
e
s
co
r
e.
co
m/jo
u
r
n
a
ls
/in
d
ex
.
p
h
p
/
I
JR
A
/in
d
ex
Desig
n
o
f
Ro
bus
t
Co
ntroller
f
o
r
H
i
g
her O
r
der In
ter
v
a
l Sys
te
m
Using
Dif
fere
n
tial Ev
o
lutiona
ry
Algo
rith
m
D.
Srini
v
a
s
a
Ra
o
1
,
M
.
Siv
a
K
u
m
a
r
2
,
M
.
Ra
m
a
lin
g
a
Ra
j
u
3
1
De
p
a
rtme
n
t
o
f
El
e
c
tri
c
a
l
a
n
d
El
e
c
tro
n
ics
En
g
i
n
e
e
rin
g
,
G
u
d
lav
a
ll
e
ru
En
g
i
n
e
e
rin
g
Co
ll
e
g
e
(A
),
G
u
d
l
a
v
a
ll
e
ru
,
A
P
,
In
d
ia
2
,
3
De
p
a
rtm
e
n
t
o
f
El
e
c
tri
c
a
l
a
n
d
El
e
c
tro
n
ics
En
g
in
e
e
rin
g
,
U
n
iv
e
rsity
C
o
ll
e
g
e
o
f
En
g
in
e
e
rin
g
,
JN
T
U Un
iv
e
rsity
,
Ka
k
in
a
d
a
,
Ka
k
in
a
d
a
,
A
P
,
In
d
ia
Art
icle
I
nfo
AB
ST
RAC
T
A
r
ticle
his
to
r
y:
R
ec
eiv
ed
J
u
l 7
,
2
0
1
8
R
ev
i
s
ed
Sep
4
,
2
0
1
8
A
cc
ep
ted
Sep
20
,
2
0
1
8
T
h
is
p
a
p
e
r
d
e
sc
rib
e
s
th
e
d
e
sig
n
o
f
th
e
ro
b
u
st P
I
/
P
ID
c
o
n
tro
l
ler
f
o
r
th
e
h
ig
h
e
r
o
rd
e
r
in
terv
a
l
s
y
ste
m
v
ia
it
s
re
d
u
c
e
d
o
rd
e
r
m
o
d
e
l
u
sin
g
t
h
e
d
iff
e
re
n
ti
a
l
e
v
o
lu
ti
o
n
(DE)
a
lg
o
rit
h
m
.
A
st
a
b
le
re
d
u
c
e
d
in
terv
a
l
m
o
d
e
l
is
g
e
n
e
r
a
ted
f
ro
m
a
h
ig
h
e
r
o
rd
e
r
i
n
terv
a
l
sy
ste
m
u
sin
g
th
e
DE
in
o
r
d
e
r
to
m
in
im
ize
th
e
c
o
st an
d
re
d
u
c
e
th
e
c
o
m
p
lex
it
y
o
f
th
e
s
y
ste
m
.
T
h
is
re
d
u
c
e
d
o
rd
e
r
in
terv
a
l
n
u
m
e
ra
to
r
a
n
d
d
e
n
o
m
in
a
to
r
p
o
ly
n
o
m
ials
a
r
e
d
e
term
in
e
d
b
y
m
in
im
izin
g
th
e
In
teg
ra
l
S
q
u
a
re
d
Err
o
r
(IS
E)
u
si
n
g
th
e
DE.
T
h
e
n
,
u
sin
g
re
d
u
c
e
d
o
rd
e
r
i
n
terv
a
l
m
o
d
e
l,
a
ro
b
u
st
P
I/
P
ID
c
o
n
tro
ll
e
r
is
d
e
sig
n
e
d
b
a
se
d
o
n
th
e
sta
b
il
it
y
c
o
n
d
it
io
n
s
f
o
r
d
e
term
in
in
g
ro
b
u
st
sta
b
il
it
y
o
f
in
terv
a
l
s
y
ste
m
.
F
in
a
ll
y
,
u
sin
g
th
e
se
sta
b
il
it
y
c
o
n
d
i
ti
o
n
s,
a
se
t
o
f
in
e
q
u
a
li
ti
e
s
i
n
term
s
o
f
c
o
n
tro
ll
e
r
p
a
ra
m
e
ters
is
o
b
tain
e
d
f
ro
m
th
e
re
d
u
c
e
d
o
rd
e
r
c
lo
se
d
lo
o
p
c
h
a
ra
c
teristic
p
o
ly
n
o
m
ial.
Th
e
n
th
e
se
in
e
q
u
a
li
ti
e
s are
so
lv
e
d
to
o
b
tain
r
o
b
u
st co
n
tro
ll
e
r
p
a
ra
m
e
ters
w
it
h
t
h
e
h
e
lp
o
f
a
DE
a
lg
o
rit
h
m
.
T
h
e
d
e
si
g
n
e
d
,
ro
b
u
st
c
o
n
tr
o
ll
e
r
f
ro
m
th
e
re
d
u
c
e
d
o
rd
e
r
in
terv
a
l
m
o
d
e
l
w
il
l
b
e
a
tt
rib
u
ted
to
th
e
h
ig
h
e
r
o
r
d
e
r
in
terv
a
l
sy
s
tem
.
T
h
e
d
e
sig
n
e
d
P
I/
P
ID
c
o
n
tro
l
ler
f
ro
m
o
u
r
p
ro
p
o
se
d
m
e
th
o
d
n
o
t
o
n
ly
sta
b
il
ize
s
th
e
re
d
u
c
e
d
o
r
d
e
r
m
o
d
e
l,
b
u
t
a
lso
s
tab
il
ize
s
th
e
o
rig
i
n
a
l
h
ig
h
e
r
o
rd
e
r
s
y
ste
m
.
T
h
e
v
iab
il
it
y
o
f
th
e
p
ro
p
o
se
d
m
e
th
o
d
o
lo
g
y
is
il
lu
stra
t
e
d
th
ro
u
g
h
t
h
e
n
u
m
e
rica
l
e
x
a
m
p
le
o
f
it
s
su
c
c
e
ss
f
u
l
im
p
le
m
e
n
tatio
n
.
T
h
e
e
ff
ic
a
c
y
o
f
th
e
p
ro
p
o
se
d
m
e
th
o
d
o
l
o
g
y
is
a
lso
e
v
a
lu
a
ted
a
g
a
in
st
th
e
a
v
a
il
a
b
le
a
p
p
r
o
a
c
h
e
s
p
re
se
n
ted
i
n
th
e
li
tera
tu
re
a
n
d
t
h
e
re
su
lt
s w
e
re
su
c
c
e
ss
f
u
ll
y
i
m
p
le
m
e
n
ted
.
K
ey
w
o
r
d
:
Dif
f
er
en
t
ial
E
v
o
l
u
ti
o
n
A
l
g
o
r
ith
m
I
n
ter
v
al
S
y
s
te
m
Kh
ar
ito
n
o
v
‟
s
T
h
eo
r
em
Mo
d
el
Or
d
er
R
ed
u
ctio
n
R
o
b
u
s
t
C
o
n
tr
o
ller
Co
p
y
rig
h
t
©
2
0
1
8
I
n
stit
u
te
o
f
Ad
v
a
n
c
e
d
En
g
in
e
e
rin
g
a
n
d
S
c
ien
c
e
.
Al
l
rig
h
ts
re
se
rv
e
d
.
C
o
r
r
e
s
p
o
nd
ing
A
uth
o
r
:
D.
Srin
i
v
asa
R
ao
,
Dep
ar
t
m
en
t o
f
E
lectr
ic
al
an
d
E
lectr
o
n
ics E
n
g
i
n
ee
r
in
g
,
Gu
d
lav
al
ler
u
E
n
g
i
n
ee
r
i
n
g
C
o
lleg
e
(
A
)
,
Gu
d
lav
al
ler
u
A
P
,
I
n
d
ia.
E
m
ail:
p
r
o
f
s
i
v
a
k
u
m
ar
@
g
m
ai
l.c
o
m
1.
I
NT
RO
D
UCT
I
O
N
I
n
m
o
s
t
o
f
co
n
tr
o
l
en
g
i
n
ee
r
in
g
p
r
o
b
le
m
s
,
d
esi
g
n
an
d
d
e
v
el
o
p
m
e
n
t
o
f
a
co
n
tr
o
ller
ar
e
a
n
i
m
p
o
r
tan
t
p
ar
t
a
m
o
n
g
th
e
r
ese
ar
ch
er
s
,
b
ec
au
s
e
th
e
s
i
m
u
latio
n
a
n
d
d
es
ig
n
o
f
a
co
n
tr
o
ller
o
f
t
h
e
h
ig
h
er
o
r
d
er
s
y
s
te
m
i
s
n
o
t
an
ea
s
y
tas
k
.
T
h
e
co
s
t
a
n
d
co
m
p
le
x
it
y
o
f
th
e
co
n
tr
o
ll
e
r
in
cr
ea
s
e
w
i
th
s
y
s
te
m
o
r
d
er
.
T
o
co
n
q
u
er
th
is
p
r
o
b
lem
o
f
s
i
m
u
latio
n
,
a
n
al
y
s
is
a
n
d
d
esig
n
o
f
a
co
n
tr
o
ller
f
o
r
th
e
h
i
g
h
er
o
r
d
er
s
y
s
te
m
,
a
r
ed
u
ce
d
o
r
d
e
r
m
o
d
el
is
ess
e
n
tia
l
f
o
r
th
e
o
r
ig
in
al
h
i
g
h
er
o
r
d
er
s
y
s
te
m
.
I
n
t
h
e
clo
s
ed
lo
o
p
s
y
s
te
m
,
a
co
n
tr
o
ller
d
esig
n
ed
f
o
r
th
is
lo
w
o
r
d
er
r
ed
u
ce
d
m
o
d
el,
w
il
l a
l
s
o
s
tab
il
ize
t
h
e
o
r
i
g
in
a
l
h
ig
h
er
o
r
d
er
s
y
s
te
m
.
Hen
ce
,
f
r
o
m
t
h
e
la
s
t
th
r
ee
d
ec
ad
es,
m
o
d
el
o
r
d
er
r
ed
u
ctio
n
o
f
t
h
e
h
ig
h
er
o
r
d
er
s
y
s
te
m
i
s
an
i
m
p
o
r
tan
t
a
s
p
ec
t
f
o
r
r
ese
ar
ch
er
s
.
Fo
r
th
i
s
r
ea
s
o
n
,
m
a
n
y
m
et
h
o
d
s
ar
e
a
v
ailab
le
f
o
r
lo
w
er
o
r
d
er
m
o
d
els
w
h
ich
ar
e
ex
p
ec
ted
to
ap
p
r
o
x
im
a
te
t
h
e
p
er
f
o
r
m
a
n
ce
o
f
a
h
ig
h
er
o
r
d
er
o
f
li
n
ea
r
t
i
m
e
i
n
v
ar
ia
n
t
s
y
s
t
e
m
.
Var
io
u
s
m
o
d
el
o
r
d
er
tech
n
iq
u
e
s
h
av
e
b
ee
n
p
r
o
p
o
s
ed
f
o
r
th
e
o
r
d
e
r
r
e
d
u
ctio
n
o
f
h
i
g
h
er
o
r
d
er
co
n
t
in
u
o
u
s
ti
m
e
as
w
ell
as
d
is
cr
ete
–
ti
m
e
s
y
s
te
m
s
.
Am
o
n
g
th
ese
m
et
h
o
d
s
,
t
h
e
f
a
m
iliar
an
d
i
m
p
o
r
ta
n
t
m
et
h
o
d
s
ar
e
d
is
cu
s
s
ed
h
er
e.
An
a
g
g
r
eg
at
io
n
m
eth
o
d
w
a
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
R
A
I
SS
N:
2089
-
4856
Desig
n
o
f ro
b
u
s
t c
o
n
tr
o
ller
fo
r
h
ig
h
er o
r
d
er in
terva
l sys
tem
u
s
in
g
d
if
feren
tia
l …
(
D.
S
r
in
iva
s
a
R
a
o
)
233
p
r
o
p
o
s
ed
co
n
tr
o
l
th
e
lar
g
e
s
ca
le
d
y
n
a
m
ic
s
y
s
te
m
[
1
]
.
P
ad
e
a
p
p
r
o
x
im
a
te
m
et
h
o
d
[
2
]
h
as
v
e
r
y
u
s
e
f
u
l
f
ea
t
u
r
e
s
f
o
r
r
esear
ch
d
u
e
to
b
ein
g
co
m
p
u
tatio
n
all
y
s
i
m
p
le.
B
u
t
t
h
e
a
p
p
r
o
x
im
a
n
t
o
b
tai
n
ed
b
y
t
h
is
m
et
h
o
d
o
f
ten
lead
s
to
b
e
u
n
s
tab
le
ev
e
n
t
h
o
u
g
h
t
h
e
o
r
ig
in
al
s
y
s
te
m
i
s
s
tab
le.
I
n
v
ie
w
to
co
n
q
u
er
t
h
e
p
r
o
b
lem
o
f
s
tab
ili
t
y
,
t
h
e
R
o
u
t
h
ap
p
r
o
x
i
m
a
tio
n
m
et
h
o
d
[
3
]
w
as
i
n
tr
o
d
u
ce
d
.
B
u
t,
t
h
e
R
o
u
t
h
ap
p
r
o
x
i
m
a
tio
n
m
et
h
o
d
f
ails
to
g
iv
e
w
el
l
ap
p
r
o
x
im
a
te
at
th
e
lo
w
f
r
eq
u
en
c
y
lev
e
l
o
f
t
h
e
h
ig
h
er
-
o
r
d
er
s
y
s
te
m
.
Se
v
er
al
m
et
h
o
d
s
[
4
]
-
[7
]
h
av
e
b
ee
n
g
iv
e
n
to
i
m
p
r
o
v
e
t
h
e
R
o
u
t
h
a
p
p
r
o
x
im
a
tio
n
m
eth
o
d
f
o
r
o
b
tain
i
n
g
a
r
ed
u
ce
d
o
r
d
er
m
o
d
el.
A
ll
t
h
ese
m
et
h
o
d
s
ar
e
av
ailab
le
f
o
r
s
y
s
te
m
s
w
it
h
f
i
x
ed
co
ef
f
icien
ts
o
n
l
y
.
Ho
w
e
v
er
,
m
a
n
y
co
n
tr
o
l
en
g
i
n
ee
r
in
g
p
r
o
b
le
m
s
li
k
e
f
le
x
ib
le
m
an
ip
u
lato
r
s
y
s
te
m
o
r
n
u
clea
r
r
ea
cto
r
s
y
s
te
m
r
ep
r
esen
ted
b
y
t
h
eir
m
ath
e
m
atica
l
m
o
d
el
in
a
w
id
e
r
an
g
e
o
f
u
n
k
n
o
w
n
o
p
er
atin
g
co
n
d
itio
n
s
ar
e
b
o
u
n
d
u
n
d
er
a
ce
r
tain
a
m
o
u
n
t
o
f
u
n
ce
r
tain
t
y
.
T
h
ese
u
n
ce
r
tai
n
p
ar
am
eter
s
ca
n
b
e
r
ep
r
esen
te
d
b
y
b
o
u
n
d
ed
in
ter
v
a
l.
T
h
is
m
o
tiv
a
tes
r
esear
c
h
er
s
f
o
r
an
i
n
v
e
n
tio
n
o
f
t
h
e
class
ical
tech
n
iq
u
es
f
o
r
i
n
ter
v
al
s
y
s
te
m
s
,
o
r
e
v
en
f
o
r
m
o
r
e
g
e
n
er
al
u
n
ce
r
tai
n
s
y
s
te
m
s
.
Sin
ce
last
d
ec
ad
e,
m
u
c
h
ef
f
o
r
t
h
as
b
ee
n
m
ad
e
i
n
t
h
e
d
es
ig
n
,
s
i
m
u
la
tio
n
an
d
co
n
tr
o
l
o
f
p
la
n
ts
w
it
h
p
ar
a
m
etr
ic
u
n
ce
r
tai
n
t
y
.
Un
f
o
r
tu
n
atel
y
,
th
er
e
ar
e
f
e
w
m
e
t
h
o
d
s
av
ailab
le
i
n
t
h
e
li
ter
atu
r
e
f
o
r
t
h
e
d
esi
g
n
o
f
p
lan
ts
w
i
th
p
ar
a
m
etr
ic
u
n
ce
r
tai
n
t
y
t
h
r
o
u
g
h
m
o
d
el
o
r
d
er
r
ed
u
ctio
n
i
n
t
h
e
e
n
tire
r
a
n
g
e
o
f
o
p
er
atio
n
.
So
m
e
o
f
t
h
e
m
o
d
el
r
ed
u
ctio
n
m
et
h
o
d
s
f
o
r
f
i
x
ed
co
-
ef
f
icie
n
ts
m
en
t
io
n
ed
ab
o
v
e
ar
e
ex
t
en
d
ed
to
o
r
d
er
th
e
r
ed
u
c
tio
n
o
f
h
i
g
h
er
o
r
d
er
in
ter
v
a
l
s
y
s
te
m
s
th
r
o
u
g
h
i
n
te
r
v
al
ar
it
h
m
etic
to
d
er
iv
e
lo
w
er
o
r
d
er
ap
p
r
o
x
i
m
ate
s
.
R
o
u
t
h
ap
p
r
o
x
i
m
atio
n
tech
n
iq
u
e
[
8
]
b
ased
o
n
th
e
d
ir
ec
t
tr
u
n
ca
tio
n
o
f
t
h
e
R
o
u
th
tab
le
f
o
r
in
ter
v
al
s
y
s
te
m
s
u
s
i
n
g
i
n
ter
v
al
ar
i
th
m
eti
c
an
d
ex
ten
d
i
n
g
th
e
R
o
u
th
-
P
ad
e
ap
p
r
o
x
i
m
atio
n
r
ed
u
ctio
n
te
ch
n
iq
u
e
[
4
]
to
th
e
in
ter
v
al
s
y
s
te
m
.
H
w
an
g
et
al.
,
[
9
]
p
o
in
ted
o
u
t
th
at
t
h
is
m
eth
o
d
h
as
s
o
m
eti
m
e
s
f
ailed
to
g
en
er
ate
s
tab
le
r
ed
u
ce
d
o
r
d
e
r
in
ter
v
al
m
o
d
els
f
o
r
s
tab
le
o
r
ig
in
al
s
y
s
te
m
s
.
T
o
im
p
r
o
v
e
t
h
e
ef
f
ec
ti
v
en
e
s
s
o
f
o
r
d
er
r
e
d
u
ctio
n
o
f
in
ter
v
al
s
y
s
te
m
s
,
m
a
n
y
m
i
x
ed
m
et
h
o
d
s
ar
e
p
r
o
p
o
s
ed
in
[
1
0
]
-
[
1
2
]
.
T
h
e
s
tab
ilit
y
a
n
al
y
s
i
s
o
f
in
ter
v
al
p
o
l
y
n
o
m
ia
l
w
a
s
p
r
esen
ted
in
[
1
3
]
u
s
in
g
H
u
r
w
itz
p
o
l
y
n
o
m
ia
l.
I
n
th
e
ar
ea
o
f
co
n
tr
o
l
en
g
i
n
ee
r
in
g
,
a
v
er
y
le
s
s
w
o
r
k
h
a
s
b
ee
n
m
ad
e
i
n
t
h
e
d
esi
g
n
o
f
co
n
tr
o
ller
f
o
r
h
ig
h
er
o
r
d
er
s
y
s
te
m
s
u
s
in
g
m
o
d
el
o
r
d
er
r
e
d
u
ctio
n
tec
h
n
iq
u
e.
Hen
ce
,
it
is
e
s
s
e
n
tial
f
o
r
r
es
ea
r
ch
er
s
to
d
esi
g
n
a
co
n
tr
o
ller
f
o
r
th
e
h
i
g
h
er
o
r
d
er
s
y
s
te
m
t
h
r
o
u
g
h
t
h
e
r
ed
u
ce
d
o
r
d
er
m
o
d
el.
Her
e
f
e
w
m
et
h
o
d
s
ar
e
in
tr
o
d
u
ce
d
in
t
h
e
d
esi
g
n
o
f
co
n
tr
o
ller
s
wh
ich
ar
e
a
v
ailab
le
i
n
t
h
e
liter
a
tu
r
e.
Fro
m
L
o
w
er
o
r
d
er
g
en
er
alize
d
ag
g
r
e
g
ated
m
o
d
el
[
1
4
]
,
a
s
tab
ilizin
g
co
n
tr
o
ller
is
d
esi
g
n
ed
f
o
r
t
h
e
r
ed
u
c
ed
o
r
d
er
m
o
d
el,
if
ap
p
lied
to
its
o
r
ig
i
n
al
h
i
g
h
e
r
o
r
d
er
s
y
s
te
m
;
it
d
o
es
n
o
t
al
wa
y
s
g
u
ar
an
tee
t
h
e
s
tab
ilit
y
o
f
its
clo
s
ed
lo
o
p
s
y
s
te
m
.
A
lo
w
er
o
r
d
er
co
n
tr
o
l
tech
n
iq
u
e
p
r
o
p
o
s
ed
[
1
5
]
u
s
in
g
a
m
o
d
el
o
r
d
er
r
ed
u
ctio
n
o
f
an
o
p
ti
m
al
clo
s
ed
lo
o
p
s
y
s
te
m
.
B
u
t
f
r
o
m
th
is
co
n
tr
o
l
tech
n
iq
u
e,
th
e
h
ig
h
er
o
r
d
er
s
y
s
te
m
s
tab
ilit
y
n
o
t
r
etain
ed
f
o
r
a
s
tab
le
o
r
ig
i
n
al
s
y
s
te
m
.
Mo
s
t
o
f
r
esear
ch
w
o
r
k
h
as
ca
r
r
ied
o
u
t
in
th
i
s
d
ir
ec
tio
n
an
d
a
f
e
w
m
e
th
o
d
s
ar
e
av
ailab
le.
A
co
n
tr
o
ll
er
s
ch
e
m
e
[
1
6
]
is
p
r
o
p
o
s
ed
,
w
h
ic
h
is
r
estricte
d
t
o
p
ar
ticu
lar
Dav
is
o
n
r
ed
u
ce
d
o
r
d
er
m
o
d
el
o
n
l
y
an
d
r
eq
u
ir
es
th
e
s
y
s
te
m
s
tate
s
to
b
e
av
ailab
le
f
o
r
f
ee
d
b
ac
k
.
C
er
tain
s
tab
ilit
y
co
n
d
itio
n
s
ar
e
m
en
tio
n
ed
i
n
[
1
7
]
th
at
th
e
c
o
n
tr
o
ller
d
esig
n
ed
is
n
o
t
o
n
l
y
s
tab
ilizes
t
h
e
r
ed
u
ce
d
o
r
d
er
m
o
d
el
b
u
t
also
s
tab
ilizes
t
h
e
o
r
i
g
i
n
al
h
i
g
h
er
o
r
d
er
s
y
s
te
m
.
He
a
ls
o
s
h
o
w
ed
th
a
t
th
e
ap
p
r
o
x
i
m
ate
in
cl
u
s
io
n
o
f
a
n
y
u
n
s
tab
le
r
ea
l
m
o
d
es
o
f
t
h
e
h
ig
h
er
o
r
d
er
s
y
s
te
m
in
t
h
e
lo
w
er
o
r
d
er
m
o
d
el
w
il
l
r
etai
n
t
h
e
e
x
is
te
n
ce
o
f
s
u
c
h
s
i
m
u
ltan
eo
u
s
s
tab
il
izin
g
co
n
tr
o
ller
.
A
s
p
ec
ial
ca
r
e
f
o
r
an
y
m
o
d
el
r
ed
u
c
tio
n
s
c
h
e
m
e
[
1
8
]
h
as
ta
k
e
n
to
o
b
tain
s
tab
iliz
in
g
co
n
tr
o
ller
s
f
r
o
m
a
r
ed
u
ce
d
o
r
d
er
m
o
d
el.
I
n
t
h
e
d
ir
ec
tio
n
o
f
A
r
n
o
L
in
n
e
m
an
n
,
a
d
o
m
in
a
n
t
p
o
le
r
ete
n
tio
n
tec
h
n
iq
u
e
[
1
9
]
is
p
r
esen
ted
in
a
co
n
tr
o
ller
d
esig
n
f
o
r
co
n
ti
n
u
o
u
s
ti
m
e
s
y
s
te
m
s
t
h
r
o
u
g
h
r
ed
u
ce
d
o
r
d
er
m
o
d
el.
A
ll
t
h
ese
a
v
ailab
le
m
et
h
o
d
s
a
r
e
u
s
ed
o
n
l
y
f
o
r
f
i
x
ed
co
ef
f
ic
ien
t
s
o
f
a
p
o
l
y
n
o
m
ial/
s
y
s
te
m
.
I
n
co
n
tr
o
l
en
g
i
n
ee
r
i
n
g
ap
p
lic
atio
n
s
,
tu
n
i
n
g
o
f
P
I
,
PID
an
d
lag
/lead
co
n
tr
o
ller
s
h
a
v
e
b
ee
n
w
id
el
y
u
s
ed
in
in
d
u
s
tr
ies
f
o
r
s
ev
er
al
d
ec
ad
es.
Ho
w
e
v
er
,
m
a
n
y
i
m
p
o
r
tan
t
r
esu
lt
s
h
a
v
e
b
ee
n
r
ec
en
tl
y
r
ep
o
r
ted
o
n
co
m
p
u
tatio
n
o
f
all
s
tab
ilizi
n
g
P
,
PI
an
d
P
I
D
c
o
n
tr
o
ller
s
in
[
2
0
]
-
[
2
2
]
.
T
h
ese
co
n
tr
o
ller
s
a
r
e
w
id
el
y
u
s
ed
i
n
v
ar
io
u
s
p
r
o
ce
s
s
co
n
tr
o
l
ap
p
licatio
n
s
.
T
h
e
co
n
tr
o
ller
p
er
f
o
r
m
an
ce
w
a
s
co
m
p
ar
ed
b
ased
o
n
s
ettli
n
g
ti
m
e,
p
er
ce
n
t
o
f
o
v
er
s
h
o
o
t
an
d
s
tab
ilit
y
an
al
y
s
is
o
f
a
g
iv
e
n
s
y
s
te
m
w
it
h
f
i
x
ed
co
ef
f
icie
n
t
s
.
B
u
t
,
m
an
y
co
n
tr
o
l
en
g
i
n
ee
r
i
n
g
p
r
o
b
le
m
s
r
e
p
r
ese
n
ted
b
y
th
eir
m
at
h
e
m
a
tical
m
o
d
el
i
n
a
w
id
e
r
an
g
e
o
f
u
n
k
n
o
w
n
o
p
er
atin
g
co
n
d
itio
n
s
ar
e
b
o
u
n
d
u
n
d
er
a
ce
r
tain
a
m
o
u
n
t
o
f
u
n
ce
r
tain
t
y
.
T
h
e
lar
g
e
u
n
ce
r
ta
in
t
y
p
r
es
en
t
in
t
h
e
co
n
tr
o
l
s
y
s
te
m
ca
u
s
e
s
d
eg
r
ad
atio
n
o
f
s
y
s
te
m
p
er
f
o
r
m
an
ce
a
n
d
d
esta
b
ilizatio
n
.
T
h
e
m
o
d
el
is
k
n
o
w
n
ap
p
r
o
x
i
m
atel
y
an
d
h
e
n
ce
it
i
s
n
ec
es
s
ar
y
to
in
co
r
p
o
r
ate
th
e
r
o
b
u
s
t
i
n
d
esi
g
n
.
T
h
er
ef
o
r
e,
r
o
b
u
s
t
co
n
tr
o
l
p
r
esen
t
in
th
e
s
e
u
n
ce
r
tai
n
tie
s
is
v
er
y
i
m
p
o
r
ta
n
t
f
o
r
p
lan
t
o
p
er
atio
n
u
n
d
er
s
tab
ilized
co
n
d
itio
n
.
T
h
is
n
ec
ess
itated
a
r
o
b
u
s
t
co
n
tr
o
ller
d
esig
n
w
h
ic
h
co
u
ld
s
tab
ilize
t
h
e
p
la
n
t
f
o
r
all
t
h
e
o
p
er
atin
g
co
n
d
itio
n
s
.
Hen
ce
d
esig
n
i
n
g
a
r
o
b
u
s
t
co
n
tr
o
ller
f
o
r
p
ar
am
etr
ic
u
n
ce
r
tain
p
lan
t
s
h
a
v
i
n
g
u
n
k
n
o
w
n
,
b
u
t
b
o
u
n
d
ed
p
ar
am
e
ter
u
n
ce
r
tai
n
tie
s
h
a
s
b
ec
o
m
e
t
h
e
p
r
o
b
lem
o
f
r
esea
r
ch
n
o
w
ad
a
y
s
.
T
o
m
i
n
i
m
ize
th
e
s
tated
u
n
ce
r
tai
n
ties
,
m
a
n
y
s
o
lu
tio
n
s
w
er
e
p
r
o
p
o
s
ed
in
th
e
liter
at
u
r
e
f
o
r
t
h
e
s
i
m
u
latio
n
,
d
esi
g
n
a
n
d
t
u
n
i
n
g
o
f
co
n
tr
o
ller
s
[
2
3
]
-
[
2
4
]
.
R
e
ce
n
tl
y
,
af
f
o
r
d
ab
le
r
esu
lt
s
h
a
v
e
b
ee
n
r
ep
o
r
ted
o
n
co
m
p
u
tatio
n
o
f
all
s
tab
ilizi
n
g
P
,
PI
an
d
P
I
D
co
n
tr
o
ller
s
.
T
h
er
ef
o
r
e,
af
ter
th
e
r
en
o
w
n
ed
t
h
eo
r
e
m
o
f
Kh
ar
ito
n
o
v
[
1
3
]
,
th
e
s
tab
ilit
y
an
al
y
s
i
s
o
f
p
o
l
y
n
o
m
ials
d
u
e
to
p
ar
am
eter
u
n
ce
r
tai
n
t
y
h
as
a
n
i
m
p
o
r
tan
t
asp
ec
ts
o
f
r
esear
ch
er
s
.
A
s
p
er
Kh
ar
i
to
n
o
v
,
th
e
in
ter
v
al
p
o
l
y
n
o
m
ial
w
h
ic
h
ass
e
s
s
e
s
r
o
b
u
s
t
s
tab
ilit
y
co
n
d
itio
n
s
t
h
at
f
o
u
r
s
p
ec
iall
y
co
n
s
tr
u
cted
ex
tr
e
m
e
p
o
ly
n
o
m
ial
s
ar
e
Hu
r
w
itz.
R
o
b
u
s
t
s
tab
ilit
y
o
f
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
9
-
4856
I
J
R
A
,
Vo
l.
7
,
No
.
4
,
Dec
em
b
er
2
0
1
8
:
2
3
2
–
2
5
0
234
in
ter
v
a
l
p
o
l
y
n
o
m
ial
is
also
d
is
cu
s
s
ed
b
y
m
a
n
y
r
esear
ch
e
r
s
'
.
I
n
[
2
5
]
,
n
ec
ess
ar
y
a
n
d
s
u
f
f
icie
n
t
s
tab
ilit
y
co
n
d
itio
n
s
f
o
r
in
ter
v
a
l p
o
l
y
n
o
m
ial
s
ar
e
p
r
o
p
o
s
ed
u
s
in
g
t
h
e
r
esu
lt
s
o
f
[
2
6
]
f
o
r
f
ix
ed
p
o
l
y
n
o
m
i
al
s
.
I
n
th
i
s
p
ap
er
,
m
o
d
el
o
r
d
er
r
e
d
u
ctio
n
o
f
in
ter
v
al
s
y
s
te
m
s
i
s
ca
r
r
ied
o
u
t
b
y
u
s
in
g
t
h
e
K
h
a
r
ito
n
o
v
‟
s
p
o
ly
n
o
m
ia
l
[
1
3
]
an
d
d
i
f
f
er
en
tial
e
v
o
lu
tio
n
[
2
7
]
u
s
in
g
th
e
I
SE
m
et
h
o
d
.
T
h
e
n
u
m
er
ato
r
an
d
d
en
o
m
in
a
to
r
p
o
ly
n
o
m
ia
ls
o
f
th
e
r
ed
u
ce
d
m
o
d
el
ar
e
o
b
tain
ed
b
y
m
i
n
i
m
izi
n
g
i
n
te
g
r
al
s
q
u
ar
ed
er
r
o
r
b
et
w
ee
n
th
e
tr
a
n
s
ie
n
t
r
esp
o
n
s
e
o
f
t
h
e
o
r
ig
in
al
h
i
g
h
er
o
r
d
er
s
y
s
te
m
a
n
d
t
h
e
r
ed
u
ce
d
o
r
d
er
m
o
d
el
p
er
tain
in
g
to
a
u
n
it
s
tep
in
p
u
t.
T
h
u
s
,
th
e
s
tab
il
it
y
i
s
g
u
ar
an
te
ed
f
o
r
th
e
r
ed
u
ce
d
o
r
d
er
s
y
s
te
m
if
t
h
e
o
r
ig
i
n
al
h
ig
h
er
o
r
d
er
s
y
s
te
m
i
s
s
tab
le
an
d
th
e
r
esp
o
n
s
e
s
m
atc
h
i
n
g
b
et
w
ee
n
o
r
ig
in
a
l
h
i
g
h
er
o
r
d
er
s
y
s
te
m
an
d
th
e
r
ed
u
ce
d
o
r
d
er
m
o
d
el.
T
h
en
a
P
I
/PID
co
n
tr
o
ller
is
d
esig
n
ed
f
o
r
r
ed
u
ce
d
lo
w
er
o
r
d
er
in
ter
v
al
p
r
o
ce
s
s
p
lan
t
b
ased
o
n
t
h
e
n
ec
e
s
s
ar
y
a
n
d
s
u
f
f
icie
n
t
s
tab
ilit
y
co
n
d
itio
n
s
[
2
8
]
.
T
h
e
s
e
co
n
d
itio
n
s
ar
e
u
s
ed
to
d
e
r
iv
e
a
s
et
o
f
in
eq
u
alities
in
ter
m
s
o
f
co
n
tr
o
ller
p
ar
am
eter
s
.
T
h
e
in
eq
u
alit
y
co
n
s
tr
ai
n
ts
i
n
t
h
e
p
o
ly
n
o
m
ial
ar
e
s
o
l
v
ed
co
n
s
eq
u
e
n
tl
y
to
o
b
tain
th
e
co
n
tr
o
ller
p
ar
a
m
eter
s
w
it
h
t
h
e
h
elp
o
f
DE
.
T
h
e
P
I
/P
I
D
co
n
tr
o
ller
s
d
esi
g
n
ed
i
n
t
h
i
s
p
r
o
p
o
s
ed
m
et
h
o
d
n
o
t
o
n
l
y
s
tab
ilizes
th
e
r
ed
u
ce
d
o
r
d
er
m
o
d
el,
b
u
t
al
s
o
th
e
o
r
i
g
in
al
h
i
g
h
er
o
r
d
er
s
y
s
te
m
.
T
h
e
ef
f
icac
y
o
f
t
h
e
p
r
o
p
o
s
ed
m
et
h
o
d
is
d
em
o
n
s
t
r
ated
b
y
i
m
p
le
m
en
tin
g
w
it
h
a
ty
p
ical
n
u
m
er
ical
ex
a
m
p
le
av
ailab
le
in
th
e
liter
atu
r
e.
I
n
co
m
p
ar
is
o
n
w
it
h
th
e
m
eth
o
d
s
a
v
ailab
le
i
n
th
e
l
i
ter
atu
r
e
[
2
3
]
,
th
e
p
r
o
p
o
s
ed
m
e
th
o
d
in
t
h
is
p
ap
er
is
s
i
m
p
le
an
d
i
n
v
o
lv
e
s
le
s
s
c
o
m
p
u
tatio
n
al
co
m
p
le
x
it
y
.
T
h
is
p
ap
er
is
o
r
g
a
n
ized
as
f
o
llo
ws:
Sect
io
n
2
g
i
v
e
s
r
o
b
u
s
t
o
f
s
tab
ilit
y
co
n
d
itio
n
s
o
f
i
n
ter
v
al
p
o
l
y
n
o
m
ial.
Secti
o
n
3
p
r
esen
t
s
t
h
e
p
r
o
b
le
m
f
o
r
m
u
la
tio
n
a
n
d
t
h
e
p
r
o
p
o
s
ed
o
r
d
e
r
r
ed
u
ctio
n
alg
o
r
ith
m
.
Sectio
n
4
p
r
o
p
o
s
es
th
e
d
esig
n
p
r
o
ce
d
u
r
e
f
o
r
r
o
b
u
s
t
s
t
ab
ilizin
g
P
I
/P
I
D
co
n
tr
o
ller
in
s
ec
tio
n
5
p
r
o
p
o
s
ed
m
e
th
o
d
is
ap
p
lied
to
d
esig
n
a
r
o
b
u
s
t
P
I
/P
I
D
co
n
tr
o
ller
f
o
r
h
i
g
h
er
o
r
d
er
in
ter
v
a
l p
r
o
c
ess
p
lan
t
w
it
h
a
s
u
itab
le
n
u
m
er
ical
ex
a
m
p
le
an
d
th
e
f
i
n
al
co
n
cl
u
s
io
n
is
g
i
v
en
in
s
ec
tio
n
6
.
2.
RO
B
UST
ST
AB
I
L
I
T
Y
CO
NDIT
I
O
NS O
F
I
N
T
E
RVA
L
P
O
L
YNO
M
I
AL
C
o
n
s
id
er
an
in
ter
v
al
p
o
l
y
n
o
m
i
al
„
P
(
s
)
‟
o
f
o
r
d
er
n
o
f
th
e
f
o
r
m
g
iv
e
n
b
elo
w
.
,
p
s
p
...
s
p
...
s
p
s
p
)
s
(
P
0
1
i
i
1
n
1
n
n
n
W
h
er
e
]
b
,
a
[
p
i
i
i
f
o
r
.
n
,.....,
3
,
2
,
1
,
0
i
A
cc
o
r
d
in
g
to
An
d
er
s
o
n
,
J
u
r
y
an
d
Ma
n
s
o
u
r
[
2
9
]
th
e
n
ec
e
s
s
ar
y
an
d
s
u
f
f
icie
n
t
co
n
d
itio
n
f
o
r
r
o
b
u
s
t
s
tab
ilit
y
o
f
in
ter
v
al
p
o
ly
n
o
m
i
als
o
f
o
r
d
er
3
n
is
p
o
s
itiv
e
lo
w
er
b
o
u
n
d
s
o
n
th
e
co
ef
f
icien
ts
o
f
a
n
in
ter
v
al
p
o
ly
n
o
m
ia
l.T
h
er
ef
o
r
e,
co
n
s
id
er
an
in
ter
v
a
l p
o
l
y
n
o
m
ial
o
f
o
r
d
er
n
=1
].
b
,
a
[
p
w
h
e
r
e
,
s
p
)
s
(
P
i
i
i
i
1
0
i
i
].
b
,
a
[
s
]
b
,
a
[
p
s
p
)
s
(
P
0
0
1
1
0
1
T
h
er
ef
o
r
e,
as p
er
A
n
d
er
s
o
n
et
al.
,
[
2
9
]
,
th
e
r
o
b
u
s
t stab
ilit
y
co
n
d
itio
n
i
s
0
a
a
n
d
0
a
0
1
i.e
.
0
a
i
f
o
r
i=0
,
1
.
Si
m
i
lar
l
y
f
o
r
o
r
d
er
n
=2
].
b
,
a
[
s
]
b
,
a
[
s
]
b
,
a
[
p
s
p
s
p
s
p
)
s
(
P
0
0
1
1
2
2
2
0
1
2
2
i
2
0
i
i
T
h
er
ef
o
r
e,
th
e
r
o
b
u
s
t stab
ilit
y
co
n
d
itio
n
is
0
a
,
0
a
1
2
an
d
0
a
0
i.e
.
0
a
i
f
o
r
i =
0
,
1
,
2
.
L
e
mm
a
2
.
1
C
o
n
s
id
er
a
r
ea
l H
u
r
w
itz
p
o
l
y
n
o
m
ial
Q(
s
)
o
f
th
e
f
o
r
m
(
1
)
0
1
i
i
1
n
1
n
n
q
s
q
....
s
q
...
s
q
s
q
n
Q
(
s
)
(
1
)
n
,...
....
,
2
,
1
,
0
i
W
h
er
e
i
q
is
r
ea
l a
n
d
p
o
s
itiv
e,
0
q
0
.
I
f
an
y
co
m
p
lex
n
u
m
b
er
Z
s
u
c
h
th
at
,
)
z
(
f
)
z
(
f
,
0
R
e
m
o
r
eo
v
er
,
,
)
z
(
f
)
z
(
f
C
on
z
C
on
z
w
h
er
e
C
i
s
a
C
lo
s
ed
co
n
to
u
r
,
th
e
n
,
ac
co
r
d
in
g
to
R
o
u
th
e
‟
s
th
eo
r
e
m
[
2
8
]
th
e
f
o
llo
win
g
t
w
o
p
o
l
y
n
o
m
ials
ca
n
b
e
f
o
r
m
u
la
ted
(2
-
3)
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
R
A
I
SS
N:
2089
-
4856
Desig
n
o
f ro
b
u
s
t c
o
n
tr
o
ller
fo
r
h
ig
h
er o
r
d
er in
terva
l sys
tem
u
s
in
g
d
if
feren
tia
l …
(
D.
S
r
in
iva
s
a
R
a
o
)
235
x
s
0
2
]
)
s
(
Q
)
s
(
Q
[
2
1
Q
(
2
)
x
s
1
2
]
)
s
(
Q
)
s
(
Q
[
s
2
1
Q
(
3
)
T
heo
re
m
2
.
1
:Fo
r
s
tab
ilit
y
o
f
)
s
(
Q
th
e
t
w
o
p
o
l
y
n
o
m
ia
ls
0
Q
an
d
1
Q
f
o
r
m
ed
b
y
th
e
alter
n
ate
co
ef
f
icie
n
t
s
o
f
a
Hu
r
w
i
tz
p
o
ly
n
o
m
ial
i
n
ac
co
r
d
an
ce
w
it
h
eq
u
atio
n
s
(
2
)
an
d
(
3
)
m
u
s
t
h
av
e
n
e
g
ati
v
e
r
ea
l
ze
r
o
s
.
T
h
e
p
r
o
o
f
o
f
th
is
is
g
i
v
e
n
in
[
2
8
]
.
2
.
1
.
Nec
ess
a
ry
co
nd
it
io
ns
f
o
r
t
he
s
t
a
bil
it
y
o
f
a
n inte
rv
a
l po
l
y
no
m
ia
l
s
C
o
n
s
id
er
an
in
ter
v
al
p
o
l
y
n
o
m
i
al
o
f
o
r
d
er
n
>
3
o
f
th
e
f
o
r
m
(
4
)
,
p
s
p
...
s
p
...
s
p
s
p
)
s
(
P
0
1
i
i
1
n
1
n
n
n
(
4
)
W
h
er
e
]
b
,
a
[
p
i
i
i
f
o
r
.
n
,.....,
3
,
2
,
1
,
0
i
T
h
e
n
ec
ess
ar
y
co
n
d
it
io
n
s
f
o
r
an
in
ter
v
al
p
o
l
y
n
o
m
ia
l to
b
e
s
tab
le
is
g
i
v
en
as
(
5
)
0
a
b
i
i
f
o
r
.
n
,......,
3
,
2
,
1
,
0
i
(
5
)
2
.
2
.
Su
f
f
icient
co
nd
it
io
ns
f
o
r
s
t
a
bil
it
y
o
f
inte
rv
a
l po
ly
no
m
ia
l
f
o
r
n > 3
A
r
ea
l c
o
ef
f
icie
n
t i
n
ter
v
al
p
o
l
y
n
o
m
ial
o
f
d
e
g
r
ee
n
is
o
f
t
h
e
f
o
r
m
(
6
)
0
1
1
n
1
n
n
n
p
s
p
......
s
p
s
p
)
s
(
P
(
6
)
W
h
er
e
4
n
,
]
b
,
a
[
p
]
,
.
.
.
.
.
.
,
b
,
a
[
p
],
b
,
a
[
p
],
b
,
a
[
p
],
b
,
a
[
p
n
n
n
3
3
3
2
2
2
1
1
1
0
0
0
T
heo
re
m
2
.
2
:
A
r
ea
l
co
ef
f
icie
n
t
i
n
ter
v
al
p
o
l
y
n
o
m
ial
o
f
d
eg
r
ee
4
n
g
i
v
en
i
n
eq
u
atio
n
(
6
)
is
s
ai
d
to
b
e
r
o
b
u
s
tl
y
s
tab
le
i
f
an
d
o
n
l
y
i
f
t
h
e
f
o
llo
w
in
g
co
n
d
itio
n
s
ar
e
s
ati
s
f
ied
.
T
h
er
ef
o
r
e,
th
e
n
ec
es
s
ar
y
a
n
d
s
u
f
f
icie
n
t
co
n
d
itio
n
s
f
o
r
s
tab
ilit
y
o
f
in
ter
v
al
p
o
l
y
n
o
m
ial
s
o
f
o
r
d
e
r
3
n
ar
e
d
er
iv
ed
f
r
o
m
[
2
8
]
an
d
th
e
y
ar
e
p
r
esen
ted
in
T
ab
le
1
.
T
ab
le
1
.
R
o
b
u
s
t Stab
ilit
y
co
n
d
itio
n
s
f
o
r
v
ar
io
u
s
h
i
g
h
er
o
r
d
er
in
ter
v
a
l p
o
l
y
n
o
m
ials
Order
o
f
t
h
e
P
o
ly
n
om
ial
Ne
c
e
s
s
ar
y
Con
d
i
t
i
on
s
S
u
f
f
icie
n
t
c
on
d
it
i
on
s
3
0
a
n
0
i
Where
i
=
0,
1,
2
,
3.
2
0
2
1
b
b
3
a
4
0
a
n
0
i
Where
i
=
0,
1
,
2,
3,
4
.
4
0
2
2
b
b
4
a
a
n
d
.
0
b
a
3
1
5
0
a
n
0
i
Where
i
=
0,
1
,
.
.
4,
5
.
4
0
2
2
b
b
4
a
a
n
d
5
1
2
3
b
b
4
a
6
0
a
n
0
i
Where
i
=
0,
1
,
.
,
5,
6
.
4
0
2
2
b
b
3
a
a
n
d
.
b
b
4
a
5
1
2
3
7
0
a
n
0
i
Where
i
=
0,
1
,
.
,
6,
.
4
0
2
2
b
b
3
a
a
n
d
5
1
2
3
b
b
3
a
.
8
0
a
n
0
i
Where
i
=
0,
1
,
,
.
.
,
7,
8
.
,
b
b
3
a
5
1
2
3
8
0
2
4
a
a
4
b
a
n
d
0
b
a
6
2
.
9
0
a
n
0
i
Where
i
=
01,
,
.
.
,
8,
9.
8
0
2
4
a
a
4
b
9
1
2
5
a
a
4
b
,
0
b
a
6
2
a
n
d
0
b
a
7
3
10
0
a
n
0
i
Where
i
=
,
1,
9,
10.
8
0
2
4
a
a
4
b
,
9
1
2
5
a
a
4
b
,
10
2
2
6
a
a
4
b
a
n
d
0
b
a
7
3
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
9
-
4856
I
J
R
A
,
Vo
l.
7
,
No
.
4
,
Dec
em
b
er
2
0
1
8
:
2
3
2
–
2
5
0
236
3.
P
ro
po
s
ed
o
rder
re
du
ct
io
n a
lg
o
rit
h
m
3
.
1
P
ro
ble
m
f
o
r
m
u
la
t
io
n
C
o
n
s
id
er
a
h
ig
h
er
o
r
d
er
co
n
tin
u
o
u
s
t
i
m
e
i
n
ter
v
al
s
y
s
te
m
g
iv
e
n
b
y
t
h
e
tr
an
s
f
er
f
u
n
ctio
n
(
7
)
:
n
n
n
1
1
0
0
m
m
m
1
1
0
0
s
]
D
,
D
[
.
.
.
.
.
.
s
]
D
,
D
[
]
D
,
D
[
s
]
C
,
C
[
.
.
.
.
.
.
.
s
]
C
,
C
[
]
C
,
C
[
)
D
,
s
(
D
)
C
,
s
(
N
)
D
,
C
,
s
(
G
n
m
(
7
)
W
h
er
e
]
C
,
C
[
i
i
f
o
r
m
,....,
2
,
1
,
0
i
ar
e
n
u
m
er
ato
r
co
ef
f
icien
ts
o
f
)
D
,
C
,
s
(
G
w
i
t
h
i
C
an
d
i
C
as
lo
w
e
r
an
d
u
p
p
er
b
o
u
n
d
s
o
f
i
n
ter
v
a
l
]
C
,
C
[
i
i
r
esp
ec
ti
v
el
y
,
an
d
]
D
,
D
[
i
i
f
o
r
n
,..
.
.,
2
,
1
,
0
i
ar
e
d
en
o
m
i
n
a
to
r
co
ef
f
icie
n
t
s
o
f
)
D
,
C
,
s
(
G
w
it
h
i
D
an
d
i
D
as lo
w
er
a
n
d
u
p
p
er
b
o
u
n
d
s
o
f
i
n
ter
v
al
]
D
,
D
[
i
i
r
esp
ec
tiv
el
y
.
I
t
is
d
esire
d
to
s
y
n
t
h
esize
a
r
o
b
u
s
t
co
n
tr
o
ller
f
o
r
th
is
h
ig
h
er
o
r
d
er
in
ter
v
al
p
lan
t.
T
h
is
ca
n
b
e
ac
h
iev
ed
b
y
ap
p
r
o
x
i
m
ate
t
h
e
h
ig
h
er
o
r
d
er
p
lan
t
in
to
th
e
r
ed
u
ce
d
o
r
d
e
r
m
o
d
el
u
s
i
n
g
th
e
p
r
o
p
o
s
ed
m
o
d
el
o
r
d
er
r
e
d
u
ctio
n
al
g
o
r
ith
m
a
n
d
a
P
I
/P
I
D
r
o
b
u
s
t
co
n
tr
o
ller
is
d
esi
g
n
ed
f
o
r
th
e
r
ed
u
ce
d
o
r
d
er
m
o
d
el
u
s
in
g
t
h
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
g
i
v
en
i
n
s
ec
tio
n
4
.
Fin
al
l
y
t
h
e
d
esig
n
ed
co
n
tr
o
ller
is
attr
ib
u
ted
to
th
e
h
i
g
h
er
o
r
d
er
s
y
s
te
m
f
o
r
r
o
b
u
s
t stab
ilit
y
.
T
h
e
k
th
o
r
d
er
r
ed
u
ce
d
in
ter
v
al
m
o
d
el
i
s
o
b
tain
ed
f
r
o
m
o
u
r
p
r
o
p
o
s
ed
m
et
h
o
d
is
g
i
v
en
a
s
(
8
)
k
k
k
1
1
0
0
1
k
1
k
1
k
1
1
0
0
k
s
]
d
,
d
[
.
.
.
.
.
.
s
]
d
,
d
[
]
d
,
d
[
s
]
c
,
c
[
.
.
.
.
.
.
.
s
]
c
,
c
[
]
c
,
c
[
)
d
,
s
(
D
)
c
,
s
(
N
)
d
,
c
,
s
(
G
(
8
)
W
h
er
e
]
c
,
c
[
i
i
f
o
r
1
k
,....,
2
,
1
,
0
i
ar
e
n
u
m
er
ato
r
co
ef
f
icie
n
ts
o
f
)
d
,
c
,
s
(
G
w
i
th
i
c
an
d
i
c
as
lo
w
er
an
d
u
p
p
er
b
o
u
n
d
s
o
f
i
n
ter
v
al
]
c
,
c
[
i
i
r
esp
ec
tiv
el
y
,
an
d
]
d
,
d
[
i
i
f
o
r
k
,
.
.
.
.
,
2
,
1
,
0
i
ar
e
d
en
o
m
in
a
t
o
r
co
ef
f
icie
n
ts
o
f
)
d
,
c
,
s
(
G
w
it
h
i
d
an
d
i
d
as lo
w
er
a
n
d
u
p
p
er
b
o
u
n
d
s
o
f
i
n
ter
v
al
]
d
,
d
[
i
i
r
esp
ec
tiv
el
y
.
3.
2
.
O
rder
Reduct
io
n O
f
I
nte
rv
a
l Sy
s
t
e
m
A
cc
o
r
d
in
g
to
th
e
Kh
ar
ito
n
o
v
‟
s
t
h
eo
r
e
m
,
th
e
in
ter
v
al
s
y
s
te
m
ca
n
b
e
r
ep
r
esen
ted
in
to
f
o
u
r
f
i
x
ed
p
ar
am
eter
K
h
ar
ito
n
o
v
tr
an
s
f
er
f
u
n
ctio
n
s
.
T
h
e
y
ar
e
g
i
v
en
a
s
(9
-
12)
:
n
n
2
2
1
0
m
m
2
2
1
0
1
1
1
s
D
.
.
.
.
.
.
.
s
D
s
D
D
s
C
.
.
.
.
.
.
.
.
.
.
s
C
s
C
C
)
D
,
s
(
D
)
C
,
s
(
N
)
D
,
C
,
s
(
G
(9
)
n
n
2
2
1
0
m
m
2
2
1
0
2
2
2
s
D
.
.
.
.
.
.
.
s
D
s
D
D
s
C
.
.
.
.
.
.
.
.
.
.
s
C
s
C
C
)
D
,
s
(
D
)
C
,
s
(
N
)
D
,
C
,
s
(
G
(
10
)
n
n
2
2
1
0
m
m
2
2
1
0
3
3
3
s
D
.
.
.
.
.
.
.
s
D
s
D
D
s
C
.
.
.
.
.
.
.
.
.
.
s
C
s
C
C
)
D
,
s
(
D
)
C
,
s
(
N
)
D
,
C
,
s
(
G
(1
1
)
n
n
2
2
1
0
m
m
2
2
1
0
4
4
4
s
D
.
.
.
.
.
.
.
s
D
s
D
D
s
C
.
.
.
.
.
.
.
.
.
.
s
C
s
C
C
)
D
,
s
(
D
)
C
,
s
(
N
)
D
,
C
,
s
(
G
(
1
2
)
T
h
e
ab
o
v
e
f
o
u
r
Kh
ar
ito
n
o
v
‟
s
t
r
an
s
f
er
f
u
n
ct
io
n
s
ar
e,
in
g
e
n
er
al
r
ep
r
esen
ted
as
(
1
3
)
)
D
,
s
(
D
)
C
,
s
(
N
)
D
,
C
,
s
(
G
i
i
i
f
o
r
i=1
,
2
,
3
,
4
(
1
3
)
T
h
ese
h
i
g
h
er
o
r
d
er
f
o
u
r
f
i
x
ed
p
ar
am
eter
Kh
ar
ito
n
o
v
‟
s
tr
an
s
f
er
f
u
n
ct
io
n
s
ar
e
ap
p
r
o
x
i
m
ated
in
to
th
e
r
ed
u
ce
d
o
r
d
e
r
m
o
d
el
u
s
i
n
g
f
o
llo
w
i
n
g
p
r
o
ce
d
u
r
e.
L
et
th
e
f
o
u
r
f
ix
ed
p
ar
a
m
eter
r
ed
u
ce
d
o
r
d
er
m
o
d
els
o
f
t
h
e
ab
o
v
e
eq
u
atio
n
s
(
9
-
1
3
)
o
b
tain
ed
b
y
th
e
p
r
o
p
o
s
ed
m
eth
o
d
ar
e
d
ef
in
ed
as
(
1
4
-
17)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
R
A
I
SS
N:
2089
-
4856
Desig
n
o
f ro
b
u
s
t c
o
n
tr
o
ller
fo
r
h
ig
h
er o
r
d
er in
terva
l sys
tem
u
s
in
g
d
if
feren
tia
l …
(
D.
S
r
in
iva
s
a
R
a
o
)
237
k
k
2
2
1
0
1
k
1
k
2
2
1
0
1
r
1
r
1
k
s
d
.
.
.
.
.
.
.
s
d
s
d
d
s
c
.
.
.
.
.
.
.
.
.
.
s
c
s
c
c
)
d
,
s
(
d
)
c
,
s
(
n
)
d
,
c
,
s
(
G
(
1
4
)
k
k
2
2
1
0
1
k
1
k
2
2
1
0
2
r
2
r
2
k
s
d
.
.
.
.
.
.
.
s
d
s
d
d
s
c
.
.
.
.
.
.
.
.
.
.
s
c
s
c
c
)
d
,
s
(
d
)
c
,
s
(
n
)
d
,
c
,
s
(
G
(
1
5
)
k
k
2
2
1
0
1
k
1
k
2
2
1
0
3
r
3
r
3
k
s
d
.
.
.
.
.
.
.
s
d
s
d
d
s
c
.
.
.
.
.
.
.
.
.
.
s
c
s
c
c
)
d
,
s
(
d
)
c
,
s
(
n
)
d
,
c
,
s
(
G
(
1
6
)
k
k
2
2
1
0
1
k
1
k
2
2
1
0
4
r
4
r
4
k
s
d
.
.
.
.
.
.
.
s
d
s
d
d
s
c
.
.
.
.
.
.
.
.
.
.
s
c
s
c
c
)
d
,
s
(
d
)
c
,
s
(
n
)
d
,
c
,
s
(
G
(
1
7
)
T
h
e
ab
o
v
e
Fo
u
r
Kh
ar
ito
n
o
v
‟
s
tr
an
s
f
er
f
u
n
c
tio
n
s
ar
e,
i
n
g
e
n
er
al
r
ep
r
esen
ted
as
(
1
8
)
:
)
d
,
s
(
d
)
c
,
s
(
n
)
d
,
c
,
s
(
G
i
k
i
k
i
k
Fo
r
i=1
,
2
,
3
,
4
.
(
1
8
)
I
n
th
is
w
o
r
k
,
t
h
e
n
u
m
er
ato
r
a
n
d
d
en
o
m
i
n
ato
r
co
ef
f
icie
n
ts
o
f
th
e
r
ed
u
ce
d
m
o
d
el
ar
e
o
b
tain
ed
b
y
m
i
n
i
m
izi
n
g
th
e
o
b
j
ec
tiv
e
f
u
n
c
tio
n
‟
J
„
,
w
h
ic
h
i
s
t
h
e
er
r
o
r
b
etw
ee
n
t
h
e
o
r
ig
in
al
h
ig
h
er
o
r
d
er
s
y
s
te
m
a
n
d
th
e
r
ed
u
ce
d
o
r
d
e
r
s
y
s
te
m
.
T
h
er
ef
o
r
e,
it is
r
ep
r
esen
ted
in
th
e
f
o
r
m
(
1
9
)
:
dt
))
t
(
y
)
t
(
y
(
J
0
2
k
(
1
9
)
Ma
th
e
m
atica
ll
y
t
h
e
I
n
te
g
r
al
S
q
u
ar
e
er
r
o
r
ca
n
b
e
r
ep
r
esen
ted
as
(
2
0
)
2
M
0
k
)]
t
(
y
)
t
(
y
[
J
(
2
0
)
W
h
er
e,
y
(
t)
i
s
t
h
e
u
n
it
s
te
p
r
e
s
p
o
n
s
e
o
f
h
ig
h
er
o
r
d
er
an
d
y
k
(
t)
is
t
h
e
u
n
it
s
tep
r
es
p
o
n
s
e
lo
w
er
o
r
d
er
s
y
s
te
m
at
t
h
e
in
s
ta
n
t
i
n
th
e
ti
m
e
in
ter
v
al
0
≤
t
≤
M,
w
h
er
e
M
is
to
b
e
ch
o
s
en
.
T
h
e
o
b
j
ec
t
iv
e
is
to
o
b
tain
a
r
ed
u
ce
d
o
r
d
e
r
m
o
d
el,
w
h
ic
h
i
s
clo
s
el
y
ap
p
r
o
x
i
m
ate
o
r
ig
i
n
al
s
y
s
te
m
.
T
h
e
o
b
j
ec
tiv
e
f
u
n
cti
o
n
is
to
m
i
n
i
m
iz
e
I
SE
b
y
u
s
i
n
g
DE
.
3
.
3
.
P
ro
po
s
ed
DE
a
lg
o
rit
h
m
f
o
r
o
rder
re
du
c
t
io
n
DE
is
a
s
to
ch
as
tic,
p
o
p
u
latio
n
b
ased
d
ir
ec
t
s
ea
r
ch
o
p
ti
m
iz
atio
n
al
g
o
r
ith
m
i
n
tr
o
d
u
ce
d
b
y
Sto
r
n
a
n
d
P
r
ice
in
1
9
9
7
[
2
7
]
.
DE
w
o
r
k
s
w
it
h
t
w
o
p
o
p
u
latio
n
s
;
o
ld
g
en
er
atio
n
an
d
n
e
w
g
e
n
er
a
tio
n
o
f
t
h
e
s
a
m
e
p
o
p
u
latio
n
.
NP
is
th
e
s
ize
o
f
t
h
e
p
o
p
u
latio
n
a
n
d
it is
ad
j
u
s
te
d
.
T
h
e
p
o
p
u
latio
n
co
n
s
i
s
ts
o
f
r
ea
l v
al
u
ed
v
ec
to
r
s
w
it
h
a
d
i
m
e
n
s
io
n
D
th
at
eq
u
als
t
h
e
n
u
m
b
er
o
f
d
esi
g
n
p
ar
a
m
eter
s
/co
n
tr
o
l
v
ar
iab
les.
T
h
e
p
o
p
u
latio
n
i
s
r
an
d
o
m
l
y
in
i
tialized
w
it
h
i
n
t
h
e
in
itial
p
ar
a
m
eter
b
o
u
n
d
s
.
T
h
e
th
r
ee
m
ai
n
o
p
er
atio
n
s
c
ar
r
y
o
p
ti
m
izatio
n
p
r
o
ce
s
s
es
ar
e:
m
u
tatio
n
,
cr
o
s
s
o
v
er
an
d
s
elec
tio
n
.
I
n
ea
c
h
g
e
n
er
atio
n
,
i
n
d
i
v
id
u
al
s
o
f
t
h
e
c
u
r
r
en
t
p
o
p
u
lat
io
n
b
ec
o
m
e
tar
g
et
v
ec
to
r
s
.
Fo
r
ea
ch
tar
g
et
v
ec
to
r
,
th
e
m
u
ta
tio
n
o
p
er
atio
n
p
r
o
d
u
ce
s
a
m
u
tan
t
v
ec
to
r
.
T
h
e
cr
o
s
s
o
v
er
o
p
er
atio
n
g
en
er
ate
s
a
n
e
w
v
ec
to
r
,
ca
lled
tr
ial
v
e
cto
r
s
,
b
y
m
i
x
i
n
g
t
h
e
p
ar
a
m
et
er
s
o
f
th
e
m
u
ta
n
t
v
ec
to
r
w
it
h
t
h
o
s
e
o
f
t
h
e
tar
g
et
v
ec
to
r
.
I
f
th
e
tr
ial
v
ec
to
r
o
b
tain
s
a
b
etter
f
it
n
es
s
v
al
u
e
t
h
an
th
e
tar
g
et
v
ec
to
r
,
th
en
t
h
e
tr
ial
v
ec
to
r
r
ep
lace
s
th
e
tar
g
et
v
ec
to
r
in
t
h
e
n
e
x
t g
e
n
er
atio
n
.
3
.
3
.
1
.
In
it
ia
liza
t
io
n:
Def
i
n
e
u
p
p
er
an
d
lo
w
er
b
o
u
n
d
s
f
o
r
ea
ch
p
ar
a
m
eter
o
f
t
h
e
r
ed
u
ce
d
o
r
d
er
m
o
d
el
(
2
1
)
,
X
X
X
U
j
1
,
i
,
j
L
j
(
2
1
)
R
an
d
o
m
l
y
s
elec
t
t
h
e
i
n
itial
p
ar
a
m
eter
v
al
u
es
u
n
i
f
o
r
m
l
y
o
n
th
e
i
n
ter
v
a
ls
],
X
,
X
[
U
j
L
j
an
d
th
e
ele
m
e
n
ts
o
f
ea
c
h
in
d
iv
id
u
al
j
0
,
i
X
ar
e
g
iv
e
n
b
y
(
2
2
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
9
-
4856
I
J
R
A
,
Vo
l.
7
,
No
.
4
,
Dec
em
b
er
2
0
1
8
:
2
3
2
–
2
5
0
238
)
X
X
.(
r
an
d
X
X
j
m
i
n
j
m
a
x
j
m
i
n
j
0
,
i
(
2
2
)
W
h
er
e
i =
1
,
2
,
…,
N
an
d
j
=
1
,
2
,
….
.
,
D.
W
h
er
e
N
is
t
h
e
p
o
p
u
latio
n
s
iz
e,
r
an
d
(
0
,
1
)
is
a
r
an
d
o
m
n
u
m
b
er
u
n
i
f
o
r
m
l
y
d
is
tr
ib
u
ted
b
etw
ee
n
0
an
d
1
,
D
i
s
th
e
n
u
m
b
er
o
f
co
n
tr
o
l v
ar
ia
b
le
s
.
3.
3
.
2
.
M
uta
t
i
o
n:
Mu
tatio
n
ex
p
an
d
s
t
h
e
s
ea
r
ch
s
p
ac
e.
DE
u
n
d
er
g
o
es
m
u
t
atio
n
o
p
er
atio
n
af
ter
in
itializ
atio
n
.
I
n
m
u
tatio
n
o
p
er
atio
n
,
it
p
r
o
d
u
ce
s
m
u
tan
t
v
ec
to
r
V
i
,
G,
w
it
h
r
es
p
ec
tiv
e
to
ea
c
h
in
d
i
v
id
u
a
l
X
i
,
G,
s
o
ca
lled
tar
g
et
v
ec
to
r
,
in
th
e
c
u
r
r
en
t p
o
p
u
lat
i
o
n
v
ia
m
u
ta
tio
n
s
tr
ateg
y
(
2
3
)
:
).
X
X
(
F
)
X
X
(
F
X
V
G
,
2
r
G
,
1
r
G
,
i
G
,
b
e
s
t
G
,
i
G
,
i
(
2
3
)
Fo
r
a
g
i
v
en
p
ar
a
m
eter
v
ec
to
r
G
,
i
X
t
w
o
v
ec
to
r
s
G
,
1
r
X
an
d
G
,
2
r
X
ar
e
s
elec
te
d
r
an
d
o
m
l
y
s
u
ch
th
a
t
th
e
in
d
ice
s
r
1
,
r
2
is
d
is
tin
ct.
T
h
e
m
u
tatio
n
f
ac
to
r
F is
a
co
n
s
t
an
t f
r
o
m
[
0
,
2
]
is
ca
lled
th
e
d
o
n
o
r
v
ec
to
r
.
3
.
3
.
3
.
Cro
s
s
o
v
er
:
C
r
o
s
s
o
v
er
in
co
r
p
o
r
ates
s
u
cc
e
s
s
f
u
l
s
o
lu
tio
n
s
f
r
o
m
th
e
p
r
ev
io
u
s
g
e
n
er
atio
n
.
A
f
ter
m
u
ta
tio
n
,
DE
u
n
d
er
g
o
e
s
cr
o
s
s
o
v
er
.
T
h
e
tr
ial
v
ec
to
r
G
,
i
U
is
d
ev
elo
p
ed
f
r
o
m
t
h
e
e
le
m
en
ts
o
f
t
h
e
tar
g
et
v
ec
to
r
,
,
X
G
,
i
an
d
th
e
ele
m
en
t
s
o
f
t
h
e
d
o
n
o
r
v
ec
t
o
r
,
:
V
G
,
i
(
2
4
)
o
t
h
e
r
w
i
s
e
X
),
j
j
(
or
)
CR
)
1
,
0
(
r
a
n
d
(
if
v
u
j
g
,
i
r
a
n
d
j
j
G
,
i
j
G
,
i
(
2
4
)
E
le
m
e
n
ts
o
f
t
h
e
d
o
n
o
r
v
ec
to
r
en
ter
th
e
tr
ial
v
ec
to
r
w
i
th
p
r
o
b
ab
ilit
y
C
R
(
cr
o
s
s
o
v
er
r
ate)
s
et
to
[
0
; 1
]
.
3.
3
.
4
.
Select
io
n:
T
h
e
n
e
w
l
y
g
e
n
er
ated
v
al
u
es
o
f
tr
ail
v
ec
to
r
s
ex
ce
ed
t
h
e
co
r
r
esp
o
n
d
in
g
u
p
p
er
an
d
lo
w
er
b
o
u
n
d
s
;
w
e
in
itial
ize
th
e
m
r
an
d
o
m
l
y
a
n
d
u
n
i
f
o
r
m
l
y
w
it
h
in
t
h
e
p
r
e
-
s
p
ec
if
ied
r
an
g
e
(
2
5
)
:
o
t
h
e
r
w
i
s
e
X
),
X
(
f
)
U
(
f
if
U
X
j
g
,
i
G
,
i
j
G
,
i
j
g
,
i
j
G
,
i
(
2
5
)
T
h
e
tr
ail
v
ec
to
r
X
i,
G
is
co
m
p
a
r
ed
w
it
h
tr
ail
v
ec
to
r
U
i,
G
a
n
d
th
e
o
n
e
w
it
h
lo
w
e
s
t
f
u
n
ctio
n
v
alu
e
i
s
ad
m
itted
to
t
h
e
n
e
x
t
g
e
n
er
atio
n
.
T
h
er
ef
o
r
e
th
e
f
o
u
r
k
th
o
r
d
er
r
ed
u
ce
d
Kh
ar
ito
n
o
v
‟
s
tr
an
s
f
er
f
u
n
ctio
n
d
en
o
m
i
n
ato
r
s
an
d
t
h
e
n
u
m
er
ato
r
s
ar
e
o
b
tain
ed
b
y
m
i
n
i
m
izin
g
i
n
teg
r
al
s
q
u
ar
e
er
r
o
r
u
s
in
g
Di
f
f
er
e
n
t
ial
E
v
o
lu
tio
n
A
l
g
o
r
ith
m
.
Fin
all
y
t
h
e
r
ed
u
ce
d
o
r
d
er
in
ter
v
al
m
o
d
el
is
o
b
tai
n
ed
b
y
t
h
e
f
o
llo
w
i
n
g
eq
u
a
tio
n
(
2
6
)
:
j
ij
k
0
j
ij
j
)
ij
1
k
0
j
ij
i
k
s
)]
a
m
ax
(
),
a
[
m
i
n
(
s
]
b
m
ax
(
),
b
[
m
i
n
(
)
d
,
c
,
s
(
G
Fo
r
i=1
,
2
,
3
,
4
.
(
2
6
)
Fo
r
th
is
r
ed
u
ce
d
o
r
d
er
m
o
d
el,
a
P
I
/PID
r
o
b
u
s
t
co
n
tr
o
ller
is
d
esig
n
ed
b
ased
o
n
th
e
m
i
n
i
m
i
za
tio
n
o
f
th
e
o
b
j
ec
tiv
e
f
u
n
ctio
n
2
0
I
I
2
0
P
P
)
K
K
(
)
K
K
(
J
f
o
r
P
I
co
n
tr
o
ller
an
d
2
0
D
D
2
0
I
I
2
0
P
P
)
K
K
(
)
K
K
(
)
K
K
(
J
f
o
r
P
I
D
co
n
tr
o
ller
u
s
in
g
th
e
s
a
m
e
DE
alg
o
r
it
h
m
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
R
A
I
SS
N:
2089
-
4856
Desig
n
o
f ro
b
u
s
t c
o
n
tr
o
ller
fo
r
h
ig
h
er o
r
d
er in
terva
l sys
tem
u
s
in
g
d
if
feren
tia
l …
(
D.
S
r
in
iva
s
a
R
a
o
)
239
4.
Desig
n P
ro
ce
du
re
f
o
r
Ro
bu
s
t
s
t
a
bil
izing
P
I
/PID
Co
ntr
o
ll
er
L
et
th
e
s
tab
ilizi
n
g
co
n
tr
o
ller
)
s
(
G
c
is
co
n
s
id
er
ed
to
b
e
a
P
I
/P
I
D
co
n
tr
o
ller
tr
an
s
f
er
f
u
n
ct
io
n
o
f
th
e
f
o
r
m
g
iv
e
n
b
elo
w
(
2
7
-
28)
.
)
s
(
D
)
s
(
N
s
K
K
)
s
(
G
c
c
I
P
c
f
o
r
P
I
co
n
tr
o
ller
(
2
7
)
)
s
(
D
)
s
(
N
s
K
s
K
K
)
s
(
G
c
c
D
I
P
c
f
o
r
P
I
D
co
n
tr
o
ller
(
2
8
)
W
h
er
e
P
K
=
P
r
o
p
o
r
tio
n
al
g
ain
,
I
K
=I
n
teg
r
al
g
ai
n
an
d
D
K
d
er
iv
ativ
e
g
a
i
n
.
No
w
t
h
e
s
y
s
te
m
w
it
h
r
o
b
u
s
t s
t
ab
ilizin
g
co
n
tr
o
ller
f
o
r
p
ar
am
e
tr
ic
u
n
ce
r
tai
n
t
y
is
as
s
h
o
w
n
i
n
Fig
u
r
e
1.
y
+
U
r
e
P
I
D
R
o
b
u
s
t
C
o
n
t
r
o
l
l
e
r
+
+
+
P
l
a
n
t
G
(
s
,
C
,
D
)
Fig
u
r
e
1
.
B
lo
ck
d
iag
r
a
m
o
f
cl
o
s
ed
lo
o
p
s
y
s
te
m
w
it
h
P
I
/P
I
D
co
n
tr
o
lle
r
T
h
en
th
e
clo
s
ed
lo
o
p
tr
an
s
f
er
f
u
n
ct
io
n
w
it
h
a
P
I
/ P
I
D
c
o
n
tr
o
ller
ca
n
b
e
d
ef
in
ed
as
(
2
9
)
)
d
,
s
(
D
)
s
(
D
)
c
,
s
(
N
)
s
(
N
)
c
,
s
(
N
)
s
(
N
)
d
,
c
,
s
(
G
)
s
(
G
1
)
d
,
c
,
s
(
G
)
s
(
G
)
s
(
T
r
c
r
c
r
c
r
c
r
c
(
2
9
)
W
h
er
e
)
c
,
s
(
N
r
an
d
)
d
,
s
(
D
r
ar
e
th
e
n
u
m
er
ato
r
an
d
d
en
o
m
i
n
ato
r
p
o
l
y
n
o
m
ial
s
o
f
th
e
r
ed
u
ce
d
o
r
d
er
in
ter
v
a
l
p
la
n
t,
)
s
(
N
c
an
d
)
s
(
D
c
ar
e
th
e
n
u
m
er
ato
r
an
d
d
en
o
m
i
n
ato
r
p
o
ly
n
o
m
ial
s
o
f
)
s
(
G
c
r
esp
ec
tiv
el
y
.
T
h
i
s
P
I
/PID
co
n
tr
o
ller
r
o
b
u
s
tl
y
s
tab
ilizes
t
h
e
in
ter
v
al
p
la
n
ts
f
a
m
il
y
,
if
f
o
r
all
C
c
an
d
D
d
,
th
en
t
h
e
ch
ar
ac
ter
is
tic
p
o
l
y
n
o
m
ial
o
f
a
clo
s
ed
lo
o
p
tr
an
s
f
er
f
u
n
ct
i
o
n
g
iv
e
n
i
n
eq
u
atio
n
(
2
9
)
h
as
all
ze
r
o
s
h
a
v
e
n
eg
at
iv
e
r
ea
l
v
al
u
es.
No
w
ap
p
l
y
t
h
e
n
ec
e
s
s
ar
y
an
d
s
u
f
f
icie
n
t
co
n
d
itio
n
s
o
f
r
o
b
u
s
t
s
tab
ili
t
y
co
n
d
itio
n
s
g
i
v
e
n
in
T
ab
le.
1
to
th
e
clo
s
ed
-
lo
o
p
ch
ar
ac
ter
is
tic
p
o
l
y
n
o
m
ial
)
d
,
s
(
D
)
s
(
D
)
c
,
s
(
N
)
s
(
N
r
c
r
c
wh
ich
lead
s
to
a
s
et
o
f
co
n
s
tr
ain
ts
i
n
ter
m
s
o
f
co
n
tr
o
ller
p
ar
am
eter
s
.
T
h
en
t
h
ese
co
n
s
tr
ain
t
s
ar
e
s
o
lv
ed
b
y
u
s
in
g
DE
s
o
as
to
m
i
n
i
m
ize
th
e
o
b
j
ec
tiv
e
f
u
n
ctio
n
2
0
I
I
2
0
P
P
)
K
K
(
)
K
K
(
J
f
o
r
P
I
co
n
tr
o
ller
an
d
2
0
D
D
2
0
I
I
2
0
P
P
)
K
K
(
)
K
K
(
)
K
K
(
J
f
o
r
P
I
D
co
n
tr
o
ller
to
o
b
tain
co
n
tr
o
ller
p
ar
a
m
eter
s
.
T
h
en
a
f
ter
o
b
tain
in
g
th
e
co
n
tr
o
ller
p
ar
a
m
eter
s
,
f
o
r
m
f
o
u
r
s
ets
o
f
K
h
a
r
ito
n
o
v
‟
s
p
o
l
y
n
o
m
ial
s
[
1
3
]
to
ch
ec
k
t
h
e
s
tab
ilit
y
an
d
th
e
clo
s
ed
-
lo
o
p
s
tep
r
esp
o
n
s
e
to
v
er
i
f
y
th
e
r
es
u
lt
s
.
A
p
p
lica
tio
n
o
f
th
e
DE
al
g
o
r
ith
m
f
o
r
d
eter
m
i
n
i
n
g
t
h
e
co
n
tr
o
ll
er
p
ar
am
eter
s
i
s
as f
o
llo
w
s
:
Step
1
: I
n
itializatio
n
:
Def
i
n
e
u
p
p
er
an
d
lo
w
er
b
o
u
n
d
s
f
o
r
ea
ch
co
n
tr
o
ller
p
ar
a
m
eter
o
f
th
e
P
I
/P
I
D
co
n
tr
o
ller
(
3
0
)
,
X
X
X
U
j
1
,
i
,
j
L
j
(
3
0
)
T
h
e
in
itial p
ar
a
m
ete
r
v
al
u
e
s
ar
e
r
an
d
o
m
l
y
s
e
lecte
d
u
n
i
f
o
r
m
l
y
w
it
h
i
n
th
e
al
lo
w
ab
le
r
a
n
g
e
(
3
1
)
],
X
,
X
[
U
j
L
j
)
X
X
.(
r
an
d
X
X
j
m
i
n
j
m
a
x
j
m
i
n
j
0
,
i
(
3
1
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
9
-
4856
I
J
R
A
,
Vo
l.
7
,
No
.
4
,
Dec
em
b
er
2
0
1
8
:
2
3
2
–
2
5
0
240
W
h
er
e
I
=
1
,
2
,
…,
N
an
d
j
=
1
,
2
,
DN
is
t
h
e
p
o
p
u
lat
io
n
s
ize,
r
an
d
(
0
;
1
)
is
a
r
an
d
o
m
n
u
m
b
er
u
n
i
f
o
r
m
l
y
d
i
s
tr
ib
u
ted
b
et
w
ee
n
0
an
d
1
,
D
is
t
h
e
n
u
m
b
er
o
f
co
n
tr
o
ller
p
ar
a
m
eter
s
.
T
h
e
m
in
i
m
u
m
an
d
m
ax
i
m
u
m
v
a
lu
e
s
o
f
t
h
e
co
n
tr
o
l v
ar
iab
les f
o
r
P
I
/P
I
D
C
o
n
tr
o
ller
i.e
.
P
K
,
I
K
an
d
D
K
ar
e
s
elec
ted
as
10
K
0
P
an
d
5
K
0
I
f
o
r
P
I
co
n
tr
o
ller
.
,
10
K
0
P
5
K
0
I
an
d
5
K
0
D
f
o
r
P
I
D
co
n
tr
o
ller
.
Step
2
: M
u
tatio
n
DE
u
n
d
er
g
o
es
m
u
tatio
n
o
p
er
atio
n
a
f
ter
i
n
it
ializatio
n
.
I
n
m
u
tatio
n
o
p
er
atio
n
,
it
p
r
o
d
u
ce
s
m
u
tan
t
v
ec
to
r
V
i,
G
,
w
it
h
r
e
s
p
ec
tiv
e
t
o
ea
ch
i
n
d
i
v
i
d
u
al
X
i,
G
,
s
o
ca
l
led
tar
g
et
v
ec
to
r
,
i
n
t
h
e
c
u
r
r
e
n
t
p
o
p
u
latio
n
v
ia
m
u
tatio
n
s
tr
ateg
y
(
3
2
)
:
).
X
X
(
F
)
X
X
(
F
X
V
G
,
2
r
G
,
1
r
G
,
i
G
,
b
e
s
t
G
,
i
G
,
i
(
3
2
)
Fo
r
a
g
iv
e
n
p
ar
a
m
e
ter
v
ec
to
r
G
,
i
X
t
w
o
v
ec
to
r
s
G
,
1
r
X
an
d
G
,
2
r
X
ar
e
s
elec
ted
r
an
d
o
m
l
y
s
u
ch
th
at
th
e
i
n
d
ices
r
1
,
r
2
is
d
is
tin
ct.
T
h
e
m
u
tatio
n
f
ac
to
r
F=1
.
2
is
ch
o
s
en
f
r
o
m
[
0
; 2
]
is
ca
lled
th
e
d
o
n
o
r
v
ec
to
r
.
Step
3
: Cro
s
s
o
v
er
Af
ter
m
u
tatio
n
,
DE
u
n
d
er
g
o
es
cr
o
s
s
o
v
er
.
T
h
e
tr
ial
v
ec
to
r
G
,
i
U
is
d
ev
elo
p
ed
f
r
o
m
t
h
e
ele
m
e
n
ts
o
f
th
e
t
ar
g
et
v
ec
to
r
,
,
X
G
,
i
an
d
th
e
ele
m
en
ts
o
f
th
e
d
o
n
o
r
v
ec
to
r
(
3
3
)
,
:
V
G
,
i
o
t
h
e
r
w
i
s
e
X
),
j
j
(
or
)
CR
)
1
,
0
(
r
a
n
d
(
if
v
u
j
g
,
i
r
a
n
d
j
j
G
,
i
j
G
,
i
(
3
3
)
E
le
m
e
n
ts
o
f
t
h
e
d
o
n
o
r
v
ec
to
r
en
ter
th
e
tr
ial
v
ec
to
r
w
i
th
p
r
o
b
ab
ilit
y
C
R
=
0
.
6
s
et
f
r
o
m
[
0
; 1
]
.
Step
4
: Sele
ctio
n
T
h
e
n
e
w
l
y
g
e
n
er
ated
v
al
u
es
o
f
tr
ail
v
ec
to
r
s
e
x
ce
ed
th
e
co
r
r
esp
o
n
d
in
g
u
p
p
er
an
d
lo
w
er
b
o
u
n
d
s
;
we
in
itial
ize
th
e
m
r
an
d
o
m
l
y
a
n
d
u
n
i
f
o
r
m
l
y
w
it
h
in
t
h
e
p
r
e
-
s
p
ec
if
ied
r
an
g
e
(
3
4
)
:
o
t
h
e
r
w
i
s
e
X
),
X
(
f
)
U
(
f
if
U
X
j
g
,
i
G
,
i
j
G
,
i
j
g
,
i
j
G
,
i
(
3
4
)
T
h
e
tr
ail
v
ec
to
r
X
i
,
g
is
co
m
p
a
r
ed
w
it
h
tr
ail
v
ec
to
r
U
i,
G
an
d
th
e
o
n
e
w
ith
lo
w
es
t
f
u
n
ctio
n
v
alu
e
i
s
ad
m
itted
to
th
e
n
e
x
t
g
e
n
er
atio
n
.
Fin
al
l
y
t
h
e
r
esu
l
ts
o
f
t
h
e
b
est
p
ar
am
e
ter
s
o
f
a
P
I
/P
I
D
co
n
tr
o
ller
an
d
f
itn
es
s
ar
e
o
b
tain
ed
5.
NUM
E
RICAL
E
XAM
P
L
E
C
o
n
s
id
er
a
w
i
n
g
air
cr
af
t [
2
3
]
w
h
o
s
e
tr
an
s
f
er
f
u
n
ct
io
n
w
it
h
p
ar
a
m
etr
ic
u
n
ce
r
ta
in
t
y
i
s
g
i
v
e
n
(
3
5
)
]
1
.
0
,
1
.
0
[
s
]
9
.
33
,
1
.
30
[
s
]
8
.
80
,
4
.
50
[
s
]
6
.
4
,
8
.
2
[
s
]
1
6
6
,
90
[
s
]
74
,
54
[
)
D
,
C
,
s
(
G
2
3
4
(
3
5
)
T
h
e
clo
s
ed
lo
o
p
tr
an
s
f
er
f
u
n
cti
o
n
o
f
th
e
ab
o
v
e
i
n
ter
v
al
s
y
s
te
m
is
g
i
v
en
b
y
(
3
6
)
]
1
.
166
9
.
89
[
s
]
9
.
107
1
.
84
[
s
]
8
.
80
4
.
50
[
s
]
6
.
4
8
.
2
[
s
]
1
1
[
]
166
90
[
s
]
74
54
[
)
s
(
T
2
3
4
(
3
6
)
T
h
is
h
i
g
h
er
o
r
d
er
clo
s
ed
lo
o
p
tr
an
s
f
er
f
u
n
ctio
n
o
f
in
ter
v
al
s
y
s
te
m
ca
n
b
e
r
ep
r
esen
ted
as
f
o
u
r
f
i
x
ed
p
ar
am
eter
K
h
ar
ito
n
o
v
tr
an
s
f
er
f
u
n
ctio
n
s
t
h
at
ar
e
g
i
v
en
a
s
(
3
7
-
40)
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
R
A
I
SS
N:
2089
-
4856
Desig
n
o
f ro
b
u
s
t c
o
n
tr
o
ller
fo
r
h
ig
h
er o
r
d
er in
terva
l sys
tem
u
s
in
g
d
if
feren
tia
l …
(
D.
S
r
in
iva
s
a
R
a
o
)
241
9
.
89
s
1
.
84
s
8
.
80
s
6
.
4
s
90
s
54
)
s
(
T
2
3
4
1
(
3
7
)
9
.
89
s
9
.
107
s
8
.
80
s
8
.
2
s
90
s
74
)
s
(
T
2
3
4
2
(
3
8
)
1
.
1
6
6
s
1
.
84
s
4
.
50
s
6
.
4
s
1
6
6
s
54
)
s
(
T
2
3
4
3
(
3
9
)
1
.
166
s
9
.
107
s
4
.
50
s
8
.
2
s
166
s
74
)
s
(
T
2
3
4
4
(
40)
I
t
is
d
if
f
ic
u
lt
to
an
al
y
ze
th
e
h
ig
h
er
o
r
d
er
s
y
s
te
m
,
b
ec
au
s
e
t
h
e
co
s
t
a
n
d
co
m
p
lex
i
t
y
o
f
t
h
e
h
ig
h
er
o
r
d
er
s
y
s
te
m
in
cr
ea
s
e
w
it
h
i
n
cr
ea
s
e
in
o
r
d
er
o
f
s
y
s
te
m
.
T
h
e
m
o
d
el
o
r
d
er
r
ed
u
ctio
n
is
r
eq
u
ir
ed
f
o
r
m
i
n
i
m
izi
n
g
t
h
e
co
s
t
a
n
d
co
m
p
lex
i
t
y
o
f
th
e
h
i
g
h
er
o
r
d
er
s
y
s
te
m
.
T
h
e
n
u
m
er
ato
r
an
d
d
en
o
m
i
n
ato
r
co
ef
f
icie
n
t
s
o
f
t
h
e
r
ed
u
ce
d
o
r
d
er
m
o
d
el
ar
e
o
b
tai
n
ed
b
y
m
i
n
i
m
izi
n
g
i
n
teg
r
al
s
q
u
ar
e
er
r
o
r
u
s
i
n
g
t
h
e
p
r
o
ce
d
u
r
e
g
iv
en
i
n
s
ec
tio
n
3
an
d
ar
e
av
ailab
le
f
r
o
m
t
h
e
f
o
llo
w
i
n
g
T
ab
les 2
an
d
3
.
T
h
e
f
o
u
r
r
ed
u
ce
d
o
r
d
e
r
Kh
ar
ito
n
o
v
‟
s
tr
an
s
f
er
f
u
n
ctio
n
s
ar
e
(
4
1
-
44)
:
7
8
7
7
.
89
s
7
1
3
1
.
81
s
4
3
6
4
.
79
1
5
1
1
.
90
s
4
0
1
4
.
53
T
2
1
k
(
4
1
)
5
8
2
9
.
89
s
9
3
7
8
.
1
0
4
s
2
5
6
9
.
77
3
0
3
2
.
90
s
2
9
5
2
.
73
T
2
2
k
(
4
2
)
7
6
5
.
1
6
9
s
80
s
50
9
4
9
4
.
1
6
9
s
9
9
6
5
.
59
T
2
3
k
(
4
3
)
9
9
9
3
.
169
s
120
s
50
2
6
7
8
.
169
s
9
9
9
4
.
84
T
2
4
k
(
4
4
)
T
h
er
ef
o
r
e
th
e
s
tep
r
esp
o
n
s
es
o
f
th
e
o
r
ig
i
n
al
an
d
r
ed
u
ce
d
o
r
d
er
Kh
ar
ito
n
o
v
‟
s
tr
an
s
f
er
f
u
n
c
tio
n
s
ar
e
s
h
o
w
n
in
F
ig
u
r
e
2
,
3
,
4
,
an
d
5
r
esp
ec
tiv
el
y
.
T
ab
le
2
.
T
y
p
ical
p
ar
am
e
ter
u
s
ed
b
y
Di
f
f
er
en
tial E
v
o
lu
t
io
n
f
o
r
f
o
u
r
Kh
ar
ito
n
o
v
‟
s
tr
an
s
f
er
f
u
n
ct
io
n
s
.
N
a
me
F
i
r
st
K
h
a
r
i
t
o
n
o
v
S
e
c
o
n
d
K
h
a
r
i
t
o
n
o
v
T
h
i
r
d
K
h
a
r
i
t
o
n
o
v
F
o
u
r
t
h
K
h
a
r
i
t
o
n
o
v
V
a
l
u
e
V
a
l
u
e
V
a
l
u
e
V
a
l
u
e
M
i
n
i
m
u
m
M
a
x
i
m
u
m
M
i
n
i
m
u
m
M
a
x
i
m
u
m
M
i
n
i
m
u
m
M
a
x
i
m
u
m
M
i
n
i
m
u
m
M
a
x
i
m
u
m
P
o
p
u
l
a
t
i
o
n
s
i
z
e
50
20
50
20
CR
0
.
8
0
.
8
0
.
8
0
.
8
F
0
.
5
0
.
5
0
.
5
0
.
5
N
u
me
r
a
t
o
r
p
a
r
a
me
t
e
r
1
50
60
70
80
50
60
80
90
N
u
me
r
a
t
o
r
p
a
r
a
me
t
e
r
2
85
95
90
1
0
0
1
6
0
1
7
0
1
7
0
1
8
0
D
e
n
o
mi
n
a
t
o
r
p
a
r
a
me
t
e
r
1
70
80
70
80
50
60
50
60
D
e
n
o
mi
n
a
t
o
r
p
a
r
a
me
t
e
r
2
80
90
1
0
0
1
1
0
80
90
1
1
0
1
2
0
D
e
n
o
mi
n
a
t
o
r
p
a
r
a
me
t
e
r
3
85
95
80
90
80
90
1
1
0
1
2
0
M
a
x
i
m
u
m
G
e
n
e
r
a
t
i
o
n
s
1
0
0
1
0
0
1
0
0
1
0
0
T
ab
le
3
.
C
o
m
p
ar
is
o
n
o
f
I
SE
f
o
r
f
o
u
r
Kh
ar
ito
n
o
v
p
o
l
y
n
o
m
ia
ls
.
N
a
me
o
f
p
o
l
y
n
o
mi
a
l
I
S
E
M
a
x
i
m
u
m
M
i
n
i
m
u
m
F
i
r
st
K
h
a
r
i
t
o
n
o
v
6
.
6
7
7
*
1
0
-
4
2
.
4
*
1
0
-
3
S
e
c
o
n
d
K
h
a
r
i
t
o
n
o
v
0
.
0
0
3
1
0
.
0
0
4
8
T
h
i
r
d
K
h
a
r
i
t
o
n
o
v
0
.
0
0
7
5
0
.
0
0
8
9
F
o
u
r
t
h
K
h
a
r
i
t
o
n
o
v
0
.
0
3
4
9
0
.
0
4
2
Evaluation Warning : The document was created with Spire.PDF for Python.