Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
8,
No.
4,
August
2018,
pp.
2180
–
2198
ISSN:
2088-8708
2180
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
Contr
olling
a
DC
Motor
thr
ough
L
ypauno
v-lik
e
Functions
and
SAB
T
echnique
Alejandr
o.
Rinc
´
on
1
,
F
abiola.
Angulo
2
,
and
Fr
edy
.
Hoy
os
3
1
F
acultad
de
Ingenier
´
ıa
y
Arquitectura,
Uni
v
ersidad
Cat
´
olica
de
Manizales,
Grupo
de
In
v
estig
aci
´
on
en
Desarrollos
T
ecnol
´
ogicos
y
Ambientales-GIDT
A,
Email:
arincons@ucm.edu.co
2
F
acultad
de
Ingeniera
y
Arquitectura
-
Departamento
de
Ingenier
´
ıa
El
´
ectrica,
Electr
´
onica
y
Computaci
´
on
-
Percepci
´
on
y
Control
Inteligente
-
Bloque
Q,
Campus
La
Nubia,
Manizales,
170003
-
Colombia,Email:
f
angulog@unal.edu.co
3
F
acultad
de
Ciencias,
Escuela
de
F
´
ısica,
Grupo
de
In
v
estig
aci
´
on
en
T
ecnolog
´
ıas
Aplicadas
-
GIT
A,
Uni
v
ersidad
Nacional
de
Colombia,
Sede
Medell
´
ın,
Email:
feho
yosv
e@unal.edu.co
Article
Inf
o
Article
history:
Recei
v
ed:
October
10,
2017
Re
vised:
March
14,
2018
Accepted:
May
2,
2018
K
eyw
ord:
Rob
ust
Adapti
v
e
Control
L
yapuno
v-lik
e
Function
Permanent
Magnet
DC
Motor
Rapid
control
prototyping
ABSTRA
CT
In
this
paper
,
state
adapti
v
e
backstepping
and
L
yapuno
v-lik
e
function
me
thods
are
used
to
design
a
rob
ust
adapti
v
e
controller
for
a
DC
motor
.
The
output
to
be
controlled
is
the
motor
speed.
It
i
s
assumed
that
the
load
torque
and
inertia
moment
e
xhibit
unkno
wn
b
ut
bounded
time-v
arying
beha
vior
,
and
that
the
measurement
of
the
motor
speed
and
motor
current
are
corrupted
by
noise.
The
controller
is
implemented
in
a
Rapid
Control
Proto-
typing
system
based
on
Digital
Signal
Processing
for
dSP
A
CE
platform
and
e
xperimental
results
agree
with
theory
.
Copyright
c
2018
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Alejandro
Rinc
´
on
Santamar
´
ıa
Uni
v
ersidad
Cat
´
olica
de
Manizales
Cra.
23
No.
60-63,
Manizales,
Colombia
T
elephone:
(57-6)
8933050
Email:
arincons@ucm.edu.co
1.
INTR
ODUCTION
DC
motors
are
commonly
used
in
industry
applications
[1],
[2].
A
significant
dif
ficulty
for
DC-motor
control
design
is
the
unkno
wn
time-v
arying
nature
of
its
parameters
[3],
[4].
An
important
frame
w
ork
for
motor
control
design
is
the
state
adapti
v
e
backstepping
(SAB)
technique
presented
in
[5],
as
can
be
noticed
from
[6],
[7].
Nussbaum
g
ain
techniques
are
usually
incorporated
in
the
SAB
control
frame
w
ork
in
order
to
handle
the
ef
fect
of
unkno
wn
time-v
arying
parameters
and
to
impro
v
e
the
rob
ustness
of
the
system.
Rob
ust
SAB
control
schemes
incorporate
a
compensation
term
in
the
control
la
w
,
and
some
modification
in
the
update
la
w
,
for
instance
the
modification,
see
[8].
Ne
v
ertheless,
upper
or
lo
wer
bounds
of
the
plant
model
coef
ficients
ha
v
e
to
be
kno
wn
to
guarantee
asymptotic
con
v
er
gence
of
the
tracking
error
to
a
residual
set
of
user
-defined
size.
In
[9]
a
non-adapti
v
e
state
backstepping
control
scheme
is
de
v
eloped.
The
resulting
time
deri
v
ati
v
e
of
the
L
yapuno
v
function
in
v
olv
es
an
unkno
wn,
time
v
arying
b
ut
bounded
term,
whose
upper
bound
is
unkno
wn,
such
that
the
backste
pp
i
ng
states
remain
bounded
and
con
v
er
ge
to
residual
set
whose
size
depends
on
plant
parameters.
Nussbaum
SAB
control
schemes
are
based
on
the
controllers
presented
in
[10],
[11],
as
can
be
noticed
from
[12],
[13],
[14].
In
turn,
the
controllers
in
[10],
[11]
are
based
on
the
Uni
v
ersal
Stabilizer
that
w
as
originally
introduced
in
[15]
and
discussed
in
[16].
The
main
dra
wback
of
the
Nussbaum
g
ain
technique
is
that
the
result-
ing
upper
bound
of
the
transient
beha
vior
of
the
tracking
error
depends
on
inte
gral
terms
that
in
v
olv
e
Nussbaum
functions
[12],
[17],
[14],
[10].
Some
controllers
that
use
this
technique
usually
present
som
e
of
the
follo
wing
dra
wbacks:
i)
some
upper
or
lo
wer
bounds
of
plant
model
parameters
ha
v
e
to
be
kno
wn
in
order
to
guarantee
the
con
v
er
gence
of
the
tracking
error
to
a
residual
set
of
user
-defined
size
[14],
[10],
b
ut
the
control
designs
in
[12],
[13]
indicate
that
a
proper
design
w
ould
relax
this
dra
wback,
and
ii)
the
control
or
update
la
ws
in
v
olv
e
signum
type
J
ournal
Homepage:
http://iaescor
e
.com/journals/inde
x.php/IJECE
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
,
DOI:
10.11591/ijece.v8i4.pp2180-2198
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
2181
signals,
as
can
be
noticed
from
[17].
In
[18],
a
linear
induction
motor
is
considered.
The
friction
force
and
unkno
wn
time-v
arying
model
parameters
lead
to
a
lumped
bounded
uncertain
term
whose
upper
bound
is
unkno
wn.
The
goal
is
to
control
the
mo
v
er
position.
The
dra
wback
is
that
the
identification
error
is
assumed
constant
in
the
definition
of
the
L
yapuno
v
function.
In
[6],
a
synchronous
motor
dri
v
en
through
A
C/DC
rectifiers
and
DC/A
C
in
v
erters
is
considered.
It
is
assumed
that
the
motor
parameters
e
xperience
unkno
wn
time-v
arying
b
ut
bounded
beha
vior
.
The
goal
is
to
control
the
motor
speed,
the
rectifier
output
v
ol
tage
and
the
d
component
of
the
stator
current;
the
tracking
error
con
v
er
ges
to
a
residual
set
whose
size
depends
on
unkno
wn
motor
parameters
and
user
-defined
controller
parameters.
There-
fore,
if
upper
bounds
of
unkno
wn
motor
parameters
are
kno
wn
and
controller
parameters
are
properly
chosen,
the
size
of
the
residual
set
can
be
user
-defined
also.
Other
w
orks
address
the
problem
of
designing
a
controller
for
dif
ferent
motors
[19,
20,
21];
despite
the
f
act
that
the
controlled
systems
operate
as
it
is
e
xpected,
the
main
disadv
antages
of
these
w
orks
are:
the
size
of
the
output
error
cannot
be
determined,
and
no
analyses
of
the
system
beha
vior
inlcuding
measurement
noise
are
presented.
In
[22]
tw
o
coupled
controllers
are
designed:
a
Linear
Quadratic
Gaussian
(LQG)
and
a
MRAS-based
Learning
Feed-F
orw
ard
Controller
(LFFC);
e
v
enthough
the
simulation
results
demonstrate
the
potential
benefits
of
the
proposed
controlled,
the
LQG
algorithm
may
f
ail
to
ensure
closed-loop
stabil
ity
when
v
ariations
in
the
uncer
-
tainties
are
lar
ge
enough.
In
[23]
a
plant
model
in
cont
rollable
form
with
unkno
wn
v
arying
b
ut
bounded
parameters
is
considered,
being
the
upper
bounds
of
such
parameters
unkno
wn.
A
SAB
control
scheme
is
de
v
eloped,
and
o
v
er
-
comes
the
main
dra
wback
of
the
Nussbaum
g
ain
method,
as
a
result,
the
transient
error
is
upper
bounded
by
an
unkno
wn
constant
that
does
not
depend
on
inte
gral
terms.
Ne
v
ertheless,
the
control
scheme
is
only
v
alid
for
plant
models
in
“companion
form”.
The
method
is
based
on
the
L
yapuno
v-lik
e
function
technique
appearing
in
[24],
[25].
In
addition
to
the
undesired
unkno
wn
time-v
arying
nature
of
plant
parameters,
other
important
issue
is
the
measurement
noise.
T
racking
performance
can
be
de
graded,
e
v
en
if
the
controller
is
rob
ust
ag
ainst
modeling
uncertainty
and
disturbances
(cf.
[26],
[27]).
Some
of
the
main
techniques
to
tackle
the
ef
fect
of
measurement
noise
are:
high
g
ain
observ
ers,
interv
al
observ
ers,
filter
theory
and
the
technique
de
v
eloped
in
[9].
High
g
ain
observ
ers
are
useful
to
estimate
system
states
and
output
deri
v
ati
v
es
(see
[28],
[29]).
In
[28],
a
nonlinear
plant
model
in
state-space
form
and
a
plant
model
in
controllable
form,
are
considered,
respecti
v
ely
.
Both
plant
models
in
v
olv
e
kno
wn
constant
coef
ficients.
The
real
output
is
defined
as
the
first
state,
and
is
measured,
whereas
the
other
states
are
not.
The
output
measurement
is
e
xpressed
as
the
sum
of
the
real
output
plus
a
bounded
measurement
noise
parameter
.
The
observ
er
depends
on
the
dif
ference
between
the
noisy
output
measurement
and
the
output
estimate.
The
stability
analysis
indicates
that
the
state
estimation
error
con
v
er
ges
to
a
residual
set
whose
upper
bound
depends
on
the
magnitude
of
both
the
measurement
noise
and
the
observ
er
parameters.
Such
upper
bound
has
a
global
minimum
for
some
v
alue
of
the
observ
er
g
ain
(see
[28]
and
[29]).
The
main
dra
wbacks
of
the
design
are:
the
size
of
the
residual
set
is
unkno
wn,
so
that
the
upper
limit
of
the
steady
st
ate
of
the
state
estimation
error
is
unkno
wn,
and
second,
the
coef
ficients
of
the
plant
model
are
required
to
be
kno
wn.
Interv
al
observ
ers
pro
vide
an
upper
and
a
lo
wer
bound
for
each
unme
asured
state
v
ariable
(s
ee
[30],
[31]).
The
main
disadv
antage
of
the
interv
al
observ
ers
is
that
se
v
eral
upper
and
lo
wer
bounds
of
the
plant
model
parameters
are
required
to
be
kno
wn.
In
[9],
it
is
assumed
that
the
measurements
of
the
plant
states
are
corrupted
by
noise
and
are
described
by
a
measurement
model,
which
consists
of
a
polynomial
with
respect
to
the
real
state
v
ector
,
with
de
gree
one
and
unkno
wn
b
ut
bounded
time-v
arying
coef
ficients,
being
the
time
deri
v
ati
v
es
of
such
coef
ficients
unkno
wn
and
bounded.
If
each
measurement
model
is
dif
ferentiated
with
respect
to
time,
the
time
deri
v
ati
v
e
of
the
state
mea-
surement
is
a
linear
polynomial
of
de
gree
one
with
respect
to
the
time
deri
v
ati
v
e
of
the
real
state
v
ector
.
The
states
resulting
from
the
backstepping
state
transform
ation
are
defined
in
terms
of
the
noisy
measurements
ins
tead
of
the
real
st
ates.
T
o
compute
the
time
deri
v
ati
v
e
of
each
quadratic
function
of
the
backstepping
procedure,
each
measurement
model
is
dif
ferentiated.
In
[32],
the
state
adapti
v
e
backstepping
(SAB)
of
[5]
is
used
as
the
basic
frame
w
ork
for
controlling
a
DC
permanent
magnet
motor
whose
v
oltage
is
supplied
by
a
b
uck
po
wer
con
v
erter
,
b
ut
the
ef
fect
of
noisy
measurement
is
not
tak
en
into
account.
In
the
present
paper
,
significant
modifications
are
incorporated
to
the
control
scheme
presented
i
n
[32],
in
order
to
handle
the
ef
fect
of
measurement
noise:
i)
a
measurement
model
is
used
to
define
the
relationship
between
the
motor
current
(
i
a
)
and
the
motor
speed
(
W
m
),
and
their
corresponding
noisy
measurements
(
i
a
j
m
and
W
m
j
m
,
ii)
the
states
of
the
backstepping
state
transformation
are
defined
in
terms
of
the
noisy
measurements,
and
Contr
olling
a
DC
Motor
thr
ough
L
ypauno
v-lik
e
Functions
and
SAB
...
(Alejandr
o
Rinc
´
on)
Evaluation Warning : The document was created with Spire.PDF for Python.
2182
ISSN:
2088-8708
iii)
the
dif
ferentiation
of
each
quadratic-lik
e
function
with
respect
to
time
in
v
olv
es
the
dif
ferentiation
of
each
mea-
surement
model.
The
controller
is
implemented
in
a
digital
platform
to
carry
out
the
real
e
xperiments.
The
main
contrib
ution
of
this
paper
respect
to
close
ones
are:
a)
the
relationship
between
the
real
and
the
measured
state
is
tak
en
into
account
in
the
control
design
procedure:
in
the
definition
and
dif
ferentiation
of
the
backstepping
states
as
well
as
in
the
definition
and
dif
ferentiation
of
the
quadratic
functions,
b)
upper
or
lo
wer
bounds
of
the
noise
model
parameters
or
their
combination
are
not
required
to
be
kno
wn
by
the
controller
,
and
c)
the
con
v
er
gence
of
the
tracking
error
to
a
residual
set
of
user
-defined
size
is
pro
v
en
in
presence
of
noisy
measurments.
The
rest
of
the
paper
is
or
g
anized
as
follo
ws.
In
section
2.,
the
plant
model
and
the
control
goal
are
described.
In
section
3.
the
controller
is
designed.
In
section
4.,
the
bounded
nature
of
the
closed
loop
signals
and
the
con
v
er
gence
of
the
tracking
error
are
pro
v
en.
In
section
5.,
numerical
and
e
xperimental
results
are
presented.
Finally
,
discussion
and
conclusions
are
presented
in
Section
6..
2.
PLANT
MODEL
AND
CONTR
OL
GO
AL
The
DC
motor
is
represented
by
the
follo
wing
plant
model
(see
[33]):
_
W
m
=
B
J
eq
W
m
+
k
t
J
eq
i
a
(
T
f
r
ic
+
T
L
)
J
eq
_
i
a
=
R
a
L
a
i
a
k
e
L
a
W
m
+
1
L
a
u
(1)
The
state
v
ariables
are:
the
armature
current
i
a
and
the
motor
speed
W
m
.
u
is
the
control
input
and
it
corresponds
to
a
v
oltage
v
alue.
The
system
output
is
y
=
W
m
.
The
model
parameters
are:
the
v
oltage
constant
k
e
[V/rad/s],
the
armature
inductance
L
a
[H],
the
armature
resistance
R
a
[
],
the
viscous
friction
coef
ficient
B
[N.m/rad/s],
the
inertia
moment
J
eq
[kg.m
2
],
the
motor
torque
constant
k
t
[N.m/A],
the
friction
torque
T
f
r
ic
[N.m],
and
the
load
torque
T
L
[N.m].
The
follo
wing
assumptions
are
made
for
the
model
(1):
Ai)
the
parameters
T
L
and
J
eq
are
time-v
arying,
unkno
wn
and
upper
bounded
by
unkn
o
wn
constants,
and
J
eq
is
positi
v
e
and
lo
wer
bounded
by
an
unkno
wn
positi
v
e
constant,
Aii)
parameters
B
,
k
t
,
R
a
,
L
a
,
k
e
are
unkno
wn,
positi
v
e
and
constant,
and
Aiii)
W
m
and
i
a
are
measured,
b
ut
their
measurements
W
m
j
m
,
i
a
j
m
are
noisy
and
satisfy:
W
m
j
m
=
a
6
W
m
+
a
7
and
i
a
j
m
=
a
8
i
a
+
a
9
(2)
where
the
parameters
a
6
,
a
7
,
a
8
,
and
a
9
are
unkno
wn,
time-v
arying
and
upper
bounded
by
unkno
wn
positi
v
e
constants,
their
time
deri
v
ati
v
es
are
unkno
wn,
time-v
arying
and
bounded
by
unkno
wn
positi
v
e
constants
and
the
parameters
a
6
and
a
8
are
positi
v
e
and
lo
wer
bounded
by
unkno
wn
positi
v
e
constants.
The
abo
v
e
e
xpressions
are
based
on
[9].
F
or
a
simpler
control
design,
the
plant
model
(1)
is
re
written
as:
_
x
1
=
a
1
x
1
+
a
2
x
2
a
3
(3)
_
x
2
=
a
4
x
1
a
5
x
2
+
bu
(4)
x
1
=
W
m
;
x
2
=
i
a
;
u
=
v
c
;
y
=
W
m
(5)
y
m
=
W
m
j
m
=
a
6
x
1
+
a
7
(6)
x
2
m
=
i
a
j
m
=
a
8
x
2
+
a
9
(7)
where
a
1
=
B
J
eq
;
a
2
=
k
t
J
eq
(8)
a
3
=
(
T
f
r
ic
+
T
L
)
J
eq
;
a
4
=
k
e
L
a
;
a
5
=
R
a
L
a
;
b
=
1
L
a
(9)
so
that
a
1
,
a
2
,
a
3
,
a
4
,
a
5
,
and
b
are
positi
v
e,
a
1
,
a
2
,
and
a
3
are
time-v
arying,
a
4
,
a
5
and
b
are
constant,
and
y
m
and
x
2
m
are
the
noisy
measurements
of
W
m
and
i
a
,
respecti
v
ely
.
Assumption
Ai
implies
that
the
parameters
a
1
,
a
2
,
a
3
are
unkno
wn
and
time
-v
arying,
b
ut
the
y
are
upper
bounded
by
unkno
wn
constants.
Assumption
Aii
implies
that
a
2
>
0
,
and
the
parameters
a
4
,
a
5
,
and
b
are
unkno
wn
and
constant.
Assumption
Aiii
impli
es
that
the
states
x
1
=
y
and
x
2
are
unkno
wn
b
ut
their
measurements
y
m
and
x
2
m
are
kno
wn,
the
parameters
a
6
,
a
7
,
a
8
,
a
9
,
_
a
6
,
_
a
7
,
_
a
8
,
and
_
a
9
are
unkno
wn
and
time-v
arying
b
ut
bounded,
and
the
parameters
a
6
and
a
8
are
positi
v
e.
In
summary
,
the
system
(1)
satisfies
the
follo
wing
properties:
Pi)
a
1
,
a
2
,
a
3
,
a
4
,
a
5
,
b
,
a
6
,
and
a
8
are
positi
v
e,
Pii)
the
parameters
IJECE
V
ol.
8,
No.
4,
August
2018:
2180
–
2198
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
2183
a
1
,
a
2
,
a
3
,
a
6
,
a
7
,
a
8
,
a
9
,
_
a
6
,
_
a
7
,
_
a
8
,
and
_
a
9
are
unkno
wn,
time-v
arying
and
upper
bounded
by
unkno
wn
constants,
Piii)
the
parameters
a
2
,
a
6
,
a
8
,
and
1
=a
8
are
positi
v
e
and
lo
wer
bounded
by
unkno
wn
positi
v
e
constants,
Pi
v)
the
parameters
a
4
,
a
5
,
and
b
are
unkno
wn,
positi
v
e
and
constant,
and
Pv)
the
v
alues
of
y
=
x
1
=
W
m
and
x
2
=
i
a
are
unkno
wn,
whereas
their
measurements
y
m
and
x
2
m
are
kno
wn.
T
aking
into
account
the
model,
the
control
goal
ca
n
be
defined
as
follo
ws.
Let
the
follo
wing
reference
model:
•
y
d
=
a
m
1
_
y
d
a
mo
y
d
+
a
mo
W
mr
ef
(10)
where
W
mr
ef
is
the
user
–defined
reference
v
alue,
and
a
m
1
and
a
mo
are
user
-defined
positi
v
e
constants.
Hence,
the
desired
output
y
d
is
pro
vided
by
(10)
subject
to
i)
a
mo
and
a
m
1
are
user
-defined
b
ut
positi
v
e
and
constant,
and
ii)
W
mr
ef
is
user
-defined
and
bounded
b
ut
it
may
be
time-v
arying.
Therefore,
equation
(10)
is
a
stable
reference
model.
The
tracking
error
is
defined
as:
e
(
t
)
=
y
m
(
t
)
y
d
(
t
)
=
W
m
j
m
y
d
(11)
e
=
f
e
:
j
e
j
C
be
g
(12)
where
y
m
is
defined
in
(6),
y
d
is
the
desired
o
ut
put
and
is
pro
vided
by
(10),
e
is
a
residual
set
whose
s
ize
is
defined
by
C
be
which
is
an
user
-defined
positi
v
e
constant.
The
goal
of
the
control
design
is
to
formulate
a
control
la
w
and
an
update
la
w
for
the
plant
model
(1),
subject
to
assumptions
Ai
to
Aiii,
such
t
hat:
CGi)
the
tracking
error
e
asymptotically
con
v
er
ges
to
the
residual
set
e
,
CGii)
the
control
and
update
la
ws
do
not
in
v
olv
e
discontinuous
signals,
CGiii)
the
control
la
w
and
the
updated
parameters
are
bounded,
and
CGi
v)
all
the
closed
loop
signals
are
bounded.
3.
CONTR
OL
DESIGN
In
this
section,
a
controller
for
the
plant
(1)
is
de
v
eloped,
it
tak
es
into
account
the
assumptions
Ai
to
Aiii,
and
the
goals
CGi
to
CGi
v
stated
pre
viously
.
The
procedure
is
similar
to
that
in
[32],
b
ut
there
are
se
v
eral
dif
fer
-
ences
due
to
the
presence
of
measurement
noise.
Therefore,
the
procedure
omits
the
steps
that
are
quite
similar
to
those
in
[32].
The
state
adapti
v
e
backstepping
(SAB)
presented
in
[5]
is
used
as
control
frame
w
ork,
b
ut
important
modifications
are
incorporated
i
n
order
to
tackle
the
ef
fect
of
unkno
wn
time-v
arying
plant
model
coef
ficients
and
measurement
noise.
The
controller
design
procedure
is
or
g
anized
in
the
follo
wing
steps:
i)
define
the
first
ne
w
state
z
1
and
dif
ferentiate
it
with
respect
to
time,
ii)
define
a
quadratic
function
V
z
1
that
depends
on
z
1
,
and
dif
ferentiate
it
with
respect
to
time,
iii)
e
xpress
the
terms
that
in
v
olv
e
time-v
arying
coef
ficients,
as
functions
of
upper
constant
bounds,
and
parameterize
such
bounds
as
function
of
parameter
and
re
gression
v
ectors,
i
v)
e
xpress
the
parameter
v
ector
in
terms
of
updating
error
v
ector
and
update
parameter
v
ector
,
and
define
the
second
ne
w
state
z
2
,
v)
dif
ferentiate
z
2
with
respect
to
time,
define
a
quadratic
function
V
z
that
depends
on
z
1
and
z
2
,
and
dif
ferentiate
it
with
respect
to
time,
vi)
e
xpress
the
terms
that
in
v
olv
e
time
v
arying
coef
ficients
as
function
of
upper
constant
bounds,
and
parameterize
in
terms
of
parameter
and
re
gression
v
ectors,
vii)
e
xpress
the
parameter
v
ector
in
terms
of
updating
error
v
ector
and
updated
parameter
v
ector
,
and
formulate
the
control
la
ws,
and
viii)
formulate
the
L
yapuno
v-lik
e
function,
dif
ferentiate
it
with
respect
to
time
and
formulate
the
update
la
ws.
Step
0.
In
this
step,
the
model
(3)-(4)
is
e
xpressed
in
terms
of
y
m
,
x
2
m
,
using
the
noise
models
(6)-(7).
Dif
ferentiating
(6)
and
using
(3),
yields:
_
y
m
=
(
_
a
6
a
6
a
1
)
x
1
+
a
2
a
6
x
2
a
3
a
6
+
_
a
7
(13)
solving
(6)
and
(7)
for
x
1
and
x
2
and
substituting
into
the
abo
v
e
e
xpression
we
obtai
n
the
basic
e
xpression
for
_
y
m
:
_
y
m
=
(
_
a
6
a
6
a
1
)
a
6
y
m
+
a
2
a
6
a
8
x
2
m
a
2
a
6
a
9
a
8
a
3
a
6
+
_
a
7
(
_
a
6
a
6
a
1
)
a
7
a
6
:
(14)
Dif
ferentiating
(7)
with
respect
to
time
and
incorporating
(4),
yields:
_
x
2
m
=
a
4
a
8
x
1
+
(
_
a
8
a
5
a
8
)
x
2
+
a
8
bu
+
_
a
9
(15)
Contr
olling
a
DC
Motor
thr
ough
L
ypauno
v-lik
e
Functions
and
SAB
...
(Alejandr
o
Rinc
´
on)
Evaluation Warning : The document was created with Spire.PDF for Python.
2184
ISSN:
2088-8708
solving
(6)
and
(7)
for
x
1
and
x
2
and
incorporating
in
the
abo
v
e
equation
we
obtai
n
the
basic
e
xpression
for
_
x
2
m
which
is
_
x
2
m
=
a
4
a
8
a
6
y
m
+
_
a
8
a
5
a
8
a
8
x
2
m
+
a
8
bu
+
a
4
a
8
a
7
a
6
+(
_
a
8
+
a
5
a
8
)
a
9
a
8
+
_
a
9
(16)
Remark
3..1
In
or
der
to
consider
the
noisy
y
m
and
x
2
m
instead
of
the
r
eal
b
ut
unknown
values
x
1
and
x
2
,
in
the
r
emaining
pr
ocedur
e
equations
(14)
and
(16)
ar
e
used
instead
of
equations
(3)
and
(4)
.
Step
1.
In
this
step,
the
first
state
v
ariable
is
defined
and
dif
ferentiated
with
respect
to
time.
The
state
v
ariable
z
1
is
defined
as
the
tracking
error:
z
1
=
e
=
y
m
y
d
(17)
where
y
d
is
pro
vided
by
(10).
Dif
ferentiating
(17)
with
respect
to
time
and
using
(14),
yields:
_
z
1
=
_
y
m
_
y
d
(18)
_
z
1
=
a
1
a
6
+
_
a
6
a
6
y
m
+
a
2
a
6
a
8
x
2
m
+
(
a
1
a
6
_
a
6
)
a
7
a
6
a
2
a
6
a
9
a
8
a
3
a
6
+
_
a
7
_
y
d
(19)
Step
2.
In
this
step,
a
quadratic
function
that
depends
on
z
1
is
defined
and
dif
ferentiated
with
respect
to
time.
Such
quadratic
form
is
defined
as:
V
z
1
=
(1
=
2)
z
2
1
(20)
Dif
ferentiating
(20)
with
respect
to
time,
using
(19)
and
adding
and
substracting
c
1
z
2
1
yields:
_
V
z
1
=
z
1
_
z
1
=
c
1
z
2
1
+
z
1
a
2
a
6
a
8
x
2
m
+
z
1
a
1
a
6
+
_
a
6
a
6
y
m
+
(
a
1
a
6
_
a
6
)
a
7
a
6
a
2
a
6
a
9
a
8
a
3
a
6
+
_
a
7
+
c
1
z
1
_
y
d
(21)
The
term
c
1
z
2
1
has
been
added
to
obtain
asymptotic
con
v
er
gence
of
the
tracking
error
later
.
The
unkno
wn
and
time
v
arying
beha
vior
of
the
bounded
parameters
a
1
,
a
2
,
a
3
,
a
6
,
_
a
6
,
a
7
,
_
a
7
,
a
8
,
and
a
9
is
a
significant
obstacle
for
the
controller
design.
F
or
this
reason,
the
terms
that
in
v
olv
e
such
coef
ficients
will
be
e
xpressed
as
function
of
upper
constant
bounds.
Step
3.
Recall
that
a
1
,
a
2
,
a
3
,
a
6
,
a
7
,
a
8
,
a
9
,
_
a
6
,
and
_
a
7
are
time-v
arying,
unkno
wn
and
bounded.
In
this
step,
the
terms
that
in
v
olv
e
such
time-v
arying
parameters
are
e
xpressed
as
function
of
upper
and
lo
wer
constant
bounds,
and
such
bounds
are
parameterized
in
terms
of
parameter
and
re
gression
v
ectors.
The
term
that
in
v
olv
es
the
brack
ets
in
(21)
yields:
z
1
a
1
a
6
+
_
a
6
a
6
y
m
+
(
a
1
a
6
_
a
6
)
a
7
a
6
a
2
a
6
a
9
a
8
a
3
a
6
+
_
a
7
+
c
1
z
1
_
y
d
10
j
y
m
jj
z
1
j
+
11
j
z
1
j
+
j
c
1
z
1
_
y
d
jj
z
1
j
(22)
where
10
,
11
are
unkno
wn
positi
v
e
constants
such
that
a
1
a
6
+
_
a
6
a
6
10
(
a
1
a
6
_
a
6
)
a
7
a
6
a
2
a
6
a
9
a
8
a
3
a
6
+
_
a
7
11
(23)
As
mentioned
in
[32],
Y
oung’
s
inequal
ity
must
be
applied
to
(22),
such
that
the
j
z
1
j
term
leads
to
z
2
1
,
to
allo
w
a
proper
definition
of
z
2
.
Arranging
(22),
and
applying
Y
oung’
s
inequality
([34]),
yields:
z
1
a
1
a
6
+
_
a
6
a
6
y
m
+
(
a
1
a
6
_
a
6
)
a
7
a
6
a
2
a
6
a
9
a
8
a
3
a
6
+
_
a
7
+
c
1
z
1
_
y
d
c
a
10
c
a
j
y
m
jj
z
1
j
+
c
a
11
c
a
j
z
1
j
+
c
a
j
c
1
z
1
_
y
d
jj
z
1
j
c
a
(24)
c
2
a
2
+
2
11
2
c
2
a
z
2
1
+
c
2
a
2
+
2
10
2
c
2
a
y
2
m
z
2
1
+
c
2
a
2
+
(
c
1
z
1
_
y
d
)
2
z
2
1
2
c
2
a
(25)
IJECE
V
ol.
8,
No.
4,
August
2018:
2180
–
2198
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
2185
where
c
a
is
a
posi
ti
v
e
constant
that
should
be
chosen
to
fulfill
certain
conditions
that
will
be
defined
later
.
Substi-
tuting
into
(21),
yields:
_
V
z
1
c
1
z
2
1
+
3
c
2
a
2
+
z
1
a
2
a
6
a
8
x
2
m
+
2
11
2
c
2
a
z
2
1
+
2
10
2
c
2
a
y
2
m
z
2
1
+
(
c
1
z
1
_
y
d
)
2
z
2
1
2
c
2
a
(26)
The
terms
that
in
v
olv
e
x
2
m
,
z
2
1
,
x
2
1
z
2
1
,
(
c
1
z
1
_
y
d
)
2
should
be
grouped
in
a
ne
w
kno
wn
state
v
ariable
z
2
,
according
to
the
procedure
in
[5].
Ne
v
ertheless,
the
unkno
wn
time-v
arying
beha
vior
of
parameter
a
2
a
6
=a
8
poses
a
significant
obstacle.
T
o
remedy
such
situation,
a
positi
v
e
constant
lo
wer
bound
of
a
2
a
6
=a
8
will
be
used.
Property
Pi
mentions
that
a
2
,
a
6
,
and
a
8
are
positi
v
e,
wher
eas
property
Piii
implies
that
a
2
,
a
6
,
and
1
=a
8
are
lo
wer
bounded
by
unkno
wn
positi
v
e
constants.
Therefore,
0
<
l
12
a
2
a
6
a
8
(27)
where
l
12
is
an
unkno
wn
positi
v
e
constant
lo
wer
bound.
Incorporating
the
constant
l
12
into
(26)
and
arranging,
yields:
_
V
z
1
c
1
z
2
1
+
3
c
2
a
2
+
z
1
a
2
a
6
a
8
x
2
m
+
1
2
c
2
a
2
11
l
12
l
12
z
2
1
+
1
2
c
2
a
2
10
l
12
l
12
y
2
m
z
2
1
+
1
2
c
2
a
1
l
12
l
12
z
2
1
(
c
1
z
1
_
y
d
)
2
(28)
=
c
1
z
2
1
+
3
c
2
a
2
+
z
1
a
2
a
6
a
8
x
2
m
+
'
>
1
1
2
c
2
a
l
12
z
2
1
(29)
where
'
1
is
a
kno
wn
re
gression
v
ector
and
1
is
an
unkno
wn
constant
parameter
v
ector
gi
v
en
by:
'
1
=
[1
;
y
2
m
;
(
c
1
z
1
_
y
d
)
2
]
>
(30)
1
=
2
11
l
12
;
2
10
l
12
;
1
l
12
>
;
(31)
Step
4.
In
this
step,
the
unkno
wn
parameter
v
ector
1
is
e
xpressed
in
terms
of
an
updating
error
v
ector
and
an
updated
parameter
v
ector
,
and
then
a
ne
w
state
v
ariable
z
2
is
defined.
The
parameter
v
ector
1
can
be
re
written
as:
1
=
^
1
~
1
(32)
where
~
=
^
1
2
11
l
12
;
2
10
l
12
;
1
l
12
>
(33)
where
^
1
is
an
updated
parameter
v
ector
pro
vided
by
an
updating
la
w
that
will
be
defined
later
,
and
~
1
is
an
updating
error
.
Substituting
(32)
into
(29),
yields:
_
V
z
1
c
1
z
2
1
+
3
c
2
a
2
+
z
1
a
2
a
6
a
8
x
2
m
+
'
>
1
^
1
1
2
c
2
a
l
12
z
2
1
'
>
1
~
1
1
2
c
2
a
l
2
z
2
1
As
can
be
noticed
from
[23]
and
[24]
the
updated
parameter
^
1
is
non-ne
g
ati
v
e,
so
that
j
^
1
j
=
^
1
for
^
1
(
t
o
)
0
.
The
accomplishment
of
this
property
will
be
sho
wn
later
.
In
vie
w
of
this
f
act
and
incorporating
the
inequality
(27)
in
the
term
'
>
1
^
1
0
:
5
c
2
a
l
12
z
2
1
and
arranging,
yields:
_
V
z
1
c
1
z
2
1
+
3
c
2
a
2
+
z
1
a
2
a
6
a
8
x
2
m
+
'
>
1
^
1
1
2
c
2
a
a
2
a
6
a
8
z
2
1
'
>
1
~
1
1
2
c
2
a
l
2
z
2
1
(34)
=
c
1
z
2
1
+
3
c
2
a
2
+
z
1
a
2
a
6
a
8
z
2
'
>
1
~
1
1
2
c
2
a
l
2
z
2
1
(35)
Contr
olling
a
DC
Motor
thr
ough
L
ypauno
v-lik
e
Functions
and
SAB
...
(Alejandr
o
Rinc
´
on)
Evaluation Warning : The document was created with Spire.PDF for Python.
2186
ISSN:
2088-8708
where
z
2
=
x
2
m
+
'
>
1
^
1
1
2
c
2
a
z
1
(36)
Step
5.
In
this
step,
the
state
v
ariable
z
2
is
dif
ferentiated
with
respect
to
time,
a
quadratic
function
V
z
is
defined
as
function
of
z
1
and
z
2
,
and
such
function
is
dif
ferentiated
with
respect
to
time.
Dif
ferentiating
(36)
with
respect
to
time,
yields:
_
z
2
=
_
x
2
m
+
_
'
>
1
^
1
1
2
c
2
a
z
1
+
'
>
1
_
^
1
1
2
c
2
a
z
1
+
'
>
1
^
1
1
2
c
2
a
_
z
1
(37)
where
_
'
1
=
[0
;
2
y
m
_
y
m
;
2(
c
1
z
1
_
y
d
)(
c
1
_
z
1
•
y
d
)]
>
(38)
Substituting
(18)
into
(37)
and
arranging,
yields:
_
z
2
=
_
x
2
m
+
'
1
b
_
y
m
+
'
1
c
(39)
where
'
1
b
=
1
2
c
2
a
2
y
m
^
1[2]
+
c
1
(
c
1
z
1
_
y
d
)
^
1[3]
z
1
+
'
>
1
^
1
(40)
'
1
c
=
2(
c
1
z
1
_
y
d
)(
c
1
_
y
d
+
•
y
d
)
^
1[3]
1
2
c
2
a
z
1
+
'
>
1
_
^
1
1
2
c
2
a
z
1
'
>
1
^
1
1
2
c
2
a
_
y
d
(41)
'
1
b
and
'
1
c
are
kno
wn
scalar
functions,
^
1[2]
and
^
1[3]
are
the
second
and
third
entries
of
the
v
ector
^
1
,
and
y
d
,
_
y
d
,
and
•
y
d
are
pro
vided
by
(10).
Substituting
(14)
and
(16)
into
(39)
and
arranging,
yields:
_
z
2
=
a
4
a
8
a
6
y
m
+
(
_
a
8
a
5
a
8
)
a
8
x
2
m
+
(
_
a
6
a
6
a
1
)
a
6
'
1
b
y
m
+
a
2
a
6
a
8
'
1
b
x
2
m
+
a
2
a
6
a
9
a
8
a
3
a
6
+
_
a
7
(
_
a
6
a
6
a
1
)
a
7
a
6
'
1
b
+
a
4
a
8
a
7
a
6
+
(
_
a
8
+
a
5
a
8
)
a
9
a
8
+
_
a
9
+
a
8
bu
+
'
1
c
(42)
The
quadratic
form
that
depends
on
z
1
and
z
2
is
defined
as:
V
z
=
(1
=
2)(
z
2
1
+
z
2
2
)
(43)
Dif
ferentiating
with
respect
to
time,
incorporating
(35)
and
(42),
and
adding
and
subtracting
c
2
z
2
2
,
yields:
_
V
z
=
z
1
_
z
1
+
z
2
_
z
2
=
_
V
z
1
+
z
2
_
z
2
c
1
z
2
1
c
2
z
2
2
+
3
2
c
2
a
+
z
2
a
2
a
6
a
8
z
1
a
4
a
8
a
6
y
m
+
_
a
8
a
5
a
8
a
8
x
2
m
+
_
a
6
a
6
a
1
a
6
'
1
b
y
m
+
a
2
a
6
a
8
'
1
b
x
2
m
+
a
2
a
6
a
9
a
8
a
3
a
6
+
_
a
7
(
_
a
6
a
6
a
1
)
a
7
a
6
'
1
b
+
a
4
a
8
a
7
a
6
+
(
_
a
8
+
a
5
a
8
)
a
9
a
8
+
_
a
9
+
c
2
z
2
+
a
8
bu
+
'
1
c
'
>
1
~
1
1
2
c
2
a
l
12
z
2
1
(44)
IJECE
V
ol.
8,
No.
4,
August
2018:
2180
–
2198
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
2187
The
control
input
u
is
defined
as
follo
ws:
u
=
u
a
+
u
b
(45)
where
u
a
is
an
user
-defined
constant.
The
constant
u
a
w
as
incorporated
in
order
to
a
v
oid
abrupt
control
input
beha
vior
at
the
be
ginning
of
the
closed
loop
operation.
Therefore,
it
should
be
chosen
as
the
open
loop
v
alue
of
the
control
input.
The
control
u
b
is
established
by
the
controller
design.
Substituting
(45)
into
(44),
yields:
_
V
z
c
1
z
2
1
c
2
z
2
2
+
3
2
c
2
a
+
z
2
a
2
a
6
a
8
(
z
1
+
'
1
b
x
2
m
)
a
4
a
8
a
6
y
m
+
_
a
8
a
5
a
8
a
8
x
2
m
+
_
a
6
a
6
a
1
a
6
'
1
b
y
m
+
a
2
a
6
a
9
a
8
a
3
a
6
+
_
a
7
(
_
a
6
a
6
a
1
)
a
7
a
6
'
1
b
+
a
4
a
8
a
7
a
6
+
(
_
a
8
+
a
5
a
8
)
a
9
a
8
+
_
a
9
+
a
8
bu
a
+
'
1
c
+
c
2
z
2
a
8
bu
b
z
2
'
>
1
~
1
1
2
c
2
a
l
12
z
2
1
(46)
Step
6.
Recall
that
a
1
,
a
2
,
a
3
,
a
6
,
a
7
,
a
8
,
a
9
,
_
a
6
,
_
a
7
,
_
a
8
are
unkno
wn
and
time-v
arying.
In
this
step,
the
terms
that
in
v
olv
e
such
parameters
are
e
xpressed
in
terms
of
upper
bounds,
and
paramet
erized
in
terms
of
parameter
and
re
gression
v
ectors.
The
term
that
in
v
olv
es
the
squared
brack
ets
can
be
re
written
as:
z
2
a
2
a
6
a
8
(
z
1
+
'
1
b
x
2
m
)
a
4
a
8
a
6
y
m
+
_
a
8
a
5
a
8
a
8
x
2
m
+
_
a
6
a
6
a
1
a
6
'
1
b
y
m
+
a
2
a
6
a
9
a
8
a
3
a
6
+
_
a
7
(
_
a
6
a
6
a
1
)
a
7
a
6
'
1
b
+
a
4
a
8
a
7
a
6
+
(
_
a
8
+
a
5
a
8
)
a
9
a
8
+
_
a
9
+
a
8
bu
a
+
'
1
c
+
c
2
z
2
j
z
2
j
[
13
j
y
m
j
+
14
j
x
2
m
j
+
15
j
'
1
b
y
m
j
+
16
j
z
1
+
'
1
b
x
2
m
j
+
17
j
'
1
b
j
+
18
+
19
j
u
a
j
+
j
'
1
c
+
c
2
z
2
j
]
(47)
where
13
,
14
,
15
,
16
,
17
,
18
,
19
are
unkno
wn
positi
v
e
constant
upper
bounds
that
satisfy:
a
4
a
8
a
6
13
;
_
a
8
a
5
a
8
a
8
14
_
a
6
a
6
a
1
a
6
15
;
a
2
a
6
a
8
16
a
2
a
6
a
9
a
8
a
3
a
6
+
_
a
7
(
_
a
6
a
6
a
1
)
a
7
a
6
17
a
4
a
8
a
7
a
6
+
(
_
a
8
+
a
5
a
8
)
a
9
a
8
+
_
a
9
18
;
j
a
8
b
j
19
(48)
From
equations
(46),
(47),
and
as
can
be
inferred
from
[32]
there
are
t
w
o
significant
obstacles.
First,
the
j
z
2
j
term
may
lead
to
discontinuous
signals
in
the
definition
of
u
b
.
This
can
be
remedied
by
using
Y
oung’
s
inequality
.
Sec-
ond,
the
unkno
wn
v
arying
beha
vior
of
a
8
b
mak
es
it
dif
ficult
for
u
to
eliminate
the
ef
fect
of
the
terms
that
in
v
olv
e
unkno
wn
time
v
arying
parameters.
This
can
be
remedied
by
incorporating
a
positi
v
e
constant
lo
wer
bound
of
a
8
b
.
From
properties
Piii
and
Pi
v
(see
page
2182)
it
follo
ws
that
a
8
and
b
are
positi
v
e,
b
is
constant
and
a
8
is
lo
wer
bounded
by
an
unkno
wn
positi
v
e
constant.
Therefore,
0
<
l
20
a
8
b
(49)
where
l
20
is
an
unkno
wn
positi
v
e
constant
lo
wer
bound.
Re
writ
ting
(47)
in
terms
of
a
parameter
v
ector
and
a
Contr
olling
a
DC
Motor
thr
ough
L
ypauno
v-lik
e
Functions
and
SAB
...
(Alejandr
o
Rinc
´
on)
Evaluation Warning : The document was created with Spire.PDF for Python.
2188
ISSN:
2088-8708
re
gression
v
ector
,
and
incorporating
(49),
yields:
z
2
a
2
a
6
a
8
(
z
1
+
'
1
b
x
2
m
)
a
4
a
8
a
6
y
m
+
_
a
8
a
5
a
8
a
8
x
2
m
+
_
a
6
a
6
a
1
a
6
'
1
b
y
m
+
a
2
a
6
a
9
a
8
a
3
a
6
+
_
a
7
(
_
a
6
a
6
a
1
)
a
7
a
6
'
1
b
+
a
4
a
8
a
7
a
6
+
(
_
a
8
+
a
5
a
8
)
a
9
a
8
+
_
a
9
+
a
8
bu
a
+
'
1
c
+
c
2
z
2
p
l
20
j
z
2
j
'
>
2
(50)
where
'
=
[
j
y
m
j
;
j
x
2
m
j
;
j
'
1
b
y
m
j
;
j
z
1
+
'
1
b
x
2
m
j
;
j
'
1
b
j
;
1
;
j
u
a
j
;
j
'
1
c
+
c
2
z
2
j
]
>
(51)
2
=
1
p
l
20
[
13
;
14
;
15
;
16
;
17
;
18
;
19
;
1]
>
;
(52)
'
is
a
re
gression
v
ector
whose
entries
are
kno
wn,
and
2
is
a
parameter
v
ector
,
whose
entries
are
positi
v
e,
constant
and
unkno
wn,
and
l
20
is
an
unkno
wn
positi
v
e
constant
lo
wer
bound.
The
constant
p
l
20
has
been
incorporated
in
order
to
handle
the
ef
fect
of
the
unkno
wn
time-v
arying
parameter
a
8
b
appearing
in
the
term
a
8
bu
b
in
(46).
Step
7.
Since
the
parameter
v
ector
2
is
unkno
wn,
in
this
step
it
is
e
xpressed
in
terms
of
an
updated
parameter
v
ector
and
an
updating
error
v
ector;
after
of
this
the
control
la
w
is
formulated.
The
parameter
2
can
be
re
written
as
2
=
^
2
~
2
(53)
where
^
2
is
an
updated
parameter
v
ector
pro
vided
by
an
update
la
w
which
is
defined
in
the
step
8,
and
~
2
is
an
updating
error
v
ector
gi
v
en
by
~
2
=
^
2
1
p
l
20
[
13
;
14
;
15
;
16
;
17
;
18
;
19
;
1]
>
(54)
Substituting
(53)
into
(50)
yields:
z
2
a
2
a
6
a
8
(
z
1
+
'
1
b
x
2
m
)
a
4
a
8
a
6
y
m
+
_
a
8
a
5
a
8
a
8
x
2
m
+
_
a
6
a
6
a
1
a
6
'
1
b
y
m
+
a
2
a
6
a
9
a
8
a
3
a
6
+
_
a
7
(
_
a
6
a
6
a
1
)
a
7
a
6
'
1
b
+
a
4
a
8
a
7
a
6
+
(
_
a
8
+
a
5
a
8
)
a
9
a
8
+
_
a
9
+
a
8
bu
a
+
'
1
c
+
c
2
z
2
p
l
20
j
z
2
j
'
>
^
2
p
l
20
j
z
2
j
'
>
~
2
(55)
incorporating
the
inequality
(49)
and
applying
Y
oung’
s
inequality
(cf.
[34]
pp.
123)
to
the
term
p
l
20
j
z
2
j
'
>
^
2
,
yields:
z
2
a
2
a
6
a
8
(
z
1
+
'
1
b
x
2
m
)
a
4
a
8
a
6
y
m
+
_
a
8
a
5
a
8
a
8
x
2
m
+
_
a
6
a
6
a
1
a
6
'
1
b
y
m
+
a
2
a
6
a
9
a
8
a
3
a
6
+
_
a
7
(
_
a
6
a
6
a
1
)
a
7
a
6
'
1
b
+
a
4
a
8
a
7
a
6
+
(
_
a
8
+
a
5
a
8
)
a
9
a
8
+
_
a
9
+
a
8
bu
a
+
'
1
c
+
c
2
z
2
c
2
c
2
+
1
2
c
2
c
a
8
bz
2
2
(
'
>
^
2
)
2
p
l
20
j
z
2
j
'
>
~
2
(56)
where
c
c
is
a
positi
v
e
constant
that
satisfies
some
some
conditions
that
will
be
defined
in
the
Step
8.
Substituting
(56)
into
(46)
and
arranging
yields:
_
V
z
c
1
z
2
1
c
2
z
2
2
+
3
2
c
2
a
+
c
2
c
2
+
a
8
bz
2
u
b
+
1
2
c
2
c
z
2
(
'
>
^
2
)
2
'
>
1
~
1
1
2
c
2
a
l
20
z
2
1
p
l
20
j
z
2
j
'
>
~
2
(57)
IJECE
V
ol.
8,
No.
4,
August
2018:
2180
–
2198
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
2189
with
the
aim
to
cancel
the
ef
fect
of
the
term
a
8
bz
2
(1
=
2)
c
2
2
z
2
(
'
>
^
2
)
2
,
the
e
xpression
for
u
b
is
chosen
as:
u
b
=
1
2
c
2
c
z
2
(
'
>
^
2
)
2
(58)
In
vie
w
of
(45),
the
control
la
w
for
u
is:
u
=
u
a
1
2
c
2
c
z
2
(
'
>
^
2
)
2
(59)
substituting
(59)
into
(57),
yields:
_
V
z
2
min
f
c
1
;
c
2
g
V
z
+
3
2
c
2
a
+
c
2
c
2
'
>
1
~
1
1
2
c
2
a
l
20
z
2
1
p
l
20
j
z
2
j
'
>
~
2
The
abo
v
e
e
xpression
implies
that
the
ti
me
deri
v
ati
v
e
of
the
L
yapuno
v
function
w
ould
contain
the
term
(3
=
2)
c
2
a
+
(1
=
2)
c
2
c
,
so
that
the
required
ne
g
ati
v
eness
properties
w
ould
be
altered.
Therefore,
the
quadratic
function
V
z
is
considered,
which
is
a
truncated
function
of
V
z
and
v
anishes
when
V
z
is
lo
wer
or
equal
than
the
constant
C
bv
z
.
The
quadratic
function
V
z
is
defined
as:
V
z
=
(1
=
2)(
p
V
z
p
C
bv
z
)
2
if
V
z
C
bv
z
0
otherwise
(60)
C
bv
z
=
(1
=
2)
C
2
be
(61)
where
V
z
is
defined
in
(43).
The
function
defined
by
(60)
and
(61)
has
the
follo
wing
properties:
V
z
0
,
V
z
3
C
bv
z
+
3
V
z
and
V
z
and
@
V
z
=@
V
z
are
locally
Lipschitz
continuous.
Dif
ferentiating
(60)
with
respect
to
time,
yields:
d
V
z
dt
=
@
V
z
@
V
z
_
V
z
(62)
where
@
V
z
@
V
z
=
(
1
2
p
V
z
p
C
bv
z
p
V
z
if
V
z
C
bv
z
0
otherwise
(63)
Combining
(62)
with
(60)
yields:
d
V
z
dt
2
min
f
c
1
;
c
2
g
V
z
@
V
z
@
V
z
+
3
2
c
2
a
+
c
2
c
2
@
V
z
@
V
z
'
>
1
~
1
1
2
c
2
a
l
20
z
2
1
@
V
z
@
V
z
p
l
20
j
z
2
j
'
>
~
2
@
V
z
@
V
z
(64)
Step
8.
In
this
step,
the
L
yapuno
v-lik
e
function
is
formulated
and
dif
ferentiated
with
respect
to
time,
and
the
update
la
ws
are
formulated.
The
L
yapuno
v-lik
e
function
is
defined
as:
V
(
x
(
t
))
=
V
z
+
V
(65)
x
(
t
)
=
[
z
1
(
t
)
;
z
2
(
t
)
;
~
>
1
;
~
>
2
]
(66)
V
=
(1
=
2)
l
20
~
>
1
1
1
~
1
+
(1
=
2)
p
l
20
~
>
2
1
2
~
2
(67)
where
~
1
and
~
2
are
defined
in
(33)
and
(54)
respecti
v
ely
,
and
V
z
is
defined
in
(60).
The
v
ector
x
(
t
)
contains
the
closed
loop
stat
es
z
1
(
t
)
,
z
2
(
t
)
,
~
>
1
,
~
>
2
.
F
or
the
sak
e
of
simplicity
,
V
(
x
(
t
))
is
represented
as
V
.
Dif
ferentiating
(65)
and
(67)
with
respect
to
time,
yields:
_
V
=
_
V
z
+
_
V
(68)
_
V
=
l
20
~
>
1
1
1
_
^
1
+
p
l
20
~
>
2
1
2
_
^
2
(69)
Incorporating
(64)
and
(69)
into
(68),
yields:
_
V
2
min
f
c
1
;
c
2
g
V
z
@
V
z
@
V
z
+
3
2
c
2
a
+
c
2
c
2
@
V
z
@
V
z
+
l
20
~
>
1
'
1
1
2
c
2
a
z
2
1
@
V
z
@
V
z
+
1
1
_
^
1
+
p
l
20
~
>
2
j
z
2
j
'
@
V
z
@
V
z
+
1
2
_
^
2
(70)
Contr
olling
a
DC
Motor
thr
ough
L
ypauno
v-lik
e
Functions
and
SAB
...
(Alejandr
o
Rinc
´
on)
Evaluation Warning : The document was created with Spire.PDF for Python.