Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications
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358
D.
C.
Theodoridis.
Y.
BOlltalis, and
M.
A. Christodoulou
5
~
hi
II
~
_:
H
lu-v
j'-\D<~~;~~'=~.~~~~~>~.~_~
__
<
__
• • <
__
._.
____
•
__
•
__
••
_____
I
40
S 20
N 0
2"5-
'0
-20
o
2
3
4 5
-~----<-----,----------,--
-------,---
_40~--------L--------L-------~--------J-------~
o 2
3
4
5
Time (sec)
Fig.
2. Evolution
of
control
input
n,
errors
c], e2
and
disturbance
d(:r;2'
'U)
for
RHONN's
(red line)
and
F-HONNF
a.pproach
(blue
line)
Pendulum
case), a significant (order
of
magnitude)
increase in
the
control
performance.
Conclusively,
the
comparison between
llHONN
and
F-HONNF'fl leads
to
a
superiority
of
F-HONNF'fl regarding
the
control abilities. Especially, if we
choose smaller learning
rate
and
reduced
number
of
high
order
terms,
then
the
difference between
the
two
methods
becomes huge.
Examples
RHONN
FHONNF
Van
del'
polA[S'Ex]
0.0024 9.5226 .
104~
M 8£:';2 0.8417 0.5930
Table
2. CompaTison
of
RHONN's
am!
F-HONNF
a.pproacbes for
the
Van del' pol
oscillator.
5
Conclusions
A direct ada.ptive control scheme was considered in
this
chapter,
'-liming
at
the
regulation
of
nonlinear
unknown
plants
of
Brunovsky
canonical form
with
the
presence
of
modeling errors.
The
approach
is based on a new Neuro-Fuzzy
Nonlinear Systems
in
Brunovsky Canonical
Form:
A Neuro-Fuzzy Algorithm 359
N
)(
*
(jj
0.6r---~--~---~--~---'-----,
-0.2
-0.4
L--
__
-'--
__
---'
___
~
__
__'_
___
~
__
.....J
o
0.5
5
0
ti(\
r,
\j
-5
.\li
V
-10
0 0.5
1.5
1.5
Time (sec)
2
2
2.5
3
2.5
3
Fig.
3.
Evolution
of
state
variable
Xl
and
X2
for
RHONN's
(red
line)
and
F-HONNF
approach
(blue
line)
Dynamical
Systems
definition,
which
uses
the
concept
of
Fuzzy
Adaptive
Systems
(FAS)
operating
in
conjunction
with
High
Order
Neural
Network
Functions
(F-HONNF's).
Since
the
plant
is
considered
unknown,
we first
propose
its
approximation
by
a
special
form
of
a
Brunovsky
type
fuzzy dy-
namical
system
(FDS)
where
the
fuzzy
rules
are
approximated
by
appropriate
HONNF's.
This
practically
transforms
the
original
unknown
system
into
a
neuro-fuzzy
model
which
is
of
known
structure,
but
contains
a
number
of
unknown
constant
value
parameters
known
as
synaptic
weights.
The
pro-
posed
scheme
does
not
require
a-priori
experts'
information
on
the
number
and
type
of
input
variable
membership
functions
making
it
less
vulnerable
to
initial
design
assumptions.
vVcight
updating
laws for
the
involved
HONNF's
are
provided,
which
guarantee
that
the
system
states
reach
zero
exponen-
tially
fast, while keeping all signals
in
the
closed
loop
bounded.
Simulations
illustrate
the
potency
of
the
method
while
the
applicability
is
tested
on
well
known
benchmarks
where
it
is
shown
that
by
following
the
proposed
proce-
dure
one
can
obtain
asymptotic
regulation
quite
well.
Compared
to
simple
RHONN's
direct
control,
proves
to
be
superior.
References
l.y.
S.
Boutalis,
D. C.
Theodoridis,
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360
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C.
Theodoridis,
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Boutalis,
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M. A. Christodoulou
i'"lJ
!:::
2000
e 1000
§ 0
~
()
, 1
o 0.5 1.5 2
2.5 3
~
~l~rrL-.~
·
L-.
'--------'----~,
l
o
0.5 1 1.5 2 2.5 3
1
JrQj~~
__
n_n~
____
~
o
0.5
1 1.5 2 2.5 3
.
.,
1
o
0.5
1.5 2 2.5 3
Time (sec)
Fig.
4.
Evolution
of
control
input
u
and
disturbance
d for
RHONN's
(red line)
and
F-HONNF
approach
(blue
line)
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and
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F.
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IEEE
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16.S.
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18.R.
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Rovithakis
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M.A.Christodoulou.
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C.
Theodoridis,
Y. S.
Boutalis,
and
M. A.
Christodoulou.
Direct
adaptive
control
of
unknown
nonlinear
systems
using
a
new
neuro-fuzzy
method
together
with
a novel
approach
of
parameter
hopping.
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24.D. C.
Theodoridis,
M. A.
Christodoulou,
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S.
Boutalis.
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high
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Chai.
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fuzzy logic
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Prentice
Hall,
NJ,
1994.
27.Y. Yang.
Direct
robust
adaptive
fuzzy
control
(drafc) for
uncertain
nonlinear
systems
using
small
gain
theorem.
Fuzzy
Sets
and
Systems,
151:79-97,
May
2004.
362
Routh-Hurwitz
Conditions
and
Lyapunov
Second
Method
for a
Nonlinear
System
Abant
Izzet Baysal University
14280 Bolu, TUrkey
(e-mail:
umuto@ibu.edu.tr
)
Omur
Umut
Abstract:
In
this
paper
, a
Lyapunov
function is
generated
to
determine
the
do-
main
of
asymptotic
stability
of
a
system
of
three
first
order
nonlinear
ordinary
differential
equ
a
tions
describing
the
behaviour
of a nuclear
spin
generator
(NSG).
The
generated
Lyapunov
function, is a simple
quadratic
form, whose coefficients
are
chosen so
that
the
Routh-Hurwitz
criteria
are
satisfied for
the
corresponding
line
ar
differential
equations.
Keywords:
Routh-Hurwitz
conditions
,
Lyapunov
method
,
Lyapunov
function,
nuclear
spin
gene
rator,
domain
of
stability,
Cartwright's
me
thod.
1
Introduction
The
well-known
Routh-Hurwitz
criteria
can
be easily applied
to
determine
the
conditions for
the
stability
of
a linear
autonomous
system.
The
simplicity
of
these conditions is a consequence
of
the
fact
that
stability, for a linear system,
is a global property.
Since
the
stability
of
nonlinear systems is
not
necessarily
global, no simple
criteria
exist for
its
determination
,
and
it
becomes necessary
to seek
domains
of
the
state
space
within
which
the
syst
em behaves in a
stable
fashion.
The
dire
ct
or
second
method
of
Lyapunov
[1]
is a general
approach
to
th
e question of stability.
The
essence
of
th
e
method
consists in
determining
sufficient conditions of
stability
not
by explicit knowledge of
the
solutions
of
the
differential
equations
describing
th
e system,
but
by using potential-like
functions, called Lyapunov functions.
Since
the
solutions
of
the
differe
ntial
e
quations
are
not
necessary,
the
method
is
particularly
well
suited
for
sta-
bility analysis
of
nonlinear systems.
The
generation
of
Lyapunov functions
appropriat
e for a
particular
problem
is
th
e critical
part
of
the
study
. A
great
deal
of
activity
has
been devoted
to
the
problem of gene
rating
function since
1950's [2]-[5].
In
this
paper,
Cartwright
method
is used for
the
determination
of
the
domain
of
asymptotic
stability
of
NSG.
2
Fundamental
concepts
The
general
nth.
order
autonomous
system
can
be
represented by
Routh-Hurwitz Conditions and Lyapul10v Second Method 363
x =
f(x),
X E R
n
(1)
The
singular
points
x = 0 represe
nt
states
of
equilibrium
of
th
e physical
system,
any
of which
can
always
be
situated
at
the
origin
of
the
state
space
by
an
appropriate
coordinate
transformation
so
that
f(O) =
O.
It
is desired
to
study
the
stability
of
a position
of
equilibrium;
that
is,
to
say,
to
inves
tigate
the
behaviour
of
th
e
system
traj
ectories
in
th
e vicinity
of
equilibrium
point
x =
O.
These
trajectori
es display only four
distinct
types
of
behaviour as
time
increases:
(a)
They
may
approach
the
origin; (b)
the
trajectories
may
tend
to
remain
within
a neighborhood
of
the
origin,
without
ever converging
to
it;
(c)
they
may
approach
a different equilibrium point;
or
(d)
th
e
trajectori
es
may
tend
to
infinity.
The
first case
repr
ese
nts
the
type
of
behaviour a designer would desire a physical
system
to
possess,
and
such a
system
would
be
called asymptotically stable.
It
is desirable
to
devise
a
method
that
will be capable
of
discriminating this case from
the
others.
Since
the
systems
to
be analyzed
ar
e nonlinear,
and
their
stability
local,
it
becomes necessary
that
the
method
used be also
capabl
e
of
yielding a domain
of
asymptotic
stability, a simply connected region of
th
e
state
spac
e co
ntaining
only
th
e equilibrium
point
x =
0,
and
such
that
any
trajectory
initiating
within
this
domain
will te
nd
to
this
equilibrium
point
as
time
increases.
The
second
or
direct
method
of
Lyapunov
is
parti
cularly
appropriate
for
these
purpos
e
s.
The
method
is based
on
the
investig
at
ion
of
sign definiteness
of
certain
functions which
determine
the
stability
of
the
equilibrium
point
,
as well as a
domain
of
asymptotic
stability.
It
therefore becomes
unn
ecessary
to
integrate
the
equations
of
motion, a
remote
possibility,
but
it
is neces-
sary
to
determine some
appropriate
functions, Lyapunov functions, for
the
differential
system
being investigated.
The
theorems
of
interest here
are
the
followings
[1]-[5].
Theorem
1.
Let
x = 0
be
an equilibrium point for (1) and D c R
n
be
a
domain containing x =
O.
Let
V : D
--7
R
be
a continuously
diff
erentiable
function
such that
V(O) = 0 and
V(
x ) > 0
in
D -
{O}
V(x)
=
grad(V(x))
.
f(x)
::;
0
in
D
Then, the origin is stable. Moreover,
if
V(x)
=
grad(V(x)).f(x)
< 0
in
D -
{O}
then the origin is asymptotically stable.
The
re
quirement
tha
t V
be
nonpositive assures
that
the
system
trajecto-
ries in D
either
remain
on
surfaces
of
constant
V
or
cross
them
toward
the
equilibrium position.
364
O.
Umut
Theorem
2.
(Barbashin and Krasovski)
Let
x = 0
be
an
equilibrium point
for (1). Let also V : D
-*
R
be
a continuously differentiable positive definite
function
on a domain D containing the origin x = 0, such that
11
:::;
0
in
D.
Let S =
{x
ED:
11
=
O}
and suppose
that
no solution can
stay
id
ent
ically
in
S other than trivial solution. Then the origin is asymptotically stable.
Theorem
2 implies
that
the
function
V(x)
cannot
have a relative maxi-
mum
inside
the
domain
D since grad(
v(
x)) is a vector
with
components V
X i
,
i =
1,2,
... , n all
of
which
ca
nnot
be
zero simultaneously in D except
at
the
equilibrium position.
3
NSG
system
NSG is a high frequency oscillator which
generates
and
controls
the
oscilla-
tions
of
th
e
motion
of
a nucl
ea
r
magneti
zat
ion vector in a
magnetic
field.
It
was first carried
out
by Sh
er
man,
[6]
in 1963.
This
system
has
the
form
X =
-;3X
+ Y
y =
-X
- ;3Y
(1-
kZ)
Z =
;3[a(1
-
Z)
-
ky2]
(2)
where
X,
Y, Z
are
the
components
of
the
nuclear
magn
et
ization vector in
the
X,
Y, Z
-directions,
respectively. a,
;3
and
k
are
parameters
where
;3a
::::
0,
and
;3
::::
0
are
linear
damping
terms,
while
the
nonlinearity
parameter
;3k
is
proportional
to
the
amplifier
gain
in
the
voltage feedback. Physical
considerations limit
the
parameter
a
to
the
range 0 < a
:::;
1.
The
equilibrium
points
of
the
system
(2)
are
found by
equating
th
e right
side
of
(2)
to
zero.
Th
e
point
A(O
,
0,1)
is
an
equilibrium
point
of
(2) for
all
parametric
values. De
pending
on
the
value
of
the
parameters
the
point
A(O,
0,1)
may
be
a spiral
or
a node,
stable
or
unstabl
e, for
exampl
e
it
is a
stable
focus when a = 0.15,
;3
= 0.25
and
k = 4 as shown
in
Fig.l
Fig.
1.
Th
e
point
A(O,
0,1)
is a
stable
focus
when
a = 0.15,
/3
= 0.25
and
k =
4.
However if,
(3)
"
Routh-Hurwitz Conditions and Lyapunov Second Method 365
the
system
(2) possesses two
additional
equilibrium points,
nam
ely
1 +
,82
B±(±al,
±,8al,
~)
where
(a[,82(k -
1)
-
I])!
a -
1 -
k,82
(4)
Notice
that
B+
and
B_
are
images
of
each
other
und
er
the
symmetry
(x,y,z)
--7
(-x,
- y
,z)
of
the
basic
system
(2).
All these equilibrium points
are
unst
able when condition in (3) holds.
Sachdev
and
Sarathy
studied
in
great
detail
the
NSC
system
in
[7
].
The
equilibrium
point
A(O,
0,
1)
is
transferred
to
th
e origin
through
the
coordinate
transformations
and
system
(2) becomes
x
= X , Y =
Y,
z = Z - 1
:i;
= - ,8x + y
iJ
=
-x
-
,8(
1 -
k)y
+
,8
ky
z
i =
-a,8z
-
,8
ky2
(5)
Now, a Lyapunov function will be
generated
for system (5) using
Cartwright's
method.
4
Cartwright's
method
This
method,
[5]
is essentially one
of
the
generalising from
the
lin
ear
to
the
nonlinear case, so
that
th
e
Routh-Hurwitz
stability
conditions
are
satisfied
for
the
linear case.
Lin
ea
rization
of
(5) is
:i;
=
-,8x
+ y
iJ
=
-x
-
,8(1
-
2k)y
i =
-a,8z
(6)
L
et
th
e following
quadratic
form
with
constant
und
etermined coefficients
k
i'
i = 1,
2,
...
,6
be
selected as a Lyapunov function for
the
system
(6)
2V
= k
1
x
2
+ k2y2 +
k3
Z2
+ k
4
xy
+ k
5XZ
+ k
6
yz
(7)
According
to
eqn (7)
. 1 2 1
)]2
V = - 2(2k
1
,8
+ k4)X + 2
[2k
2
,8(
2k - 1 +
k4
y -
(8)
k3a
,8z2
-
[k2
- kl + k
4
,8(1
-
k)]
xy -
1 1
2
[k
5
,8(
1 +
a)
+
k6]XZ
+ 2[k
5
+
k6
,8
(2k - 1 -
a)]yz
366
O.
Umut
Let
11
be
constrained
to
be a
n.d
(or n.s.d) function
of
the
stat
e variables
x
and
z.
This
results in
the
following
set
of
equations
2k
2
(3(2k
- 1) +
k4
= 0
kl-k2-(3(I
- k)k4 0
k5(3(1
+
0:)
-
k6
0
k5
-
(3(1
+
0:
- 2k)k6 0
The
last
two e
quations
imply
k5
=
k6
=
O.
11
can
be
constrained
as a
n.s.d function
of
the
state
variables x
and
z by
setting
k2
= 1
and
k3
=
1,
arbitrarily. So
the
solution
of
the
above
eq
uations
yields kl = 1 +
2(32(1
-
k)(1 - 2k)
and
k4
=
2(3(1
- 2k).
Substitution
of
these
values in (7)
and
(8)
yields
and
(10)
Inspection of eqns (9)
and
(10) shows
that
the
stability requirements
of
the
1 +
(32
Lyapunov
th
eo
rem
are
satisfied for
0:
>
0,
(3
> 0
and
0 < k <
~.
This
is
not
difficult
to
und
ersta
nd, since
stability
is a global
property
for a
linear
system
. For linear
systems
Parks,
[9]
has
derived
the
Routh-Hurwitz
stability
criterion using
th
e second
method
of
Lyapunov.
For
the
nonlinear
system
(5), a Lyapunov function
and
its
time
derivative
are
111
V = 2(1+,82[2-3
k(l+z)+k2(1+z)2])x2+2y2+2z2+,8[1
- k(1+z)]
xy
(11)
and
11
= -[(3(1 +
,82(1
-
k)(l
- 2k)) + 2,8(1 -
k(l
+ z)) -
~0:,83kz
-
~,83k2y2
+
2 2
,83k
3
(1
+
Z)y2]X
2
-
[o:,8z
+ ,8ky2 +
0:,82k2(1
+
z)x
2
-
0:,82kxy]z
V is
n.d
in a
certain
neighborhood
of
the
origin
and
V is
p.d
in
the
same
neighborhood.
If
Theor
em 2 is applied
to
the
Lyapunov function (11), a
boundary
for
the
stable
domain
is determined.
The
gradient
of
the
function is
[
(1
+
,82[2
- 3k(1 + z) + k
2
(1
+
Z)2)]X
+
,8[1-
k(l
+ Z)]y]
\7V
= y +
,8[1
- k(1 + z)]x
z
+
~[,82(
-3k
+ 2k2(1 + z)]x
2
- ,8kxy
which vanishes
at
the
origin
(0,0,0)
and
at
the
points
( ±
k~2
)2(1
+
(1
-
k),82],
±
k~3
)2[1
+
(1
-
k)(32],
1
:,8~2
-
1)
Routh-Hunvitl. Conditions and Lyapunov Second Method 367
Thus,
I>
is identically
equal
to
zero
at
these
three
points.
vVhen
0:
= 0.15,
f3
= 0.25
and
/;;
= 4 nontrivial critical
points
are
13
(±J26, ±4v26. 4 )
and
Lyapunov
fUIlction is
It
is found
that
the
largest
value
of
c
represents
a
dosed
surface is 5.28125.
Hence, a
domain
of
asymptotic
stability
for
the
system
(5)
[0.5625-0.6667(1+
+0.5(1+z)2]x
2
+O.5
y
2+0.
5z
2 -
(.z+O.75):z:y
= 5.28125
(
1~~)
A
three
dimensional
portrail
of
this
surface is given
in
Fig.2
Fig.
2. Stabilit.y
domain
Since
I>
is negative everywhere, all
trajectories
inside
the
bounded
do-
main
tend
toward
the
origin
and
the
solutions
are
asymptotically
stable,
Fig.a.
Outside
of
the
domain,
the
solutions
mayor
Hl.ay
not
tend
toward
equilibrium, Fig.4, As
with
the
original criterion,
Theorem
2 gives only suf-
ficient conditions for stability,
Another
Lyapunov
function
may
give
another
domain
of
stability.
2040
fiO
80
1.0'
1:~
_::.:[.,;;;
:;M~o:-"g~Cl;~)
-J~O
t
~-'---'-~~'~'~'
~.
-..-.I-
t-I.O~l
/,.-"
~O.5
r:
20
40 60 80
100
120
~
0.[,5
'.1
j/
( ,
~.O
8
100
120
--
1.0,
~25'v
Fig
.
3.
The
time
variations
of
X',
II
and
,:;;
for
the
initial
conditions
inside
the
stability
region,
Xo
= 1.3,
yo
= 1.5,
Zo
= 1.0