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Sl M J.
ONTROL
N OPTIMIZATION
Vol.
21, No. 2,
March
1983
1983
Society
for Industrial and
Applied Mathematics
0363-0129/83/2102-0006
01.25/0
NEW CLASS
OF
STABILIZING
CONTROLLERS
FOR
UN ERT IN
DYN MI L
SYSTEMS
B. R.
BARMISH?,
M.
CORLESS
ND
G.
LEITMANN
Abstract.
This
paper
is
concerned
with the
problem
of
designing
a
stabilizing controller
for a class
of
uncertain
dynamical
systems. The vector of
uncertain
parameters
q .) is
time-varying,
and
its
values
q t
li e
within a
prespecified bounding
set
Q
in
R
p
Furthermore,
no statistical
description
of
q .)
is
assumed,
and
the controller
is
shown to
render
the
closed
loop
system
practically
stable
in
a
so-called guaranteed
sense;
that
is,
the desired
stability
properties
are
assured
no
matter what
admissible
uncertainty q .)
is
realized. Within
the
perspective
of
previous research in
this
area,
this
paper contains
one
salient feature:
the
class of
stabilizing
controllers which
we
characterize
is shown
to
include linear
controllers when the
nominal
system happens
to
be linear and
time-invariant. In
contrast,
in
much
of
the
previous
literature
see, for
example,
[1], [2], [7],
and
[9] ,
a
linear
system
is stabilized
via
nonlinear
control. Another
feature
of
this paper is
the
fact that
the methods of
analysis
and
design
do
not
rely
on
transforming
the
system
into
a more
convenient canonical
form; e.g., see
[3].
It
is also
interesting
to
note that
a
linear
stabilizing
controller
can
sometimes be
constructed even
when
the
system
dynamics
are nonlinear.
This
is
illustrated
with
an
example.
Key
words,
stability,
uncertain
dynamical systems,
guaranteed
performance
1.
Introduction.
During
recent
years,
a
number
of
papers
have
appeared
which
deal
with
the
design
of
stabilizing
controllers for
uncertain
dynamical
systems;
e.g.,
see
[1]-[7].
In
these
papers
the
uncertain
quantities
are
described
only
in
terms
of
bounds on their
possible
sizes;
that
is
no
statistical
description
is assumed.
Within
this
framework,
the
objective
is
to find
a
class
of
controllers
which
guarantee
stable
operation
for
all
possible
variations
of
the uncertain
quantities.
Roughly speaking,
the
results to date
fall into
two
categories.
There are
those
results which
might appropriately
be termed
structural in
nature;
e.g.,
see
[1]-[3],
[6].
By
this
we
mean
that
the
uncertainty
cannot
enter
arbitrarily
into
the
state
equations;
certain
preconditions
must
be met
regarding
the locations of the
uncertainty
within
the
system
description.
Such conditions are often referred
to
as matching
assumptions.
We
note
that
in
this situation
uncertainties
can be
tolerated
with
an
arbitrarily
large
prescribed
bound.
second
body
of
results
might
appropriately
be
termed
nonstruc-
rural
in
nature;
e.g.,
see
[4]
and
[5].
Instead
of
imposing matching assumptions
on
the
system,
these
authors
permit
more
general
uncertainties at
the
expense
of
sufficient smallness
assumptions
on
the allowable sizes
of
the
uncertainties.
This
work falls within the class of structural
results
mentioned above.
Our
motivation comes
from a
simple
observation.
Namely, given
a
theory
which
yields
stabilizing
controllers for
a
class
of
uncertain
nonlinear
systems,
it
is
often desirable
for
this
theory
to have the
following
property:
upon
specializing
the
recipe
for
controller
construction
from
nonlinear
to linear
systems,
one of the
possible
stabilizing
control laws should
be linear
in
form. It
is
of
importance
to note
that
existing
results
do
not have this
property.
Upon
specialization
to
the
linear
case,
one
typically
obtains
controllers
of
the
discontinuous
bang-bang variety; e.g.,
see
[1]
and
[2].
One
can
often
approximate
these
controllers
using
a
so-called saturation
nonlinearity; e.g.,
see
*
Received
by
the
editors
March
4,
1981, and
in
revised
form January
15 ,
1982.
Department of Electrical
Engineering,
University
of
Rochester,
Rochester,
New
York 14627.
The
work
of
this
author
was
supported
by
the
U.S. Department
of
Energy
under
contract no. ET-78-S-01-3390.
Department
of
Mechanical
Engineering,
University
of
California
at
Berkeley,
Berkeley, California
94720.
The
work of
these
authors
was
supported
by
the National
Science
Foundation
under
grant
ENG
78-13931.
246
D o w n l o a d e d 0 1 / 0 3 / 1 3 t o 1 2 8 . 1 4 8 . 2 5 2 . 3 5 . R e d i s t r i b u t i o n s u b j e c t t o S
I A M l i c e n s e o r c o p y r i g h t ; s e e
h t t p : / / w w w . s i a m . o r g / j o u r n a l s / o j s a . p h p
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STABILIZING CONTROLLERS FOR
UNCERTAIN
DYNAMICAL
SYSTEMS
247
[7].
Such
an
approach
leads to
uniform
ultimate boundedness of the
state
to
an
arbitrarily
small
neighborhood
of
the
origin;
this
type
of behavior
might
be termed
practical
stability.
Our
desire in
this
paper
is
to
develop
a controller
which is linear when
the
system
dynamics
are
linear.
By
taking
known
results
such
as
in
[3])
which were
developed
exclusively
for
linear
systems,
one
encounters a
fundamental
difficulty
when
attempting
to
generalize
2
to a
class
of
nonlinear
systems; namely,
it is
no
longer possible
to
transform
the
system dynamics
to
a
more convenient
canonical form. The
subsequent
analysis
is free of
such
transformations.
2.
Systems,
assumptions
and
the
concept
of
practical stability.
We
consider
an
uncertain
dynamical
system
described
by
the
state
equation
2 t)
=f x t),
t)+
Af x
t ,
q t),
t
+[B x t),
t)+ AB x t),
q t),
t)]u t),
where
x(t)R
is the
state,
u(t)R
is
the
control,
q(t)R
p
is the
uncertainty
and
f x,
t ,
Af x,
q,
t ,
B
x,
t
and
AB
x,
q,
t
are
matrices
of
appropriate
dimensions
which
depend
on
the
structure
of the
system.
Furthermore,
it
is assumed that the
uncertainty,
q .):R-->R
p
is
Lebesgue
measurable and its values q t)
lie within
a
prespecified
bounding set
Q
cR for all R We denote this
by writing
q(.)M(Q).
As mentioned
in the
introduction,
given
that
stabilization is
the
goal,
we
must
impose
additional
conditions
on the
manner in which
q t)
enters
structurally
into the
state
equations.
We
refer
to
such
conditions
as
matching
assumptions.
Assumption
1.
There
are
mappings
h(.):R xRPxRR
and
E( ):R xRPxR-R
such
that
Af x,
q,
t B x,
t)h
x, q,
t ,
AB x,
q,
t)=
B x, t)E x, q,
t ,
liE
x, q,
t ll
<
1
for
all
x
R
q
O
and R.
We
note that this
assumption
can
sometimes be
weakened. For
example,
in
[9]
a
certain
measure
of
mis-match is introduced and
results
are
obtained
under
the
proviso
that
this
measure
does
not
exceed
a
certain
critical
level
termed
the
mis-match
threshold.
Our
second
assumption
reflects
the
fact
that the uncertainties
must
be
bounded
in
order to
permit
one
to
guarantee
stability.
Assumption
2.
The set
O
c
R
is
compact.
Our next
assumption
is introduced
to
guarantee
the
existence of solutions
of
the
state
equations.
Assumption
3.
The
mappings
f .
:
R
x
R
R
n
B
):
R
x
R R
mxn,
h
.)
and
E .) see Assumption 1)
are continuous.
3
This
notion
is
no t
to
be interpreted
in the
sense
of
Lasalle
and
Lefschetz
[12]
but
as defined
subsequently.
That
is ,
one
begins
with
a
linear
control
law for
a
linear
system
and
generalizes
the
controller in
such
a
way
that
is
applies
to a
class
of nonlinear
systems.
In fact,
one
can
modify
the
analysis
to
follow
so
as
to allow
mappings which
are
Carath6odory
and
satisfy
certain
integrability
conditions.
See,
for
example,
Corless and
Leitmann
[7].
All the
results
of
this
paper
still
hold
under
this
weakening
of
hypotheses.
D o w n l o a d e d 0 1 / 0 3 / 1 3 t o 1 2 8 . 1 4 8 . 2 5 2 . 3 5 . R e d i s t r i b u t i o n s u b j e c t t o S
I A M l i c e n s e o r c o p y r i g h t ; s e e
h t t p : / / w w w . s i a m . o r g / j o u r n a l s / o j s a . p h p
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248
B.
R.
BARMISH
M.
CORLESS
AND G.
LEITMANN
In
order
to
satisfy
our
final
assumption,
one
may
need
to
precompensate
the
so-called
nominal
system,
that
is ,
the
system
with
Af x,
q,
t)=--O
and
AB(x,
q,
t)----0;
e.g.,
see
[2].
Thus,
prior
to
controlling
the
effects
of
the
uncertainty,
it
may
be
necessary
to
employ
a
portion
of
the control
to
obtain
an
uncontrolled
nominal
system
(UC)
(t)=f(x(t),t)
that has
certain
stability
properties
embodied in
the
next
assumption.
Assumption
4.
f 0,
t)-
0
for
all
R
and,
moreover,
there
exist a C function
V(.):R
xR
[0, o0)
and
strictly
increasing
continuous
functions
y(.),
y2 ),
y3 ):
[0,
00)
[0, 00) satisfying
4
yl(0)
y2(0)--y3(0)
0 and limr_.
yl r)
limr_.
TE r)
lim_.oo
ya r) o, such
that for
all
x, t)
e R
R,
2.2)
v (llxll)
Moreover,
defining
the
Lyapunov
derivative
o( ):
R
R
R
by
2.3)
o(X,
t)
a
0
V(x,
t)
=+V xV(x,t)f(x,t),
at
where
V
denotes the
transpose
of the
gradient operation,
we also
require
that
t)
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STABILIZING
CONTROLLERS
FOR
UNCERTAIN DYNAMICAL SYSTEMS 249
(ii)
Given
any
r>0
and
any
solution
x(.):[to, tl]-->R
n,
X to =Xo,
of
(2.4)
with
Ilxoll
0
such
that
IIx
t l l
R
can
be
continued over
[to,
oo).
(iv)
Given
any
d
>_-
_d,
any
r >
0 and
any
solution
x
(.)
[to,
oo)
-- >
R
,
x
(to)
Xo,
of
(2.4)
with
Ilxoll--_d
and
any
solution
x( ): [to, oo)-->R , X(to)
=Xo,
of
(2.4),
there
is
a
constant 6
a >
0
such
that
Ilxoll-=
to .
3.
Controller construction.
We
take
_d
>
0 as
given
and
proceed
to
construct
a
control
law
p
_e(
which
will
later
be shown
to
satisfy
conditions
(i)-(v)
in
the
definition
of practical
stabilizability.
Construction
of
pa_( ).
The first
step
is to
select
functions
AI .
and
A2(.) R
R
-->
R
satisfying
(3.1) A x,
t)-->_max
Ilh(x,
q,
t ll,
(3.2)
1
> A).(x,
t)
>-max
liE(x,
q,
t l[.
q
The
standing Assumptions
1-4
assure that
there
is
a
A2(x,
t)
such that
1)
A2(x, t)
[0,
oo) satisfying
a(x,t)
(3.3)
g(x,
t)>-4[1_
AE x t ][C2-Cl.o X,
t ]
where
and
C2
are
any
designer chosen
nonnegative
constants such
that
a)
C1<
1;
b)
either
C1
0
or
C2
0;
(3.4)
c)
C2
0
whenever
limx_,0
[A(x,
t /o X,
t) ]
does
not
exist;
v
v _d .
)
1
C
Note that these conditions can indeed be satisfied because
of
continuity
of
the
3 .
and
the fact
that lim_,o
y(r)=
0
for
1, 2,
3.
This
construction
then
enables
one to let
(3.5)
Pa(X,
t)a---y(x,
t)B (x, t VV x,
t) .
Remark.
In fact,
(3.3)
and
(3.5)
describe
a
class
of
controllers
yielding
practical
stability.
It
will
be
shown
in
5
that
this
class
includes
linear
controllers
when
the
nominal
system
happens
to
be linear
and
time-invariant.
4.
Main
result
and
stability
estimates.
The theorem
below and its
proof
differ
from
existing
results
(see
[1],
[2]
and
[6])
in
one
fundamental
way:
The
control
p_a( )
which
leads
to the
satisfaction
of
the conditions
for
practical
stabilizability
degenerates
D o w n l o a d e d 0 1 / 0 3 / 1 3 t o 1 2 8 . 1 4 8 . 2 5 2 . 3 5 . R e d i s t r i b u t i o n s u b j e c t t o S
I A M l i c e n s e o r c o p y r i g h t ; s e e
h t t p : / / w w w . s i a m . o r g / j o u r n a l s / o j s a . p h p
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250 B.
R.
BARMISH,
M.
CORLESS
AND
G. LEITMANN
into
a linear
controller whenever the
nominal
system,
obtained
by
setting
Af(x(t),
q(t),
t)=--O
and
AB(x(t),
q(t),
t)=-O
in
(2.1),
is linear
and time-invariant.
This
will be demonstrated
in
the
sequel.
In
fact,
even
for
certain
nonlinear nominal
systems,
the
controller
turns out
to
be linear. This
phenomenon
will
be
illustrated
with an
example
of a nonlinear
pendulum.
Central
to
the
proof
of
the
theorem below
is
one
fundamental
concept
a
system satisfying
Assumptions
1-4
admits
a
control such
that
the
Lyapunov
function for the
nominal
system (UC)
is also
a
Lyapunov
function
for
the
uncertain
system (2.1).
THEOREM 1.
Subfect
to
Assumptions 1-4,
the
uncertain
dynamical system
(2.1)
is
practically
stabilizable.
Proof.
For a
given
_d>0
and
a
given
uncertainty
q . M Q ,
the
Lyapunov
derivative
.) Rn
R
R
for
the closed
loop
system
obtained
with the feedback
control
(3.5)
is
given by
x,
t)a_
o X,
t)+
V’V(x,
t){Af(x,
q(t),
t)
(4.1)
+[B
(x, t)+
aB
(x,
q(t),
t)]pa_(x,
t)}.
By
using
the
matching
assumptions
in
conjunction
with
(3.5), (4.1)
becomes
q’(x,
t)=
f 0(x,
t)-3/(x,
t)llB x,
t)VxV(x,
t ll
+
V’V(x, t)B(x,
t)[h(x,
q(t),
t)
-y(x,
t)E(x,
q(t),
t)B (x,
t)VxV(x,
t)].
Letting
(.)
R R
-- >
R be
given
by
(x,
t)
a---B x,
t)VxV(x,
t) ,
and
recalling
the
definition
of
AI(.
and
Az(. ),
a
straightforward computation
yields
x, t)_-< 0 x, t)-[1-A2(x,
t)]y(x,
t)ll
(x,
t l[
=
+
+/-(x,
t)ll
(x,
t [I
Now
there are
two
cases
to
consider.
Case
1.
The
pair
(x,
t)
is such
that
Al(x, t)
0.
It
then
follows
from
the
preceding
inequality
that
x,
t)
0.
Moreover, in
view
of
(3.3)
and
the conditions
on
the
Ci ,
o (x, t)_
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STABILIZING
CONTROLLERS
FOR
UNCERTAIN DYNAMICAL
SYSTEMS
251
Combining
Cases
1 and
2,
and
noting
that
C1
< 1,
we conclude
(as
a
consequence
of
Assumption 4)
that
(4.2)
x, t) _-_d,
using
the
estimates
provided
in
[7],
one can
define
0
if r
_-<
(yl
yl)(a),
(4.3)
T(d-,
r)
A
y2(r)
Z(’]/_l
yl
y1)(d)
otherwise,
(1
C1)( y3
-
1) =
C2
and
in accordance with
[7],
the
desired
uniform
ultimate
boundedness
condition
(iv)
holds with
the
proviso
that
(4.4)
(1
C1)(y3
yl
yl)(a)-
C2
>0.
Note
that
this
requirement
is
implied
by
the satisfaction
of
condition
(d)
of
(3.4)
which
entered
into
the
construction
of
the
controller.
Finally,
to
complete
the
proof,
it
remains to
establish the
desired
uniform
stability
property. Indeed,
le t
d->_d
be
specified
and
notice that if
6(d)=
R,
the
following
property
will
hold
Given
any
solution
x( ) [t0,
)R ,
X(to)=Xo
of
(2.4)
with
tlx011=
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252
B.
R. BARMISH
M.
CORLESS
AND
G.
LEITMANN
where D
(.),
E
(.)
and
v
(.)
have
appropriate
dimensions
and
depend continuously
on
their
arguments.
In
accordance with
Assumption
4,
the
matrix
A
must
be
asymptoti-
cally
stable.
To
obtain
a
Lyapunov
function
for the
uncontrolled
nominal
system,
we
select
simply
an
n
x
n positive-definite
symmetric
matrix
H
and solve
the
equation
(5.3)
A P
+
PA H
for P which
is
positive-definite;
see
[11].
Then
we have
(5.4)
V(x, t) x’Px
and
(5.5) 0 x, t)=-x’Hx.
It is clear from
(5.4)
and
(5.5)
that one
can
take
the
bounding
functions
yi(’)
to
be
(5.6)
yl(r) h
min[e]r
2,
y2
(r)
-
h
max[.P]r
2,
y3
(r)
h
min[H]r
2,
where
h
max(min)[
denotes
the
operation
of
taking
the largest
(smallest)
eigenvalue.
Construction
of
the
controller. We
take
_d
>0 as
prescribed
and construct
the
controller
p_d( given
in
3.
Using
the
notation
above,
we
define first
6
(5.7)
po
a--maxl[O(q)[I,
pz
a-maxl[E(q)ll-
4(1
-pE)Chmin[H]
Case
2.
po
O, p
>
O.
Clearly,
it suffices
to
take
C
0 and
(5.12)
y(x,
t)_=
y0
02
4(1
p)C2
where
C2
is
required
to
satisfy
condition
d)
of
(3.4). Using
the
descriptions
of the
yi(.
given
in
(5.6),
this
amounts
to
restricting
C2
by
C2
X
min[e]
<
Amin[H]_d
2
(5.13)
1
C1
Amax[P]
with
C1
0
in
the
above.
6
One
ca n
in
fact
us e
overestimates
to
and
tSE
for
Po
and
pE
as long as
the
inequality
tSz
< is
satisfied.
D o w n l o a d e d 0 1 / 0 3 / 1 3 t o 1 2 8 . 1 4 8 . 2 5 2 . 3 5 . R e d i s t r i b u t i o n s u b j e c t t o S
I A M l i c e n s e o r c o p y r i g h t ; s e e
h t t p : / / w w w . s i a m . o r g / j o u r n a l s / o j s a . p h p
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STABILIZING
CONTROLLERS
FOR UNCERT IN
DYN MIC L
SYSTEMS 253
Case 3.
Oo
>0,
p
>0.
Now,
in order to
satisfy
(5.9),
we select
Ca
(0,
1),
C2
satisfying
(5.13)
and
(5.14) y(x,
t
=-
yo
>max
(por
+p)2
}
4(1-pE)[Clhmin[n]r
2-t-
C2]
Letting
f(r)
denote the bracketed
quantity
in
(5.14)
above,
a
straightforward
but
lengthy
differentiation
yields
(5.15)
maxf(r)=
1
020
P }
_->0
4(1-Oz)
Clhmin[H]
Hence,
any
3 o
equal
to or
exceeding
this
maximum value
will
be
appropriate
in
(5.14).
6. Illustrative
example.
We
consider
now
the
simple
pendulum
which was
analyzed in
[7]. However,
here
it
will
be
shown
that
the desired
practical
stability
can
actually
be
achieved
via a
linear
control. This
may
seem
somewhat
surprising
in
light
of
the
fact
that
the
nominal
system
dynamics
are nonlinear.
pendulum
of
length
is
subjected
to
a
control
moment u
(.) (per
unit
mass).
The
point
of
support
is
subject
to an uncertain
acceleration
q(.),
with
[q t [
0
is
a
given
constant. In
order
to
satisfy
the
assumptions
of
2 one must
assure
a
uniformly
asymptotically
stable
equilibrium
for
(UC),
the
uncontrolled
nominal
system.
Hence,
for
a
given _d
>
0,
we
propose
a
controller
of the
form
(6.2)
u(t)
=-bx(t)-CXE(t)+pa_(x(t), t ,
where b and
c are
positive
constants
and
p_a(.)
will
be
specified
later
in
accordance
with
the
results
of 3. The
linear
portion
of
the controller
(6.2)
is
used
to
obtain
a
stable nominal
system.
Substitution
of
(6.2)
into
(6.1)
now
yields
the state
equation
2
t
f
(x
t , t
+
B [p
(X t ,
t +
h (x
t ,
q(t),
t ],
6.3)
where
(6.4)
B
1
f(x,
t
-bx-cxE-a
sinx
h(x,
q, t
-q
cos
xl
suitable Lyapunov function
for the
uncontrolled
nominal
system
(with
x
0
as
equilibrium)
is
2\
2
(6.5)
V x,t = b+gc
x
+cxxE+x
+2a 1-cosx ,
and, provided
b is
sufficiently
large,
the
associated
3 (.
are
given by
3 (r)=Alr
2,
2r
+
4a if
r >
/3(r)
X3r
2
D o w n l o a d e d 0 1 / 0 3 / 1 3 t o 1 2 8 . 1 4 8 . 2 5 2 . 3 5 . R e d i s t r i b u t i o n s u b j e c t t o S
I A M l i c e n s e o r c o p y r i g h t ; s e e
h t t p : / / w w w . s i a m . o r g / j o u r n a l s / o j s a . p h p
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9/10
254
B.
R. BARMISH,
M.
CORLESS
AND
G.
LEITMANN
A
,5
Xmax[P],
3
min
{/,
c
}
6.7)
6.8)
P=[
b+c2
]
b+amin
si
x
>0.
C
x
Following
the
procedure
described
in
3 for
the
construction of
the
controller
p
.),
we select
first
6.9) l(X,t)=lcosxl,
=(x, t
0.
Inequality 3.3)
can then
be
assured by requiring
2
p
COS
X
6.
0)
e x,
t
4[
Given our desire
for a linear
feedback,
one can select
C
0 and
satisfy 6.10) by
choosing
6.11)
x,
t)v0>
=4C
To
complete
the
design,
C
must be selected to
satisfy
condition
d)
of
3.4).
The
analysis
must
account for two
cases,
depending
on
the
size
of the
given
radius
>
0.
Case 1.
a>
h
4a. The
required
conditions
on
C
are
_
h3
d
6.12) 0
0
is
chosen
suNciently
small
so that
6.13)
h2C2
a
1-cos
0. As the
radius
decreases,
C
decreases,
which
in
turn implies that
T0
increases.
In
contrast,
the
nonlinear
saturation
controller
of
[7]
remains
bounded
by
the
bound
of the
uncertainty,
and
the radius
can
be
decreased
by
increasing
the nonlinear
gain; i.e.,
by
approaching
a discontinuous
control.
.
eels This
paper
addresses
the so-called problem
of
practical
stabiliza-
bility
for
a
class
of uncertain
dynamical
systems.
In
contrast
to
previous
work
on
problems
of
this
sort,
the
main
emphasis
here
is
on the
structure
of
the controller.
It
is
shown
that
by choosing
the function
T
in
a
special
way,
the
resultant
control
law
can
often
be
realized as
a
linear time-invariant
feedback.
REFERENCES
[1]
G.
LEITMANN,
Guaranteed
asymptotic
stability
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some linear
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with
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J.
Dynamic
Systems, Meaurement
and
Control,
101
(1979),
pp.
212-216.
[2]
S.
GUTMAN, Uncertain
dynamical
systems,
a
Lyapunov rain-max
approach,
IEEE
Trans. Automat.
Control,
AC-24
(1979),
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25
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D o w n l o a d e d 0 1 / 0 3 / 1 3 t o 1 2 8 . 1 4 8 . 2 5 2 . 3 5 . R e d i s t r i b u t i o n s u b j e c t t o S
I A M l i c e n s e o r c o p y r i g h t ; s e e
h t t p : / / w w w . s i a m . o r g / j o u r n a l s / o j s a . p h p
-
8/17/2019 Journal on Lyapunov Stability Theory Based Control of Uncertain Dynamical Systems
10/10
STABILIZING
ONTROLLERS
FOR
UN ERT IN DYN MI L
SYSTEMS
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THORP
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BARMISH
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L. H NG N
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VINKLER
N J
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Guidance and
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449-456.
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P
MOLANDER Stabilization
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M.
CORLESS
N G.
LEITMANN
Continuous
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1139-1144.
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A.
VINKLER ND
J WOOD
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B. R. BARMISH
N
G.
LEITMANN
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BERGE
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D o w n l o a d e d 0 1 / 0 3 / 1 3 t o 1 2 8 . 1 4 8 . 2 5 2 . 3 5 . R e d i s t r i b u t i o n s u b j e c t t o S
I A M l i c e n s e o r c o p y r i g h t ; s e e
h t t p : / / w w w . s i a m . o r g / j o u r n a l s / o j s a . p h p