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±
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Massachusetts
Institute
of
Technology
Department
of
Electrical
Engineering
and
Computer
Science
6.243j
(Fall
2003):
DYNAMICS
OF
NONLINEAR
SYSTEMS
by
A.
Megretski
Problem
Set
8
1
Problem
8.1
Autonomous
system
equations
have
the
form
y
(
t
)
(8.1)
y
¨(
t
)
=
y
y
˙(
(
t
t
)
)
Q
y
˙(
t
)
,
where
y
is
the
scalar
output,
and
Q
=
Q
�
is
a
given
symmetric
2by2
matrix
with
real
coeﬃcients.
(a)
Find
all
Q
for
which
there
exists
a
C
�
bijection
�
:
R
2
≥�
R
2
,
matrices
A,
C
,
and
a
C
�
function
�
:
R
≥�
R
2
such
that
z
=
�
(
y, y
˙)
satisfies
the
ODE
z
˙(
t
)
=
Az
(
t
)
+
�
(
y
(
t
))
,
y
(
t
)
=
Cz
(
t
)
whenever
y
(
·
)
satisfies
(8.1).
(b)
For those
Q
found
in
(a),
construct
C
�
functions
F
=
F
Q
:
R
2
×
R
≥�
R
2
and
H
=
H
Q
:
R
2
≥�
R
such
that
H
Q
(
�
(
t
))
−
y
˙(
t
)
�
0
as
t
� →
whenever
y
:
[0
,
→
)
≥�
R
is
a
solution
of
(8.1),
and
�
˙(
t
)
=
F
Q
(
�
(
t
)
,
y
(
t
))
.
1
Posted
November
19,
2003.
Due
date
November
26,
2003
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�
2
Problem
8.2
A
linear
constrol
system
x
˙
1
(
t
)
=
x
2
(
t
)
+
w
1
(
t
)
,
x
˙
2
(
t
)
=
−
x
1
(
t
)
−
x
2
(
t
)
+
u
+
w
2
(
t
)
is
equipped
with
the
nonlinear
sensor
y
(
t
)
=
x
1
(
t
)
+
sin(
x
2
(
t
))
+
w
3
(
t
)
,
where
w
i
(
·
)
represent
plant
disturbances
and
sensor
noise
satisfying
a
uniform
bound

w
i
(
t
)
 �
d
.
Design
an
observer
of
the
form
�
˙(
t
)
=
F
(
�
(
t
)
, y
(
t
)
, u
(
t
))
and
constants
d
0
>
0
and
C
>
0
such
that

�
(
t
)
−
x
(
t
)

�
Cd
�
t
∀
0
whenever
�
(0)
=
x
(0)
and
d
<
d
0
.
(Try
to
make
d
0
as
large
as
possible,
and
C
as
small
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 Spring '09
 AlexandreMegretski
 Dynamics, Derivative, Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, DYNAMICS OF NONLINEAR SYSTEMS, valid control Lyapunov

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