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6.079/6.975,
Fall
2009-10
S.
Boyd
&
P.
Parrilo
Homework
7
additional
problems
1.
Identifying
a
sparse
linear
dynamical
system.
A
linear
dynamical
system
has
the
form
x
(
t
+ 1)
=
Ax
(
t
) +
Bu
(
t
) +
w
(
t
)
,
t
= 1
, . . . , T
−
1
,
where
x
(
t
)
∈
R
n
is
the
state,
u
(
t
)
∈
R
m
is
the
input
signal,
and
w
(
t
)
∈
R
n
is
the
process
noise,
at
time
t
.
We
assume
the
process
noises
are
IID
N
(0
, W
),
where
W
�
0
is
the
covariance
matrix.
The
matrix
A
∈
R
n
×
n
is
called
the
dynamics
matrix
or
the
state
transition
matrix,
and
the
matrix
B
∈
R
n
×
m
is
called
the
input
matrix.
You
are
given
accurate
measurements
of
the
state
and
input
signal,
i.e.
,
x
(1)
, . . . , x
(
T
),
u
(1)
, . . . , u
(
T
−
1),
and
W
is
known.
Your
job
is
to
find
a
state
transition
matrix
A
ˆ
and
input
matrix
B
ˆ
from
these
data,
that
are
plausible,
and
in
addition
are
sparse,
i.e.
,
have
many
zero
entries.
(The
sparser
the
better.)
By
doing
this,
you
are
effectively
estimating
the
structure
of
the
dynamical
system,
i.e.
,
you
are
determining
which
components
of
x
(
t
)
and
u
(
t
)
affect
which
components
of
x
(
t
+
1).
In
some
applications,
this
structure
might
be
more
interesting
than
the
actual
values
of
the
(nonzero)
coeﬃcients
in
A
ˆ
and
B
ˆ
.
By
plausible,
we
mean
that
T
−
1
�
=1
t
2
W
−
1
/
2
x
(
t
+
1)
−
ˆ
Bu
(
t
)
Ax
(
t
)
−
ˆ
2
∈
n
(
T
−
1)
±
2
2
n
(
T
−
1)
,
where
a
±
b
means
the
interval
[
a
−
b, a
+
b
].
(You
can
just
take
this
as
our
definition
of
plausible.
But
to
explain
this
choice,
we
note
that
when
A
ˆ
=
A
and
B
ˆ
=
B
,
the
left-hand
side
is
χ
2
,
with
n
(
T
−
1)
degrees
of
freedom,
and
so
has
mean
n
(
T
−
1)
and
standard
deviation
2
n
(
T
−
1).)
(a)
Describe
a
method
for
finding
A