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[xOut, info] = ckbs_affine_singular(...
z, g, h, ...
dg, dh, q, r)
Q
and
R
may be singular.
\[
\begin{array}{rcl}
G^{-1} (w - Q G^{-T}H^T\Phi^{-1}(HG^{-1}w-v))
\end{array}
\]
where
\[
\Phi = HG^{-1}QG^{-T}H^T + R ,
\]
and the matrices
R_k
and
Q_k
are
symmetric positive semidefinite.
Note that
g_1
is the initial state estimate
and
Q_1
is the corresponding covariance.
z
is a two dimensional array,
for
k = 1 , \ldots , N
\[
z_k = z(:, k)
\]
and
z
has size
m \times N
.
g
is a two dimensional array,
for
k = 1 , \ldots , N
\[
g_k = g(:, k)
\]
and
g
has size
n \times N
.
The value
g_1
serves as the initial state estimate.
h
is a two dimensional array,
for
k = 1 , \ldots , N
\[
h_k = h(:, k)
\]
and
h
has size
m \times N
.
dg
is a three dimensional array,
for
k = 1 , \ldots , N
\[
G_k = dg(:,:,k)
\]
and
dg
has size
n \times n \times N
.
The initial state estimate
g_1
does not depend on the value of
x_0
, hence
G_1
should be zero.
dh
is a three dimensional array,
for
k = 1 , \ldots , N
\[
H_k = dh(:,:,k)
\]
and
dh
has size
m \times n \times N
.
q
is a three dimensional array,
for
k = 1 , \ldots , N
\[
Q_k = q(:,:, k)
\]
and
q
has size
n \times n \times N
.
The value of
Q_k
is the variance of the initial state
estimate
g_1
.
r
is a three dimensional array,
for
k = 1 , \ldots , N
\[
R_k = r(:,:, k)
\]
and
r
has size
m \times m \times N
.
It is ok to signify a missing data value by setting the corresponding
row and column of
r
to infinity. This also enables use
to interpolate the state vector
x_k
to points where
there are no measurements.
xOut
contains the optimal sequence
( x_1 , \ldots , x_N )
.
For
k = 1 , \ldots , N
\[
x_k = xOut(:, k)
\]
and
xOut
is a two dimensional array with size
n \times N
.
ckbs_affine_singular.
It returns true if ckbs_affine_singular passes the test
and false otherwise.