Aerospace and Electronic Systems Magazine May 2017 - 23
Crouse
ple model filtering is demonstrated in the demoMultipleModelFiltering function.
A small number of dynamic models are provided for the filters in the Dynamic Models folder. These are mostly functions
that can be passed to the filters, or for discrete linear filters, they
produce state transition matrices and covariance matrices. A
number of the functions generalize what is in the literature. For
example, the functions FGaussMarkov and QGaussMarkov produce the discrete state transition matrices and covariance matrices associated with Gauss-Markov dynamic models of arbitrary
orders. A first-order model is the integrated Ornstein-Uhlenbeck
process in the literature, and a second-order model is the wellknown Singer dynamic model. A generalization to an arbitrary
order might be suitable for a graduate-level or advanced undergraduate-level homework problem in certain statistics classes.
In such an instance, one has a linear continuous-time dynamic
model of the form
dxt = Axt dt + Dd β t
(2)
where xt is the target state at time t and dβt is differential multivariate white noise. The drift and diffusion matrices, respectively A
and D, for one-dimensional motion of an nth-order model are
1
− τ
A = 0n ,1
0
for n > 0
01, n −1, − 1
τ
1
0
D = 1n ,1 q 2 ,
(3)
(4)
where q is a process noise power spectral density and τ is the decorrelation time. The state transition matrix F and the process noise
covariance matrix Q are found by finding the mean and covariance
of the propagated state after a time duration T, as described in [18,
Ch. 4.8, 4.9], [1, Ch. 4.3, 6.2], and [20, Ch. 1.7, 2.4]. The elements
in row r and column c of F and Q, designated Fr,c and Qr,c, for
one-dimensional motion, with the row and column indices running
from 0 to n, are given by the expressions
Fr , c
T c−r
if r ≤ c and c < n
(c − r )!
i
T
T n − r −1 −
−
τ
if c = n
= (−τ ) n − r e τ −
!
i
i =0
0
otherwise
MAY 2017
T
−2
1
− q (−τ ) 2 n − r − c +1 1 − e τ
2
(6)
i+ j
+ q(−τ ) 2 n − r − c
n − r −1 n − c −1
(−1)
i =0
i+ j
j =0
T
T
τ
i ! j !(i + j + 1)
where if the upper limit of a sum is less than the lower limit, then
the value of the sum is zero. The solution in d dimensions involves
taking a Kronecker product of the one-dimensional matrices and a
d-dimensional identity matrix.
Though the previous example of a generalization may seem
rather complicated, all functions that generalized dynamic models
are commented so that the user can see what the generalization is
and where one might look to rederive the generalization.
COORDINATE CONVERSIONS
for n = 0 (position only)
I n, n
Qr , c
k
T
T
n − r −1
i
−
τ
= q (−τ ) 2 n − r − c +1 (−1)i 1 − e τ
k!
i =0
k =0
k
T
T i
n − c −1
−
τ
+ q(−τ ) 2 n − r − c +1 (−1)i 1 − e τ
k!
i =0
k =0
(5)
The library contains a great many functions related to coordinatesystem conversions. For example, ruv2Cart and Cart2Ruv convert
between monostatic or bistatic range/direction cosines and Cartesian coordinates, and ellips2Cart is useful for converting geodetic latitude, longitude, and altitude into Cartesian coordinates.
To simplify local/global coordinate conversions, the function
findTransParam is particularly useful, as one can give it direction
vectors in two coordinate systems, and it will provide a rotation
matrix or, if appropriate, a rotation matrix, a translation vector,
and a scale constant. The function findRFTransParam is useful
for constructing simulations, as it allows one to obtain a rotation
matrix for a radar face in terms of its direction with respect to
north, and the local tangent to the Earth's reference ellipsoid. For
example, one might want a radar to point 10° east of north at 15°
up from the horizontal.
In addition to those basic conversions, the Celestial and Terrestrial Systems subfolder contains high-precision conversions between precise definitions of Earth-centered Earth-fixed coordinate
systems by using the IAU's Standards of Fundamental Astronomy
library and allowing for the inclusion of Earth orientation parameters. Such coordinate systems are summarized in [8].
Of particular interest for target tracking are coordinate conversions of noisy measurements, preserving the first two moments
(i.e., measurement conversion with consistent covariances). Thus,
the Conversions with Covariances folder contains functions for
unbiased measurement conversion routines for a number of coordinate systems. Both cubature methods, as well as Taylor-series
approaches, are given. Some files offer multiple approaches. For
IEEE A&E SYSTEMS MAGAZINE
23
Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine May 2017
No label
Aerospace and Electronic Systems Magazine May 2017 - No label
Aerospace and Electronic Systems Magazine May 2017 - Cover2
Aerospace and Electronic Systems Magazine May 2017 - 1
Aerospace and Electronic Systems Magazine May 2017 - 2
Aerospace and Electronic Systems Magazine May 2017 - 3
Aerospace and Electronic Systems Magazine May 2017 - 4
Aerospace and Electronic Systems Magazine May 2017 - 5
Aerospace and Electronic Systems Magazine May 2017 - 6
Aerospace and Electronic Systems Magazine May 2017 - 7
Aerospace and Electronic Systems Magazine May 2017 - 8
Aerospace and Electronic Systems Magazine May 2017 - 9
Aerospace and Electronic Systems Magazine May 2017 - 10
Aerospace and Electronic Systems Magazine May 2017 - 11
Aerospace and Electronic Systems Magazine May 2017 - 12
Aerospace and Electronic Systems Magazine May 2017 - 13
Aerospace and Electronic Systems Magazine May 2017 - 14
Aerospace and Electronic Systems Magazine May 2017 - 15
Aerospace and Electronic Systems Magazine May 2017 - 16
Aerospace and Electronic Systems Magazine May 2017 - 17
Aerospace and Electronic Systems Magazine May 2017 - 18
Aerospace and Electronic Systems Magazine May 2017 - 19
Aerospace and Electronic Systems Magazine May 2017 - 20
Aerospace and Electronic Systems Magazine May 2017 - 21
Aerospace and Electronic Systems Magazine May 2017 - 22
Aerospace and Electronic Systems Magazine May 2017 - 23
Aerospace and Electronic Systems Magazine May 2017 - 24
Aerospace and Electronic Systems Magazine May 2017 - 25
Aerospace and Electronic Systems Magazine May 2017 - 26
Aerospace and Electronic Systems Magazine May 2017 - 27
Aerospace and Electronic Systems Magazine May 2017 - 28
Aerospace and Electronic Systems Magazine May 2017 - 29
Aerospace and Electronic Systems Magazine May 2017 - 30
Aerospace and Electronic Systems Magazine May 2017 - 31
Aerospace and Electronic Systems Magazine May 2017 - 32
Aerospace and Electronic Systems Magazine May 2017 - 33
Aerospace and Electronic Systems Magazine May 2017 - 34
Aerospace and Electronic Systems Magazine May 2017 - 35
Aerospace and Electronic Systems Magazine May 2017 - 36
Aerospace and Electronic Systems Magazine May 2017 - 37
Aerospace and Electronic Systems Magazine May 2017 - 38
Aerospace and Electronic Systems Magazine May 2017 - 39
Aerospace and Electronic Systems Magazine May 2017 - 40
Aerospace and Electronic Systems Magazine May 2017 - 41
Aerospace and Electronic Systems Magazine May 2017 - 42
Aerospace and Electronic Systems Magazine May 2017 - 43
Aerospace and Electronic Systems Magazine May 2017 - 44
Aerospace and Electronic Systems Magazine May 2017 - 45
Aerospace and Electronic Systems Magazine May 2017 - 46
Aerospace and Electronic Systems Magazine May 2017 - 47
Aerospace and Electronic Systems Magazine May 2017 - 48
Aerospace and Electronic Systems Magazine May 2017 - 49
Aerospace and Electronic Systems Magazine May 2017 - 50
Aerospace and Electronic Systems Magazine May 2017 - 51
Aerospace and Electronic Systems Magazine May 2017 - 52
Aerospace and Electronic Systems Magazine May 2017 - 53
Aerospace and Electronic Systems Magazine May 2017 - 54
Aerospace and Electronic Systems Magazine May 2017 - 55
Aerospace and Electronic Systems Magazine May 2017 - 56
Aerospace and Electronic Systems Magazine May 2017 - 57
Aerospace and Electronic Systems Magazine May 2017 - 58
Aerospace and Electronic Systems Magazine May 2017 - 59
Aerospace and Electronic Systems Magazine May 2017 - 60
Aerospace and Electronic Systems Magazine May 2017 - 61
Aerospace and Electronic Systems Magazine May 2017 - 62
Aerospace and Electronic Systems Magazine May 2017 - 63
Aerospace and Electronic Systems Magazine May 2017 - 64
Aerospace and Electronic Systems Magazine May 2017 - Cover3
Aerospace and Electronic Systems Magazine May 2017 - Cover4
http://www.brightcopy.net/allen/aesm/34-2s
http://www.brightcopy.net/allen/aesm/34-2
http://www.brightcopy.net/allen/aesm/34-1
http://www.brightcopy.net/allen/aesm/33-12
http://www.brightcopy.net/allen/aesm/33-11
http://www.brightcopy.net/allen/aesm/33-10
http://www.brightcopy.net/allen/aesm/33-09
http://www.brightcopy.net/allen/aesm/33-8
http://www.brightcopy.net/allen/aesm/33-7
http://www.brightcopy.net/allen/aesm/33-5
http://www.brightcopy.net/allen/aesm/33-4
http://www.brightcopy.net/allen/aesm/33-3
http://www.brightcopy.net/allen/aesm/33-2
http://www.brightcopy.net/allen/aesm/33-1
http://www.brightcopy.net/allen/aesm/32-10
http://www.brightcopy.net/allen/aesm/32-12
http://www.brightcopy.net/allen/aesm/32-9
http://www.brightcopy.net/allen/aesm/32-11
http://www.brightcopy.net/allen/aesm/32-8
http://www.brightcopy.net/allen/aesm/32-7s
http://www.brightcopy.net/allen/aesm/32-7
http://www.brightcopy.net/allen/aesm/32-6
http://www.brightcopy.net/allen/aesm/32-5
http://www.brightcopy.net/allen/aesm/32-4
http://www.brightcopy.net/allen/aesm/32-3
http://www.brightcopy.net/allen/aesm/32-2
http://www.brightcopy.net/allen/aesm/32-1
http://www.brightcopy.net/allen/aesm/31-12
http://www.brightcopy.net/allen/aesm/31-11s
http://www.brightcopy.net/allen/aesm/31-11
http://www.brightcopy.net/allen/aesm/31-10
http://www.brightcopy.net/allen/aesm/31-9
http://www.brightcopy.net/allen/aesm/31-8
http://www.brightcopy.net/allen/aesm/31-7
https://www.nxtbookmedia.com