where
F
!
(
ˆ
!
k
M
k
)=(
@=@!
)
F
ct
(
!
)
M
!
=
ˆ
!
k
M
k
. If (66) is
used, howe ver, these models have an identical state
prediction because
E
[
x
k
+1
M
z
k
]=
E
[
F
ct
(
!
)
x
k
M
z
k
]=
F
ct
(
ˆ
!
k
M
k
)
ˆ
x
k
M
k
(70)
where
z
k
stands for all measurements through time
t
k
. This follows from
E
[
¯
!
M
z
k
]=
E
[
!
k
+1
M
z
k
]=
E
[
!
k
M
z
k
]=
ˆ
!
k
M
k
, a consequence of (66). Nevertheless,
the corresponding covariances differ because different
models are used. That is why these models result
in very similar tracking performance, as reported
in [72] without explanation, for two IMM-EKF
(interacting multiple model) algorithms using the first
two schemes above with real ATC data. The state
prediction would also be different if model (68) is
used.
We point out that it is theoretically more appealing
to obtain a discrete-time model for state vector
x
=[
x
,
_
x
,
y
,
_
y
,
!
]
by discretizing the following
continuous-time equations
jointly
:
[
_
x
,
¨
x
,
_
y
,
¨
y
]
(
t
)=(
_
x
,
!
_
y
,
_
y
,
!
_
x
)
(
t
)+
w
(
t
) (71)
_
!
(
t
)=
®!
(
t
)+
w
!
(
t
) (72)
where
®
=0or 1
=¿
!
. It is reasonable to expect an
improvement in performance with this alternative at
the price of an increased complexity.
A continuous-time model is more accurate
than its discrete-time versions, but the latter are
needed for most applications. In the case where
the continuous-time model is nonlinear, such as
the CT models with unknown turn rate, there are
in general at least two approaches to acquiring its
discrete-time linear approximate models. Such models
are needed in the application of a linear (e.g., Kalman)
filter based nonlinear filtering technique. The first
is to linearize the nonlinear equation for the state
first and then discretize the linearized differential
equation (by finding the solution via integration).
This approach is more commonly used because it
is easy. An alternati ve is to discretize the nonlinear
state-space equation first and then linearize the
discrete-time model. These two approaches are
referred to as
discretized linearization
and
linearized
discretization
, respectively, in [73, 64]. The second
approach appears more accurate in general than the
first since linearization usually lose more accuracy
than discretization and thus should be done later.
However, the second approach may not be tractable
in general because discretization requires solving
the nonlinear dif ferential equation, which is often a
great challenge. Fortunately , for the CT motion with
an unknown turn rate, the corresponding differential
equations for the state are simple and the solution
can be readily obtained [73, 64]. The equations of the
nearly CT model, obtained by the first approach, and
some performance comparison results are also gi ven
in [73, 64].
As stated before, the choice of the state vector is
not trivial for the turn models. Essentially two classes
have been proposed. They differ in the representation
of the velocity vector: in the Cartesian and polar
coordinates, respectively.
1)
CT Models with Cartesian Velocity
:In
this model, the state vector is chosen to be
x
=
[
x
,
_
x
,
y
,
_
y
,
!
]
; that is, the velocity vector (
_
x
,
_
y
)
is represented in the Cartesian coordinates.
Consequently, a direct discrete-time version of the
model is given by (see [67, 2]):
x
k
+1
=
F
ct
(
!
)0
0
¯
x
k
+diag[
G
2
,
G
2
,1]
w
k
(73)
where
¯
=
e
®T
,
!
=
!
k
,
!
k
+1
,
¯
!
, or something
similar,
G
2
was given in (14),
w
k
=[
w
x
,
w
y
,
w
!
]
k
is
zero-mean white noise with suitable statistics—they
are noise terms for acceleration in the
x
and
y
directions and for turn rate.
This model is known to be used successfully
as one of models in numerous multiple-model
configurations (see, e.g., [74, 67, 2, 75, 4]).
2)
CT Models with Polar Velocity
: Obviously, the
velocity vector can also be represented in the polar
coordinates as [
v
,
Á
]
in terms of speed
v
=
T
_
x
2
+
_
y
2
and (velocity) heading angle
Á
=tan
1
(
_
y=
_
x
). The state
vector is thus
x
=[
x
,
y
,
v
,
Á
,
!
]
. The corresponding
differential equation is given by (5) with
f
(
x
,
u
,
t
)=
[
v
cos
Á
,
v
sin
Á
,0,
!
,0]
;thatis
_
x
(
t
)=[
v
cos
Á
,
v
sin
Á
,0,
!
,0]
+
w
(
t
) (74)
which follows immediately from (57)–(60) by setting
!
and
v
constant. By linearization first and then
discretization, its discrete-time linearized model is
given in [76] (see also [1]):
x
k
+1
=
x
+(2
=!
)
v
sin(
!T=
2)cos(
Á
+
!T=
2)
y
+(2
=!
)
v
,sin(
!T=
2)sin(
Á
+
!T=
2)
v
Á
+
!T
!
k
+
w
k
(75)
where
w
k
is white noise with covariance
Q
=diag
00
00
,
T
2
¾
2
_
v
,
T
3
¾
2
_
!
=
3
T
2
¾
2
_
!
=
2
T
2
¾
2
_
!
=
2
T
2
¾
2
_
!
:
(76)
This model was successfully used as a building block
of a multiple-model algorithm for aircraft tracking
application in an air defense system [77, 78].
The above model uses (65) for the turn rate
per se. Instead, it may be better to use (67). The
corresponding discrete-time model need be modified
accordingly.
1348 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 39, NO. 4 OCTOBER 2003
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