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The matrices Q(t) and R(t) are the driving noise covariance in the system dynamics
and the measurement noise covariance, respectively. The matrix
Q.t/
must be
nonnegative definite and
R.t/
must be positive definite. The initial state estimate
x
0
and its covariance
P
0
describe the prior knowledge of the true state at the
beginning of the process. The Kalman filter “represents the most widely applied and
demonstrable useful result to emerge from the state variable approach of modern
control theory” (
Sorenson 1985
). It can be found in many textbooks on control
theory, for instance (
Gelb 1974
;
Brown and Hwang 1997
). A drawback of EKF is
that the convergence is, in general, not guaranteed. Simple examples can be found
in which an EKF estimation process diverges (
Krener 2004
). Various proofs of its
local convergence exist in the literature. Interested readers are referred to
Krener
(
2003b
) and references therein.
An EKF requires the linearization of system models, which may not be easily
available during real-time operations. In addition, the linearization is changed if the
model is modified or updated. A different approach is to use the Unscented Kalman
Filter (UKF) which does not require the online computation of the linearization.
Following
Julier and Uhlmann
(
2004
), consider a discrete time nonlinear system
x
k
D
f.x
k
1
;
w
k
1
/
y
k
D
h.x
k
1
;
v
k
1
/
where
v
and
w
are the state, measurement, process noise and measurement
noise respectively. The UKF is “founded on the intuition that it is easier to approx-
imate a probability distribution than it is to approximate an arbitrary nonlinear
function or transformation” (
Julier and Uhlmann 2004
). The UKF assumes that at
every sampling instance, the state
x;y;
is always a normally distributed variable. The
mean and the covariance information of this random variable can be stored in a set
of specially chosen points called sigma points. One simple choice of such sigma
points is given below (
Julier and Uhlmann 2004
)
i
D
E.x/
˙
p
x
nP; i
D
1;2;:::;n
where
E.x/
is the mean of the random variable
x
,
P
is the covariance matrix and
n
is the dimension of
. It can be shown that the nonlinear transformation of the sigma
points preserves statistics up to second order in a Taylor serious expansion (
Julier
and Uhlmann 2004
). Based on this fact, a prediction of the state and the covariance
matrices in the filter algorithm can be carried out as follows:
x
•
Based on the previous-step estimation of the state,
x
k
1
, and the covariance
P
xx
k
1
matrix,
, calculate a set of sigma points as
q
n P
xx
i
D
x
k
1
˙
k
1
;i
D
1;2;:::;n
I
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