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28.2 Other Kalman Filter Formulations

In addition to the LKF, there are several other formulations of the Kalman filter that may be employed to
more closely follow the characteristics of specific state observation scenarios. The LKF may be varied according
to the temporal nature of the dynamic and measurement systems to be continuous in dynamics and mea-
surements or continuous in dynamics and discrete in measurements [12 ]. Also, there are applications when
the dynamic system is energetic or the measurement quality is poor that may cause the reference state in the
LKF to quickly leave the region of linearity about the environment state. In such systems, the reference state
can be updated through addition of the filter state into an implementation known as the extended Kalman
filter (EKF). The EKF is highly suited to real-time applications but is nonlinear in the sense that the reference
state is essentially reinitialized at the time of each measurement update. Both the continuous–discrete LKF
and EKF will be developed in the following sections.
The Continuous–Discrete Linear Kalman Filter

There may quite naturally arise an application where the reference state, filter state, and state error
covariance are more suitably propagated in a continuous fashion than through the linear application of
the state transition matrix. Also, it is common for the measurement system to deliver discrete-time
observations even when the dynamics are best modeled continuously. In such a situation the update
mechanization is unchanged from the previous LKF derivation while the propagation between updates
is carried out through continuous integration. Without loss of generality, it may be stated that the
reference dynamics of a continuous Kalman filter may be represented by
˜ ˙
(
˜
X t() = f X t(),αα αα,t) (28.35)
Furthermore, by taking time derivatives of the filter state and covariance propagation (Eqs. (28.11) and
(28.21)) and substituting in Eq. (28.13) for the derivative of the state transition matrix, the continuous-
time filter state and covariance relations are found to be

(
(
˜
˜
˜
x ˆ ˙ − () t () = f X t(),αα αα,t) + FX t(),αα αα,t)[x ˆ − () t () X t()] (28.36)
–
˙
T
P t() = F t()P t() + P t()F t() + Q t() (28.37)
where (t) is the spectral density of the dynamic process noise at time t and the explicit functional
Q
dependency of F was dropped for notational convenience. In this mechanization of the LKF, the state
transition matrix need not be calculated as the dynamics are included directly via the partial derivative
matrix and the reference state, filter state, and error covariance are propagated continuously.
The process and measurement noise representations in this formulation are continuous and discrete
for the respective models, and are again assumed to be zero mean processes governed by the continuous
dynamic process noise covariance
[
(
T
E w t()w t()] = Q t()d t t) (28.38)
–
and the discrete measurement noise covariance
T
E v k v j = R k d kj (28.39)
It is also assumed here that the process and measurement noises are uncorrelated so that
T =
E w t()v k 0 (28.40)

although the formulation can be modified to accommodate process and measurement noise correlations
if necessary [7].

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