6 Fundamentals of Filtering
203
˙
x = F (t)x + G(t)w
y = H (t)x +
x 0 ∼ N x 0 (ˆ x 0 , P 0 ) .
(6.57)
At time t 0 , before any observation, the conditional distribution coincides with the
prior distribution
p(x 0 |y 0 ) = p(x 0 ) = N x 0 (ˆ x 0 , P 0 ) ,
(6.58)
where y 0 has been introduced for notation’s consistency, but it is a fictitious quantity.
For generality, the derivation process will be carried out starting from a generic
time t k−1 and distribution p(x k−1 |y 1:k−1 ) after an observation has been processed,
which could be also the initial time for k = 1 thanks to the fictitious observation
introduced.
The density distribution can be propagated to represent the state probability
distribution at a given time of interest. In the general case, the Kolmogorov partial
differential equation should be used (see Eq. (6.21)). However, in Sect. 6.2.1, it
has been shown how the first two moments evolve according to simple ordinary
differential equations in the linear case. As the initial condition is given by a
normally distributed density, the first two moments fully capture the statistics of the
conditional density. Therefore, when propagating at the time of the next observation,
the conditional probability is [29]:
p(x k |y 1:k−1 ) = N x k (ˆ x
−
k , P
−
k ) ,
(6.59)
where the superscript {·} − has been introduced to describe a quantity at an
infinitesimal time before t k , i.e. just before the observation update. Similarly, the
superscript {·} + will be used to identify a quantity at an infinitesimal time after
t k , i.e. immediately after the update with a new measurement. The moments ˆ
x
−
k
and P
−
k can be obtained by numerical propagation of the ordinary differential
equations (6.33)–(6.34), respectively, with initial conditions ˆ
x k−1 and P k−1 .
When an observation is available, this new knowledge is combined with the
dynamically propagated distribution to obtain a better estimate of the state. The
Kalman filter updates the conditional state distribution through Bayes’ rule, in the
sequential filtering form of Eq. (6.14). The second probability in the numerator
is computed as in Eq. (6.59). The density of the observation, conditional on the
state immediately before the observation, is given by Eq. (6.36), written here as
p(y k |x k ) = N y k (H (t k )x k , R k ). Lastly, the denominator could be decomposed as in
Eq. (6.18). However, instead of computing the quantity p(y k |y 1:k−1 ) by integration,
we can first exploit a well-known property for computing the joint distribution of
two Gaussian random variables with conditional dependencies [51]:
203
˙
x = F (t)x + G(t)w
y = H (t)x +
x 0 ∼ N x 0 (ˆ x 0 , P 0 ) .
(6.57)
At time t 0 , before any observation, the conditional distribution coincides with the
prior distribution
p(x 0 |y 0 ) = p(x 0 ) = N x 0 (ˆ x 0 , P 0 ) ,
(6.58)
where y 0 has been introduced for notation’s consistency, but it is a fictitious quantity.
For generality, the derivation process will be carried out starting from a generic
time t k−1 and distribution p(x k−1 |y 1:k−1 ) after an observation has been processed,
which could be also the initial time for k = 1 thanks to the fictitious observation
introduced.
The density distribution can be propagated to represent the state probability
distribution at a given time of interest. In the general case, the Kolmogorov partial
differential equation should be used (see Eq. (6.21)). However, in Sect. 6.2.1, it
has been shown how the first two moments evolve according to simple ordinary
differential equations in the linear case. As the initial condition is given by a
normally distributed density, the first two moments fully capture the statistics of the
conditional density. Therefore, when propagating at the time of the next observation,
the conditional probability is [29]:
p(x k |y 1:k−1 ) = N x k (ˆ x
−
k , P
−
k ) ,
(6.59)
where the superscript {·} − has been introduced to describe a quantity at an
infinitesimal time before t k , i.e. just before the observation update. Similarly, the
superscript {·} + will be used to identify a quantity at an infinitesimal time after
t k , i.e. immediately after the update with a new measurement. The moments ˆ
x
−
k
and P
−
k can be obtained by numerical propagation of the ordinary differential
equations (6.33)–(6.34), respectively, with initial conditions ˆ
x k−1 and P k−1 .
When an observation is available, this new knowledge is combined with the
dynamically propagated distribution to obtain a better estimate of the state. The
Kalman filter updates the conditional state distribution through Bayes’ rule, in the
sequential filtering form of Eq. (6.14). The second probability in the numerator
is computed as in Eq. (6.59). The density of the observation, conditional on the
state immediately before the observation, is given by Eq. (6.36), written here as
p(y k |x k ) = N y k (H (t k )x k , R k ). Lastly, the denominator could be decomposed as in
Eq. (6.18). However, instead of computing the quantity p(y k |y 1:k−1 ) by integration,
we can first exploit a well-known property for computing the joint distribution of
two Gaussian random variables with conditional dependencies [51]: