206
C. Greco and M. Vasile
where the integral term has zero-mean. The state transition matrix evolution is given
by:
˙
Φ(t, t k−1 ) = F (t)Φ(t, t k−1 ), with Φ(t k−1 , t k−1 ) = I .
(6.69)
In this formulation, Eqs. (6.33) and (6.34) can be reformulated as [58]:
ˆ
x k = Φ(t k , t k−1 )ˆ x k−1
P x (t k ) = Φ(t k , t k−1 )P x (t k−1 )Φ
T (t k , t k−1 )
+
t k
t k−1
Φ(t k , τ )G(τ )Q(τ )G
T (τ )Φ
T (t k , τ )dτ .
(6.70)
When w is approximated to be a random sequence, i.e. piecewise constant w(t) =
w k for t k−1 ≥ t ≥ t k , with covariance Q k−1 , Eq. (6.68) can be written as:
x k = Φ(t k , t k−1 )x k−1 + Γ (t k , t k−1 )w k−1 ,
(6.71)
with Γ (t k , t k−1 ) =
t k
t k−1
Φ(t k , τ )G(τ )dτ which can be computed by quadrature.
The second term of Eq. (6.70) becomes:
P x (t k ) = Φ(t k , t k−1 )P x (t k−1 )Φ
T (t k , t k−1 ) + Γ (t k , t k−1 )Q k−1 Γ
T (t k , t k−1 ) ,
(6.72)
where Γ (t k , t k−1 ) is called process noise transition matrix [58].
The corresponding algorithm for the state transition matrix approach is schematised in Algorithm 2. This procedure and its corresponding algorithm show that a
linear filtering problem with continuous dynamics and discrete observations can be
translated into an equivalent fully discrete linear filtering problem. This equivalence
in linear filtering problems stands as an intuitive, although not formal, proof of
the applicability of the techniques for nonlinear transformation approximation
introduced in the previous section to the continuous-discrete filtering problem.
As a final note of the discussion on the Kalman filter, it is worth discussing the
nature of the noise term in the practical applications. Indeed, since in aerospace
applications the dynamics is regarded as deterministic, although not completely
known or modelled, the term w is not a proper Gaussian noise. Nonetheless, in such
cases, it is just a useful tool for taking into account errors arising from unmodelled
terms, neglected nonlinearities, numerical errors and so on [29].
In statistical derivations, this term is often left out, resulting in the covariance
matrix evolution being determined exclusively by the deterministic terms (step 4 of
Algorithm 1, step 5 of Algorithm 2). Then, a rightful doubt could arise whether the
term w is actually necessary for similar applications. Clearly, it introduces analytical
difficulties in the filter derivation and numerical complexity in the algorithm.
Nevertheless, it turns out that the additive term in the covariance propagation
helps the filter accuracy and general performance. Indeed, without the random
C. Greco and M. Vasile
where the integral term has zero-mean. The state transition matrix evolution is given
by:
˙
Φ(t, t k−1 ) = F (t)Φ(t, t k−1 ), with Φ(t k−1 , t k−1 ) = I .
(6.69)
In this formulation, Eqs. (6.33) and (6.34) can be reformulated as [58]:
ˆ
x k = Φ(t k , t k−1 )ˆ x k−1
P x (t k ) = Φ(t k , t k−1 )P x (t k−1 )Φ
T (t k , t k−1 )
+
t k
t k−1
Φ(t k , τ )G(τ )Q(τ )G
T (τ )Φ
T (t k , τ )dτ .
(6.70)
When w is approximated to be a random sequence, i.e. piecewise constant w(t) =
w k for t k−1 ≥ t ≥ t k , with covariance Q k−1 , Eq. (6.68) can be written as:
x k = Φ(t k , t k−1 )x k−1 + Γ (t k , t k−1 )w k−1 ,
(6.71)
with Γ (t k , t k−1 ) =
t k
t k−1
Φ(t k , τ )G(τ )dτ which can be computed by quadrature.
The second term of Eq. (6.70) becomes:
P x (t k ) = Φ(t k , t k−1 )P x (t k−1 )Φ
T (t k , t k−1 ) + Γ (t k , t k−1 )Q k−1 Γ
T (t k , t k−1 ) ,
(6.72)
where Γ (t k , t k−1 ) is called process noise transition matrix [58].
The corresponding algorithm for the state transition matrix approach is schematised in Algorithm 2. This procedure and its corresponding algorithm show that a
linear filtering problem with continuous dynamics and discrete observations can be
translated into an equivalent fully discrete linear filtering problem. This equivalence
in linear filtering problems stands as an intuitive, although not formal, proof of
the applicability of the techniques for nonlinear transformation approximation
introduced in the previous section to the continuous-discrete filtering problem.
As a final note of the discussion on the Kalman filter, it is worth discussing the
nature of the noise term in the practical applications. Indeed, since in aerospace
applications the dynamics is regarded as deterministic, although not completely
known or modelled, the term w is not a proper Gaussian noise. Nonetheless, in such
cases, it is just a useful tool for taking into account errors arising from unmodelled
terms, neglected nonlinearities, numerical errors and so on [29].
In statistical derivations, this term is often left out, resulting in the covariance
matrix evolution being determined exclusively by the deterministic terms (step 4 of
Algorithm 1, step 5 of Algorithm 2). Then, a rightful doubt could arise whether the
term w is actually necessary for similar applications. Clearly, it introduces analytical
difficulties in the filter derivation and numerical complexity in the algorithm.
Nevertheless, it turns out that the additive term in the covariance propagation
helps the filter accuracy and general performance. Indeed, without the random