3.2 Extended EIV Kalman Filtering
A drawback of the linear ﬁlter described in Subsec-
tion 3.1, is that it relies on exact information of the
noise characteristics and an exact model of the pro-
cess generating the data. In an attempt to compensate
for the latter requirement, an extended EIV Kalman
ﬁlter (EIVeKf), based on the EIV Kalman ﬁlter given
in (Guidorzi et al., 2003), has been proposed in (Vin-
sonneau et al., 2005). Instead of an exact process
representation, the EIVeKf requires only an approx-
imate parametrisation of a linear default model, char-
acterised by θ
d
, to achieve acceptable results. Under
certain conditions, use of the EIVeKf can lead to a
superior ﬁlter performance with respect to the linear
counterpart in cases where the system parametrisation
is only approximately known. Moreover, the EIVeKf
is also able to accommodate, to a certain degree, the
case of LTV systems.
The idea of the EIVeKf is very similar to the jeKf
approach; the state vector is augmented with the com-
pensating parameters θ
c
such that the new state vector
becomes
ξ
k
θ
c
k
(61)
where ξ
k
is the original state vector in (34) for the cal-
culation of the residual sequence
{
γ
k
}
. The resulting
system equations are thus nonlinear and the EIVKf
ﬁlter can be modiﬁed using ﬁrst order Taylor approx-
imations for the predicting step, which results to the
EIVeKf equations.
4 EIV EXTENDED KALMAN
FILTER FOR JOINT
PARAMETER ESTIMATION
Since the EIV ﬁltering problem can be solved by the
means of a standard Kalman ﬁlter, as outlined in Sec-
tion 3.1, one could apply well known modiﬁcations
of traditional Kalman ﬁltering techniques to estimate
u
0
k
and y
0
k
. The approach proposed here is to ap-
ply the idea of joint state and parameter estimation,
as summarised in Section 2, to EIV systems. This is
expected lead to a similar ﬁlter as the one presented in
(Vinsonneau et al., 2005) with the difference that the
estimate
ˆ
θ
k
is not only used for the prediction step, but
in the overall ﬁlter equations. In addition, the changes
in the parameters can be tracked by the means of Σ
d
deﬁned in (7).
4.1 Algorithm
Assuming the data is generated by a Eiv system of
structure (36)-(39) and the assumed model structure
is given by
x
k+1
= Ax
k
+ Bu
k
+ v
k
(62)
y
k
= Cx
k
+ Du
k
+ e
k
(63)
with v
k
and e
k
as deﬁned in (49)-(53). Then the EIV
extended Kalman ﬁlter for joint parameter estimation
(EIVjeKf) is readily given by (23)-(30) together with
the estimated inputs and outputs as deﬁned in (59) and
(60), whereas A, B, C, D are replaced by A
k
, B
k
, C
k
,
D
k
, respectively.
However, it it found in simulations, that this form
of the EIVjeKf can suffer from outliers in terms of
overall EIV ﬁlter performance as illustrated in Section
5. Therefore, a slight different formulation will be
preferred in the subsequent: while the Kalman gain is
still to be estimated, in order to assure the existence of
the terms
h
∂
∂θ
K(θ)
i
θ=
ˆ
θ
k
¯
ε
k
within F
k
, these estimates
are not further utilised in the algorithm but rather K
k
as given by (11)-(16). For clariﬁcation, the algorithm
is summarised as follows.
Algorithm 4.1 Assuming an EIV system of the form
(36)-(39) and the model structure of (62)-(63) with
noise characteristics (49)-(53), the EIVjeKf is given
by
1. Augment the state vector x
k
with the model param-
eters θ
k
and Kalman gain K
k
2. Determine the time-varying model matrices A
k
,
B
k
, C
k
and D
k
as given in (8)
3. Compute the innovation given by (25) and its co-
variance matrix as deﬁned in (30)
4. Determine the Jacobians F
k
and H
k
as given in
(31) and (18), respectively
5. Determine the reformulated covariance matrices
(51)-(53) with B and D replaced by B
k
and D
k
6. Compute ˆx
k+1
and
ˆ
θ
k+1
using (9)-(16)
7. Determine ˆu
0
k
and ˆu
0
k
given by (59) and (60),
where D is replaced with D
k
8. Increment k and continue with step 2
Remark 4.1 As outlined in Section 2, the parame-
ter estimator resulting from the EIVjeKf can be inter-
preted as an recursive prediction error method with
the correction inspired by the eKf algorithm. How-
ever, it is known, that the application of standard pre-
diction error methods to EIV systems does not yield
consistent estimates as demonstrated in (S
¨
oderstr
¨
om,
1981). Therefore, the estimated
ˆ
θ
m
is expected to be
biased with respect to the true parametrisation.
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