FILTERING |
Measurement Model
Considering the parametric case, a vector of observations l, with variance-covariance matrix Ql l, is related to the parameters by:
 |
(7.4-5a) |
The weight matrix depends entirely on the stochastic model of the observations and is defined by P = Ql l-1. The matrix P is not necessarily diagonal. When measurements are made at time t, a predicted state vector is computed. This is often the best estimate of the parameters available at this time. Hence, it is logical to linearise the measurements around the predicted state, and eqn (7.4-5) is slightly changed:
 |
(7.4-5b) |
In this expression, the vector of predicted residuals 
is computed as
l - 
(
), consequently the increment vector 
is related to
the predicted state: = (x - ). This situation is often encountered
in practice, but it is preferable to make a clear distinction here between
and 
, because they represent different concepts and have different
stochastic behaviours, even if the parameters contained in both state vectors
are the same. Eqn (7.4-5) is therefore the more general expression.
Basic Least Squares Filter
The requirements for a Least Squares filter can now be explicitly stated. Both the predicted state (through the kinematic model) and the measurements can be fully described by their functional and stochastic models, a total of four models. Eqns (7.4-4) and (7.4-5) can be grouped into a system of parametric equations:
 |
(7.4-6) |
This system can be rewritten in the usual form of parametric equations (section 7.1.1), by simply replacing each matrix such that:
 |
The solution follows as usual:
 |
Back-substituting the matrices A*, P*
and 
* through their original expressions in eqn
(7.4-6):
 |
(7.4-7a) |
This is now the expression for the filtered estimate. The minimisation
of the extended quadratic form VTP v + TP
would yield identical results (MERMINOD
& RIZOS, 1988). For most practical applications is chosen equal
to , hence simplifying eqn (7.4-7) as = 0. The use of the
predicted state for linearisation is made apparent by replacing the quantity
by . Thus, the most commonly used expression for the computation of
the filtered increment, though not the most general, is:
 |
(7.4-7b) |
An important restriction on the use of this simplified formula is in
the case of solution iterations. It is straightforward to select the set
of parameters = for the initial linearisation. For subsequent iterations
however , is changed and the complete eqn (7.4-7a) should be used.
This considerably reduces the efficiency of the filtering algorithm, hence
iterations should be avoided as far as possible.
The covariance matrix of the filtered estimate can be computed by applying
the Law of Propagation of Variances to eqn (7.4-7), assuming the stochastic
independence of and l :
 |
(7.4-8) |
The variance-covariance matrix of the residuals is obtained by the same method:
 |
(7.4-9) |
This expression is identical to the standard parametric case, though
the definition of Q
is somewhat
different. In this form, the algorithm is called the Bayes filter.
The contributions of the predicted and observed components are easily identified.
Comparing eqn (7.4-7) with the standard solution for the parametric case
(section 7.1.1):
 |
It appears that the filtered estimate is fairly easy to compute. It is
sufficient to simply add the weight of the predicted estimate to the classical
normal matrix ATPA, add the weighted predicted
estimate to the constant term ATP and
proceed as usual. The drawback of this procedure is that it always requires
the inversion of a normal matrix of dimension u x u, where u is the size
of the state vector, even if only one observation is added. This computational
problem makes this filter inappropriate for many applications.
The Kalman Form
The Kalman form is obtained by using a matrix identity described in many textbooks of linear algebra, or more specifically in the estimation literature:
 |
(7.4.10) |
Substituting in the Bayes eqns (7.4-5) and (7.4-7), the filtered estimate of x becomes:
 |
(7.4.11a) |
where:
 |
(7.4.12) |
At first sight, this new expression for appears more complicated
than the Bayes form. However, the only matrix to invert has dimension n'
x n', where n' is the number of new observations. This is an important
computational advantage. In addition, if the observations are linearised
around the predicted state, eqn (7.4-11a) reduces to:
 |
(7.4.11b) |
K is referred to as the Kalman gain matrix. The term is used to describe the function of feedback loops in electrical engineering, where the output of a system is used to improve the separation of the input signal from the noise within the system. In the form eqn (7.4-11), K represents the propagation of the "unexpected" part of the new measurements into an improvement of the predicted estimate.
Example: n independent
measurements l of a distance x
are available. The standard deviation of each measurement is s. The Least
Squares estimate n is the mean n of all measurements, with
variance s2/n. The expected value of a new similar measurement ln+1 is n.
The new Least Squares estimate of the distance is the mean of the n+1 measurements:
 |
The last expression may be regarded as the Kalman form, where 1/n+1 is the gain matrix. This example is simply a degenerate case, completely described by:
 the measurement model |
A = 1, Ql l =  2 |
| the kinematic model |
= 1, Qww = 0 |
This can be checked by replacing the terms in eqns (7.4-2), (7.4-3),
(7.4-11) and (7.4-12). Using eqn (7.4-11), the variance-covariance matrix
of the Kalman filtered estimate can be computed. According to the Law of
Propagation of Variances and the assumed stochastic independence of and
l :
 |
(7.4.13) |
This expression can be simplified to:
 |
(7.4.14) |
The variance-covariance of the vector of residuals is identical to eqn (7.4-9) obtained for the Bayes filter, that is:
 |
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© Chris Rizos, SNAP-UNSW, 1999